Introducing the n^th-Order Features Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (n^th-FASAM-N): II. Illustrative Example

Dan Gabriel Cacuci

Center for Nuclear Science and Energy, University of South Carolina, Columbia, USA.

DOI: 10.4236/ajcm.2024.141003                

Abstract

This work highlights the unparalleled efficiency of the “n^th-Order Function/ Feature Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (n^th-FASAM-N) by considering the well-known Nordheim-Fuchs reactor dynamics/safety model. This model describes a short-time self-limiting power excursion in a nuclear reactor system having a negative temperature coefficient in which a large amount of reactivity is suddenly inserted, either intentionally or by accident. This nonlinear paradigm model is sufficiently complex to model realistically self-limiting power excursions for short times yet admits closed-form exact expressions for the time-dependent neutron flux, temperature distribution and energy released during the transient power burst. The n^th-FASAM-N methodology is compared to the extant “n^th-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (n^th-CASAM-N) showing that: (i) the 1^st-FASAM-N and the 1^st-CASAM-N methodologies are equally efficient for computing the first-order sensitivities; each methodology requires a single large-scale computation for solving the “First-Level Adjoint Sensitivity System” (1^st-LASS); (ii) the 2^nd-FASAM-N methodology is considerably more efficient than the 2^nd-CASAM-N methodology for computing the second-order sensitivities since the number of feature-functions is much smaller than the number of primary parameters; specifically for the Nordheim-Fuchs model, the 2^nd-FASAM-N methodology requires 2 large-scale computations to obtain all of the exact expressions of the 28 distinct second-order response sensitivities with respect to the model parameters while the 2^nd-CASAM-N methodology requires 7 large-scale computations for obtaining these 28 second-order sensitivities; (iii) the 3^rd-FASAM-N methodology is even more efficient than the 3^rd-CASAM-N methodology: only 2 large-scale computations are needed to obtain the exact expressions of the 84 distinct third-order response sensitivities with respect to the Nordheim-Fuchs model’s parameters when applying the 3^rd-FASAM-N methodology, while the application of the 3^rd-CASAM-N methodology requires at least 22 large-scale computations for computing the same 84 distinct third-order sensitivities. Together, the n^th-FASAM-N and the n^th-CASAM-N methodologies are the most practical methodologies for computing response sensitivities of any order comprehensively and accurately, overcoming the curse of dimensionality in sensitivity analysis.

Keywords

Nordheim-Fuchs Reactor Safety Model, Feature Functions of Model Parameters, High-Order Response Sensitivities to Parameters, Adjoint Sensitivity Systems

1. Introduction

As presented in the accompanying work by Cacuci [1] , the “n^th-Order Function/ Feature Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (abbreviated as “n^th-FASAM-N”) enables the most efficient computation of exactly-determined expressions of arbitrarily high-order sensitivities of generic nonlinear system responses with respect to functions (features) of model parameters, which enables, in turn, the most efficient computation of the corresponding high-order sensitivities to primary model parameters, uncertain boundaries and internal interfaces in the model’s phase-space. It was shown in [1] that the n^th-FASAM-N methodology requires the fewest possible number of large-scale computations of any method for determining exactly the respective high-order sensitivities, being even more efficient than the well-known “n^th-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (n^th-CASAM-N) [2] . The unparalleled efficiency of the n^th-FASAM-N methodology stems from its formulation in the phase-space of “feature-functions,” which is always smaller than the phase-space of model parameters. The application of the nth-FASAM-N is illustrated in this work by considering a well-known paradigm model that describes a short-time self-limiting power excursion in a nuclear reactor system having a negative temperature coefficient in which a large amount of reactivity is suddenly inserted, either intentionally or by accident. In the textbook by Lamarsh [3] , this model is called the “Fuchs model” while in the textbook by Hetrick [4] this model is called the “Nordheim-Fuchs model.” This nonlinear paradigm model is sufficiently complex to realistically model self-limiting power excursions for short times while admitting closed-form exact expressions for the time-dependent neutron flux, temperature distribution and energy released during the transient power burst.

This work is structured as follows: Section 2 presents the balance equations which underly the Nordheim-Fuchs phenomenological model describing a prompt-critical reactor transient. Section 3 illustrates the computation of first-order sensitivities of the Nordheim-Fuchs model’s response (which is chosen to be the total energy released during the modeled power-burst) with respect to the parameters underlying this model by applying the well-known 1^st-CASAM-N methodology (in Subsection 3.1) and subsequently comparing (in Subsection 3.2) this methodology with the 1^st-FASAM-N methodology. Both methodologies require just one large-scale adjoint computation to obtain the exact analytical expressions of the first-order sensitivities of the model’s response with respect to the model’s parameters.

Section 4 illustrates the computation of the second-order response sensitivities with respect to the Nordheim-Fuchs model’s parameters by first showing that applying the 2^nd-FASAM-N (in Subsection 4.1) requires just 2 large-scale computations, while applying the 2^nd-CASAM-N (in Subsection 4.2) requires 7 large-scale computations, to obtain all of the 7 distinct second-order sensitivities.

Section 5 illustrates the computation of the third-order response sensitivities with respect to the Nordheim-Fuchs model’s parameters by applying the 3^rd-FASAM-N versus applying the 3^rd-CASAM-N. It is shown (in Subsection 5.1) that the 3^rd-FASAM-N requires just 2 large-scale computations to obtain all of the 84 distinct third-order sensitivities. By comparison, applying the 3^rd-CASAM-N (in Subsection 5.2) requires at least 22 computations large-scale for determining the 84 distinct third-order sensitivities. Evidently, the 3^rd-FASAM methodology is significantly more efficient for computing the second-and higher-order sensitivities than the 3^rd-CASAM-N methodology. Both the 3^rd-FASAM-N and the 3^rd-CASAM-N methodologies yield exact values for the expressions of the third-order sensitivities

The concluding discussion presented in Section 6 commences by noting that when no feature functions of parameters can be identified, the mathematical frameworks of the n^th-FASAM-N and the n^th-CASAM-N methodologies coincide. When feature-functions of parameters can be identified within the model, the n^th-FASAM-N methodology requires the least number of large-scale computations of any practical methodology for computing exact expressions of second- and higher-order sensitivities. In general, the number of large-scale computations required when applying the n^th-FASAM-N is proportional to the number of feature-functions underlying the model being analyzed, while the number of large-scale computations required when applying the n^th-CASAM-N is proportional to the number of the parameters underlying the respective model. Since the number of feature-functions is necessarily smaller than the number of model parameters, it follows that the n^th-FASAM-N is computationally more efficient than the n^th-CASAM-N. Both the n^th-FASAM-N and the n^th-CASAM-N are vastly more efficient computationally than finite-difference schemes, particularly for computing sensitivities of order higher than first for large-scale models with many parameters. Furthermore, the finite difference-schemes are approximate, while the n^th-FASAM-N and the n^th-CASAM-N accurately compute exact expressions of the arbitrarily-high order sensitivities of model responses with respect to model’s parameters. Altogether, the n^th-FASAM-N and the n^th-CASAM-N methodologies remain the most practical methodologies for computing response sensitivities comprehensively and accurately, overcoming the curse of dimensionality in sensitivity analysis of nonlinear systems.

2. The Nordheim-Fuchs Phenomenological Reactor Dynamics/Safety Model

The Nordheim-Fuchs phenomenological model (Lamarsh, 1966; Hetrick, 1993) describes a short-time self-limiting power transient in a nuclear reactor system having a negative temperature coefficient in which a large amount of reactivity is suddenly inserted, either intentionally or by accident. The response of such a reactor system can be estimated by considering that the reactivity insertion is sufficiently large and the time-span of the transient phenomena under consideration is of the order of the life-time of prompt-neutrons, which is sufficiently small to neglect the effects of delayed neutrons. For such short times, the local spatial variations of the neutron distribution in the reactor are negligible, and the heat generated during the transient remains within the reactor.

Using the notation of Lamarsh (1966), the Nordheim-Fuchs paradigm model describing the aforementioned self-limiting power transient comprises the following balance equations:

1) The time-dependent neutron balance (point kinetics) equation for the neutron flux $\phi \left(t\right)$ :

$\frac{\text{d}\phi \left(t\right)}{\text{d}t}=\frac{k\left(t\right)-1}{l_p}\phi \left(t\right),t>0$ , (1)

$\phi \left(0\right)={\phi }_0,t=0$ , (2)

where $l_p$ denotes the prompt-neutron lifetime, $k\left(t\right)$ denotes the reactor’s multiplication factor, and ${\phi }_0$ denotes the initial (i.e., extant flux) prior to initiating the transient at time $t=0$ .

2) The energy conservation equation:

$c_p\left[T\left(t\right)-T_0\right]=E\left(t\right)$ , (3)

where $E\left(t\right)$ denotes the total energy released (per cm³) at time t in the reactor since the onset of reactivity change; $c_p$ denotes the specific heat (per cm3) of the reactor.

3) The energy production equation:

$E\left(t\right)=\gamma {\Sigma }_f{\displaystyle \underset{0}{\overset{t}{\int }}\phi \left(x\right)}\text{d}x$ , (4)

where $\gamma$ denotes the recoverable energy per fission; ${\Sigma }_f\triangleq {\sigma }_f{N}_f$ denotes the reactor’s effective macroscopic fission cross section, where ${\sigma }_f$ denotes the reactor’s equivalent microscopic fission cross section while $N_f$ denotes the reactor’s equivalent atomic number density.

4) The reactivity-temperature feedback equation: $k\left(t\right)=k_0-{\alpha }_T{k}_0\left[T\left(t\right)-T_0\right]$ , where $k_0\triangleq k\left(0\right)\ge 1$ denotes the changed multiplication factor following the reactivity insertion at $t=0$ , ${\alpha }_T$ denotes the magnitude of the negative temperature coefficient, $T\left(t\right)$ denotes the reactor’s temperature, and $T_0$ denotes the reactor’s initial temperature at time $t=0$ . For illustrating the application of the 1st-FASAM methodology, it suffices to consider the special case of a “prompt critical transient”, when the reactor becomes prompt critical after the reactivity insertion, i.e., when $k_0=1$ , so that the reactivity-temperature feedback equation takes on the following particular form:

$k\left(t\right)=1-{\alpha }_T\left[T\left(t\right)-T_0\right]$ . (5)

The Nordheim-Fuchs model, comprising Equations (1)-(5), is representative of the types of equations that underly large-scale computational models, including a combination of algebraic, differential, and integral equations. Typically, such equations are discretized into systems of nonlinear algebraic and/or differential equations, which are then solved numerically by using standard solvers for such systems of equations. Equations (1)-(5) can be transformed into the following system of nonlinear differential equations:

$\frac{\text{d}\phi \left(t\right)}{\text{d}t}=-\frac{{\alpha }_T}{l_p{c}_p}E\left(t\right)\phi \left(t\right),t>0$ . $\phi \left(0\right)={\phi }_0,t=0$ (6)

$\frac{\text{d}E\left(t\right)}{\text{d}t}=\gamma {\sigma }_f{N}_f\phi \left(t\right),E\left(0\right)=0$ , (7)

$\frac{\text{d}T\left(t\right)}{\text{d}t}=\frac{\gamma {\sigma }_f{N}_f}{c_p}\phi \left(t\right);T\left(0\right)=T_0$ . (8)

The Nordheim-Fuchs model described by Equations (6)-(8) can be solved analytically to obtain closed-form expression for the state functions $\phi \left(t\right)$ , $E\left(t\right)$ , and $T\left(t\right)$ . Thus, eliminating the function $\phi \left(t\right)$ from Equations (6) and (7) yields a nonlinear equation that can be integrated directly to obtain the following relation:

$\phi \left(t\right)=-\frac{{\alpha }_T}{2l_p{c}_p\gamma {\sigma }_f{N}_f}{E}^2\left(t\right)+{\phi }_0$ . (9)

Using Equation (9) in Equation (7) yields the following nonlinear equation for the released energy $E\left(t\right)$ :

$\frac{\text{d}E\left(t\right)}{\text{d}t}=-\frac{{\alpha }_T}{2l_p{c}_p}{E}^2\left(t\right)+{\phi }_0\gamma {\sigma }_f{N}_f,E\left(0\right)=0$ . (10)

The most important quantity of interest (i.e., “model response”) for the Nordheim-Fuchs model is the total energy per cm³, $E\left(\tau \right)$ , released at a user-chosen “final time” instance denoted as $t=\tau$ , after the initiation at $t=0$ of the prompt-critical power transient. This response can be defined mathematically in several equivalent ways, the simplest of which is as follows:

$E\left(\tau \right)={\displaystyle {\int }_0^{\tau }E\left(t\right)\delta \left(t-\tau \right)\text{d}t}$ , (11)

where $\delta \left(t-\tau \right)$ denotes the Dirac-delta functional.

The response $E\left(\tau \right)$ defined in Equation (11) is an implicit function of seven uncertain parameters $i=1,\cdots ,7=TP$ primary model parameters (where “TP” denotes the “total number of primary model parameters”) which are considered to be the components of a “vector of model parameters” denoted as $\alpha$ and defined as follows:

$\alpha \triangleq {\left({\alpha }_1,\cdots ,{\alpha }_7\right)}^{†}\triangleq {\left({\alpha }_T,l_p,c_p,{\phi }_0,\gamma ,{\sigma }_f,N_f\right)}^{†}$ . (12)

In this work, all vectors are considered to be column vectors and the dagger symbol ( $†$ ) will be used to denote “transposition.” The model parameters are considered to be uncertain (i.e., imprecisely known), but have known nominal values which will be denoted using a superscript “zero,” as follows:

${\alpha }^0\triangleq {\left({\alpha }_1^0,\cdots ,{\alpha }_7^0\right)}^{†}\triangleq {\left({\alpha }_T^0,l_p^0,c_p^0,{\phi }_0^0,{\gamma }^0,{\sigma }_f^0,N_f^0\right)}^{†}$ . (13)

For further reference, the closed-form solution of Equation (10) has the following form:

$E\left(t\right)=K\left(\alpha \right)\tanh \left[t\theta \left(\alpha \right)\right]$ , (14)

where:

$K\left(\alpha \right)\triangleq {\left[\frac{2{\phi }_0\gamma {\sigma }_f{N}_f{l}_p{c}_p}{{\alpha }_T}\right]}^{1/2};\theta \left(\alpha \right)\triangleq {\left[\frac{{\alpha }_T{\phi }_0\gamma {\sigma }_f{N}_f}{2l_p{c}_p}\right]}^{1/2}$ . (15)

The closed-form expression of $\phi \left(t\right)$ is obtained by replacing Equation (15) into Equation (9) to obtain:

$\phi \left(t\right)={\phi }_0\left\{1-{\tanh }^2\left[t\theta \left(\alpha \right)\right]\right\}=\frac{{\phi }_0}{{\cosh }^2\left[t\theta \left(\alpha \right)\right]}$ . (16)

The closed-form expression of $T\left(t\right)$ is obtained by replacing Equation (15) into Equation (3) to obtain:

$T\left(t\right)=T_0+\frac{K\left(\alpha \right)}{c_p}\tanh \left[t\theta \left(\alpha \right)\right]$ . (17)

The remainder of this work will use the model response $E\left(\tau \right)$ to illustrate the advantages of applying the nth-FASAM-N versus the nth-CASAM-N for the higher-order sensitivity analysis of this response to the underlying model parameters.

3. Computation of First-Order Sensitivities of the Model Response to Model Parameters: Application of the Conventional 1^st-CASAM-N Versus the 1^st-FASAM-N Methodology

This Section presents the computation of the first-order sensitivities of the selected model response by following two alternative pathways. Thus, Subsection 3.1 presents the application of the conventional 1st-CASAM methodology, while Subsection 3.2 presents the application of the novel 1st-FASAM methodology; the advantages of applying the 1st-FASAM methodology versus the conventional 1st-CASAM methodology are summarized in Subsection 3.1

3.1. Application of the 1^st-CASAM-N to Obtain the First-Order Sensitivities of the Response $E\left(\tau \right)$ Directly with Respect to the Model Parameters

The first-order sensitivities of $E\left(\tau \right)$ with respect to variations in the model parameters are obtained by determining the first-order Gateaux (G)-differential $\delta E\left(\tau \right)$ of $E\left(\tau \right)$ for known parameter variations $\delta \alpha \triangleq \alpha -{\alpha }^0$ around the nominal values $\left(E^0;{\alpha }^0\right)$ . Considering that the final observation time is perfectly well known, the first-order Gateaux (G)-differential $\delta E\left(\tau \right)$ is obtained, by definition, as follows:

$\delta E\left(\tau \right)=\frac{\text{d}}{\text{d}\epsilon }{\left\{{\displaystyle {\int }_0^{\tau }\left[E^0+\epsilon \delta E\left(t\right)\right]\delta \left(t-\tau \right)}\right\}}_{\epsilon =0}\text{d}t={\displaystyle {\int }_0^{\tau }\delta E\left(t\right)\delta \left(t-\tau \right)}\text{d}t$ . (18)

The variational function $\delta E\left(t\right)$ is the solution of the first-order G-differential of Equation (10), which is obtained, by definition, as follows:

$\begin{array}{l}\frac{\text{d}}{\text{d}\epsilon }{\left\{\frac{\text{d}\left[E\left(t\right)+\epsilon \delta E\left(t\right)\right]}{\text{d}t}\right\}}_{\epsilon =0}\\ =\frac{\text{d}}{\text{d}\epsilon }\{-\frac{\left({\alpha }_T+\epsilon \delta {\alpha }_T\right)}{2\left(l_p+\epsilon \delta l_p\right)\left(c_p+\epsilon \delta c_p\right)}{\left[E\left(t\right)+\epsilon \delta E\left(t\right)\right]}^2\\ {+\left({\phi }_0+\epsilon \delta {\phi }_0\right)\left(\gamma +\epsilon \delta \gamma \right)\left({\sigma }_f+\epsilon \delta {\sigma }_f\right)\left(N_f+\epsilon \delta N_f\right)\}}_{\epsilon =0},\end{array}$ (19)

$\frac{\text{d}}{\text{d}\epsilon }{\left\{{\left[E\left(t\right)+\epsilon \delta E\left(t\right)\right]}_{t=0}\right\}}_{\epsilon =0}=0$ . (20)

Performing the operations indicated in Equations (19) and (20) yields the following differential equation, which constitutes the 1^st-Level Variational Sensitivity System (1^st-LVSS) for the 1^st-level variational function $\delta E\left(t\right)$ :

$\begin{array}{l}{\left\{\left[\frac{\text{d}}{\text{d}t}+\frac{{\alpha }_T}{l_p{c}_p}E\left(t\right)\right]\delta E\left(t\right)\right\}}_{{\alpha }^0}\\ ={\left\{-\frac{\delta {\alpha }_T}{2l_p{c}_p}+\frac{{\alpha }_T}{2l_p{\left(c_p\right)}^2}\delta c_p+\frac{{\alpha }_T}{2{\left(l_p\right)}^2{c}_p}\delta l_p\right\}}_{{\alpha }^0}{E}^2\left(t\right)\\ +{\left\{{\phi }_0{\sigma }_f{N}_f\left(\delta \gamma \right)+{\phi }_0\gamma N_f\left(\delta {\sigma }_f\right)+{\phi }_0\gamma {\sigma }_f\left(\delta N_f\right)+\gamma {\sigma }_f{N}_f\left(\delta {\phi }_0\right)\right\}}_{{\alpha }^0},t>0,\end{array}$ (21)

$\delta E\left(0\right)=0,t=0.$ (22)

In Equation (21), the parameter variations are known and the notation ${\left\{\right\}}_{{\alpha }^0}$ indicates that the quantity enclosed within the braces is to be evaluated at the nominal parameter values ${\alpha }^0$ . For every parameter variation, the 1^st-LVSS would need to be solved anew. This need for repeatedly solving the 1^st-LVSS can be avoided by constructing the corresponding 1^st-Level Adjoint Sensitivity System (1^st-LASS), by applying the 1^st-CASAM-N methodology. Thus, the Hilbert space appropriate for the construction of the 1^st-LASS corresponding to the above 1^st-LVSS is endowed with the following inner product, denoted as $\langle u\left(t\right),v\left(t\right)\rangle$ , between two square integrable functions $u\left(t\right)$ and $v\left(t\right)$ :

$\langle u\left(t\right),v\left(t\right)\rangle \triangleq {\displaystyle {\int }_0^{\tau }u\left(t\right)v\left(t\right)}\text{d}t$ . (23)

Forming the inner product of Equation (21) with a yet undefined function $a^{\left(1\right)}\left(t\right)$ yields the following relation:

$\begin{array}{l}{\left\{{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)\left[\frac{\text{d}}{\text{d}t}+\frac{{\alpha }_T}{l_p{c}_p}E\left(t\right)\right]\delta E\left(t\right)\text{d}t}\right\}}_{{\alpha }^0}\\ =\{{\phi }_0{\sigma }_f{N}_f\left(\delta \gamma \right)+{\phi }_0\gamma N_f\left(\delta {\sigma }_f\right)+{{\phi }_0\gamma {\sigma }_f\left(\delta N_f\right)+\gamma {\sigma }_f{N}_f\left(\delta {\phi }_0\right)\}}_{{\alpha }^0}{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)\text{d}t}\\ +{\left\{-\frac{\delta {\alpha }_T}{2l_p{c}_p}+\frac{{\alpha }_T}{2l_p{\left(c_p\right)}^2}\delta c_p+\frac{{\alpha }_T}{2{\left(l_p\right)}^2{c}_p}\delta l_p\right\}}_{{\alpha }^0}{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)E^2\left(t\right)\text{d}t}.\end{array}$ (24)

Integrating by parts the left side of Equation (24) yields the following relation:

$\begin{array}{c}{\left\{{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)\left[\frac{\text{d}}{\text{d}t}+\frac{{\alpha }_T}{l_p{c}_p}E\left(t\right)\right]\delta E\left(t\right)\text{d}t}\right\}}_{{\alpha }^0}\\ =a^{\left(1\right)}\left(\tau \right)\delta E\left(\tau \right)-a^{\left(1\right)}\left(0\right)\delta E\left(0\right)\\ +{\left\{{\displaystyle {\int }_0^{\tau }\delta E\left(t\right)\left[-\frac{\text{d}{a}^{\left(1\right)}\left(t\right)}{\text{d}t}+\frac{{\alpha }_T}{l_p{c}_p}E\left(t\right)a^{\left(1\right)}\left(t\right)\right]\text{d}t}\right\}}_{{\alpha }^0}.\end{array}$ (25)

Identifying the left-side of Equation (25) with the G-differential $\delta E\left(\tau \right)$ of the response $E\left(\tau \right)$ and eliminating the unknown value $\delta E\left(\tau \right)$ from the right-side of Equation (25) by setting $a^{\left(1\right)}\left(\tau \right)=0$ yields the following 1^st-Level Adjoint Sensitivity System (1^st-LASS) for the 1^st-level adjoint sensitivity function $a^{\left(1\right)}\left(t\right)$ :

${\left\{\left[-\frac{\text{d}}{\text{d}t}+\frac{{\alpha }_T}{l_p{c}_p}E\left(t\right)\right]a^{\left(1\right)}\left(t\right)\right\}}_{{\alpha }^0}=\delta \left(t-\tau \right),\text{}\text{}\text{}t>0$ , (26)

$a^{\left(1\right)}\left(\tau \right)=0,t=\tau$ . (27)

For further reference, solving the above 1^st-LASS yields the following closed-form expression for the 1st-level adjoint sensitivity function $a^{\left(1\right)}\left(t\right)$ :

$a^{\left(1\right)}\left(t\right)=H\left(\tau -t\right){\left\{\frac{\cosh \left[t\theta \left(\alpha \right)\right]}{\cosh \left[\tau \theta \left(\alpha \right)\right]}\right\}}^2$ , (28)

where $H\left(t-t_f\right)$ denotes the Heaviside functional.

Using the relations provided by the 1^st-LVSS and the 1^st-LASS in Equation (24), and recalling Equation (18), yields the following alternate expression of $\delta E\left(\tau \right)$ in terms of the 1^st-Level adjoint sensitivity function $a^{\left(1\right)}\left(t\right)$ :

$\begin{array}{l}\delta E\left(\tau \right)={\left\{-\frac{\delta {\alpha }_T}{2l_p{c}_p}+\frac{{\alpha }_T}{2l_p{\left(c_p\right)}^2}\delta c_p+\frac{{\alpha }_T}{2{\left(l_p\right)}^2{c}_p}\delta l_p\right\}}_{{\alpha }^0}{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)E^2\left(t\right)\text{d}t}\\ +{\left\{{\phi }_0{\sigma }_f{N}_f\left(\delta \gamma \right)+{\phi }_0\gamma N_f\left(\delta {\sigma }_f\right)+{\phi }_0\gamma {\sigma }_f\left(\delta N_f\right)+\gamma {\sigma }_f{N}_f\left(\delta {\phi }_0\right)\right\}}_{{\alpha }^0}{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)\text{d}t}.\end{array}$

(29)

It follows from Equation (29) that the first-order sensitivities of the response $E\left(\tau \right)$ with respect to the model parameters have the following expressions in terms of the 1^st-Level adjoint sensitivity function $a^{\left(1\right)}\left(t\right)$ :

$\begin{array}{c}\frac{\partial E\left(\tau \right)}{\partial {\alpha }_T}=-\frac{1}{2l_p{c}_p}{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)E^2\left(t\right)\text{d}t}\\ =-\frac{K^2\left(\alpha \right)}{2l_p{c}_p}\left\{\frac{\tanh \left[\tau \theta \left(\alpha \right)\right]}{2\theta \left(\alpha \right)}-\frac{\tau }{2{\cosh }^2\left[\tau \theta \left(\alpha \right)\right]}\right\}\\ =\frac{\tau {\phi }_0\gamma {\sigma }_f{N}_f}{\left(2{\alpha }_T\right){\cosh }^2\left[\tau \theta \left(\alpha \right)\right]}-{\left[\frac{2{\phi }_0\gamma {\sigma }_f{N}_f{l}_p{c}_p}{{\alpha }_T}\right]}^{1/2}\frac{\tanh \left[\tau \theta \left(\alpha \right)\right]}{2{\alpha }_T};\end{array}$ (30)

$\begin{array}{c}\frac{\partial E\left(\tau \right)}{\partial l_p}=\frac{{\alpha }_T}{2{\left(l_p\right)}^2{c}_p}{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)E^2\left(t\right)\text{d}t}\\ ={\left[\frac{{\phi }_0\gamma {\sigma }_f{N}_f{c}_p}{2{\alpha }_T{l}_p}\right]}^{1/2}\tanh \left[\tau \theta \left(\alpha \right)\right]-\frac{\tau {\phi }_0\gamma {\sigma }_f{N}_f}{\left(2l_p\right){\cosh }^2\left[\tau \theta \left(\alpha \right)\right]};\end{array}$ (31)

$\begin{array}{c}\frac{\partial E\left(\tau \right)}{\partial c_p}=\frac{{\alpha }_T}{2l_p{\left(c_p\right)}^2}{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)E^2\left(t\right)\text{d}t}\\ ={\left[\frac{{\phi }_0\gamma {\sigma }_f{N}_f{l}_p}{2{\alpha }_T{c}_p}\right]}^{1/2}\tanh \left[\tau \theta \left(\alpha \right)\right]-\frac{\tau {\phi }_0\gamma {\sigma }_f{N}_f}{\left(2c_p\right){\cosh }^2\left[\tau \theta \left(\alpha \right)\right]};\end{array}$ (32)

$\frac{\partial E\left(\tau \right)}{\partial {\phi }_0}=\gamma {\sigma }_f{N}_f{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)\text{d}t}=\gamma {\sigma }_f{N}_f\left\{\frac{\tanh \left[\tau \theta \left(\alpha \right)\right]}{2\theta \left(\alpha \right)}+\frac{\tau }{2{\cosh }^2\left[\tau \theta \left(\alpha \right)\right]}\right\}$ (33)

$\frac{\partial E\left(\tau \right)}{\partial \gamma }={\phi }_0{\sigma }_f{N}_f{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)\text{d}t}={\phi }_0{\sigma }_f{N}_f\left\{\frac{\tanh \left[\tau \theta \left(\alpha \right)\right]}{2\theta \left(\alpha \right)}+\frac{\tau }{2{\cosh }^2\left[\tau \theta \left(\alpha \right)\right]}\right\}$ (34)

$\frac{\partial E\left(\tau \right)}{\partial {\sigma }_f}={\phi }_0\gamma N_f{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)\text{d}t}=N_f{\phi }_0\gamma \left\{\frac{\tanh \left[\tau \theta \left(\alpha \right)\right]}{2\theta \left(\alpha \right)}+\frac{\tau }{2{\cosh }^2\left[\tau \theta \left(\alpha \right)\right]}\right\}$ (35)

$\frac{\partial E\left(\tau \right)}{\partial N_f}={\phi }_0\gamma {\sigma }_f{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)\text{d}t}={\sigma }_f{\phi }_0\gamma \left\{\frac{\tanh \left[\tau \theta \left(\alpha \right)\right]}{2\theta \left(\alpha \right)}+\frac{\tau }{2{\cosh }^2\left[\tau \theta \left(\alpha \right)\right]}\right\}$ (36)

The following formulas have been used to obtain the expressions in Equations (30)-(33): ${\displaystyle \int {\sinh }^2\left(ax\right)\text{d}x}=\sinh \left(2ax\right)/4a-x/2$ and ${\displaystyle \int {\cosh }^2\left(ax\right)\text{d}x}=\sinh \left(2ax\right)/4a+x/2$ .

3.2. Application of the 1^st-FASAM-N to Obtain the First-Order Sensitivities of the Response $E\left(\tau \right)$ with Respect to the Features and Model Parameters

The form of Equation (10) indicates that the “features” (i.e., functions) of model parameters characterizing this balance equation can be chosen as follows:

$f_1\left(\alpha \right)\triangleq \frac{{\alpha }_T}{2l_p{c}_p};f_2\left(\alpha \right)\triangleq {\phi }_0\gamma {\sigma }_f{N}_f;f\left(\alpha \right)\triangleq {\left[f_1\left(\alpha \right),f_2\left(\alpha \right)\right]}^{†}.$ (37)

Consequently, Equation (10) can alternatively be written in terms of the “feature function” $f\left(\alpha \right)\triangleq {\left[f_1\left(\alpha \right),f_2\left(\alpha \right)\right]}^{†}$ as follows:

$\frac{\text{d}E\left(t\right)}{\text{d}t}=-f_1\left(\alpha \right)E^2\left(t\right)+f_2\left(\alpha \right),E\left(0\right)=0$ . (38)

In terms of the feature function $f\left(\alpha \right)\triangleq {\left[f_1\left(\alpha \right),f_2\left(\alpha \right)\right]}^{†}$ , the solution of Equation (38) has the following form:

$E\left(t\right)={\left[\frac{f_2\left(\alpha \right)}{f_1\left(\alpha \right)}\right]}^{1/2}\tanh \left[tg\left(\alpha \right)\right];g\left(\alpha \right)\triangleq \sqrt{f_1\left(\alpha \right)f_2\left(\alpha \right)}$ . (39)

Taking the G-differential of Equation (38) yields the following 1st-Level Variational Sensitivity System (1^st-LVSS) for the variational function $\delta E\left(t\right)$ :

$\frac{\text{d}}{\text{d}\epsilon }{\left\{\frac{\text{d}\left[E\left(t\right)+\epsilon \delta E\left(t\right)\right]}{\text{d}t}+\left[f_1+\epsilon {\left(\delta f\right)}_1\right]{\left[E\left(t\right)+\epsilon \delta E\left(t\right)\right]}^2-\left[f_2+\epsilon \left(\delta f_2\right)\right]\right\}}_{\epsilon =0}=0$

(40)

$\frac{\text{d}}{\text{d}\epsilon }{\left\{{\left[E\left(t\right)+\epsilon \delta E\left(t\right)\right]}_{t=0}\right\}}_{\epsilon =0}=0$ . (41)

Performing the operations indicated in Equations (40) and (41) yields the following expressions for the 1st-LVSS:

${\left\{\left[\frac{\text{d}}{\text{d}t}+2f_{1}E\left(t\right)\right]\delta E\left(t\right)\right\}}_{f^0}={\left\{-\delta f_1{E}^2\left(t\right)+\delta f_2\right\}}_{f^0},t>0$ , (42)

$\delta E\left(0\right)=0,t=0.$ (43)

In Equation (42), the notation ${\left\{\right\}}_{f^0}$ indicates that the quantity enclosed within the braces is to be evaluated at the nominal values $f^0\triangleq {\left(f_1^0,f_2^0\right)}^{†}$ , $f_1^0\triangleq f_1\left({\alpha }^0\right)$ , $f_2^0\triangleq f_2\left({\alpha }^0\right)$ , of the components of the feature function $f\left(\alpha \right)$ . The 1^st-LVSS would need to be solved anew for all variations $\delta f_1$ , $\delta f_2$ , in the components of the feature function $f\left(\alpha \right)$ . This need for repeatedly solving the 1^st-LVSS can be avoided by constructing the corresponding 1^st-Level Adjoint Sensitivity System (1^st-LASS). Note that the left-side of Equation (42) is the same as the left-side of Equation (21). Therefore, the Hilbert space appropriate for the construction of the 1^st-LASS corresponding to Equation (42) is the same as for the application of the 1^st-CASAM-N, being endowed with the inner product defined in Equation (23). It is therefore also expected that the left-side of the 1^st-LASS to be constructed for Equation (42) will be the same as the left-side of Equation (26). Thus, forming the inner product of Equation(42) with a yet undefined function $a^{(1)}\left(t\right)$ yields the following relation:

$\begin{array}{l}{\left\{{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)\left[\frac{\text{d}}{\text{d}t}+2f_{1}E\left(t\right)\right]\delta E\left(t\right)\text{d}t}\right\}}_{f^0}\\ ={\left\{-\left(\delta f_1\right){\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)E^2\left(t\right)\text{d}t}+\left(\delta f_2\right){\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)\text{d}t}\right\}}_{f^0}.\end{array}$ (44)

Integrating by parts the left side of Equation (44) yields the following relation:

$\begin{array}{l}{\left\{{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)\left[\frac{\text{d}}{\text{d}t}+2f_{1}E\left(t\right)\right]\delta E\left(t\right)\text{d}t}\right\}}_{{\alpha }^0}\\ =a^{\left(1\right)}\left(\tau \right)\delta E\left(\tau \right)-a^{\left(1\right)}\left(0\right)\delta E\left(0\right)+{\left\{{\displaystyle {\int }_0^{\tau }\delta E\left(t\right)\left[-\frac{\text{d}{a}^{\left(1\right)}\left(t\right)}{\text{d}t}+2f_{1}E\left(t\right)a^{\left(1\right)}\left(t\right)\right]\text{d}t}\right\}}_{{\alpha }^0}.\end{array}$

(45)

Identifying the left-side of Equation (45) with the G-differential $\delta E\left(\tau \right)$ of the response $E\left(\tau \right)$ obtained in Equation (18), and eliminating the unknown value $\delta E\left(\tau \right)$ from the right-side of Equation (45) by setting $a^{\left(1\right)}\left(\tau \right)=0$ yields the following 1^st-Level Adjoint Sensitivity System (1st-LASS) for the 1^st-Level adjoint sensitivity function $a^{\left(1\right)}\left(t\right)$ :

${\left\{\left[-\frac{\text{d}}{\text{d}t}+2f_{1}E\left(t\right)\right]a^{\left(1\right)}\left(t\right)\right\}}_{f^0}=\delta \left(t-\tau \right),\text{}\text{}\text{}t>0$ , (46)

$a^{\left(1\right)}\left(\tau \right)=0,t=\tau$ . (47)

Note that the above 1^st-LASS, comprising Equations (46) and (47), is the same as the 1^st-LASS obtained for determining the sensitivities of the response with respect to the model parameters, i.e., Equations (26) and (27). This outcome is expected since the 1^st-LASS is independent of any variations in the model parameters and hence, variations in the feature functions. Therefore, the use of the same symbol, $a^{\left(1\right)}\left(t\right)$ , for the 1^st-level adjoint sensitivity function, which is the solution of either of these 1^st-LASS, is justified. In terms of the feature function $f\left(\alpha \right)$ , the 1^st-level adjoint sensitivity function $a^{\left(1\right)}\left(t\right)$ has the following closed-form expression:

$a^{\left(1\right)}\left(t\right)=H\left(\tau -t\right){\left\{\frac{\cosh \left[t{\left(f_1{f}_2\right)}^{1/2}\right]}{\cosh \left[\tau {\left(f_1{f}_2\right)}^{1/2}\right]}\right\}}^2$ , (48)

where $H\left(t-\tau \right)$ denotes the Heaviside functional.

Using Equations (45)-(47) in Equation (44) yields the following expression for the first-order total G-differential $\delta E\left(\tau \right)$ of the response $E\left(\tau \right)$ in terms of the 1st-level adjoint function $a^{\left(1\right)}\left(t\right)$ :

$\delta E\left(\tau \right)={\left\{-\left(\delta f_1\right){\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)E^2\left(t\right)\text{d}t}+\left(\delta f_2\right){\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)\text{d}t}\right\}}_{{\alpha }^0}$ . (49)

It follows from Equations (49), (48) and (39) that the two sensitivities of the response $E\left(\tau \right)$ with respect to the two components of the feature function $f\triangleq {\left(f_1,f_2\right)}^{†}$ have the following expressions:

$\begin{array}{c}\frac{\partial E\left(\tau \right)}{\partial f_1}=-{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)E^2\left(t\right)}\text{d}t\\ =\frac{1}{2}{\left[\frac{f_2\left(\alpha \right)}{f_1\left(\alpha \right)}\right]}^{1/2}\left\{\frac{\tau }{{\cosh }^2\left[\tau g\left(\alpha \right)\right]}-\frac{\tanh \left[\tau g\left(\alpha \right)\right]}{g\left(\alpha \right)}\right\};\end{array}$ (50)

$\frac{\partial E\left(\tau \right)}{\partial f_2}={\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)\text{d}t}=\frac{1}{2g\left(\alpha \right)}\tanh \left[\tau g\left(\alpha \right)\right]+\frac{\tau }{2{\cosh }^2\left[\tau g\left(\alpha \right)\right]}$ . (51)

The above expressions are to be evaluated at the nominal parameter values but the notation ${\left\{\right\}}_{{\alpha }^0}$ has been omitted, for simplicity. The expressions obtained in Equations (50) and (51) can be verified by differentiating the expression provided in Equation (39), evaluated at a user-chosen time $t=\tau$ within the interval $0<\tau <\infty$ .

The sensitivities of the response $E\left(\tau \right)$ with respect to the model parameters are obtained by using the general relationship:

$\frac{\partial E\left(\tau ;f_1;f_2\right)}{\partial {\alpha }_i}=\frac{\partial E\left(\tau \right)}{\partial f_1}\frac{\partial f_1\left(\alpha \right)}{\partial {\alpha }_i}+\frac{\partial E\left(\tau \right)}{\partial f_2}\frac{\partial f_2\left(\alpha \right)}{\partial {\alpha }_i};i=1,\cdots ,7.$ (52)

Using Equations (50) and (51) while recalling the definitions of the feature functions $f_1\left(\alpha \right)$ and $f_2\left(\alpha \right)$ defined in Equation (37) yields the explicit formulas for the particular cases of Equation (52), as follows:

$\frac{\partial E\left(\tau \right)}{\partial {\alpha }_T}=\frac{\partial E\left(\tau \right)}{\partial f_1}\frac{\partial f_1}{\partial {\alpha }_T}+\frac{\partial E\left(\tau \right)}{\partial f_2}\frac{\partial f_2}{\partial {\alpha }_T}=\frac{1}{2l_p{c}_p}\frac{\partial E\left(\tau \right)}{\partial f_1}$ ; (53)

$\frac{\partial E\left(\tau \right)}{\partial l_p}=\frac{\partial E\left(\tau \right)}{\partial f_1}\frac{\partial f_1}{\partial l_p}+\frac{\partial E\left(\tau \right)}{\partial f_2}\frac{\partial f_2}{\partial l_p}=-\frac{{\alpha }_T}{2{\left(l_p\right)}^2{c}_p}\frac{\partial E\left(\tau \right)}{\partial f_1}$ ; (54)

$\frac{\partial E\left(\tau \right)}{\partial c_p}=\frac{\partial E\left(\tau \right)}{\partial f_1}\frac{\partial f_1}{\partial c_p}+\frac{\partial E\left(\tau \right)}{\partial f_2}\frac{\partial f_2}{\partial c_p}=-\frac{{\alpha }_T}{2{\left(c_p\right)}^2{l}_p}\frac{\partial E\left(\tau \right)}{\partial f_1}$ ; (55)

$\frac{\partial E\left(\tau \right)}{\partial {\phi }_0}=\frac{\partial E\left(\tau \right)}{\partial f_1}\frac{\partial f_2}{\partial {\phi }_0}+\frac{\partial E\left(\tau \right)}{\partial f_2}\frac{\partial f_2}{\partial {\phi }_0}=\gamma {\sigma }_f{N}_f\frac{\partial E\left(\tau \right)}{\partial f_2}$ ; (56)

$\frac{\partial E\left(\tau \right)}{\partial \gamma }=\frac{\partial E\left(\tau \right)}{\partial f_1}\frac{\partial f_1}{\partial \gamma }+\frac{\partial E\left(\tau \right)}{\partial f_2}\frac{\partial f_2}{\partial \gamma }={\phi }_0{\sigma }_f{N}_f\frac{\partial E\left(\tau \right)}{\partial f_2}$ ; (57)

$\frac{\partial E\left(\tau \right)}{\partial {\sigma }_f}=\frac{\partial E\left(\tau \right)}{\partial f_1}\frac{\partial f_1}{\partial {\sigma }_f}+\frac{\partial E\left(\tau \right)}{\partial f_2}\frac{\partial f_2}{\partial {\sigma }_f}={\phi }_0\gamma N_f\frac{\partial E\left(\tau \right)}{\partial f_2}$ ; (58)

$\frac{\partial E\left(\tau \right)}{\partial N_f}=\frac{\partial E\left(\tau \right)}{\partial f_1}\frac{\partial f_1}{\partial N_f}+\frac{\partial E\left(\tau \right)}{\partial f_2}\frac{\partial f_2}{\partial N_f}={\phi }_0\gamma {\sigma }_f\frac{\partial E\left(\tau \right)}{\partial f_2}$ . (59)

3.3. Comparative Discussion: Applying the 1^st-CASAM-N versus the 1^st-FASAM-N for Computing the First-Order Response Sensitivities to Model Parameters

Both the 1^st-CASAM-N and the 1^st-FASAM-N require the solution of the same 1^st-LASS. Hence, the application of the 1^st-CASAM-N necessitates a single large-scale computation (for solving the 1^st-LASS) to obtain all of the 7 first-order sensitivities for the instant energy response to the model parameters. The application of the 1^st-FASAM-N also necessitates a single large-scale computation (for solving the same 1^st-LASS as for the 1^st-CASAM-N) to obtain the two first-order sensitivities of the model’s response to the model’s chosen feature functions. This equivalence between the application of the 1^st-FASAM-N and the 1^st-CASAM-N is as expected, since the 1^st-LASS is independent of parameter variations or, equivalently, of variations in the feature functions. After the 1^st-level adjoint function has been computed, the computation of the sensitivities to the model parameters using the 1^st-FASAM-N additionally requires two quadratures to compute $\partial E\left(\tau \right)/\partial f_1$ and $\partial E\left(\tau \right)/\partial f_2$ using Equations (50) and (51), respectively, followed by simple differentiations of the feature functions with respect to the component parameters, as shown in Equations (53)-(59). Alternatively, computing the response sensitivities to the same model parameters using the 1^st-CASAM-N additionally requires 7 integrations (quadratures), as shown in Equations (30)-(36). Neither these differentiations nor these quadratures require “large-scale” computations, so the differences in the computational resources needed to apply the 1^st-FASAM-N versus applying the 1^st-CASAM-N are minimal, with a slight advantage towards the 1^st-FASAM-N, since the respective differentiations are computationally somewhat less demanding than the integrations/quadratures required by the application of the 1^st-CASAM-N.

4. Computation of the Second-Order Response Sensitivities with Respect to Model Parameters: Applying the 2^nd-FASAM-N Versus the 2^nd-CASAM-N

The fundamental principle underlying both the 2^nd-FASAM-N and the 2^nd-CASAM-N methodologies is to determine the second-order sensitivities by employing their definition of being the “first-order sensitivities of the first-order sensitivities.” Thus, each first-order sensitivity is treated as a “model response,” and the G-differential of each of these “model responses” subsequently provides the partial second-order sensitivities that stem from the respective first-order sensitivity. As will be highlighted in Section 4.1, below, the computation of the 49 second-order sensitivities (of which 28 are distinct) of the response $E\left(\tau \right)$ with respect to the 7 model parameters will require just two large-scale computations (the solving the corresponding two 2^nd-Level Adjoint Sensitivity Systems) when using the 2^nd-FASAM-N methodology, since there are only two sensitivities of the response $E\left(\tau \right)$ with respect to the “feature functions” $f_1\left(\alpha \right)$ and $f_2\left(\alpha \right)$ . In contradistinction, as will be highlighted in Section 4.2, applying the conventional 2^nd-CASAM-N methodology requires 7 large-scale computations for solving the seven 2^nd-Level Adjoint Sensitivity Systems (one 2^nd-LASS for each of the 7 first-order sensitivities with respect to the primary model parameters) for obtaining all second-order sensitivities of the response $E\left(\tau \right)$ with respect to the model parameters.

4.1. Computation of Second-Order Sensitivities Using the 2^nd-FASAM-N

The determination of the second-order sensitivities that stem from the first-order sensitivities $\partial E\left(\tau \right)/\partial f_1$ of the response $E\left(\tau \right)$ with respect to the “feature function” $f_1\left(\alpha \right)$ is presented in Subsection 4.1.1, while the second-order sensitivities that arise from the first-order sensitivities $\partial E\left(\tau \right)/\partial f_2$ of the response $E\left(\tau \right)$ with respect to the “feature function” $f_2\left(\alpha \right)$ will be presented in Subsection 4.1.2.

4.1.1. Computation of Second-Order Sensitivities Stemming from the First-Order Sensitivity $\partial E\left(\tau \right)/\partial f_1$

The 2^nd-order sensitivities which stem from the 1^st-order sensitivity $\partial E\left(\tau \right)/\partial f_1$ defined in Equation (50) will be obtained by determining the first-order G-differential $\delta \left\{\partial E\left(\tau \right)/\partial f_1\right\}$ of $\partial E\left(\tau \right)/\partial f_1$ . By definition, the first-order G-differential of $\partial E\left(\tau \right)/\partial f_1$ is obtained as follows:

$\begin{array}{c}\delta \left\{\frac{\partial E\left(\tau ;f_1,f_2\right)}{\partial f_1}\right\}=-{\left\{\frac{\text{d}}{\text{d}\epsilon }\left[{\displaystyle {\int }_0^{\tau }\left(a^{\left(1\right)}+\epsilon \delta a^{\left(1\right)}\right){\left[E\left(t\right)+\epsilon \delta E\left(t\right)\right]}^2\text{d}t}\right]\right\}}_{\epsilon =0}\\ =-{\displaystyle {\int }_0^{\tau }\left[2a^{\left(1\right)}\left(t\right)E\left(t\right)\delta E\left(t\right)+\delta a^{\left(1\right)}\left(t\right)E^2\left(t\right)\right]\text{d}t}\\ =\frac{{\partial }^{2}E\left(\tau ;f_1,f_2\right)}{\partial f_1\partial f_1}\delta f_1+\frac{{\partial }^{2}E\left(\tau ;f_1,f_2\right)}{\partial f_2\partial f_1}\delta f_2.\end{array}$ (60)

The variational function $\delta a^{\left(1\right)}\left(t\right)$ is the solution of the system of equations obtained by G-differentiating the 1^st-LASS defined in Equations (46) and (47). Performing the G-differentiation of this 1^st-LASS yields the following equations:

$\begin{array}{l}{\left\{\left[-\frac{\text{d}}{\text{d}t}+2f_{1}E\left(t\right)\right]\delta a^{\left(1\right)}\left(t\right)+2f_1{a}^{\left(1\right)}\left(t\right)\left[\delta E\left(t\right)\right]\right\}}_{f^0}\\ =-2{\left\{\left(\delta f_1\right)a^{\left(1\right)}\left(t\right)E\left(t\right)\right\}}_{f^0},0<t<\tau ,\end{array}$ (61)

$\delta a^{\left(1\right)}\left(\tau \right)=0,t=\tau$ . (62)

Concatenating Equations (61) and (62) with the 1^st-LVSS for $\delta E\left(t\right)$ defined in Equations (42) and (43) yields the following 2^nd-Level Variational Sensitivity System (2^nd-LVSS) for the 2^nd-Level variational function

$V^{\left(2\right)}\left(2;t\right)\triangleq {\left[v^{\left(2\right)}\left(1;t\right),v^{\left(2\right)}\left(2;t\right)\right]}^{†}\triangleq {\left[\delta E\left(t\right),\delta a^{\left(1\right)}\left(t\right)\right]}^{†}$ :

${\left\{V{M}^{\left(2\right)}\left[2\times 2;f\right]V^{\left(2\right)}\left(2;t\right)\right\}}_{f^0}={\left\{Q_V^{\left(2\right)}\left[2;f;\delta f\right]\right\}}_{f^0},0<t<\tau ,$ (63)

${\left\{B_V^{\left(2\right)}\left[2;V^{\left(2\right)}\left(2;t\right);f;\delta f\right]\right\}}_{f^0}=0\left[2\right],0\left[2\right]\triangleq {\left[0,0\right]}^{†},$ (64)

where

$V{M}^{\left(2\right)}\left[2\times 2;f\right]\triangleq \left(\begin{array}{cc}\frac{\text{d}}{\text{d}t}+2f_{1}E\left(t\right)& 0\\ 2f_1{a}^{\left(1\right)}\left(t\right)& -\frac{\text{d}}{\text{d}t}+2f_{1}E\left(t\right)\end{array}\right)$ ; (65)

$Q_V^{\left(2\right)}\left[2;f;\delta f\right]\triangleq \left[\begin{array}{c}-\left(\delta f_1\right)E^2\left(t\right)+\left(\delta f_2\right)\\ -2\left(\delta f_1\right)a^{\left(1\right)}\left(t\right)E\left(t\right)\end{array}\right]$ ; (66)

$B_V^{\left(2\right)}\left[2;V^{\left(2\right)}\left(2;t\right);f;\delta f\right]\triangleq \left(\begin{array}{c}\delta E\left(0\right)\\ \delta a^{\left(1\right)}\left(t_f\right)\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right)$ . (67)

The need for solving the 2^nd-LVSS is circumvented by deriving an alternative expression for the first-order G-differential $\delta \left\{\partial E\left(\tau \right)/\partial f_1\right\}$ defined in Equation (60), in which the variational function $V^{\left(2\right)}\left(2;t\right)\triangleq {\left[v^{\left(2\right)}\left(1;t\right),v^{\left(2\right)}\left(2;t\right)\right]}^{†}\triangleq {\left[\delta E\left(t\right),\delta a^{\left(1\right)}\left(t\right)\right]}^{†}$ is replaced by a 2^nd-level adjoint function which will be denoted as $A^{\left(2\right)}\left(2;1;t\right)\triangleq {\left[a^{\left(2\right)}\left(1;1;t\right),a^{\left(2\right)}\left(2;1;t\right)\right]}^{†}\in H_2$ . The notation for $A^{\left(2\right)}\left(2;1;t\right)\triangleq {\left[a^{\left(2\right)}\left(1;1;t\right),a^{\left(2\right)}\left(2;1;t\right)\right]}^{†}\in H_2$ has the following significance: (i) the bold letter “A” indicates a vector-valued “adjoint” function; (ii) the superscript “(2)” indicates “second-level”; (iii) the first argument, denoted as “2”, in $A^{\left(2\right)}\left(2;1;t\right)$ indicates that this vector has 2 components, denoted as $a^{\left(2\right)}\left(i;1;t\right)$ , $i=1,2$ , each of which is a scalar-valued function of time; (iv) the second argument of $A^{\left(2\right)}\left(2;1;t\right)$ , denoted as “1”, indicates that this 2nd-level adjoint function corresponds to the first-order sensitivity $\partial E\left(\tau \right)/\partial f_1$ of the response with respect to the “first feature function”, $f_1\left(\alpha \right)$ . In Equation (63) and in the remainder of this work, matrices will be denoted by using two bold capital letters.

The 2^nd-level adjoint function $A^{\left(2\right)}\left(2;1;t\right)$ will be the solution of a 2^nd-Level Adjoint Sensitivity System (2^nd-LASS) to be constructed by applying the 2^nd-FASAM-N methodology. This 2^nd-LASS is constructed in a Hilbert space, denoted as $H_2$ , which comprises as elements block-vectors of the same form as $V^{\left(2\right)}\left(2;t\right)$ , and is endowed with the following inner product of two vectors

${\Psi }^{\left(2\right)}\left(2;t\right)\triangleq {\left[{\psi }^{\left(2\right)}\left(1;t\right),{\psi }^{\left(2\right)}\left(2;t\right)\right]}^{†}\in H_2$ and

${\Phi }^{\left(2\right)}\left(t\right)\triangleq {\left[{\phi }^{\left(2\right)}\left(1;t\right),{\phi }^{\left(2\right)}\left(2;t\right)\right]}^{†}\in H_2$ :

${\langle {\Psi }^{\left(2\right)}\left(2;t\right),{\Phi }^{\left(2\right)}\left(2;t\right)\rangle }_2\triangleq {\displaystyle \underset{i=1}{\overset{2}{\sum }}{\displaystyle {\int }_0^{\tau }{\psi }^{\left(2\right)}\left(i;t\right){\phi }^{\left(2\right)}\left(i;t\right)\text{d}t}}$ . (68)

The inner product defined in Equation (68) is now used to construct the 2^nd-Level Adjoint Sensitivity System (2^nd-LASS) for the 2^nd-level adjoint function $A^{\left(2\right)}\left(2;t\right)\triangleq {\left[a^{\left(2\right)}\left(1;t\right),a^{\left(2\right)}\left(2;t\right)\right]}^{†}\in H_2$ , as follows:

i) Using the inner product defined in Equation (68), form the inner product of $A^{\left(2\right)}\left(2;1;t\right)\triangleq {\left[a^{\left(2\right)}\left(1;1;t\right),a^{\left(2\right)}\left(2;1;t\right)\right]}^{†}\in H_2$ with Equation (63), and subsequently integrate by parts the left-side of the resulting equation to obtain the following relation:

$\begin{array}{l}{\left\{{\langle A^{\left(2\right)}\left(2;1;t\right),V{M}^{\left(2\right)}\left[2\times 2;f\right]V^{\left(2\right)}\left(2;t\right)\rangle }_2\right\}}_{f^0}\\ ={\left\{{\langle A^{\left(2\right)}\left(2;1;t\right),Q_V^{\left(2\right)}\left(2;f;\delta f\right)\rangle }_2\right\}}_{f^0}\\ ={\left\{a^{\left(2\right)}\left(1;1;t\right)\delta E\left(t\right)-a^{\left(2\right)}\left(2;1;t\right)\delta a^{\left(1\right)}\left(t\right)\right\}}_{t=0}^{t=\tau }\\ +{\left\{{\langle V^{\left(2\right)}\left(2;t\right),A{M}^{\left(2\right)}\left[2\times 2;f\right]A^{\left(2\right)}\left(2;1;t\right)\rangle }_2\right\}}_{f^0}.\end{array}$ (69)

where the operator $A{M}^{\left(2\right)}\left[2\times 2;f\right]$ represents the formal adjoint of the operator $V{M}^{\left(2\right)}\left(2\times 2;f\right)$ , i.e., $A{M}^{\left(2\right)}\left[2\times 2;f\right]\triangleq {\left[V{M}^{\left(2\right)}\left(2\times 2;f\right)\right]}^{*}$ , and is defined as follows:

$\begin{array}{c}A{M}^{\left(2\right)}\left[2\times 2;f\right]\triangleq {\left[V{M}^{\left(2\right)}\left(2\times 2;f\right)\right]}^{*}\\ \triangleq \left(\begin{array}{cc}-\frac{\text{d}}{\text{d}t}+2f_1\left(\alpha \right)E\left(t\right)& 2f_1\left(\alpha \right)a^{\left(1\right)}\left(t\right)\\ 0& \frac{\text{d}}{\text{d}t}+2f_1\left(\alpha \right)E\left(t\right)\end{array}\right).\end{array}$ (70)

ii) Eliminate the boundary terms on the right-side of the second equality in Equation (69) and require the last term on the right-side of the second equality in Equation (69) to represent the right-side of Equation (60) by imposing the following relations:

${\left\{A{M}^{\left(2\right)}\left[2\times 2;f\right]A^{\left(2\right)}\left(2;1;t\right)\right\}}_{f^0}={\left\{\left(\begin{array}{c}-2a^{\left(1\right)}\left(t\right)E\left(t\right)\\ -E^2\left(t\right)\end{array}\right)\right\}}_{f^0},0<t<\tau ,$ (71)

${\left\{B_A^{\left(2\right)}\left[2;A^{\left(2\right)}\left(2;1;t\right);\alpha \right]\right\}}_{f^0}\triangleq {\left(\begin{array}{c}{a}^{\left(2\right)}\left(1;1;\tau \right)\\ a^{\left(2\right)}\left(2;1;0\right)\end{array}\right)}_{f^0}=\left(\begin{array}{c}0\\ 0\end{array}\right)$ . (72)

The relations represented by Equations (71) and (72) constitute the 2^nd-LASS for the 2^nd-Level adjoint function $A^{\left(2\right)}\left(2;1;t\right)$ . Notably, the 2^nd-LASS is independent of variations in the feature functions (and/or parameter variations), so it needs to be solved just once to obtain the 2^nd-Level adjoint function $A^{\left(2\right)}\left(2;1;t\right)$ . Furthermore, the 2^nd-LASS is an upper-triangular system, so the equations need not solved simultaneously, but can be solved sequentially, first for the component $a^{\left(2\right)}\left(2;1;t\right)$ and subsequently for the component $a^{\left(2\right)}\left(1;1;t\right)$ .

Solving Equations (71) and (72) yields the following closed-form expressions for the components of the 2^nd-Level adjoint sensitivity function $A^{\left(2\right)}\left(2;1;t\right)$ :

$\begin{array}{l}{a}^{\left(2\right)}\left(1;1;t\right)=\frac{{\displaystyle {\int }_t^{\tau }\left[2a^{\left(1\right)}\left(x\right)E\left(x\right)-2f_1{a}^{\left(2\right)}\left(2;1;x\right)\right]{\cosh }^2\left[xg\left(\alpha \right)\right]\text{d}x}}{{\cosh }^2\left[tg\left(\alpha \right)\right]}\\ ={\cosh }^{-2}\left[tg\left(\alpha \right)\right]\{\frac{2}{f_1\left(\alpha \right)}\left[1-\frac{\cosh \left[tg\left(\alpha \right)\right]}{\cosh \left[\tau g\left(\alpha \right)\right]}\right]\\ -\frac{2}{3f_1\left(\alpha \right)}[\frac{1}{2}{\cosh }^2\left[\tau g\left(\alpha \right)\right]-\frac{1}{2}{\cosh }^2\left[tg\left(\alpha \right)\right]-\ln \frac{\cosh \left[\tau g\left(\alpha \right)\right]}{\cosh \left[tg\left(\alpha \right)\right]}]\};\end{array}$ (73)

$\begin{array}{c}{a}^{\left(2\right)}\left(2;1;t\right)={\cosh }^2\left[tg\left(\alpha \right)\right]{\displaystyle {\int }_0^t{E}^2\left(x\right){\cosh }^{-2}\left[xg\left(\alpha \right)\right]\text{d}x}\\ =\frac{1}{3g\left(\alpha \right)}\frac{f_2\left(\alpha \right)}{f_1\left(\alpha \right)}\tanh \left[tg\left(\alpha \right)\right]{\sinh }^2\left[tg\left(\alpha \right)\right].\end{array}$ (74)

iii) Use the relations provided by the 2^nd-LVSS and the 2^nd-LASS in Equation (69) to obtain the following expression for the variation $\delta \left\{\partial E\left(\tau \right)/\partial f_1\right\}$ in terms of the 2^nd-level adjoint function $A^{\left(2\right)}\left(2;1;t\right)$ :

$\begin{array}{l}\delta \left\{\frac{\partial E\left(\tau \right)}{\partial f_1}\right\}=\frac{{\partial }^{2}E\left(\tau ;f_1,f_2\right)}{\partial f_1\partial f_1}\delta f_1+\frac{{\partial }^{2}E\left(\tau ;f_1,f_2\right)}{\partial f_2\partial f_1}\delta f_2\\ ={\left\{{\langle A^{\left(2\right)}\left(2;1;t\right),Q_V^{\left(2\right)}\left(2;f;\delta f\right)\rangle }_2\right\}}_{f^0}\\ ={\displaystyle {\int }_0^{\tau }\text{d}t\left\{a^{\left(2\right)}\left(1;1;t\right)\left[-\left(\delta f_1\right)E^2\left(t\right)+\left(\delta f_2\right)\right]-a^{\left(2\right)}\left(2;1;t\right)2\left(\delta f_1\right)a^{\left(1\right)}\left(t\right)E\left(t\right)\right\}}.\end{array}$ (75)

It follows from Equation (75) that:

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial f_1\partial f_1}=-{\displaystyle {\int }_0^{\tau }\left[a^{\left(2\right)}\left(1;1;t\right)E\left(t\right)+2a^{\left(2\right)}\left(2;1;t\right)a^{\left(1\right)}\left(t\right)\right]}E\left(t\right)\text{d}t;$ (76)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial f_2\partial f_1}={\displaystyle {\int }_0^{\tau }{a}^{\left(2\right)}\left(1;1;t\right)\text{d}t}$ . (77)

4.1.2. Computation of Second-Order Sensitivities Stemming from the First-Order Sensitivity $\partial E\left(\tau \right)/\partial f_2$

The 2^nd-order sensitivities which stem from the 1^st-order sensitivity $\partial E\left(\tau \right)/\partial f_2$ defined in Equation (51) will be obtained from the first-order G-differential $\delta \left\{\partial E\left(\tau \right)/\partial f_2\right\}$ of $\partial E\left(\tau \right)/\partial f_2$ . By definition, the first-order G-differential $\delta \left\{\partial E\left(\tau \right)/\partial f_2\right\}$ is obtained as follows:

$\begin{array}{c}\delta \left\{\frac{\partial E\left(\tau ;f_1,f_2\right)}{\partial f_2}\right\}={\left\{\frac{\text{d}}{\text{d}\epsilon }\left[{\displaystyle {\int }_0^{\tau }\left(a^{\left(1\right)}+\epsilon \delta a^{\left(1\right)}\right)\text{d}t}\right]\right\}}_{\epsilon =0}={\displaystyle {\int }_0^{\tau }\delta a^{\left(1\right)}\left(t\right)\text{d}t}\\ =\frac{{\partial }^{2}E\left(\tau ;f_1,f_2\right)}{\partial f_1\partial f_2}\delta f_1+\frac{{\partial }^{2}E\left(\tau ;f_1,f_2\right)}{\partial f_2\partial f_2}\delta f_2.\end{array}$ (78)

The variational function $\delta a^{\left(1\right)}\left(t\right)$ is the solution of Equations (61) and (62). Notably, the right-side of Equation (78) depends only on the variational function $\delta a^{\left(1\right)}\left(t\right)$ , but does not depend directly on the variational function $\delta E\left(t\right)$ . Nevertheless, since the variational function $\delta a^{\left(1\right)}\left(t\right)$ is related to the variational function $\delta E\left(t\right)$ through Equations (61) and (62), the 2^nd-level adjoint function that will be constructed in order to eliminate the appearance of $\delta a^{\left(1\right)}\left(t\right)$ on the right-side of Equation (78) will be the solution of a 2^nd-LASS which will correspond to the 2^nd-LVSS defined by Equations (63) and (64). The construction of the 2^nd-LASS that will be used to eliminate the appearance of the variational function $\delta a^{\left(1\right)}\left(t\right)$ from Equation (78) follows the same steps as in Subsection 4.1, above. The 2^nd-level adjoint function that will be defined for this purpose will be denoted as $A^{\left(2\right)}\left(2;2;t\right)\triangleq {\left[a^{\left(2\right)}\left(1;2;t\right),a^{\left(2\right)}\left(2;2;t\right)\right]}^{†}\in H_2$ , where the notation has the following significance: (i) the bold letter “A” indicates a vector-valued “adjoint” function within the 2^nd-FASAM-N formalism; (ii) the superscript “(2)” indicates “second-level”; (iii) the first argument, i.e., “2”, in $A^{\left(2\right)}\left(2;2;t\right)$ indicates that this vector has 2 components, denoted as $a^{\left(2\right)}\left(i;2;t\right)$ , $i=1,2$ , each of which is a scalar-valued function of time; (iv) the (second) argument of $A^{\left(2\right)}\left(2;2;t\right)$ , denoted as “2”, indicates that this 2^nd-level adjoint function corresponds to the first-order sensitivity $\partial E\left(\tau \right)/\partial f_2$ of the response with respect to the “second feature function”, i.e., $f_2\left(\alpha \right)$ .

The inner product defined in Equation (68) is now used to construct the 2^nd-Level Adjoint Sensitivity System (2^nd-LASS) for the 2^nd-Level adjoint function $A^{\left(2\right)}\left(2;2;t\right)$ by following the same sequence of steps as used in Subsection 4.1, above, but using the expression provided in Equation (78) to determine the right-side (“source”) for the 2^nd-LASS. This procedure leads to the following 2^nd-LASS for the 2^nd-Level adjoint function $A^{\left(2\right)}\left(2;2;t\right)$ :

${\left\{A{M}^{\left(2\right)}\left[2\times 2;f\right]A^{\left(2\right)}\left(2;2;t\right)\right\}}_{f^0}={\left\{\left(\begin{array}{c}0\\ 1\end{array}\right)\right\}}_{f^0},\text{}\text{}\text{}\text{}0<t<\tau ,$ (79)

${\left\{B_A^{\left(2\right)}\left[2;A^{\left(2\right)}\left(2;2;t\right);\alpha \right]\right\}}_{f^0}\triangleq {\left\{\left(\begin{array}{c}{a}^{\left(2\right)}\left(1;2;\tau \right)\\ a^{\left(2\right)}\left(2;2;0\right)\end{array}\right)\right\}}_{f^0}=\left(\begin{array}{c}0\\ 0\end{array}\right)$ . (80)

Solving Equations (79) and (80) yields the following closed-form expressions for the components of the 2^nd-Level adjoint sensitivity function $A^{\left(2\right)}\left(2;2;t\right)$ :

$a^{\left(2\right)}\left(1;2;t\right)=\frac{1}{2f_2\left(\alpha \right)}\left\{-{\cosh }^2\left[tg\left(\alpha \right)\right]+\frac{{\cosh }^4\left[\tau g\left(\alpha \right)\right]}{{\cosh }^2\left[tg\left(\alpha \right)\right]}\right\}$ ; (81)

$a^{\left(2\right)}\left(2;2;t\right)=-\frac{1}{2g\left(\alpha \right)}\sinh \left[2tg\left(\alpha \right)\right]$ . (82)

The expressions in Equations (81) and (82) are to be evaluated at the nominal values of the feature functions (and, implicitly, at the nominal parameter values) but the notation ${\left\{\right\}}_{f^0}$ has been omitted for simplicity.

Using the relations provided by the 2^nd-LVSS and the 2^nd-LASS provided in Equations (79) and (80) for the of the 2^nd-Level adjoint function $A^{\left(2\right)}\left(2;2;t\right)$ leads to the following expression for the variation $\delta \left\{\partial E\left(\tau \right)/\partial f_2\right\}$ in terms of the 2^nd-Level adjoint function $A^{\left(2\right)}\left(2;2;t\right)$ :

$\begin{array}{l}\delta \left\{\frac{\partial E\left(\tau \right)}{\partial f_2}\right\}=\frac{{\partial }^{2}E\left(\tau ;f_1,f_2\right)}{\partial f_1\partial f_2}\delta f_1+\frac{{\partial }^{2}E\left(\tau ;f_1,f_2\right)}{\partial f_2\partial f_2}\delta f_2\\ ={\left\{{\langle A^{\left(2\right)}\left(2;2;t\right),Q_V^{\left(2\right)}\left(2;f;\delta f\right)\rangle }_2\right\}}_{f^0}\\ ={\displaystyle {\int }_0^{\tau }\text{d}t\left\{a^{\left(2\right)}\left(1;2;t\right)\left[-\left(\delta f_1\right)E^2\left(t\right)+\left(\delta f_2\right)\right]-a^{\left(2\right)}\left(2;2;t\right)2\left(\delta f_1\right)a^{\left(1\right)}\left(t\right)E\left(t\right)\right\}}.\end{array}$ (83)

It follows from Equation (83) that:

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial f_1\partial f_2}=-{\displaystyle {\int }_0^{\tau }\left[a^{\left(2\right)}\left(1;2;t\right)E\left(t\right)+2a^{\left(2\right)}\left(2;2;t\right)a^{\left(1\right)}\left(t\right)\right]}E\left(t\right)\text{d}t;$ (84)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial f_2\partial f_2}={\displaystyle {\int }_0^{\tau }{a}^{\left(2\right)}\left(1;2;t\right)\text{d}t}$ . (85)

It is important to note that the mixed second-order partial derivative ${\partial }^{2}E\left(\tau \right)/\partial f_1\partial f_2$ can be obtained by using either Equation (84) or Equation (77). The equivalence between the two respective expressions provides a stringent verification of the accuracy of solving the two 2^nd-LASS, one for $A^{\left(2\right)}\left(2;1;t\right)$ comprising Equations (71) and (72), and the other 2^nd-LASS, comprising Equations (79) and (80) for $A^{\left(2\right)}\left(2;2;t\right)$ .

The second-order sensitivities of the response $E\left(\tau \right)$ with respect to the primary model parameters are obtained by using the parameter-dependencies of the functions $f_1\left(\alpha \right)$ and $f_2\left(\alpha \right)$ , cf. Equation (37), in conjunction with the expressions obtained in Equations (76), (77), (84) and (85) by using the following general formula, which is obtained by taking the total differential of the expression provided in Equation (52):

$\begin{array}{l}\frac{{\partial }^{2}E\left(\tau ;f_1;f_2\right)}{\partial {\alpha }_j\partial {\alpha }_i}=\left[\frac{{\partial }^{2}E\left(\tau \right)}{\partial f_1\partial f_1}\frac{\partial f_1\left(\alpha \right)}{\partial {\alpha }_j}+\frac{{\partial }^{2}E\left(\tau \right)}{\partial f_2\partial f_1}\frac{\partial f_2\left(\alpha \right)}{\partial {\alpha }_j}\right]\frac{\partial f_1\left(\alpha \right)}{\partial {\alpha }_i}+\frac{\partial E\left(\tau \right)}{\partial f_1}\frac{{\partial }^2{f}_1\left(\alpha \right)}{\partial {\alpha }_j\partial {\alpha }_i}\\ +\left[\frac{{\partial }^{2}E\left(\tau \right)}{\partial f_1\partial f_2}\frac{\partial f_1\left(\alpha \right)}{\partial {\alpha }_j}+\frac{{\partial }^{2}E\left(\tau \right)}{\partial f_2\partial f_2}\frac{\partial f_2\left(\alpha \right)}{\partial {\alpha }_j}\right]\frac{\partial f_2\left(\alpha \right)}{\partial {\alpha }_i}+\frac{\partial E\left(\tau \right)}{\partial f_2}\frac{{\partial }^2{f}_2\left(\alpha \right)}{\partial {\alpha }_j\partial {\alpha }_i};\\ i,j=1,\cdots ,7.\end{array}$ (86)

For example, the second-order sensitivities of the response $E\left(\tau \right)$ with respect to the parameter ${\alpha }_T$ are obtained as follows:

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial {\alpha }_T\partial {\alpha }_T}=\frac{\partial }{\partial {\alpha }_T}\left(\frac{\partial E\left(\tau \right)}{\partial f_1}\frac{\partial f_1}{\partial {\alpha }_T}\right)=\frac{{\partial }^{2}E\left(\tau \right)}{\partial f_1\partial f_1}{\left(\frac{\partial f_1}{\partial {\alpha }_T}\right)}^2={\left(\frac{1}{2l_p{c}_p}\right)}^2\frac{{\partial }^{2}E\left(\tau \right)}{\partial f_1\partial f_1}$ ; (87)

$\begin{array}{c}\frac{{\partial }^{2}E\left(\tau \right)}{\partial c_p\partial {\alpha }_T}=\frac{\partial }{\partial c_p}\left(\frac{\partial E\left(\tau \right)}{\partial f_1}\frac{\partial f_1}{\partial {\alpha }_T}\right)=\frac{{\partial }^{2}E\left(\tau \right)}{\partial f_1\partial f_1}\frac{\partial f_1}{\partial c_p}\frac{\partial f_1}{\partial {\alpha }_T}+\frac{\partial E\left(\tau \right)}{\partial f_1}\frac{{\partial }^2{f}_1}{\partial c_p\partial {\alpha }_T}\\ =-\frac{{\alpha }_T}{4{\left(l_p\right)}^2{\left(c_p\right)}^3}\frac{{\partial }^{2}E\left(\tau \right)}{\partial f_1\partial f_1}-\frac{1}{2l_p{\left(c_p\right)}^2}\frac{\partial E\left(\tau \right)}{\partial f_1}.\end{array}$ (88)

$\begin{array}{c}\frac{{\partial }^{2}E\left(\tau \right)}{\partial l_p\partial {\alpha }_T}=\frac{\partial }{\partial l_p}\left(\frac{\partial E\left(\tau \right)}{\partial f_1}\frac{\partial f_1}{\partial {\alpha }_T}\right)=\frac{{\partial }^{2}E\left(\tau \right)}{\partial f_1\partial f_1}\frac{\partial f_1}{\partial l_p}\frac{\partial f_1}{\partial {\alpha }_T}+\frac{\partial E\left(\tau \right)}{\partial f_1}\frac{{\partial }^2{f}_1}{\partial l_p\partial {\alpha }_T}\\ =-\frac{{\alpha }_T}{4{\left(c_p\right)}^2{\left(l_p\right)}^3}\frac{{\partial }^{2}E\left(\tau \right)}{\partial f_1\partial f_1}-\frac{1}{2c_p{\left(l_p\right)}^2}\frac{\partial E\left(\tau \right)}{\partial f_1}.\end{array}$ (89)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial {\phi }_0\partial {\alpha }_T}=\frac{\partial }{\partial {\phi }_0}\left(\frac{\partial E\left(\tau \right)}{\partial f_1}\frac{\partial f_1}{\partial {\alpha }_T}\right)=\frac{1}{2l_p{c}_p}\frac{{\partial }^{2}E\left(\tau \right)}{\partial f_1\partial f_2}\frac{\partial f_2}{\partial {\phi }_0}=\frac{\gamma {\sigma }_f{N}_f}{2l_p{c}_p}\frac{{\partial }^{2}E\left(\tau \right)}{\partial f_1\partial f_2}$ ; (90)

$\frac{{\partial }^{2}E\left(t_f\right)}{\partial \gamma \partial {\alpha }_T}=\frac{\partial }{\partial \gamma }\left(\frac{\partial E\left(t_f\right)}{\partial f_1}\frac{\partial f_1}{\partial {\alpha }_T}\right)=\frac{1}{2l_p{c}_p}\frac{{\partial }^{2}E\left(t_f\right)}{\partial f_1\partial f_2}\frac{\partial f_2}{\partial \gamma }=\frac{{\phi }_0{\sigma }_f{N}_f}{2l_p{c}_p}\frac{{\partial }^{2}E\left(t_f\right)}{\partial f_1\partial f_2}$ ; (91)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial {\sigma }_f\partial {\alpha }_T}=\frac{\partial }{\partial {\sigma }_f}\left(\frac{\partial E\left(\tau \right)}{\partial f_1}\frac{\partial f_1}{\partial {\alpha }_T}\right)=\frac{1}{2l_p{c}_p}\frac{{\partial }^{2}E\left(\tau \right)}{\partial f_1\partial f_2}\frac{\partial f_2}{\partial {\sigma }_f}=\frac{\gamma {\phi }_0{N}_f}{2l_p{c}_p}\frac{{\partial }^{2}E\left(\tau \right)}{\partial f_1\partial f_2}$ ; (92)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial N_f\partial {\alpha }_T}=\frac{\partial }{\partial N_f}\left(\frac{\partial E\left(\tau \right)}{\partial f_1}\frac{\partial f_1}{\partial {\alpha }_T}\right)=\frac{1}{2l_p{c}_p}\frac{{\partial }^{2}E\left(\tau \right)}{\partial f_1\partial f_2}\frac{\partial f_2}{\partial N_f}=\frac{\gamma {\phi }_0{\sigma }_f}{2l_p{c}_p}\frac{{\partial }^{2}E\left(\tau \right)}{\partial f_1\partial f_2}$ . (93)

Because of symmetry, the mixed second-order sensitivities ${\partial }^{2}E\left(\tau \right)/\partial f_1\partial f_2$ or ${\partial }^{2}E\left(\tau \right)/\partial f_2\partial f_1$ can be obtained by using distinct but equivalent expressions in terms of the 2^nd-Level adjoint functions $A^{\left(2\right)}\left(2;1;t\right)$ and $A^{\left(2\right)}\left(2;2;t\right)$ , since the expressions obtained in Equations (77) and (84) represent the same quantity, because ${\partial }^{2}E\left(\tau \right)/\partial f_1\partial f_2={\partial }^{2}E\left(\tau \right)/\partial f_2\partial f_1$ by definition. Notably, only two “large-scale computations” are necessary for solving the two distinct 2^nd-LASS for obtaining the 2^nd-Level adjoint functions $A^{\left(2\right)}\left(2;1;t\right)$ and $A^{\left(2\right)}\left(2;2;t\right)$ involved in the computation of the three distinct second-order response sensitivities i.e., ${\partial }^{2}E\left(\tau \right)/\partial f_1\partial f_1$ , ${\partial }^{2}E\left(\tau \right)/\partial f_1\partial f_2$ and ${\partial }^{2}E\left(\tau \right)/\partial f_2\partial f_2$ , with respect to the two feature functions $f_1$ and $f_2$ . The subsequent use of Equation (86) to obtain the 49 second-order sensitivities ${\partial }^{2}E\left(\tau \right)/\partial {\alpha }_i\partial {\alpha }_j$ of the response with respect to the primary model parameters involves only inexpensive differentiations that are performed exactly, analytically, since the exact dependence of the feature functions on the model parameters is explicitly known. For verification purposes, all of the mixed second-order sensitivities ${\partial }^{2}E\left(\tau \right)/\partial {\alpha }_i\partial {\alpha }_j$ , $i,j=1,\cdots ,7$ , with respect to the model parameters can be computed twice, using distinct expressions in terms of the 2^nd-Level adjoint functions $A^{\left(2\right)}\left(2;1;t\right)$ and $A^{\left(2\right)}\left(2;2;t\right)$ .

4.2. Computation of Second-Order Sensitivities Using the 2^nd-CASAM-N

The conventional 2^nd-CASAM-N methodology applies the same fundamental principle (namely that the second-order sensitivities are the “first-order sensitivities of the first-order sensitivities”) as the 2^nd-FASAM. However, this principle is applied within the 2^nd-CASAM-N methodology directly to the first-order sensitivities with respect to the primary model parameters, as opposed to applying this principle to the first-order response sensitivities with respect to the feature functions, as implemented within the application of the 2^nd-FASAM-N methodology. This Section illustrates the application of the conventional 2^nd-CASAM-N methodology to obtain the second-order response sensitivities from the seven first-order sensitivities with respect to the underlying model parameters (as obtained in Subsection 3.1). It will be shown in this Section that the application of the 2^nd-CASAM-N methodology will require solving seven distinct 2^nd-Level Adjoint Sensitivity Systems (2^nd-LASS), each system comprising a distinct source term which corresponds to one of the seven distinct first-order sensitivities, in order to obtain all of the second-order sensitivities. In contradistinction, the application of the 2^nd-FASAM-N requires solving only two 2^nd-LASS, as will be shown in Section 6, in the sequel.

4.2.1. Computation of Second-Order Sensitivities Stemming from the First-Order Sensitivity $\partial E\left(\tau \right)/\partial {\alpha }_T$

The second-order sensitivities which stem from the first-order sensitivity $\partial E\left(\tau \right)/\partial {\alpha }_T$ are the components of the first-order G-differential of Equation (30). By definition, the first-order G-differential of Equation (30) is obtained as follows:

$\begin{array}{l}\delta {\left\{\partial E\left(\tau \right)/\partial {\alpha }_T\right\}}_{{\alpha }^0}\triangleq {\left\{\delta \left[\partial E\left(\tau \right)/\partial {\alpha }_T\right]\right\}}_{dir}+{\left\{\delta \left[\partial E\left(\tau \right)/\partial {\alpha }_T\right]\right\}}_{ind}\\ \triangleq \{-\frac{\text{d}}{\text{d}\epsilon }{\left\{{\left[\frac{1}{2\left(l_p+\epsilon \delta l_p\right)\left(c_p+\epsilon \delta c_p\right)}{\displaystyle {\int }_0^{\tau }\left(a^{\left(1\right)}+\epsilon \delta a^{\left(1\right)}\right){\left[E\left(t\right)+\epsilon \delta E\left(t\right)\right]}^2\text{d}t}\right]}_{{\alpha }^0}\right\}}_{\epsilon =0},\end{array}$ (94)

where the “direct-effect” term ${\left\{\delta \left[\partial E\left(\tau \right)/\partial {\alpha }_T\right]\right\}}_{dir}$ can be determined immediately and is defined as follows:

${\left\{\delta \left[\partial E\left(\tau \right)/\partial {\alpha }_T\right]\right\}}_{dir}\triangleq {\left\{\left[\frac{\delta l_p}{2{\left(l_p\right)}^2{c}_p}+\frac{\delta c_p}{2l_p{\left(c_p\right)}^2}\right]{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)E^2\left(t\right)\text{d}t}\right\}}_{{\alpha }^0}$ (95)

and where the “indirect-effect” term ${\left\{\delta \left[\partial E\left(\tau \right)/\partial {\alpha }_T\right]\right\}}_{ind}$ is defined as follows:

${\left\{\delta \left[\partial E\left(\tau \right)/\partial {\alpha }_T\right]\right\}}_{ind}\triangleq -{\left\{\frac{1}{2l_p{c}_p}{\displaystyle {\int }_0^{\tau }\left[\delta a^{\left(1\right)}\left(t\right)E^2\left(t\right)+2a^{\left(1\right)}\left(t\right)E\left(t\right)\delta E\left(t\right)\right]\text{d}t}\right\}}_{{\alpha }^0}$ .

(96)

The variational function $\delta E\left(t\right)$ in Equation (96) is the solution of the 1^st-LVSS provided in Equations (21) and (22). The variational function $\delta a^{\left(1\right)}\left(t\right)$ in Equation (96) is the solution of Equations (61) and (62), but written in terms of the primary model parameters and variations thereof, as follows:

$\begin{array}{l}{\left\{\left[-\frac{\text{d}}{\text{d}t}+\frac{{\alpha }_T}{l_p{c}_p}E\left(t\right)\right]\delta a^{\left(1\right)}\left(t\right)+\frac{{\alpha }_T}{l_p{c}_p}\left[\delta E\left(t\right)\right]\right\}}_{{\alpha }^0}\\ ={\left\{\left[\frac{\delta {\alpha }_T}{l_p{c}_p}-\frac{{\alpha }_T}{l_p{\left(c_p\right)}^2}\delta c_p-\frac{{\alpha }_T}{{\left(l_p\right)}^2{c}_p}\delta l_p\right]a^{\left(1\right)}\left(t\right)E\left(t\right)\right\}}_{{\alpha }^0},0<t<\tau ,\end{array}$ (97)

$\delta a^{\left(1\right)}\left(\tau \right)=0,t=\tau$ . (98)

Concatenating Equations (97) and (98) with the 1^st-LVSS for $\delta E\left(t\right)$ represented by Equations (21) and (22) yields the following 2^nd-LVSS for the 2^nd-level variational function

$V^{\left(2\right)}\left(2;t\right)\triangleq {\left[v^{\left(2\right)}\left(1;t\right),v^{\left(2\right)}\left(2;t\right)\right]}^{†}\triangleq {\left[\delta E\left(t\right),\delta a^{\left(1\right)}\left(t\right)\right]}^{†}$ :

${\left\{V{M}^{\left(2\right)}\left[2\times 2;\alpha \right]V^{\left(2\right)}\left(2;t\right)\right\}}_{{\alpha }^0}={\left\{S_V^{\left(2\right)}\left[2;\alpha ;\delta \alpha \right]\right\}}_{{\alpha }^0},0<t<\tau ,$ (99)

${\left\{B_V^{\left(2\right)}\left[2;V^{\left(2\right)}\left(2;t\right);\alpha ;\delta \alpha \right]\right\}}_{{\alpha }^0}=0\left[2\right],0\left[2\right]\triangleq {\left[0,0\right]}^{†},$ (100)

where

$V{M}^{\left(2\right)}\left[2\times 2;\alpha \right]\triangleq \left(\begin{array}{cc}\frac{\text{d}}{\text{d}t}+\frac{{\alpha }_T}{l_p{c}_p}E\left(t\right)& 0\\ \frac{{\alpha }_T}{l_p{c}_p}& -\frac{\text{d}}{\text{d}t}+{\frac{{\alpha }_T}{l_p{c}_p}}_{1}E\left(t\right)\end{array}\right)$ ; (101)

$S_V^{\left(2\right)}\left[2;\alpha ;\delta \alpha \right]\triangleq \left[\begin{array}{c}{s}_V^{\left(2\right)}\left(1;\alpha ;\delta \alpha \right)\\ s_V^{\left(2\right)}\left(2;\alpha ;\delta \alpha \right)\end{array}\right]$ ; (102)

$\begin{array}{l}{s}_V^{\left(2\right)}\left(1;\alpha ;\delta \alpha \right)\triangleq {\left\{-\frac{\delta {\alpha }_T}{2l_p{c}_p}+\frac{{\alpha }_T}{2{\left(l_p\right)}^2{c}_p}\delta l_p+\frac{{\alpha }_T}{2l_p{\left(c_p\right)}^2}\delta c_p\right\}}_{{\alpha }^0}{E}^2\left(t\right)\\ +{\left\{\gamma {\sigma }_f{N}_f\left(\delta {\phi }_0\right)+{\phi }_0{\sigma }_f{N}_f\left(\delta \gamma \right)+{\phi }_0\gamma N_f\left(\delta {\sigma }_f\right)+{\phi }_0\gamma {\sigma }_f\left(\delta N_f\right)\right\}}_{{\alpha }^0};\end{array}$ (103)

$s_V^{\left(2\right)}\left(2;\alpha ;\delta \alpha \right)\triangleq {\left\{-\frac{\delta {\alpha }_T}{l_p{c}_p}+\frac{{\alpha }_T}{{\left(l_p\right)}^2{c}_p}\delta l_p+\frac{{\alpha }_T}{l_p{\left(c_p\right)}^2}\delta c_p\right\}}_{{\alpha }^0}{a}^{\left(1\right)}\left(t\right)E\left(t\right)$ ; (104)

$B_V^{\left(2\right)}\left[2;V^{\left(2\right)}\left(2;t\right);\alpha ;\delta \alpha \right]\triangleq \left(\begin{array}{c}\delta E\left(0\right)\\ \delta a^{\left(1\right)}\left(\tau \right)\end{array}\right)$ . (105)

Except for the distinct notation, the 2^nd-LVSS defined by Equations (99) and (100) is identical with the 2^nd-LVSS defined by Equations (63) and (64), which is the reason for having used for both systems of equations the same notation for the respective 2^nd-Level variational function, namely $V^{\left(2\right)}\left(2;t\right)\triangleq {\left[v^{\left(2\right)}\left(1;t\right),v^{\left(2\right)}\left(2;t\right)\right]}^{†}\triangleq {\left[\delta E\left(t\right),\delta a^{\left(1\right)}\left(t\right)\right]}^{†}$ , which is the solution of this 2^nd-LVSS. However, the dependence on the components of the vector of parameters, $\alpha \triangleq {\left(\gamma ,{\sigma }_f,N_f,{\phi }_0,l_p,{\alpha }_T,c_p\right)}^{†}$ , is emphasized in Equations (99) and (100), because this explicit dependence is necessary to distinguish the developments of the 2^nd-Level Adjoint Sensitivity Systems to follow, which will be distinct from each other depending on the specific expression of each of the seven first-order sensitivities of the response with respect to the primary model parameters.

As discussed in Subsection 4.1, the computationally expensive path of solving the 2^nd-LVSS repeatedly for every possible parameter variation will be avoided by replacing the variational function $V^{\left(2\right)}\left(2;t\right)$ in the expression of the “indirect-effect” term defined in Equation by a corresponding 2^nd-level adjoint function, which will be denoted as $C^{\left(2\right)}\left(2;1;t\right)\triangleq {\left[c^{\left(2\right)}\left(1;1;t\right),c^{\left(2\right)}\left(2;1;t\right)\right]}^{†}\in H_2$ . This vector-valued function will be the solution of a 2^nd-Level Adjoint Sensitivity System (2^nd-LASS) to be constructed by applying the 2^nd-CASAM-N. The notation used for $C^{\left(2\right)}\left(2;1;t\right)\triangleq {\left[c^{\left(2\right)}\left(1;1;t\right),c^{\left(2\right)}\left(2;1;t\right)\right]}^{†}\in H_2$ has the following significance: (i) the bold letter “C” indicates a vector-valued “adjoint” function within the 2^nd-CASAM-N formalism; (ii) the superscript “(2)” indicates “second-level”; (iii) the first argument, namely “2”, in $C^{\left(2\right)}\left(2;1;t\right)$ indicates that this vector has 2 components, denoted as $c^{\left(2\right)}\left(i;2;t\right)$ , $i=1,2$ , each of which is a scalar-valued function of time; (iv) the second argument of $C^{\left(2\right)}\left(2;1;t\right)$ is denoted as “1” and indicates that this 2nd-level adjoint function corresponds to the first-order sensitivity $\partial E\left(\tau \right)/\partial {\alpha }_T$ of the response with respect to the first component of the vector of model parameters $\alpha \triangleq {\left({\alpha }_1,\cdots ,{\alpha }_7\right)}^{†}\triangleq {\left({\alpha }_T,l_p,c_p,{\phi }_0,\gamma ,{\sigma }_f,N_f\right)}^{†}$ , namely ${\alpha }_T$ .

The 2^nd-LASS for the function $C^{\left(2\right)}\left(2;1;t\right)\triangleq {\left[c^{\left(2\right)}\left(1;1;t\right),c^{\left(2\right)}\left(2;1;t\right)\right]}^{†}\in H_2$ is constructed in the same Hilbert space which was denoted as $H_2$ in the previous Subsection, and which is endowed with the inner product defined in Equation (68). This inner product is used to construct the 2^nd-Level Adjoint Sensitivity System (2^nd-LASS) for the 2^nd-level adjoint function $C^{\left(2\right)}\left(2;1;t\right)\triangleq {\left[c^{\left(2\right)}\left(1;1;t\right),c^{\left(2\right)}\left(2;1;t\right)\right]}^{†}\in H_2$ , as follows:

i) Using Equation (68), form the inner product of $C^{\left(2\right)}\left(2;1;t\right)$ with Equation (99) to obtain the following relation which has the same form as shown in Equation (69), namely:

$\begin{array}{l}{\left\{{\langle C^{\left(2\right)}\left(2;1;t\right),V{M}^{\left(2\right)}\left[2\times 2;\alpha \right]V^{\left(2\right)}\left(2;t\right)\rangle }_2\right\}}_{{\alpha }^0}\\ ={\left\{{\langle C^{\left(2\right)}\left(2;1;t\right),S_V^{\left(2\right)}\left(2;\alpha ;\delta \alpha \right)\rangle }_2\right\}}_{{\alpha }^0}\\ ={\left\{c^{\left(2\right)}\left(1;1;t\right)\delta E\left(t\right)-c^{\left(2\right)}\left(2;1;t\right)\delta a^{\left(1\right)}\left(t\right)\right\}}_{t=0}^{t=\tau }\\ +{\left\{{\langle V^{\left(2\right)}\left(2;t\right),A{M}^{\left(2\right)}\left[2\times 2;\alpha \right]C^{(2)}\left(2;1;t\right)\rangle }_2\right\}}_{{\alpha }^0},\end{array}$ (106)

where the adjoint operator $A{M}^{\left(2\right)}\left[2\times 2;\alpha \right]$ is the same as defined in Equation (70).

ii) Eliminate the boundary terms on the right side of Equation (106) and require the term on the right-side of the second equality in Equation (106) to represent the “indirect-effect” term defined in Equation (96) by imposing the following relations:

${\left\{A{M}^{\left(2\right)}\left[2\times 2;\alpha \right]C^{\left(2\right)}\left(2;1;t\right)\right\}}_{{\alpha }^0}=\left(\begin{array}{c}-\frac{a^{\left(1\right)}\left(t\right)E\left(t\right)}{l_p{c}_p}\\ -\frac{E^2\left(t\right)}{2l_p{c}_p}\end{array}\right),\text{}\text{}\text{}\text{}0<t<\tau ,$ (107)

${\left\{B_A^{\left(2\right)}\left[2;C^{\left(2\right)}\left(2;1;t\right);\alpha \right]\right\}}_{{\alpha }^0}\triangleq {\left(\begin{array}{c}{c}^{\left(2\right)}\left(1;1;\tau \right)\\ c^{\left(2\right)}\left(2;1;0\right)\end{array}\right)}_{{\alpha }^0}=\left(\begin{array}{c}0\\ 0\end{array}\right)$ . (108)

The relations represented by Equations (107) and (108) constitute the 2^nd-LASS for the 2^nd-level adjoint function $C^{\left(2\right)}\left(2;1;t\right)\triangleq {\left[c^{\left(2\right)}\left(1;1;t\right),c^{\left(2\right)}\left(2;1;t\right)\right]}^{†}\in H_2$ .

Inserting the equations underlying the 2^nd-LVSS, i.e. Equations (99) and (100), together with those underlying the 2^nd-LASS, i.e., Equations (107) and (108), into Equation (106) and recalling Equation (96) yields the following expression for the “indirect-effect” term as a function of $C^{\left(2\right)}\left(2;1;t\right)$ :

${\left\{\delta \left[\partial E\left(\tau \right)/\partial {\alpha }_T\right]\right\}}_{ind}={\left\{{\langle C^{\left(2\right)}\left(2;1;t\right),S_V^{\left(2\right)}\left(2;\alpha ;\delta \alpha \right)\rangle }_2\right\}}_{{\alpha }^0}$ . (109)

Inserting the result for the indirect-effect term obtained in Equation (109) together with the expression for the direct-effect term shown in Equation (95) into Equation (94) yields the following expression for the first-order G-differential $\delta {\left\{\partial E\left(\tau \right)/\partial {\alpha }_T\right\}}_{{\alpha }^0}$ :

$\begin{array}{c}\delta {\left\{\partial E\left(\tau \right)/\partial {\alpha }_T\right\}}_{{\alpha }^0}={\left\{\left[\frac{\delta l_p}{2{\left(l_p\right)}^2{c}_p}+\frac{\delta c_p}{2l_p{\left(c_p\right)}^2}\right]{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)E^2\left(t\right)\text{d}t}\right\}}_{{\alpha }^0}\\ +{\left\{{\langle C^{\left(2\right)}\left(2;1;t\right),S_V^{\left(2\right)}\left(2;\alpha ;\delta \alpha \right)\rangle }_2\right\}}_{{\alpha }^0}\\ ={\left\{{\displaystyle \underset{i=1}{\overset{7}{\sum }}\left[{\partial }^{2}E\left(\tau \right)/\partial {\alpha }_i\partial {\alpha }_T\right]\delta {\alpha }_i}\right\}}_{{\alpha }^0}.\end{array}$ (110)

Inserting into Equation (110) the expressions provided in Equation (103) and (104) for the respective components of the source $S_V^{\left(2\right)}\left(2;\alpha ;\delta \alpha \right)$ and collecting the terms that multiply the respective parameter variations yields the following expressions for the second-order partial sensitivities that stem from

$\delta {\left\{\partial E\left(\tau \right)/\partial {\alpha }_T\right\}}_{{\alpha }^0}$ :

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial {\alpha }_T\partial {\alpha }_T}=-\frac{1}{2l_p{c}_p}{\displaystyle {\int }_0^{\tau }\left[c^{\left(2\right)}\left(1;1;t\right)E^2\left(t\right)+2c^{\left(2\right)}\left(2;1;t\right)a^{\left(1\right)}\left(t\right)E\left(t\right)\right]\text{d}t}$ ; (111)

$\begin{array}{c}\frac{{\partial }^{2}E\left(\tau \right)}{\partial l_p\partial {\alpha }_T}=\frac{1}{2{\left(l_p\right)}^2{c}_p}{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)E^2\left(t\right)\text{d}t}\\ +\frac{{\alpha }_T}{2{\left(l_p\right)}^2{c}_p}{\displaystyle {\int }_0^{\tau }\left[c^{\left(2\right)}\left(1;1;t\right)E^2\left(t\right)+2c^{\left(2\right)}\left(2;1;t\right)a^{\left(1\right)}\left(t\right)E\left(t\right)\right]\text{d}t};\end{array}$ (112)

$\begin{array}{c}\frac{{\partial }^{2}E\left(\tau \right)}{\partial c_p\partial {\alpha }_T}=\frac{1}{2{\left(c_p\right)}^2{l}_p}{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)E^2\left(t\right)\text{d}t}\\ +\frac{{\alpha }_T}{2{\left(c_p\right)}^2{l}_p}{\displaystyle {\int }_0^{\tau }\left[c^{\left(2\right)}\left(1;1;t\right)E^2\left(t\right)+2c^{\left(2\right)}\left(2;1;t\right)a^{\left(1\right)}\left(t\right)E\left(t\right)\right]\text{d}t};\end{array}$ (113)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial {\phi }_0\partial {\alpha }_T}=\gamma {\sigma }_f{N}_f{\displaystyle {\int }_0^{\tau }{c}^{\left(2\right)}\left(1;1;t\right)\text{d}t}$ ; (114)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial \gamma \partial {\alpha }_T}={\phi }_0{\sigma }_f{N}_f{\displaystyle {\int }_0^{\tau }{c}^{\left(2\right)}\left(1;1;t\right)\text{d}t}$ ; (115)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial {\sigma }_f\partial {\alpha }_T}={\phi }_0\gamma N_f{\displaystyle {\int }_0^{\tau }{c}^{\left(2\right)}\left(1;1;t\right)\text{d}t}$ ; (116)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial N_f\partial {\alpha }_T}={\phi }_0\gamma {\sigma }_f{\displaystyle {\int }_0^{\tau }{c}^{\left(2\right)}\left(1;1;t\right)\text{d}t}$ . (117)

4.2.2. Computation of Second-Order Sensitivities Stemming from the First-Order Sensitivity $\partial E\left(\tau \right)/\partial l_p$

The second-order sensitivities which stem from the first-order sensitivity $\partial E\left(\tau \right)/\partial l_p$ are the components of the first-order G-differential of Equation (31). By definition, the first-order G-differential of Equation (31) is obtained as follows:

$\begin{array}{l}\delta {\left\{\partial E\left(\tau \right)/\partial l_p\right\}}_{{\alpha }^0}\triangleq {\left\{\delta \left[\partial E\left(\tau \right)/\partial l_p\right]\right\}}_{dir}+{\left\{\delta \left[\partial E\left(\tau \right)/\partial l_p\right]\right\}}_{ind}\\ \triangleq \{\frac{\text{d}}{\text{d}\epsilon }{\left\{{\left[\frac{{\alpha }_T+\epsilon \delta {\alpha }_T}{2{\left(l_p+\epsilon \delta l_p\right)}^2\left(c_p+\epsilon \delta c_p\right)}{\displaystyle {\int }_0^{\tau }\left(a^{\left(1\right)}+\epsilon \delta a^{\left(1\right)}\right){\left[E\left(t\right)+\epsilon \delta E\left(t\right)\right]}^2\text{d}t}\right]}_{{\alpha }^0}\right\}}_{\epsilon =0},\end{array}$ (118)

where the “direct-effect” term ${\left\{\delta \left[\partial E\left(\tau \right)/\partial l_p\right]\right\}}_{dir}$ can be determined immediately and is defined as follows:

$\begin{array}{l}{\left\{\delta \left[\partial E\left(\tau \right)/\partial l_p\right]\right\}}_{dir}\\ \triangleq {\left\{\left[\frac{\delta {\alpha }_T}{2{\left(l_p\right)}^2{c}_p}-\frac{{\alpha }_T\delta l_p}{{\left(l_p\right)}^3{c}_p}-\frac{{\alpha }_T\delta c_p}{2{\left(l_p{c}_p\right)}^2}\right]{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)E^2\left(t\right)\text{d}t}\right\}}_{{\alpha }^0}\end{array}$ (119)

and where the “indirect-effect” term ${\left\{\delta \left[\partial E\left(\tau \right)/\partial l_p\right]\right\}}_{ind}$ is defined as follows:

$\begin{array}{l}{\left\{\delta \left[\partial E\left(\tau \right)/\partial l_p\right]\right\}}_{ind}\\ \triangleq {\left\{\frac{{\alpha }_T}{2{\left(l_p\right)}^2{c}_p}{\displaystyle {\int }_0^{\tau }\left[\delta a^{\left(1\right)}\left(t\right)E^2\left(t\right)+2a^{\left(1\right)}\left(t\right)E\left(t\right)\delta E\left(t\right)\right]\text{d}t}\right\}}_{{\alpha }^0}\end{array}$ . (120)

Just as in Subsection 4.2.1, the 2^nd-Level variational function

$V^{\left(2\right)}\left(2;t\right)\triangleq {\left[\delta E\left(t\right),\delta a^{\left(1\right)}\left(t\right)\right]}^{†}$ , which is needed to evaluate the “indirect-effect” term ${\left\{\delta \left[\partial E\left(\tau \right)/\partial l_p\right]\right\}}_{ind}$ , is the solution of the 2^nd-LVSS defined by Equations (99) and (100). The computationally expensive path of solving the 2^nd-LVSS repeatedly for every possible parameter variation is avoided by replacing the dependence of the “indirect-effect” term defined in Equation (120) on the variational function $V^{\left(2\right)}\left(2;t\right)$ by a dependence on a corresponding 2^nd-Level adjoint function, which will be denoted as $C^{\left(2\right)}\left(2;2;t\right)\triangleq {\left[c^{\left(2\right)}\left(1;2;t\right),c^{\left(2\right)}\left(2;2;t\right)\right]}^{†}\in H_2$ , where the notation has the following significance: (i) the bold letter “C” indicates a vector-valued “adjoint” function within the 2nd-CASAM-N formalism; (ii) the superscript “(2)” indicates “second-level”; (iii) the first argument, namely “2”, in $C^{\left(2\right)}\left(2;2;t\right)$ indicates that this vector has 2 components, denoted as $c^{\left(2\right)}\left(i;2;t\right)$ , $i=1,2$ , each of which is a scalar-valued function of time; (iv) the second argument of $C^{\left(2\right)}\left(2;2;t\right)$ is denoted as “2” and indicates that this 2^nd-Level adjoint function corresponds to the first-order sensitivity $\partial E\left(\tau \right)/\partial l_p$ of the response with respect to the second component of the vector of model parameters $\alpha \triangleq {\left({\alpha }_1,\cdots ,{\alpha }_7\right)}^{†}\triangleq {\left({\alpha }_T,l_p,c_p,{\phi }_0,\gamma ,{\sigma }_f,N_f\right)}^{†}$ , namely $l_p$ .

The 2^nd-LASS for the function $C^{\left(2\right)}\left(2;2;t\right)\triangleq {\left[c^{\left(2\right)}\left(1;2;t\right),c^{\left(2\right)}\left(2;2;t\right)\right]}^{†}\in H_2$ is constructed by following the same procedure as in Subsection 4.1.1, except for the source term (i.e., right-side) of the 2^nd-LASS; this source-term now corresponds to the “indirect-effect” term ${\left\{\delta \left[\partial E\left(\tau \right)/\partial l_p\right]\right\}}_{ind}$ defined in Equation (120). This procedure leads to the following 2nd-LASS for $C^{\left(2\right)}\left(2;2;t\right)$ :

${\left\{A{M}^{\left(2\right)}\left[2\times 2;\alpha \right]C^{(2)}\left(2;2;t\right)\right\}}_{{\alpha }^0}=\left(\begin{array}{c}\frac{{\alpha }_T}{{\left(l_p\right)}^2{c}_p}{a}^{\left(1\right)}\left(t\right)E\left(t\right)\\ \frac{{\alpha }_T}{2{\left(l_p\right)}^2{c}_p}{E}^2\left(t\right)\end{array}\right),\text{}\text{}\text{}\text{}0<t<\tau ,$ (121)

${\left\{B_A^{\left(2\right)}\left[2;C^{\left(2\right)}\left(2;2;t\right);\alpha \right]\right\}}_{{\alpha }^0}\triangleq {\left(\begin{array}{c}{c}^{\left(2\right)}\left(1;2;\tau \right)\\ c^{\left(2\right)}\left(2;2;0\right)\end{array}\right)}_{{\alpha }^0}=\left(\begin{array}{c}0\\ 0\end{array}\right)$ . (122)

Furthermore, the 2^nd-order partial sensitivities that are obtained in terms of the components of $C^{\left(2\right)}\left(2;2;t\right)$ have expressions that are formally similar to those obtained in Subsection 4.1.1, above, except for the contributions stemming from the direct effect term defined in Equation (119). Omitting these repetitive derivations, the final expressions for the 2^nd-order partial sensitivities that stem from $\partial E\left(\tau \right)/\partial l_p$ are as follows:

$\begin{array}{c}\frac{{\partial }^{2}E\left(\tau \right)}{\partial {\alpha }_T\partial l_p}=\frac{1}{2{\left(l_p\right)}^2{c}_p}{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)E^2\left(t\right)\text{d}t}\\ -\frac{1}{2l_p{c}_p}{\displaystyle {\int }_0^{\tau }\left[c^{\left(2\right)}\left(1;2;t\right)E^2\left(t\right)+2c^{\left(2\right)}\left(2;2;t\right)a^{\left(1\right)}\left(t\right)E\left(t\right)\right]\text{d}t;}\end{array}$ (123)

$\begin{array}{c}\frac{{\partial }^{2}E\left(\tau \right)}{\partial l_p\partial l_p}=-\frac{{\alpha }_T}{{\left(l_p\right)}^3{c}_p}{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)E^2\left(t\right)\text{d}t}\\ +\frac{{\alpha }_T}{2{\left(l_p\right)}^2{c}_p}{\displaystyle {\int }_0^{\tau }\left[c^{\left(2\right)}\left(1;2;t\right)E^2\left(t\right)+2c^{\left(2\right)}\left(2;2;t\right)a^{\left(1\right)}\left(t\right)E\left(t\right)\right]\text{d}t};\end{array}$ (124)

$\begin{array}{c}\frac{{\partial }^{2}E\left(\tau \right)}{\partial c_p\partial l_p}=-\frac{{\alpha }_T}{2{\left(l_p{c}_p\right)}^2}{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)E^2\left(t\right)\text{d}t}\\ +\frac{{\alpha }_T}{2{\left(c_p\right)}^2{l}_p}{\displaystyle {\int }_0^{\tau }\left[c^{\left(2\right)}\left(1;2;t\right)E^2\left(t\right)+2c^{\left(2\right)}\left(2;2;t\right)a^{\left(1\right)}\left(t\right)E\left(t\right)\right]\text{d}t};\end{array}$ (125)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial {\phi }_0\partial l_p}=\gamma {\sigma }_f{N}_f{\displaystyle {\int }_0^{\tau }{c}^{\left(2\right)}\left(1;2;t\right)\text{d}t}$ ; (126)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial \gamma \partial l_p}={\phi }_0{\sigma }_f{N}_f{\displaystyle {\int }_0^{\tau }{c}^{\left(2\right)}\left(1;2;t\right)\text{d}t}$ ; (127)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial {\sigma }_f\partial l_p}={\phi }_0\gamma N_f{\displaystyle {\int }_0^{\tau }{c}^{\left(2\right)}\left(1;2;t\right)\text{d}t}$ ; (128)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial N_f\partial l_p}={\phi }_0\gamma {\sigma }_f{\displaystyle {\int }_0^{\tau }{c}^{\left(2\right)}\left(1;2;t\right)\text{d}t}$ . (129)

4.2.3. Computation of Second-Order Sensitivities Stemming from the First-Order Sensitivity $\partial E\left(\tau \right)/\partial c_p$

The second-order sensitivities which stem from the first-order sensitivity $\partial E\left(\tau \right)/\partial c_p$ are the components of the first-order G-differential of Equation (32), which has by definition the following expression:

$\begin{array}{l}\delta {\left\{\partial E\left(\tau \right)/\partial c_p\right\}}_{{\alpha }^0}\triangleq {\left\{\delta \left[\partial E\left(\tau \right)/\partial c_p\right]\right\}}_{dir}+{\left\{\delta \left[\partial E\left(\tau \right)/\partial c_p\right]\right\}}_{ind}\\ \triangleq \{\frac{\text{d}}{\text{d}\epsilon }{\left\{{\left[\frac{{\alpha }_T+\epsilon \delta {\alpha }_T}{2\left(l_p+\epsilon \delta l_p\right){\left(c_p+\epsilon \delta c_p\right)}^2}{\displaystyle {\int }_0^{\tau }\left(a^{\left(1\right)}+\epsilon \delta a^{\left(1\right)}\right){\left[E\left(t\right)+\epsilon \delta E\left(t\right)\right]}^2\text{d}t}\right]}_{{\alpha }^0}\right\}}_{\epsilon =0},\end{array}$

(130)

where the “direct-effect” term ${\left\{\delta \left[\partial E\left(\tau \right)/\partial c_p\right]\right\}}_{dir}$ can be determined immediately and is defined as follows:

$\begin{array}{l}{\left\{\delta \left[\partial E\left(\tau \right)/\partial c_p\right]\right\}}_{dir}\\ \triangleq {\left\{\left[\frac{\delta {\alpha }_T}{2l_p{\left(c_p\right)}^2}-\frac{{\alpha }_T\delta l_p}{2{\left(l_p{c}_p\right)}^2}-\frac{{\alpha }_T\delta c_p}{l_p{\left(c_p\right)}^3}\right]{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)E^2\left(t\right)\text{d}t}\right\}}_{{\alpha }^0}\end{array}$ , (131)

and where the “indirect-effect” term ${\left\{\delta \left[\partial E\left(\tau \right)/\partial c_p\right]\right\}}_{ind}$ is defined as follows:

$\begin{array}{l}{\left\{\delta \left[\partial E\left(\tau \right)/\partial c_p\right]\right\}}_{ind}\\ \triangleq {\left\{\frac{{\alpha }_T}{2l_p{\left(c_p\right)}^2}{\displaystyle {\int }_0^{\tau }\left[\delta a^{\left(1\right)}\left(t\right)E^2\left(t\right)+2a^{\left(1\right)}\left(t\right)E\left(t\right)\delta E\left(t\right)\right]\text{d}t}\right\}}_{{\alpha }^0}\end{array}$ . (132)

Just as in the previous Subsections of Section 4.2, the 2^nd-Level variational function $V^{\left(2\right)}\left(2;t\right)\triangleq {\left[\delta E\left(t\right),\delta a^{\left(1\right)}\left(t\right)\right]}^{†}$ , which is needed to evaluate the “indirect-effect” term ${\left\{\delta \left[\partial E\left(\tau \right)/\partial c_p\right]\right\}}_{ind}$ , is the solution of the 2^nd-LVSS defined by Equations (99) and (100). The computationally expensive path of solving the 2^nd-LVSS repeatedly for every possible parameter variation is avoided by replacing the dependence of the “indirect-effect” term defined in Equation (132) on the variational function $V^{\left(2\right)}\left(2;t\right)$ by a dependence on a corresponding 2^nd-Level adjoint function, which will be denoted as $C^{\left(2\right)}\left(2;3;t\right)\triangleq {\left[c^{\left(2\right)}\left(1;3;t\right),c^{\left(2\right)}\left(2;3;t\right)\right]}^{†}\in H_2$ , where the notation is as in the previous Subsections, except that the second argument of $C^{\left(2\right)}\left(2;3;t\right)$ is denoted as “3” and indicates that this 2^nd-level adjoint function corresponds to the first-order sensitivity $\partial E\left(\tau \right)/\partial c_p$ of the response with respect to the third component of the vector of model parameters

$\alpha \triangleq {\left({\alpha }_1,\cdots ,{\alpha }_7\right)}^{†}\triangleq {\left({\alpha }_T,l_p,c_p,{\phi }_0,\gamma ,{\sigma }_f,N_f\right)}^{†}$ , namely $c_p$ .

The 2^nd-LASS for the function $C^{\left(2\right)}\left(2;3;t\right)\triangleq {\left[c^{\left(2\right)}\left(1;3;t\right),c^{\left(2\right)}\left(2;3;t\right)\right]}^{†}\in H_2$ is constructed by following the same procedure as in the previous Subsections, except for the source term (i.e., right-side) of the 2^nd-LASS which corresponds to the “indirect-effect” term ${\left\{\delta \left[\partial E\left(\tau \right)/\partial c_p\right]\right\}}_{ind}$ defined in Equation (132). This procedure leads to the following 2^nd-LASS for $C^{\left(2\right)}\left(2;3;t\right)$ :

${\left\{A{M}^{\left(2\right)}\left[2\times 2;\alpha \right]C^{\left(2\right)}\left(2;3;t\right)\right\}}_{{\alpha }^0}=\left(\begin{array}{c}\frac{{\alpha }_T}{l_p{\left(c_p\right)}^2}{a}^{\left(1\right)}\left(t\right)E\left(t\right)\\ \frac{{\alpha }_T}{2l_p{\left(c_p\right)}^2}{E}^2\left(t\right)\end{array}\right)\text{},0<t<\tau ,$ (133)

${\left\{B_A^{\left(2\right)}\left[2;C^{\left(2\right)}\left(2;3;t\right);\alpha \right]\right\}}_{{\alpha }^0}\triangleq {\left(\begin{array}{c}{c}^{\left(2\right)}\left(1;3;\tau \right)\\ c^{\left(2\right)}\left(2;3;0\right)\end{array}\right)}_{{\alpha }^0}=\left(\begin{array}{c}0\\ 0\end{array}\right)$ . (134)

The 2^nd-order partial sensitivities are obtained in terms of the components of $C^{\left(2\right)}\left(2;3;t\right)$ by applying the same procedure as in Subsections 4.2.1 and 4.2.2, and will have expressions that are formally similar to those obtained in the previous Subsections except for the contributions stemming from the direct effect term defined in Equation (131). The final expressions for the 2^nd-order partial sensitivities that stem from $\partial E\left(\tau \right)/\partial c_p$ are as follows:

$\begin{array}{c}\frac{{\partial }^{2}E\left(\tau \right)}{\partial {\alpha }_T\partial c_p}=\frac{1}{2l_p{\left(c_p\right)}^2}{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)E^2\left(t\right)\text{d}t}\\ -\frac{1}{2l_p{c}_p}{\displaystyle {\int }_0^{\tau }\left[c^{\left(2\right)}\left(1;3;t\right)E^2\left(t\right)+2c^{\left(2\right)}\left(2;3;t\right)a^{\left(1\right)}\left(t\right)E\left(t\right)\right]\text{d}t};\end{array}$ (135)

$\begin{array}{c}\frac{{\partial }^{2}E\left(\tau \right)}{\partial l_p\partial c_p}=-\frac{{\alpha }_T}{2{\left(l_p{c}_p\right)}^2}{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)E^2\left(t\right)\text{d}t}\\ +\frac{{\alpha }_T}{2{\left(l_p\right)}^2{c}_p}{\displaystyle {\int }_0^{\tau }\left[c^{\left(2\right)}\left(1;3;t\right)E^2\left(t\right)+2c^{\left(2\right)}\left(2;3;t\right)a^{\left(1\right)}\left(t\right)E\left(t\right)\right]\text{d}t};\end{array}$ (136)

$\begin{array}{c}\frac{{\partial }^{2}E\left(\tau \right)}{\partial c_p\partial c_p}=-\frac{{\alpha }_T}{l_p{\left(c_p\right)}^3}{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)E^2\left(t\right)\text{d}t}\\ +\frac{{\alpha }_T}{2{\left(c_p\right)}^2{l}_p}{\displaystyle {\int }_0^{\tau }\left[c^{\left(2\right)}\left(1;3;t\right)E^2\left(t\right)+2c^{\left(2\right)}\left(2;3;t\right)a^{\left(1\right)}\left(t\right)E\left(t\right)\right]\text{d}t};\end{array}$ (137)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial {\phi }_0\partial c_p}=\gamma {\sigma }_f{N}_f{\displaystyle {\int }_0^{\tau }{c}^{\left(2\right)}\left(1;3;t\right)\text{d}t}$ ; (138)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial \gamma \partial c_p}={\phi }_0{\sigma }_f{N}_f{\displaystyle {\int }_0^{\tau }{c}^{\left(2\right)}\left(1;3;t\right)\text{d}t}$ ; (139)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial {\sigma }_f\partial c_p}={\phi }_0\gamma N_f{\displaystyle {\int }_0^{\tau }{c}^{\left(2\right)}\left(1;3;t\right)\text{d}t}$ ; (140)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial N_f\partial c_p}={\phi }_0\gamma {\sigma }_f{\displaystyle {\int }_0^{\tau }{c}^{\left(2\right)}\left(1;3;t\right)\text{d}t}$ . (141)

4.2.4. Computation of Second-Order Sensitivities Stemming from the First-Order Sensitivity $\partial E\left(\tau \right)/\partial {\phi }_0$

The second-order sensitivities which stem from the first-order sensitivity $\partial E\left(\tau \right)/\partial {\phi }_0$ are the components of the first-order G-differential of Equation (33), which has by definition the following expression:

$\begin{array}{l}\delta {\left\{\partial E\left(\tau \right)/\partial {\phi }_0\right\}}_{{\alpha }^0}\triangleq {\left\{\delta \left[\partial E\left(\tau \right)/\partial {\phi }_0\right]\right\}}_{dir}+{\left\{\delta \left[\partial E\left(\tau \right)/\partial {\phi }_0\right]\right\}}_{ind}\\ \triangleq \{\frac{\text{d}}{\text{d}\epsilon }{\left\{{\left[\left(\gamma +\epsilon \delta \gamma \right)\left({\sigma }_f+\epsilon \delta {\sigma }_f\right)\left(N_f+\epsilon \delta N_f\right){\displaystyle {\int }_0^{\tau }\left(a^{\left(1\right)}+\epsilon \delta a^{\left(1\right)}\right)\text{d}t}\right]}_{{\alpha }^0}\right\}}_{\epsilon =0},\end{array}$ (142)

where the “direct-effect” term ${\left\{\delta \left[\partial E\left(\tau \right)/\partial {\phi }_0\right]\right\}}_{dir}$ can be determined immediately and is defined as follows:

$\begin{array}{l}{\left\{\delta \left[\partial E\left(\tau \right)/\partial {\phi }_0\right]\right\}}_{dir}\\ \triangleq {\left\{\left[\left(\delta \gamma \right){\sigma }_f{N}_f+\left(\delta {\sigma }_f\right)\gamma N_f+\left(\delta N_f\right)\gamma {\sigma }_f\right]{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)}\text{d}t\right\}}_{{\alpha }^0}\end{array}$ , (143)

and where the “indirect-effect” term ${\left\{\delta \left[\partial E\left(\tau \right)/\partial {\phi }_0\right]\right\}}_{ind}$ is defined as follows:

${\left\{\delta \left[\partial E\left(\tau \right)/\partial {\phi }_0\right]\right\}}_{ind}\triangleq {\left\{\gamma {\sigma }_f{N}_f{\displaystyle {\int }_0^{\tau }\delta a^{\left(1\right)}\left(t\right)\text{d}t}\right\}}_{{\alpha }^0}$ . (144)

Just as in the previous Subsections of Section 4.2, the 2^nd-Level variational function $V^{\left(2\right)}\left(2;t\right)\triangleq {\left[\delta E\left(t\right),\delta a^{\left(1\right)}\left(t\right)\right]}^{†}$ , which is needed to evaluate the “indirect-effect” term ${\left\{\delta \left[\partial E\left(\tau \right)/\partial {\phi }_0\right]\right\}}_{ind}$ , is the solution of the 2^nd-LVSS defined by Equations (99) and (100). The computationally expensive path of solving the 2^nd-LVSS repeatedly for every possible parameter variation is avoided by replacing the dependence of the “indirect-effect” term defined in Equation (144) on the variational function $V^{\left(2\right)}\left(2;t\right)$ by a dependence on a corresponding 2^nd-Level adjoint function, which will be denoted as $C^{\left(2\right)}\left(2;4;t\right)\triangleq {\left[c^{\left(2\right)}\left(1;4;t\right),c^{\left(2\right)}\left(2;4;t\right)\right]}^{†}\in H_2$ , where the notation is as in the previous Subsections, except that the second argument of $C^{\left(2\right)}\left(2;4;t\right)$ is denoted as “4” and indicates that this 2^nd-level adjoint function corresponds to the first-order sensitivity $\partial E\left(\tau \right)/\partial {\phi }_0$ of the response with respect to the fourth component of the vector of model parameters

$\alpha \triangleq {\left({\alpha }_1,\cdots ,{\alpha }_7\right)}^{†}\triangleq {\left({\alpha }_T,l_p,c_p,{\phi }_0,\gamma ,{\sigma }_f,N_f\right)}^{†}$ , namely ${\phi }_0$ .

The 2^nd-LASS for the function $C^{\left(2\right)}\left(2;4;t\right)\triangleq {\left[c^{\left(2\right)}\left(1;4;t\right),c^{\left(2\right)}\left(2;4;t\right)\right]}^{†}\in H_2$ is constructed by following the same procedure as in the previous Subsections, except for the source term (i.e., right-side) of the 2^nd-LASS which corresponds to the “indirect-effect” term ${\left\{\delta \left[\partial E\left(\tau \right)/\partial {\phi }_0\right]\right\}}_{ind}$ defined in Equation (144). This procedure leads to the following 2^nd-LASS for $C^{\left(2\right)}\left(2;4;t\right)$ :

${\left\{A{M}^{\left(2\right)}\left[2\times 2;\alpha \right]C^{\left(2\right)}\left(2;4;t\right)\right\}}_{{\alpha }^0}=\left(\begin{array}{c}0\\ \gamma {\sigma }_f{N}_f\end{array}\right),0<t<\tau ,$ ; (145)

${\left\{B_A^{\left(2\right)}\left[2;C^{\left(2\right)}\left(2;4;t\right);\alpha \right]\right\}}_{{\alpha }^0}\triangleq {\left(\begin{array}{c}{c}^{\left(2\right)}\left(1;4;\tau \right)\\ c^{\left(2\right)}\left(2;4;0\right)\end{array}\right)}_{{\alpha }^0}=\left(\begin{array}{c}0\\ 0\end{array}\right)$ . (146)

The 2^nd-order partial sensitivities are obtained in terms of the components of $C^{\left(2\right)}\left(2;4;t\right)$ by applying the same procedure as in Subsections 4.2.1 and 4.2.2, and will have expressions that are formally similar to those obtained in the previous Subsections except for the contributions stemming from the direct-effect term defined in Equation (143). The final expressions for the 2^nd-order partial sensitivities that stem from $\partial E\left(\tau \right)/\partial {\phi }_0$ are as follows:

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial {\alpha }_T\partial {\phi }_0}=-\frac{1}{2l_p{c}_p}{\displaystyle {\int }_0^{\tau }\left[c^{\left(2\right)}\left(1;4;t\right)E^2\left(t\right)+2c^{\left(2\right)}\left(2;4;t\right)a^{\left(1\right)}\left(t\right)E\left(t\right)\right]\text{d}t};$ (147)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial l_p\partial {\phi }_0}=\frac{{\alpha }_T}{2{\left(l_p\right)}^2{c}_p}{\displaystyle {\int }_0^{\tau }\left[c^{\left(2\right)}\left(1;4;t\right)E^2\left(t\right)+2c^{\left(2\right)}\left(2;4;t\right)a^{\left(1\right)}\left(t\right)E\left(t\right)\right]\text{d}t};$ (148)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial c_p\partial {\phi }_0}=\frac{{\alpha }_T}{2{\left(c_p\right)}^2{l}_p}{\displaystyle {\int }_0^{\tau }\left[c^{\left(2\right)}\left(1;4;t\right)E^2\left(t\right)+2c^{\left(2\right)}\left(2;4;t\right)a^{\left(1\right)}\left(t\right)E\left(t\right)\right]\text{d}t};$ (149)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial {\phi }_0\partial {\phi }_0}=\gamma {\sigma }_f{N}_f{\displaystyle {\int }_0^{\tau }{c}^{\left(2\right)}\left(1;4;t\right)\text{d}t}$ ; (150)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial \gamma \partial {\phi }_0}={\sigma }_f{N}_f{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)}\text{d}t+{\phi }_0{\sigma }_f{N}_f{\displaystyle {\int }_0^{\tau }{c}^{\left(2\right)}\left(1;4;t\right)\text{d}t}$ ; (151)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial {\sigma }_f\partial {\phi }_0}=\gamma N_f{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)}\text{d}t+{\phi }_0\gamma N_f{\displaystyle {\int }_0^{\tau }{c}^{\left(2\right)}\left(1;4;t\right)\text{d}t}$ ; (152)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial N_f\partial {\phi }_0}=\gamma {\sigma }_f{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)}\text{d}t+{\phi }_0\gamma {\sigma }_f{\displaystyle {\int }_0^{\tau }{c}^{\left(2\right)}\left(1;4;t\right)\text{d}t}$ . (153)

4.2.5. Computation of Second-Order Sensitivities Stemming from the First-Order Sensitivity $\partial E\left(\tau \right)/\partial \gamma$

The second-order sensitivities which stem from the first-order sensitivity $\partial E\left(\tau \right)/\partial \gamma$ are the components of the first-order G-differential of Equation (34), which has by definition the following expression:

$\begin{array}{l}\delta {\left\{\partial E\left(\tau \right)/\partial \gamma \right\}}_{{\alpha }^0}\triangleq {\left\{\delta \left[\partial E\left(\tau \right)/\partial \gamma \right]\right\}}_{dir}+{\left\{\delta \left[\partial E\left(\tau \right)/\partial \gamma \right]\right\}}_{ind}\\ \triangleq \{\frac{\text{d}}{\text{d}\epsilon }{\left\{{\left[\left({\phi }_0+\epsilon \delta {\phi }_0\right)\left({\sigma }_f+\epsilon \delta {\sigma }_f\right)\left(N_f+\epsilon \delta N_f\right){\displaystyle {\int }_0^{\tau }\left(a^{\left(1\right)}+\epsilon \delta a^{\left(1\right)}\right)\text{d}t}\right]}_{{\alpha }^0}\right\}}_{\epsilon =0},\end{array}$ (154)

where the “direct-effect” term ${\left\{\delta \left[\partial E\left(\tau \right)/\partial \gamma \right]\right\}}_{dir}$ can be determined immediately and is defined as follows:

$\begin{array}{l}{\left\{\delta \left[\partial E\left(\tau \right)/\partial \gamma \right]\right\}}_{dir}\\ \triangleq {\left\{\left[\left(\delta {\phi }_0\right){\sigma }_f{N}_f+\left(\delta {\sigma }_f\right){\phi }_0{N}_f+\left(\delta N_f\right){\phi }_0{\sigma }_f\right]{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)}\text{d}t\right\}}_{{\alpha }^0}\end{array}$ , (155)

and where the “indirect-effect” term ${\left\{\delta \left[\partial E\left(\tau \right)/\partial \gamma \right]\right\}}_{ind}$ is defined as follows:

${\left\{\delta \left[\partial E\left(\tau \right)/\partial \gamma \right]\right\}}_{ind}\triangleq {\left\{{\phi }_0{\sigma }_f{N}_f{\displaystyle {\int }_0^{\tau }\delta a^{\left(1\right)}\left(t\right)\text{d}t}\right\}}_{{\alpha }^0}$ . (156)

Just as in the previous Subsections of Section 4.2, the 2^nd-level variational

function $V^{\left(2\right)}\left(2;t\right)\triangleq {\left[\delta E\left(t\right),\delta a^{\left(1\right)}\left(t\right)\right]}^{†}$ , which is needed to evaluate the “indirect-effect” term ${\left\{\delta \left[\partial E\left(\tau \right)/\partial \gamma \right]\right\}}_{ind}$ , is the solution of the 2^nd-LVSS defined by Equations (99) and (100). The computationally expensive path of solving the 2^nd-LVSS repeatedly for every possible parameter variation is avoided by replacing the dependence of the “indirect-effect” term defined in Equation (144) on the variational function $V^{\left(2\right)}\left(2;t\right)$ by a dependence on a corresponding 2^nd-Level adjoint function, which will be denoted as $C^{\left(2\right)}\left(2;5;t\right)\triangleq {\left[c^{\left(2\right)}\left(1;5;t\right),c^{\left(2\right)}\left(2;5;t\right)\right]}^{†}\in H_2$ , where the notation is as in the previous Subsections, except that the second argument of $C^{\left(2\right)}\left(2;5;t\right)$ is denoted as “5” and indicates that this 2^nd-level adjoint function corresponds to the first-order sensitivity $\partial E\left(\tau \right)/\partial \gamma$ of the response with respect to the fifth component of the vector of model parameters $\alpha \triangleq {\left({\alpha }_1,\cdots ,{\alpha }_7\right)}^{†}\triangleq {\left({\alpha }_T,l_p,c_p,{\phi }_0,\gamma ,{\sigma }_f,N_f\right)}^{†}$ , namely $\gamma$ .

The 2^nd-LASS for the function $C^{\left(2\right)}\left(2;5;t\right)\triangleq {\left[c^{\left(2\right)}\left(1;5;t\right),c^{\left(2\right)}\left(2;5;t\right)\right]}^{†}\in H_2$ is constructed by following the same procedure as in the previous Subsections, except for the source term (i.e., right-side) of the 2^nd-LASS, which now corresponds to the “indirect-effect” term ${\left\{\delta \left[\partial E\left(\tau \right)/\partial \gamma \right]\right\}}_{ind}$ defined in Equation (156). This procedure leads to the following 2^nd-LASS for $C^{\left(2\right)}\left(2;5;t\right)$ :

${\left\{A{M}^{\left(2\right)}\left[2\times 2;\alpha \right]C^{\left(2\right)}\left(2;5;t\right)\right\}}_{{\alpha }^0}=\left(\begin{array}{c}0\\ {\phi }_0{\sigma }_f{N}_f\end{array}\right),\text{}0<t<\tau ,$ (157)

${\left\{B_A^{\left(2\right)}\left[2;C^{\left(2\right)}\left(2;5;t\right);\alpha \right]\right\}}_{{\alpha }^0}\triangleq {\left(\begin{array}{c}{c}^{\left(2\right)}\left(1;5;\tau \right)\\ c^{\left(2\right)}\left(2;5;0\right)\end{array}\right)}_{{\alpha }^0}=\left(\begin{array}{c}0\\ 0\end{array}\right)$ . (158)

The 2^nd-order partial sensitivities are obtained in terms of the components of $C^{\left(2\right)}\left(2;5;t\right)$ by applying the same procedure as in Subsections 4.2.1-4.2.4, and will have expressions that are formally similar to those obtained in the previous Subsections except for the contributions stemming from the direct-effect term defined in Equation (155). The final expressions for the 2^nd-order partial sensitivities that stem from $\partial E\left(\tau \right)/\partial \gamma$ are as follows:

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial {\alpha }_T\partial \gamma }=-\frac{1}{2l_p{c}_p}{\displaystyle {\int }_0^{\tau }\left[c^{\left(2\right)}\left(1;5;t\right)E^2\left(t\right)+2c^{\left(2\right)}\left(2;5;t\right)a^{\left(1\right)}\left(t\right)E\left(t\right)\right]\text{d}t};$ (159)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial l_p\partial \gamma }=\frac{{\alpha }_T}{2{\left(l_p\right)}^2{c}_p}{\displaystyle {\int }_0^{\tau }\left[c^{\left(2\right)}\left(1;5;t\right)E^2\left(t\right)+2c^{\left(2\right)}\left(2;5;t\right)a^{\left(1\right)}\left(t\right)E\left(t\right)\right]\text{d}t};$ (160)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial c_p\partial \gamma }=\frac{{\alpha }_T}{2{\left(c_p\right)}^2{l}_p}{\displaystyle {\int }_0^{\tau }\left[c^{\left(2\right)}\left(1;5;t\right)E^2\left(t\right)+2c^{\left(2\right)}\left(2;5;t\right)a^{\left(1\right)}\left(t\right)E\left(t\right)\right]\text{d}t};$ (161)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial {\phi }_0\partial \gamma }={\sigma }_f{N}_f{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)}\text{d}t+\gamma {\sigma }_f{N}_f{\displaystyle {\int }_0^{\tau }{c}^{\left(2\right)}\left(1;5;t\right)\text{d}t}$ ; (162)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial \gamma \partial \gamma }={\phi }_0{\sigma }_f{N}_f{\displaystyle {\int }_0^{\tau }{c}^{\left(2\right)}\left(1;5;t\right)\text{d}t}$ ; (163)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial {\sigma }_f\partial \gamma }={\phi }_0{N}_f{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)}\text{d}t+{\phi }_0\gamma N_f{\displaystyle {\int }_0^{\tau }{c}^{\left(2\right)}\left(1;5;t\right)\text{d}t}$ ; (164)

$\frac{{\partial }^{2}E\left(\tau \right)}{\partial N_f\partial \gamma }={\phi }_0{\sigma }_f{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)}\text{d}t+{\phi }_0\gamma {\sigma }_f{\displaystyle {\int }_0^{\tau }{c}^{\left(2\right)}\left(1;5;t\right)\text{d}t}$ . (165)

4.2.6. Computation of Second-Order Sensitivities Stemming from the First-Order Sensitivity $\partial E\left(\tau \right)/\partial {\sigma }_f$

The second-order sensitivities which stem from the first-order sensitivity $\partial E\left(\tau \right)/\partial {\sigma }_f$ are the components of the first-order G-differential of Equation (35), which has by definition the following expression:

$\begin{array}{l}\delta {\left\{\partial E\left(\tau \right)/\partial {\sigma }_f\right\}}_{{\alpha }^0}\triangleq {\left\{\delta \left[\partial E\left(\tau \right)/\partial {\sigma }_f\right]\right\}}_{dir}+{\left\{\delta \left[\partial E\left(\tau \right)/\partial {\sigma }_f\right]\right\}}_{ind}\\ \triangleq \{\frac{\text{d}}{\text{d}\epsilon }{\left\{{\left[\left({\phi }_0+\epsilon \delta {\phi }_0\right)\left(\gamma +\epsilon \delta \gamma \right)\left(N_f+\epsilon \delta N_f\right){\displaystyle {\int }_0^{\tau }\left(a^{\left(1\right)}+\epsilon \delta a^{\left(1\right)}\right)\text{d}t}\right]}_{{\alpha }^0}\right\}}_{\epsilon =0},\end{array}$ (166)

where the “direct-effect” term ${\left\{\delta \left[\partial E\left(\tau \right)/\partial {\sigma }_f\right]\right\}}_{dir}$ can be determined immediately and is defined as follows:

$\begin{array}{l}{\left\{\delta \left[\partial E\left(\tau \right)/\partial {\sigma }_f\right]\right\}}_{dir}\\ \triangleq {\left\{\left[\left(\delta {\phi }_0\right)\gamma N_f+\left(\delta \gamma \right){\phi }_0{N}_f+\left(\delta N_f\right){\phi }_0\gamma \right]{\displaystyle {\int }_0^{\tau }{a}^{\left(1\right)}\left(t\right)}\text{d}t\right\}}_{{\alpha }^0}\end{array}$ , (167)

and where the “indirect-effect” term ${\left\{\delta \left[\partial E\left(\tau \right)/\partial {\sigma }_f\right]\right\}}_{ind}$ is defined as follows:

${\left\{\delta \left[\partial E\left(\tau \right)/\partial {\sigma }_f\right]\right\}}_{ind}\triangleq {\left\{{\phi }_0\gamma N_f{\displaystyle {\int }_0^{\tau }\delta a^{\left(1\right)}\left(t\right)\text{d}t}\right\}}_{{\alpha }^0}$ . (168)

Just as in the previous Subsections of Section 4.2, the dependence of the “indirect-effect” term defined in Equation (168) on the variational function $V^{\left(2\right)}\left(2;t\right)$ is replaced by a dependence on a corresponding 2^nd-Level adjoint function, which will be denoted as $C^{\left(2\right)}\left(2;6;t\right)\triangleq {\left[c^{\left(2\right)}\left(1;6;t\right),c^{\left(2\right)}\left(2;6;t\right)\right]}^{†}\in H_2$ , where the notation is as in the previous Subsections, except that the second argument of $C^{\left(2\right)}\left(2;6;t\right)$ is denoted as “6” and indicates that this 2^nd-Level adjoint function corresponds to the first-order sensitivity $\partial E\left(\tau \right)/\partial {\sigma }_f$ of the response with respect to the sixth component of the vector of model parameters $\alpha \triangleq {\left({\alpha }_1,\cdots ,{\alpha }_7\right)}^{†}\triangleq {\left({\alpha }_T,l_p,c_p,{\phi }_0,\gamma ,{\sigma }_f,N_f\right)}^{†}$ , namely ${\sigma }_f$ .

The 2^nd-LASS for the function $C^{\left(2\right)}\left(2;6;t\right)\triangleq {\left[c^{\left(2\right)}\left(1;6;t\right),c^{\left(2\right)}\left(2;6;t\right)\right]}^{†}\in H_2$ is constructed by following the same procedure as in the previous Subsections, except for the source term (i.e., right-side) of the 2^nd-LASS which corresponds to the “indirect-effect” term ${\left\{\delta \partial E\left(\tau \right)/\partial {\sigma }_f\right\}}_{ind}$ defined in Equation (168). This procedure leads to the following 2^nd-LASS for $C^{\left(2\right)}\left(2;6;t\right)$ :

${\left\{A{M}^{\left(2\right)}\left[2\times 2;\alpha \right]C^{\left(2\right)}\left(2;6;t\right)\right\}}_{{\alpha }^0}=\left(\begin{array}{c}0\\ {\phi }_0\gamma N_f\end{array}\right),\text{}0<t<\tau ,$ (169)

${\left\{B_A^{\left(2\right)}\left[2;C^{\left(2\right)}\left(2;6;t\right);\alpha \right]\right\}}_{{\alpha }^0}\triangleq {\left(\begin{array}{c}{c}^{\left(2\right)}\left(1;6;\tau \right)\\ c^{\left(2\right)}\left(2;6;0\right)\end{array}\right)}_{{\alpha }^0}=\left(\begin{array}{c}0\\ 0\end{array}\right)$ . (170)

The 2^nd-order partial sensitivities are obtained in terms of the components of $C^{}$

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

References

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