The n^th-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (n^th-CASAM-N): Mathematical Framework

Abstract

This work presents the n^th-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (n^th-CASAM-N), which enables the most efficient computation of exactly determined expressions of arbitrarily high-order sensitivities of generic nonlinear system responses with respect to model parameters, uncertain boundaries, and internal interfaces in the model’s phase space. The mathematical framework underlying the n^th-CASAM-N is proven to be correct by using mathematical induction. The n^th-CASAM-N is formulated in linearly increasing higher-dimensional Hilbert spaces—as opposed to exponentially increasing parameter-dimensional spaces—thus overcoming the curse of dimensionality in sensitivity analysis of nonlinear systems.

1. Introduction

The high-order sensitivities (i.e., functional derivatives) of model responses (i.e., results of interest produced by models) with respect to model parameters are notoriously difficult to compute for large-scale models involving many parameters. Furthermore, the computation of higher-order sensitivities by conventional methods is subject to the curse of dimensionality, a term coined by Belmann [1] to describe phenomena in which the number of computations increases exponentially in the respective phase space. In the particular case of sensitivity analysis, the number of large-scale computations increases exponentially in the parameter phase space as the order of sensitivities increases. It is known that the adjoint method of sensitivity analysis is the most efficient method for computing exactly first-order sensitivities. The idea underlying the computation of response sensitivities with respect to model parameters using adjoint operators was first used by Wigner [2] to analyze first-order perturbations in linear problems of interest to nuclear reactor physics. Cacuci [3,4] is credited (see, e.g., [5,6]) for having conceived the rigorous first-order adjoint sensitivity analysis methodology for generic large-scale nonlinear (as opposed to linearized) systems involving generic operator responses and having introduced these principles to the earth, atmospheric, and other sciences.

The general mathematical framework for the second-order adjoint sensitivity analysis methodology for linear and nonlinear systems, respectively, was conceived by Cacuci [7,8]. The unparalleled efficiency of the second-order adjoint sensitivity analysis methodology for linear systems was demonstrated [9] by applying this methodology to compute exactly the 21,976 first-order sensitivities and 482,944,576 second-order sensitivities (of which 241,483,276 are distinct from each other) for an OECD/NEA reactor physics benchmark [10], which is representative of a large-scale system that involves many (21,976, in this illustrative example) parameters. This paradigm large-scale system cannot be analyzed comprehensively by any other (including statistical) methods. The results obtained in [9] indicated that many second-order sensitivities were much larger than the largest first-order ones (contrary to the widely held belief to the contrary, for reactor physics systems), which motivated the investigation [11] of the largest third-order and, subsequently, fourth-order sensitivities. The evident conclusions that resulted from these works [9,11] are that the consideration of only the first-order sensitivities is insufficient for making credible predictions regarding the expected values and uncertainties (variances, covariances, skewness) of calculated and predicted/adjusted responses. At the very least, the second-order sensitivities must also be computed in order to enable the quantitative assessment of their impact on the predicted model responses. Consequently, Cacuci [12] has conceived the mathematical framework of the n^th-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems (abbreviated as “n^th-CASAM-L”). The n^th-CASAM-L enables the efficient computation of exactly determined expressions of arbitrarily high-order sensitivities of a generic system response—which can depend on both the forward and adjoint state functions—with respect to all of the parameters that characterize the physical system. The qualifier “comprehensive” is employed in order to highlight that the model parameters considered within the framework of the n^th-CASAM-L include the system’s uncertain boundaries and internal interfaces in phase space. The n^th-CASAM-L mathematical framework was developed specifically for linear systems because the most important model responses produced by such systems are various Lagrangian functionals, which depend simultaneously on both the forward and adjoint state functions governing the respective linear system. Included among such functionals are the Raleigh quotient for computing eigenvalues and/or separation constants when solving linear partial differential equations, and the Schwinger and Rousopoulos functionals (see, e.g., [13,14,15,16]), which play a fundamental role in optimization and control procedures, and the derivation of numerical methods for solving equations (differential, integral, integro-differential). Responses that depend on both the forward and adjoint state functions underlying the respective model can occur only for linear systems, because responses in nonlinear systems can only depend on the system’s forward state functions (since nonlinear operators do not admit adjoint operators). Consequently, the sensitivity analysis of responses that simultaneously involve both forward and adjoint state functions makes it necessary to treat linear models/systems in their own right, rather than treating them as particular cases of nonlinear systems.

The n^th-CASAM-L provides the tools for the exact and efficient computation of higher-order sensitivities for response-coupled forward/adjoint linear systems while overcoming the curse of dimensionality that has hindered their computation thus far. In parallel, Cacuci [17,18] has also developed the 5^th-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (5^th-CASAM-N), thereby generalizing the original mathematical framework [3,4] for nonlinear systems. The 5^th-CASAM-N enables the efficient and exact computation of response sensitivities, up to the fifth order, with respect to imprecisely known (i.e., uncertain) model boundaries and/or internal interfaces. The material to be presented in this work generalizes the 5^th-CASAM-N to enable the exact and efficient computation of sensitivities of arbitrarily high order. This general methodology is called the n^th-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (n^th-CASAM-N) and is presented in Section 2 of this work. The n^th-CASAM-N is formulated in linearly increasing higher-dimensional Hilbert spaces (as opposed to exponentially increasing parameter-dimensional spaces), thus overcoming the curse of dimensionality in sensitivity analysis of nonlinear systems, enabling the most efficient computation of exactly determined expressions of arbitrarily high-order sensitivities of generic nonlinear system responses with respect to model parameters, uncertain boundaries, and internal interfaces in the model’s phase space. The mathematical framework underlying the n^th-CASAM-N is proven to be correct by using mathematical induction. Section 3 concludes this work by discussing the significance of the n^th-CASAM-N.

2. The n^th-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (n^th-CASAM-N)

The general framework of the n^th-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems (n^th-CASAM-L) is constructed based on the pattern that emerges from the 4^th-CASAM-N [17] and the 5^th-CASAM-N [18]. The generic mathematical model of a nonlinear system, which comprises uncertain (i.e., imprecisely known) model parameters, boundaries, and interfaces, is detailed in Appendix A, since it is the same as that used in [17,18]. The validity of mathematical methodology underlying the n^th-CASAM-L is established in this section by using a proof by mathematical induction comprising the usual steps, as follows:

• Establish the general pattern underlying the n^th-CASAM-N for an arbitrarily high-order $n$.
• Prove that the general pattern underlying the n^th-CASAM-N is valid for the lowest values of $n$, i.e., $n=1$ and $n=2$.
• Assuming that that the pattern is valid for an arbitrarily high-order $n$, prove that the pattern is also valid for $n+1$.

2.1. The n^th-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (n^th-CASAM-N)

The n^th-order sensitivity $R^{\left(n\right)}\left[j_n;\dots ;j_1;U^{\left(n\right)};A^{(n)};\alpha \right]\equiv {\partial }^{n}R\left[u\left(x\right);\alpha \right]/\partial {\mathsf{\alpha }}_{j_n}\dots \partial {\mathsf{\alpha }}_{j_1}$ of the response $R\left[u\left(x\right);\alpha \right]$ with respect to the parameters ${\mathsf{\alpha }}_{j_1},\dots ,{\mathsf{\alpha }}_{j_n}$ is obtained, for each index $j_1,\dots ,j_n=1,\dots ,TP$, from the expression of the total first-order G-differential of each of the (n − 1)^th-order sensitivities, each of which is expected to have the following expression for $j_1,\dots ,j_{n-1}=1,\dots ,TP$:

$$\begin{array}{l}{R}^{\left(n-1\right)}\left[j_{n-1};\dots ;j_1;U^{\left(n-1\right)};A^{(n-1)};\alpha \right]\equiv {\partial }^{n-1}R\left[u\left(x\right);\alpha \right]/\partial {\mathsf{\alpha }}_{j_{n-1}}\dots \partial {\mathsf{\alpha }}_{j_1}\\ \triangleq {\displaystyle \underset{{\mathsf{\lambda }}_1\left(\alpha \right)}{\overset{{\mathsf{\omega }}_1\left(\alpha \right)}{\int }}\dots {\displaystyle \underset{{\mathsf{\lambda }}_{TI}\left(\alpha \right)}{\overset{{\mathsf{\omega }}_{TI}\left(\alpha \right)}{\int }}{S}^{\left(n-1\right)}\left[j_{n-1};\dots ;j_1;U^{\left(n-1\right)};A^{(n-1)};\alpha \right]d{x}_1\dots d{x}_{TI}}}.\end{array}$$

(1)

Generalizing the pattern established in Appendix B for the n^th-CASAM-N for $n=1,2,3$, the total first-order G-differential of the (n − 1)^th-order sensitivity $R^{\left(n-1\right)}\left[j_{n-1};\dots ;j_1;U^{\left(n-1\right)};A^{(n-1)};\alpha \right]$ is expected to have the following expression:

$$\begin{array}{l}{\left\{\mathsf{\delta }{R}^{\left(n-1\right)}\left[j_{n-1};\dots ;j_1;U^{\left(n-1\right)};A^{(n-1)};\alpha \right]\right\}}_{{\alpha }^0}\triangleq {\displaystyle \sum _{j_n=1}^{TP}{\left\{\frac{\partial R^{\left(n-1\right)}\left[j_{n-1};\dots ;j_1;U^{\left(n-1\right)};A^{(n-1)};\alpha \right]}{\partial {\mathsf{\alpha }}_{j_n}}\right\}}_{{\alpha }^0}}\mathsf{\delta }{\mathsf{\alpha }}_{j_n}\\ +{\left\{\mathsf{\delta }{R}^{\left(n-1\right)}\left[j_{n-1};\dots ;j_1;U^{\left(n-1\right)};A^{(n-1)};\alpha ;V^{\left(n-1\right)};\mathsf{\delta }{A}^{(n-1)}\right]\right\}}_{ind}\end{array}$$

(2)

where the quantity ${\left\{\mathsf{\delta }{R}^{\left(n-1\right)}\left[j_{n-1};\dots ;j_1;U^{\left(n-1\right)};A^{(n-1)};\mathsf{\alpha };V^{\left(n-1\right)};\mathsf{\delta }{A}^{(n-1)}\right]\right\}}_{ind}$ denotes the indirect-effect term, which is defined as follows:

$$\begin{array}{l}{\left\{\mathsf{\delta }{R}^{\left(n-1\right)}\left[j_{n-1};\dots ;j_1;U^{\left(n-1\right)};A^{(n-1)};V^{\left(n-1\right)};\mathsf{\delta }{A}^{(n-1)};\alpha \right]\right\}}_{ind}\\ \triangleq {\displaystyle \underset{{\mathsf{\lambda }}_1\left(\alpha \right)}{\overset{{\mathsf{\omega }}_1\left(\alpha \right)}{\int }}\dots {\displaystyle \underset{{\mathsf{\lambda }}_{TI}\left(\mathsf{\alpha }\right)}{\overset{{\mathsf{\omega }}_{TI}\left(\mathsf{\alpha }\right)}{\int }}{\left\{\frac{\partial S^{\left(n-1\right)}}{\partial U^{\left(n-1\right)}\left(x\right)}{V}^{\left(n-1\right)}\left(2^{n-2};x\right)+\frac{\partial S^{\left(n-1\right)}}{\partial A^{(n-1)}\left(x\right)}\mathsf{\delta }{A}^{(n-1)}\left(2^{n-2};x\right)\right\}}_{{\mathsf{\alpha }}^0}d{x}_1\dots d{x}_{TI}}}.\end{array}$$

(3)

The vectors $V^{\left(n-1\right)}\left(2^{n-2};j_{n-3};\dots ;j_1;x\right)\triangleq \mathsf{\delta }{U}^{\left(n-1\right)}\left(2^{n-2};j_{n-3};\dots ;j_1;x\right)$ and $\mathsf{\delta }{A}^{(n-1)}\left(2^{n-2};j_{n-2};\dots ;j_1;x\right)$ are the solution of the following n^th-Level Variational Sensitivity System (n^th-LVSS), which is obtained by concatenating the (n − 1)^th-LVSS together with the G-differentiated (n − 1)^th-LASS for $j_1,\dots ,j_{n-1}=1,\dots ,TP$:

$$\begin{array}{l}{\left\{V{M}^{\left(n\right)}\left[2^{n-1}\times 2^{n-1};U^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right);\alpha \right]V^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right)\right\}}_{{\mathsf{\alpha }}^0}\\ ={\left\{Q_V^{\left(n\right)}\left[2^{n-1};U^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right);\alpha ;\mathsf{\delta }\alpha \right]\hspace{0.17em}\right\}}_{{\mathsf{\alpha }}^0},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x\in {\mathsf{\Omega }}_x,\end{array}$$

(4)

$${\left\{B_V^{\left(n\right)}\left[2^{n-1};U^{\left(n\right)}\left(2^{n-1};x\right);V^{\left(n\right)}\left(2^{n-1};x\right);\alpha ;\mathsf{\delta }\alpha \right]\right\}}_{{\alpha }^0}=0\left[2^{n-1}\right];\hspace{0.17em}\hspace{0.17em}x\in \partial {\mathsf{\Omega }}_x\left({\alpha }^0\right),$$

(5)

The variational matrix $V{M}^{\left(n\right)}\left[2^{n-1}\times 2^{n-1};U^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right);\alpha \right]$ comprises $\left(2^{n-1}\times 2^{n-1}\right)$ block-matrices, each comprising $T{D}^2$ components/elements; thus, the matrix $V{M}^{\left(n\right)}\left[2^{n-1}\times 2^{n-1};U^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right);\alpha \right]$ comprises a total of $\left(2^{n-1}\times 2^{n-1}\right)T{D}^2$ components/elements. Each of the vectors $V^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right)$, $Q_V^{\left(n\right)}\left[2^{n-1};U^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right);\alpha ;\mathsf{\delta }\alpha \right]$, and $B_V^{\left(n\right)}\left[2^{n-1};U^{\left(n\right)}\left(2^{n-1};x\right);V^{\left(4\right)}\left(2^{n-1};x\right);\alpha ;\mathsf{\delta }\alpha \right]$ comprises $2^{n-1}$ $TD$-dimensional vectors, so that each of these vectors comprises $\left(2^{n-1}\times TD\right)$ components/elements. The quantity $0\left[2^{n-1}\right]$ denotes a block-vector with $2^{n-1}$ components, with each component being a $TD$-dimensional vector with identically zero components. The various quantities appearing in Equations (4) and (5) are defined as follows:

$$V{M}^{\left(n\right)}\left[2^{n-1}\times 2^{n-1};x;\alpha \right]\hspace{0.17em}\triangleq \left(\begin{array}{cc}V{M}^{\left(n-1\right)}\left[2^{n-2}\times 2^{n-2};x\right]& 0\left[2^{n-2}\times 2^{n-2}\right]\\ V{M}_{21}^{\left(n\right)}\left(2^{n-2}\times 2^{n-2};x\right)& V{M}_{22}^{\left(n\right)}\left(2^{n-2}\times 2^{n-2};x\right)\end{array}\right);$$

(6)

$$U^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right)\triangleq \left(\begin{array}{c}{U}^{\left(n-1\right)}\left(2^{n-2};j_{n-3};\dots ;j_1;x\right)\\ A^{(n-1)}\left(2^{n-2};j_{n-2};\dots ;j_1;x\right)\end{array}\right);$$

(7)

$$\begin{array}{l}{V}^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right)\triangleq \mathsf{\delta }{U}^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right)=\left(\begin{array}{c}{V}^{\left(n-1\right)}\left(2^{n-2};j_{n-3};\dots ;j_1;x\right)\\ \mathsf{\delta }{A}^{(n-1)}\left(2^{n-2};j_{n-2};\dots ;j_1;x\right)\end{array}\right)\\ =\left[v^{\left(1\right)}\left(x\right),\mathsf{\delta }{a}^{\left(1\right)}\left(x\right),\mathsf{\delta }{a}^{\left(2\right)}\left(1;j_1;x\right),\mathsf{\delta }{a}^{\left(2\right)}\left(2;j_1;x\right),\right.\dots \\ {\left.\dots ,\mathsf{\delta }{a}^{(n-1)}\left(1;j_{n-2};\dots ;j_1;x\right),\dots ,\mathsf{\delta }{a}^{(n-1)}\left(2^{n-2};j_{n-2};\dots ;j_1;x\right)\right]}^{†};\end{array}$$

(8)

$$\begin{array}{l}V{M}_{21}^{\left(n\right)}\left(2^{n-2}\times 2^{n-2};j_{n-2};\dots ;j_1;x\right)\triangleq \hspace{0.17em}\\ -\frac{\partial Q_A^{\left(n-1\right)}\left[2^{n-2};j_{n-2};\dots ;j_1;U^{\left(n-2\right)}\left(2^{n-2};j_{n-3};\dots ;j_1;x\right);\right]}{\partial U^{\left(n-1\right)}\left(2^{n-2};j_{n-3};\dots ;j_1;x\right)}\\ +\frac{\partial \left\{A{M}^{\left(n-1\right)}\left[2^{n-2}\times 2^{n-2};U^{\left(n-1\right)}\left(2^{n-2};x\right);\alpha \right]A^{(n-1)}\left(2^{n-2};j_{n-2};\dots ;j_1;x\right)\right\}}{\partial U^{\left(n-1\right)}\left(2^{n-2};j_{n-3};\dots ;j_1;x\right)};\end{array}$$

(9)

$$V{M}_{22}^{\left(n\right)}\hspace{0.17em}\left(2^{n-2}\times 2^{n-2};x\right)\triangleq A{M}^{\left(n-1\right)}\left[2^{n-2}\times 2^{n-2};U^{\left(n-1\right)}\left(2^{n-2};j_{n-3};\dots ;j_1;x\right);\alpha \right];$$

(10)

$$\begin{array}{l}{Q}_V^{\left(n\right)}\left[2^{n-1};j_1;U^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right);\alpha ;\mathsf{\delta }\alpha \right]\triangleq \left(\begin{array}{c}{Q}_V^{\left(n-2\right)}\left[2^{n-2};U^{\left(n-1\right)}\left(2^{n-2};x\right);\alpha ;\mathsf{\delta }\alpha \right]\\ Q_2^{\left(n\right)}\left[2^{n-2};U^{\left(n\right)}\left(2^{n-1};x\right);\alpha ;\mathsf{\delta }\alpha \right]\end{array}\right)\\ \triangleq {\left\{q_V^{\left(n\right)}\left[1;U^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right);\alpha ;\mathsf{\delta }\alpha \right],\dots ,q_V^{\left(n\right)}\left[2^{n-1};U^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right);\alpha ;\mathsf{\delta }\alpha \right]\right\}}^{†};\end{array}$$

(11)

$$q_V^{\left(n\right)}\left[i;U^{\left(n\right)}\left(x\right);\alpha ;\mathsf{\delta }\alpha \right]\equiv {\displaystyle \sum _{j_n=1}^{TP}{s}_V^{\left(n\right)}\left[i;j_{n-2};\dots ;j_1;U^{\left(n\right)}\left(x\right);\mathsf{\alpha }\right]\mathsf{\delta }{\mathsf{\alpha }}_{j_n}};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1,\dots ;2^{n-1};$$

(12)

$$\begin{array}{l}{Q}_2^{\left(n\right)}\left[2^{n-2};U^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right);\alpha ;\mathsf{\delta }\alpha \right]\triangleq \frac{\partial Q_A^{\left(n-1\right)}\left[2^{n-2};j_1;U^{\left(n-1\right)}\left(x\right);\alpha \right]}{\partial \alpha }\partial \alpha \\ -\frac{\partial \left\{A{M}^{\left(n-1\right)}\left[2^{n-2}\times 2^{n-2};U^{\left(n-1\right)}\left(2^{n-2};x\right);\alpha \right]A^{(n-1)}\left(2^{n-2};x\right)\right\}}{\partial \alpha }\partial \alpha ;\end{array}$$

(13)

$$\begin{array}{l}{B}_V^{\left(n\right)}\left[2^{n-1};U^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right);V^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right);\alpha ;\mathsf{\delta }\alpha \right]\\ \triangleq \left(\begin{array}{c}{B}_V^{\left(n-1\right)}\left[2^{n-2};U^{\left(n-1\right)}\left(2^{n-2};j_{n-3};\dots ;j_1;x\right);V^{\left(n-1\right)}\left(2^{n-2};j_{n-3};\dots ;j_1;x\right);\alpha ;\mathsf{\delta }\alpha \right]\\ \mathsf{\delta }{B}_A^{\left(n-1\right)}\left[2^{n-2};U^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right);V^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right);\alpha ;\mathsf{\delta }\alpha \right]\end{array}\right).\end{array}$$

(14)

Solving the n^th-LVSS would require $O\left(T{P}^n\right)$ large-scale computations, which is unrealistic for large-scale systems comprising many parameters. The n^th-CASAM-N circumvents the need to solve the n^th-LVSS by deriving an alternative expression for the indirect-effect term defined in Equation (3), in which the function $V^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right)$ is replaced by a n^th-level adjoint function, denoted as $A^{(n)}\left(2^{n-1};j_{n-1};\dots ;j_1;x\right)\triangleq {\left[\hspace{0.17em}{a}^{(n)}\left(1;j_{n-1};\dots ;j_1;x\right),\dots ,a^{(n)}\left(2^{n-1};j_{n-1};\dots ;j_1;x\right)\right]}^{†}\in {\mathrm{H}}_n\left({\mathsf{\Omega }}_x\right)$, which is independent of parameter variations. The elements of the Hilbert space ${\mathrm{H}}_n\left({\mathsf{\Omega }}_x\right)$ are block-vectors of the form ${\Psi }^{(n)}\left(2^{n-1};j_{n-1};\dots ;j_1;x\right)\triangleq {\left[{\psi }^{(n)}\left(1;j_{n-1};\dots ;j_1;x\right),\dots ,{\psi }^{(n)}\left(2^{n-1};j_{n-1};\dots ;j_1;x\right)\right]}^{†}$, comprising $2^{n-1}$ $TD$-dimensional vectors of the form ${\psi }^{\left(n\right)}\left(i;x\right)\triangleq {\left[{\mathsf{\psi }}_1^{\left(n\right)}\left(i;x\right),\dots ,{\psi }_{TD}^{\left(n\right)}\left(i;x\right)\right]}^{†}\in {\mathrm{H}}_1\left({\mathsf{\Omega }}_x\right)$, $i=1,\dots ,2^n$. The inner product of two vectors ${\Psi }^{(n)}\left(2^{n-1};j_{n-1};\dots ;j_1;x\right)$ and ${\Phi }^{(n)}\left(2^{n-1};j_{n-1};\dots ;j_1;x\right)$ in the Hilbert space ${\mathrm{H}}_n\left({\mathsf{\Omega }}_x\right)$ is denoted as ${〈{\Psi }^{(n)}\left(2^{n-1};x\right),{\Phi }^{(n)}\left(2^{n-1};x\right)〉}_n$ and defined as follows:

$${〈{\Psi }^{(n)}\left(2^{n-1};x\right),{\Phi }^{(n)}\left(2^{n-1};x\right)〉}_n\triangleq \hspace{0.17em}{\displaystyle \sum _{i=1}^{2^{n-1}}{〈{\psi }^{(n)}\left(i;x\right),{\phi }^{(n)}\left(i;x\right)〉}_1}.$$

(15)

The n^th-Level Adjoint Sensitivity System (n^th-LASS) for the n^th-level adjoint function $A^{(n)}\left(2^{n-1};j_{n-1};\dots ;j_1;x\right)$ is obtained by using the inner product defined in Equation (15) as follows:

$$\begin{array}{l}{\left\{{〈A^{(n)}\left(2^{n-1};x\right),V{M}^{\left(n\right)}\left(2^{n-1};x\right)〉}_2\right\}}_{{\alpha }^0}={\left\{{\left[P^{\left(n\right)}\left(U^{\left(n\right)};A^{(n)};V^{\left(n\right)};\alpha \right)\right]}_{\partial {\mathsf{\Omega }}_x}\right\}}_{{\alpha }^0}\\ +{\left\{{〈V^{\left(n\right)}\left(2^{n-1};x\right),A{M}^{\left(n\right)}\left[2^{n-1}\times 2^{n-1};U^{\left(n\right)}\left(2^{n-1};x\right);\alpha \right]A^{(n)}\left(2^{n-1};x\right)〉}_n\right\}}_{{\alpha }^0},\end{array}$$

(16)

where the quantity ${\left\{{\left[P^{\left(n\right)}\left(U^{\left(n\right)};A^{(n)};V^{\left(n\right)};\alpha \right)\right]}_{\partial {\mathsf{\Omega }}_x}\right\}}_{{\alpha }^0}$ denotes the corresponding bilinear concomitant on the domain’s boundary, evaluated at the nominal values for the parameters and respective state functions.

In terms of the n^th-level adjoint function $A^{(n)}\left(2^{n-1};j_{n-1};\dots ;j_1;x\right)$, the indirect-effect term defined by Equation (3) will have the following expression:

$$\begin{array}{l}{\left\{\mathsf{\delta }{R}^{\left(n-1\right)}\left[j_{n-1};\dots ;j_1;U^{\left(n-1\right)};A^{(n-1)};\alpha ;V^{\left(n-1\right)};\mathsf{\delta }{A}^{(n-1)}\right]\right\}}_{ind}=-{\left\{{\left[{\widehat{P}}^{\left(n\right)}\left(U^{\left(n\right)};A^{(n)};\mathsf{\delta }\alpha \right)\right]}_{\partial {\mathsf{\Omega }}_x}\right\}}_{{\alpha }^0}\\ +{\left\{{〈A^{(n)}\left(2^{n-1};j_{n-1};\dots ;j_1;x\right),Q_V^{\left(n\right)}\left[2^{n-1};U^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right);\alpha ;\mathsf{\delta }\alpha \right]〉}_n\right\}}_{{\alpha }^0}.\end{array}$$

(17)

where $A^{(n)}\left(2^{n-1};j_{n-1};\dots ;j_1;x\right)$ is the solution of the following n^th-Level Adjoint Sensitivity System (n^th-LASS):

$$\begin{array}{l}A{M}^{\left(n\right)}\left[2^{n-1}\times 2^{n-1};U^{\left(n\right)}\left(2^{n-1};x\right);\alpha \right]A^{(n)}\left(2^{n-1};j_{n-1};\dots ;j_1;x\right)\\ =Q_A^{\left(n\right)}\left[2^{n-1};j_{n-1};\dots ;j_1;U^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right);\alpha \right]\hspace{0.17em},\end{array}$$

(18)

subject to boundary conditions represented in operator form as follows:

$$\begin{array}{l}{\left\{B_A^{\left(n\right)}\left[2^{n-1};U^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right);A^{(n)}\left(2^{n-1};j_{n-1};\dots ;j_1;x\right);\alpha \right]\right\}}_{{\alpha }^0}=0\left[2^{n-1}\right],\\ x\in \partial {\mathsf{\Omega }}_x\left({\alpha }^0\right);\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{j}_1=1,\dots ,TP;\hspace{0.17em}\hspace{0.17em}{j}_2=1,\dots ,j_1;\hspace{0.17em}\dots ;\hspace{0.17em}{j}_{n-1}=1,\dots ,j_{n-2}.\end{array}$$

(19)

In Equation (17), the quantity ${\left\{{\left[{\widehat{P}}^{\left(n\right)}\left(U^{\left(n\right)};A^{(n)};\mathsf{\delta }\alpha \right)\right]}_{\partial {\mathsf{\Omega }}_x}\right\}}_{{\alpha }^0}$ denotes residual boundary terms that may have not vanished automatically after having used the boundary conditions provided in Equations (5) and (19) to eliminate from Equation (17) all unknown values of the n^th-level variational function $V^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right)$.

The quantities that appear in the definition of the n^th-LASS, cf. Equations (18) and (19), are defined as follows:

$$\begin{array}{l}{\mathrm{AM}}^{\left(n\right)}\left[2^{n-1}\times 2^{n-1};U^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;\mathrm{AM}\right);\alpha \right]\triangleq {\left[{\mathrm{VM}}^{\left(n\right)}\left(2^{n-1}\times 2^{n-1};U^{\left(n\right)};\alpha \right)\right]}^{*}\\ =\left(\begin{array}{cc}{\left\{{\left[{\mathrm{VM}}^{\left(n-1\right)}\left(2^{n-2}\times 2^{n-2}\right)\right]}^{*}\right\}}^{†}& {\left\{{\left[{\mathrm{VM}}_{21}^{\left(n\right)}\left(2^{n-2}\times 2^{n-2}\right)\right]}^{*}\right\}}^{†}\\ 0\left[2^{n-2}\times 2^{n-2}\right]& {\left\{{\left[{\mathrm{VM}}_{22}^{\left(n\right)}\left(2^{n-2}\times 2^{n-2}\right)\right]}^{*}\right\}}^{†}\end{array}\right),\end{array}$$

(20)

$$\begin{array}{l}{Q}_A^{\left(n\right)}\left[2^{n-1};j_{n-1};\dots ;j_1;U^{\left(n\right)}\left(2^{n-1};x\right);\alpha \right]\triangleq \left\{q_A^{\left(n\right)}\left[1;j_{n-1};\dots ;j_1;U^{\left(n\right)}\left(2^{n-1};x\right);\alpha \right]\right.,\dots \\ {\left.\dots ,q_A^{\left(n\right)}\left[2^{n-1};j_{n-1};\dots ;j_1;U^{\left(n\right)}\left(2^{n-1};x\right);\alpha \right]\right\}}^{†},\hspace{0.17em}\hspace{0.17em}{j}_1=1,\dots ,TP;\hspace{0.17em}\hspace{0.17em}\dots ;\hspace{0.17em}{j}_{n-1}=1,\dots ,j_{n-2};\end{array}$$

(21)

$$q_A^{\left(n\right)}\left[1;j_{n-1};\dots ;j_1;U^{\left(n\right)}\left(x\right);\alpha \right]\triangleq \partial S^{\left(n-1\right)}\left[j_{n-1};\dots ;j_1;U^{\left(n\right)};\alpha \right]/\partial u\left(x\right);$$

(22)

$$q_A^{\left(n\right)}\left[2;j_{n-1};\dots ;j_1;U^{\left(n\right)}\left(x\right);\alpha \right]\triangleq \partial S^{\left(n-1\right)}\left[j_{n-1};\dots ;j_1;U^{\left(n\right)};\alpha \right]/\partial a^{\left(1\right)}\left(x\right);$$

(23)

$$\begin{array}{l}for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}n\ge 3\hspace{0.17em}:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{q}_A^{\left(n\right)}\left[2^k+i;j_{n-1};\dots ;j_1;U^{\left(n\right)}\left(x\right);\alpha \right]\\ \triangleq \frac{\partial S^{\left(n-1\right)}\left[j_{n-1};\dots ;j_1;U^{\left(n\right)};\alpha \right]}{\partial a^{(k+1)}\left(i;j_k;\dots ;j_1;x\right)};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}k=1,\dots ,n-2;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1,\dots ,2^k.\end{array}$$

(24)

The final expression of the total differential expressed by Equation (2) is obtained by inserting into Equation (2) the expression for the indirect-effect term obtained in Equation (17), which yields the following expression for the n^th-order sensitivities $R^{\left(n\right)}\left[j_n;\dots ;j_1;U^{\left(n\right)};A^{(n)};\alpha \right]\equiv {\partial }^{n}R\left[u\left(x\right);\alpha \right]/\partial {\mathsf{\alpha }}_{j_n}\dots \partial {\mathsf{\alpha }}_{j_1}$ of the response $R\left[u\left(x\right);\alpha \right]$ with respect to the parameters ${\mathsf{\alpha }}_{j_1},\dots ,{\mathsf{\alpha }}_{j_n}$, for $j_1=1,\dots ,TP;\hspace{0.17em}\hspace{0.17em}\dots ;\hspace{0.17em}{j}_{n-1}=1,\dots ,j_{n-2}$:

$$\begin{array}{l}{R}^{\left(n\right)}\left[j_n;\dots ;j_1;U^{\left(n\right)};A^{\left(n\right)};\alpha \right]\equiv {\partial }^{n}R\left[u\left(x\right);\alpha \right]/\partial {\alpha }_{j_n}\dots \partial {\alpha }_{j_1}=\\ \frac{\partial R^{\left(n-1\right)}\left[j_{n-1};\dots ;j_1;U^{\left(n-1\right)}\left(x\right);A^{\left(n-1\right)}\left(x\right);\alpha \right]}{\partial {\alpha }_{j_n}}-\frac{{\left[\partial {\widehat{P}}^{\left(n\right)}\left(U^{\left(n\right)};A^{\left(n\right)};\mathsf{\delta }\alpha \right)\right]}_{\partial {\mathsf{\Omega }}_x}}{\partial {\alpha }_{j_n}}\\ +{\displaystyle {\displaystyle \sum }_{i=1}^{2^{n-1}}}\langle a^{\left(n\right)}\left(i;j_{n-1};\dots ;j_1;x\right),s_V^{\left(n\right)}{\left[i;j_n;\dots ;j_1;U^{\left(n\right)}\left(x\right);\alpha \right]\rangle }_1.\\ \triangleq \underset{{\mathsf{\lambda }}_1\left(\alpha \right)}{\overset{{\mathsf{\omega }}_1\left(\alpha \right)}{{\displaystyle \int }}}\dots \underset{{\mathsf{\lambda }}_{TI}\left(\alpha \right)}{\overset{{\mathsf{\omega }}_{TI}\left(\alpha \right)}{{\displaystyle \int }}}{S}^{\left(n\right)}\left[j_n;\dots ;j_1;U^{\left(n\right)}\left(2^{n-1};x\right);A^{\left(n\right)}\left(2^{n-1};x\right);\alpha \right]d{x}_1\dots d{x}_{TI}.\end{array}$$

(25)

2.2. The n^th-CASAM-N for $n=1$ and $n=2$

Setting $n=1$ in the mathematical framework of the n^th-CASAM-N conjectured above in Section 2.1 yields the following expressions for the particular case $n=1$:

(i)

Expression of the model response

$$R^{\left(0\right)}\left[U^{\left(0\right)};\alpha \right]\equiv R\left[u\left(x\right);\alpha \right]\triangleq {\displaystyle \underset{{\mathsf{\lambda }}_1\left(\alpha \right)}{\overset{{\mathsf{\omega }}_1\left(\alpha \right)}{\int }}\dots {\displaystyle \underset{{\mathsf{\lambda }}_{TI}\left(\alpha \right)}{\overset{{\mathsf{\omega }}_{TI}\left(\alpha \right)}{\int }}S\left[u\left(x\right);\alpha \right]d{x}_1\dots d{x}_{TI}}},$$

(26)

(ii)

Expression of the first-order response sensitivities for $j_1=1,\dots ,TP$:

$$\begin{array}{l}{R}^{\left(1\right)}\left[j_1;U^{\left(1\right)}\left(2^0;x\right);A^{(1)}\left(2^0;x\right);\alpha \right]=\frac{\partial R\left[u\left(x\right);\alpha \right]}{\partial {\mathsf{\alpha }}_{j_1}}\\ =-\frac{{\left[\partial {\widehat{P}}^{\left(1\right)}\left(U^{\left(1\right)};A^{(1)};\mathsf{\delta }\alpha \right)\right]}_{\partial {\mathsf{\Omega }}_x}}{\partial {\mathsf{\alpha }}_{j_1}}+{〈a^{(1)}\left(x\right),s_V^{\left(1\right)}\left[U^{\left(1\right)}\left(x\right);\alpha \right]〉}_1\hspace{0.17em}.\end{array}$$

(27)

The various quantities that appear in Equation (27) take on the following particular forms for $n=1$:

$$U^{\left(1\right)}\left(2^0;x\right)\equiv u\left(x\right);\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{A}^{(1)}\left(2^0;x\right)\triangleq a^{(1)}\left(x\right);\hspace{0.17em}\hspace{0.17em}{s}_V^{\left(1\right)}\left(j_1;u;\alpha \right)\triangleq \frac{\partial \left[Q\left(\alpha \right)-N\left(\hspace{0.17em}u;\alpha \right)\right]}{\partial {\mathsf{\alpha }}_{j_1}}.$$

(28)

The first-level adjoint sensitivity function $A^{\left(1\right)}\left(2^0;x\right)\equiv a^{\left(1\right)}\left(x\right)$ is the solution of the following 1st-LASS obtained by setting $n=1$ in Equations (18) and (19):

$${\left\{A{M}^{\left(1\right)}\left[2^0\times 2^0;U^{\left(1\right)}\left(2^0;x\right);\alpha \right]A^{\left(1\right)}\left(2^0;x\right)\right\}}_{{\alpha }^0}={\left\{Q_A^{\left(1\right)}\left[2^0;U^{\left(1\right)}\left(2^0;x\right);\alpha ;\mathsf{\delta }\alpha \right]\right\}}_{{\alpha }^0},\hspace{0.17em}\hspace{0.17em}x\in {\mathsf{\Omega }}_x,$$

(29)

$${\left\{B_A^{\left(1\right)}\left[2^0;U^{\left(1\right)}\left(2^0;x\right);V^{\left(1\right)}\left(2^0;x\right);\alpha ;\mathsf{\delta }\alpha \right]\right\}}_{{\alpha }^0}=0,\hspace{1em}x\in \partial {\mathsf{\Omega }}_x\left({\alpha }^0\right).$$

(30)

where

$$A{M}^{\left(1\right)}\left[2^0\times 2^0;U^{\left(1\right)}\left(2^0;x\right);\alpha \right]\equiv {\left[N^{\left(1\right)}\left(u;\alpha \right)\right]}^{*};$$

(31)

$$Q_A^{\left(1\right)}\left[2^0;U^{\left(1\right)}\left(2^0;x\right);\alpha ;\mathsf{\delta }\alpha \right]\equiv q_A^{\left(1\right)}\left[u\left(x\right);\alpha \right]\triangleq {\left\{\partial S\left(u;\alpha \hspace{0.17em}\right)/\partial u\right\}}_{{\alpha }^0};$$

(32)

$$B_V^{\left(1\right)}\left[2^0;U^{\left(1\right)}\left(2^0;x\right);V^{\left(1\right)}\left(2^0;x\right);\alpha ;\mathsf{\delta }\alpha \right]\equiv b_A^{\left(1\right)}\left(u;a^{\left(1\right)};\alpha \right).$$

(33)

Comparing the expressions obtained above in Equations (26)–(33) with the expressions obtained in Appendix B.3 reveals that the corresponding expressions are identical to each other, which proves the correctness for the particular case $n=1$ of the conjectured general expressions underlying the n^th-CASAM-N.

Setting $n=2$ in the mathematical framework of the n^th-CASAM-N conjectured above in Section 2.1 yields the following expressions for the second-order response sensitivities for $j_1,j_2=1,\dots ,TP$:

$$\begin{array}{l}{R}^{\left(2\right)}\left[j_2;j_1;U^{\left(2\right)}\left(2;x\right);A^{\left(2\right)}\left(2;j_1;x\right);\alpha \right]\equiv \frac{{\partial }^{\left(2\right)}R\left[u\left(x\right);\alpha \right]}{\partial {\alpha }_{j_2}\partial {\alpha }_{j_1}}=\frac{\partial R^{\left(1\right)}\left[j_1;u\left(x\right);a^{\left(1\right)}\left(x\right);\alpha \right]}{\partial {\alpha }_{j_2}}\\ -\frac{{\left\{\partial {\widehat{P}}^{\left(2\right)}\left[U^{\left(2\right)}\left(2;x\right);A^{\left(2\right)}\left(2;j_1;x\right);\alpha \right]\right\}}_{\partial {\mathsf{\Omega }}_x}}{\partial {\alpha }_{j_2}}+{\displaystyle {\displaystyle \sum }_{i=1}^2}\langle a^{\left(2\right)}\left(i;j_1;x\right),s_V^{\left(2\right)}{\left[i;j_2;U^{\left(2\right)}\left(2;x\right);\alpha \right]\rangle }_1,\end{array}$$

(34)

where $A^{(2)}\left(2;j_1;x\right)\triangleq {\left[a^{(2)}\left(1;j_1;x\right),a^{(2)}\left(2;j_1;x\right)\right]}^{†}\in {\mathrm{H}}_2\left({\mathsf{\Omega }}_x\right)$ is the solution of the following 2nd-Level Adjoint Sensitivity System (2nd-LASS) for each $j_1=1,\dots ,TP$:

$$\begin{array}{l}{\left\{A{M}^{\left(2\right)}\left[2\times 2;U^{\left(2\right)}\left(2;x\right);\alpha \right]A^{(2)}\left(2;j_1;x\right)\right\}}_{{\alpha }^0}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}={\left\{Q_A^{\left(2\right)}\left[2;j_1;U^{\left(2\right)}\left(2;x\right);\alpha \right]\right\}}_{{\alpha }^0},\hspace{0.17em}\hspace{0.17em}{j}_1=1,\dots ,TP;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x\in {\mathsf{\Omega }}_x,\end{array}$$

(35)

$${\left\{B_A^{\left(2\right)}\left[2;U^{\left(2\right)}\left(2;x\right);A^{(2)}\left(2;j_1;x\right);\alpha \right]\right\}}_{{\alpha }^0}=0\left[2\right];\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{j}_1=1,\dots ,TP;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x\in \partial {\mathsf{\Omega }}_x\left({\alpha }^0\right).$$

(36)

$$\begin{array}{l}{Q}_A^{\left(2\right)}\left[2;j_1;U^{\left(2\right)}\left(2;x\right);\alpha \right]\triangleq \left(\begin{array}{c}{q}_A^{\left(2\right)}\left(1;j_1;U^{\left(2\right)};\alpha \right)\\ q_A^{\left(2\right)}\left(2;j_1;U^{\left(2\right)};\alpha \right)\end{array}\right)\\ \triangleq \left(\begin{array}{c}\partial S^{\left(1\right)}\left[j_1;u\left(x\right);a^{\left(1\right)}\left(x\right);\alpha ;v^{\left(1\right)}\left(x\right)\right]/\partial u\\ \partial S^{\left(1\right)}\left[j_1;u\left(x\right);a^{\left(1\right)}\left(x\right);\alpha ;v^{\left(1\right)}\left(x\right)\right]/\partial a^{\left(1\right)}\end{array}\right),\hspace{0.17em}\hspace{0.17em}{j}_1=1,\dots ,TP.\end{array}$$

(37)

Comparing the expressions obtained in Equations (26)–(33) with the expressions obtained in Appendix B.3 reveals that the corresponding expressions are identical to each other, which proves the correctness for the particular case $n=2$ of the conjectured general expressions underlying the n^th-CASAM-N.

2.3. The (n + 1)^th- CASAM-N

The n^th-order sensitivities $R^{\left(n\right)}\left[j_n;\dots ;j_1;U^{\left(n\right)};A^{(n)};\alpha \right]\equiv {\partial }^{n}R\left[u\left(x\right);\alpha \right]/\partial {\mathsf{\alpha }}_{j_n}\dots \partial {\mathsf{\alpha }}_{j_1}$$R^{\left(4\right)}\left[j_4;j_3;j_2;j_1;U^{\left(4\right)};A^{(4)};\alpha \right]$ will be assumed to satisfy the conditions stated in Equations (A10) and (A11) in Appendix B for each $j_1,\dots ,j_n=1,\dots ,TP$. Hence, the first-order total G-differential of $R^{\left(n\right)}\left[j_n;\dots ;j_1;U^{\left(n\right)};A^{(n)};\alpha \right]$ will exist and will be linear in the variations $V^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right)\triangleq \mathsf{\delta }{U}^{\left(n-1\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right)$ and $\mathsf{\delta }{A}^{(n)}\left(2^{n-1};j_{n-1};\dots ;j_1;x\right)$ in a neighborhood around the nominal values of the parameters and the respective state functions. By definition, the first-order total G-differential of $R^{\left(n\right)}\left[j_n;\dots ;j_1;U^{\left(n\right)};A^{(n)};\alpha \right]$ is given by the following expression:

$$\begin{array}{l}{\left\{\mathsf{\delta }{R}^{\left(n\right)}\left[j_n;\dots ;j_1;U^{\left(n\right)};A^{\left(n\right)};\alpha \right]\right\}}_{{\alpha }^0}\triangleq \frac{d}{d\epsilon }\left\{R^{\left(n\right)}\right.\left[j_n;\dots ;j_1;U^{\left(n\right)}+\epsilon V^{\left(n\right)};\right.\\ {\left.\left.A^{\left(n\right)}+\epsilon \mathsf{\delta }{A}^{\left(n\right)};\alpha +\epsilon \mathsf{\delta }\alpha \right]\right\}}_{\epsilon =0}\triangleq {\displaystyle {\displaystyle \sum }_{j_{n+1}=1}^{TP}}{\left\{\frac{\partial R^{\left(n\right)}\left[\dots ;U^{\left(n\right)};A^{\left(n\right)};\alpha \right]}{\partial {\alpha }_{j_{n+1}}}\right\}}_{{\alpha }^0}\mathsf{\delta }{\alpha }_{j_{n+1}}\\ +{\left\{\mathsf{\delta }{R}^{\left(n\right)}\left[j_n;\dots ;j_1;U^{\left(n\right)};A^{\left(n\right)};\alpha ;V^{\left(n\right)}\left(x\right);\mathsf{\delta }{A}^{\left(n\right)}\left(x\right)\right]\right\}}_{ind},\end{array}$$

(38)

where the quantity ${\left\{\mathsf{\delta }{R}^{\left(n\right)}\left[j_n;\dots ;j_1;U^{\left(n\right)};A^{(n)};\alpha ;V^{\left(n\right)};\mathsf{\delta }{A}^{(n)}\right]\right\}}_{ind}$ denotes the indirect-effect term, which is defined as follows:

$$\begin{array}{l}{\left\{\mathsf{\delta }{R}^{\left(n\right)}\left[j_n;\dots ;j_1;U^{\left(n\right)};A^{(n)};\alpha ;V^{\left(n\right)};\mathsf{\delta }{A}^{(n)}\right]\right\}}_{ind}\\ \triangleq {\displaystyle \underset{{\mathsf{\lambda }}_1\left(\alpha \right)}{\overset{{\mathsf{\omega }}_1\left(\alpha \right)}{\int }}\dots {\displaystyle \underset{{\mathsf{\lambda }}_{TI}\left(\alpha \right)}{\overset{{\mathsf{\omega }}_{TI}\left(\alpha \right)}{\int }}{\left\{\frac{\partial S^{\left(n\right)}}{\partial U^{\left(n\right)}\left(x\right)}{V}^{\left(n\right)}\left(2^{n-1};x\right)+\frac{\partial S^{\left(n\right)}}{\partial A^{(n)}\left(x\right)}\mathsf{\delta }{A}^{(n)}\left(2^{n-1};x\right)\right\}}_{{\alpha }^0}d{x}_1\dots d{x}_{TI}}}\end{array}$$

(39)

The vectors $V^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right)\triangleq \mathsf{\delta }{U}^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right)$ and $\mathsf{\delta }{A}^{(n)}\left(2^{n-1};j_{n-1};\dots ;j_1;x\right)$, which are needed in order to evaluate the indirect-effect term ${\left\{\mathsf{\delta }{R}^{\left(n\right)}\left[j_n;\dots ;j_1;U^{\left(n\right)};A^{(n)};\alpha ;V^{\left(n\right)};\mathsf{\delta }{A}^{(n)}\right]\right\}}_{ind}$, are the solutions of the following (n + 1)^th-Level Variational Sensitivity System, which is obtained by concatenating the n^th-LVSS defined by Equations (4) and (5) together with the G-differentiated n^th-LASS for $j_1,\dots ,j_n=1,\dots ,TP$:

$$\begin{array}{l}{\left\{V{M}^{\left(n+1\right)}\left[2^n\times 2^n;U^{\left(n+1\right)}\left(2^n;j_{n-1};\dots ;j_1;x\right);\alpha \right]V^{\left(n+1\right)}\left(2^n;j_{n-1};\dots ;j_1;x\right)\right\}}_{{\alpha }^0}\\ ={\left\{Q_V^{\left(n+1\right)}\left[2^n;U^{\left(n\right)}\left(2^n;j_{n-2};\dots ;j_1;x\right);\alpha ;\mathsf{\delta }\alpha \right]\hspace{0.17em}\right\}}_{{\alpha }^0},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x\in {\mathsf{\Omega }}_x,\end{array}$$

(40)

$${\left\{B_V^{\left(n+1\right)}\left[2^n;U^{\left(n+1\right)}\left(2^n;x\right);V^{\left(n+1\right)}\left(2^n;x\right);\alpha ;\mathsf{\delta }\alpha \right]\right\}}_{{\alpha }^0}=0\left[2^n\right];\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x\in \partial {\mathsf{\Omega }}_x\left({\alpha }^0\right),$$

(41)

where

$$U^{\left(n+1\right)}\left(2^n;j_{n-1};\dots ;j_1;x\right)\triangleq \left(\begin{array}{c}{U}^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right)\\ A^{(n)}\left(2^{n-1};j_{n-1};\dots ;j_1;x\right)\end{array}\right);$$

(42)

$$\begin{array}{l}{V}^{\left(n+1\right)}\left(2^n;j_{n-1};\dots ;j_1;x\right)\triangleq \mathsf{\delta }{U}^{\left(n+1\right)}\left(2^n;j_{n-1};\dots ;j_1;x\right)=\left(\begin{array}{c}{V}^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right)\\ \mathsf{\delta }{A}^{(n)}\left(2^{n-1};j_{n-1};\dots ;j_1;x\right)\end{array}\right)\\ =\left[v^{\left(1\right)}\left(x\right),\mathsf{\delta }{a}^{\left(1\right)}\left(x\right),\mathsf{\delta }{a}^{\left(2\right)}\left(1;j_1;x\right),\mathsf{\delta }{a}^{\left(2\right)}\left(2;j_1;x\right),\right.\dots \\ {\left.\dots ,\mathsf{\delta }{a}^{(n)}\left(1;j_{n-1};\dots ;j_1;x\right),\dots ,\mathsf{\delta }{a}^{(n)}\left(2^{n-1};j_{n-1};\dots ;j_1;x\right)\right]}^{†};\end{array}$$

(43)

$$\begin{array}{l}V{M}^{\left(n+1\right)}\left[2^n\times 2^n;U^{\left(n+1\right)}\left(2^n;j_{n-1};\dots ;j_1;x\right);\alpha \right]\hspace{0.17em}\\ \triangleq \left(\begin{array}{cc}V{M}^{\left(n\right)}\left[2^{n-1}\times 2^{n-1};x;\alpha \right]& 0\left[2^{n-1}\times 2^{n-1}\right]\\ V{M}_{21}^{\left(n+1\right)}\left(2^{n-1}\times 2^{n-1};x;\alpha \right)& V{M}_{22}^{\left(n+1\right)}\left(2^{n-1}\times 2^{n-1};x;\alpha \right)\end{array}\right);\end{array}$$

(44)

$$\begin{array}{l}V{M}_{21}^{\left(n+1\right)}\left(2^{n-1}\times 2^{n-1};j_{n-1};\dots ;j_1;x;\alpha \right)\triangleq \\ -\frac{\partial Q_A^{\left(n\right)}\left[2^{n-1};j_{n-1};\dots ;j_1;U^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right);\alpha \right]}{\partial U^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right)}\\ +\frac{\partial \left\{A{M}^{\left(n\right)}\left[2^{n-1}\times 2^{n-1};U^{\left(n\right)}\left(2^{n-1};x\right);\alpha \right]A^{(n)}\left(2^{n-1};j_{n-1};\dots ;j_1;x\right)\right\}}{\partial U^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right)};\end{array}$$

(45)

$$V{M}_{22}^{\left(n+1\right)}\left(2^{n-1}\times 2^{n-1};x;\alpha \right)\triangleq A{M}^{\left(n\right)}\left[2^{n-1}\times 2^{n-1};U^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right);\alpha \right];\hspace{0.17em}$$

(46)

$$\begin{array}{l}{Q}_V^{\left(n+1\right)}\left[2^n;j_1;U^{\left(n+1\right)}\left(2^n;j_{n-1};\dots ;j_1;x\right);\alpha ;\mathsf{\delta }\alpha \right]\\ \triangleq \left(\begin{array}{c}{Q}_V^{\left(n\right)}\left[2^{n-1};U^{\left(n\right)}\left(2^{n-1};x\right);\alpha ;\mathsf{\delta }\alpha \right]\\ Q_2^{\left(n+1\right)}\left[2^{n-1};U^{\left(n+1\right)}\left(2^n;x\right);\alpha ;\mathsf{\delta }\alpha \right]\end{array}\right)\triangleq \left\{q_V^{\left(n+1\right)}\left[1;U^{\left(n+1\right)}\left(2^n;j_{n-1};\dots ;j_1;x\right);\alpha ;\mathsf{\delta }\alpha \right]\right.\\ \hspace{1em}\hspace{1em}{\left.,\dots ,q_V^{\left(n+1\right)}\left[2^n;U^{\left(n+1\right)}\left(2^n;j_{n-1};\dots ;j_1;x\right);\alpha ;\mathsf{\delta }\alpha \right]\right\}}^{†};\end{array}$$

(47)

$$q_V^{\left(n\right)}\left[i;U^{\left(n\right)}\left(x\right);\alpha ;\mathsf{\delta }\alpha \right]\equiv {\displaystyle \sum _{j_n=1}^{TP}{s}_V^{\left(n\right)}\left[i;j_{n-2};\dots ;j_1;U^{\left(n\right)}\left(x\right);\alpha \right]\mathsf{\delta }{\mathsf{\alpha }}_{j_n}};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1,\dots ;2^{n-1};$$

(48)

$$\begin{array}{l}{Q}_2^{\left(n+1\right)}\left[2^{n-1};U^{\left(n+1\right)}\left(2^n;j_{n-1};\dots ;j_1;x\right);\alpha ;\mathsf{\delta }\alpha \right]\\ \triangleq \frac{\partial Q_A^{\left(n\right)}\left[2^{n-1};j_{n-1};\dots ;j_1;U^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right);\alpha \right]}{\partial \alpha }\partial \alpha \\ -\frac{\partial \left\{A{M}^{\left(n\right)}\left[2^{n-1}\times 2^{n-1};U^{\left(n\right)}\left(2^{n-1};x\right);\mathsf{\alpha }\right]A^{(n)}\left(2^{n-1};j_{n-1};\dots ;j_1;x\right)\right\}}{\partial \alpha }\partial \alpha ;\end{array}$$

(49)

$$\begin{array}{l}{B}_V^{\left(n+1\right)}\left[2^n;U^{\left(n+1\right)}\left(2^n;j_{n-1};j_1;x\right);V^{\left(n+1\right)}\left(2^n;j_{n-1};\dots ;j_1;x\right);\alpha ;\mathsf{\delta }\alpha \right]\\ \triangleq \left(\begin{array}{c}{B}_V^{\left(n\right)}\left[2^{n-1};U^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right);V^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right);\alpha ;\mathsf{\delta }\alpha \right]\\ \mathsf{\delta }{B}_A^{\left(n\right)}\left[2^{n-1};U^{\left(n\right)}\left(2^{n-1};j_{n-2};\dots ;j_1;x\right);A^{\left(n\right)}\left(2^{n-1};j_{n-1};\dots ;j_1;x\right);\alpha \right]\end{array}\right).\end{array}$$

(50)

Solving the (n + 1)^th-LVSS would require $O\left(T{P}^{n+1}\right)$ large-scale computations, which is unrealistic for large-scale systems comprising many parameters. The (n + 1)^th-CASAM-N circumvents the need to solve the (n + 1)^th-LVSS by deriving an alternative expression for the indirect-effect term defined in Equation (39), in which the function $V^{\left(n+1\right)}\left(2^n;j_{n-1};\dots ;j_1;x\right)$ is replaced by a (n + 1)^th-level adjoint function, denoted as $A^{(n+1)}\left(2^n;j_n;\dots ;j_1;x\right)\triangleq {\left[\hspace{0.17em}{a}^{(n+1)}\left(1;j_n;\dots ;j_1;x\right),\dots ,a^{(n+1)}\left(2^n;j_n;\dots ;j_1;x\right)\right]}^{†}\in {\mathrm{H}}_{n+1}\left({\mathsf{\Omega }}_x\right)$, which is independent of parameter variations. The elements of the Hilbert space ${\mathrm{H}}_{n+1}\left({\mathsf{\Omega }}_x\right)$ are block-vectors of the form ${\Psi }^{(n+1)}\left(2^n;j_n;\dots ;j_1;x\right)\triangleq {\left[\hspace{0.17em}{\psi }^{(n+1)}\left(1;j_n;\dots ;j_1;x\right),\dots ,{\psi }^{(n+1)}\left(2^n;j_n;\dots ;j_1;x\right)\right]}^{†}$, comprising $2^n$ $TD$-dimensional vectors of the form ${\psi }^{\left(n+1\right)}\left(i;x\right)\triangleq {\left[{\mathsf{\psi }}_1^{\left(n+1\right)}\left(i;x\right),\dots ,{\mathsf{\psi }}_{TD}^{\left(n+1\right)}\left(i;x\right)\right]}^{†}\in {\mathrm{H}}_1\left({\mathsf{\Omega }}_x\right)$, $i=1,\dots ,2^{n+1}$. The inner product of two vectors ${\Psi }^{(n+1)}\left(2^n;j_n;\dots ;j_1;x\right)$ and ${\Phi }^{(n+1)}\left(2^n;j_n;\dots ;j_1;x\right)$ in the Hilbert space ${\mathrm{H}}_{n+1}\left({\mathsf{\Omega }}_x\right)$ is denoted as ${〈{\Psi }^{(n+1)}\left(2^n;x\right),{\Phi }^{(n+1)}\left(2^n;x\right)〉}_{n+1}$ and defined as follows:

$${〈{\Psi }^{(n+1)}\left(2^n;x\right),{\Phi }^{(n+1)}\left(2^n;x\right)〉}_{n+1}\triangleq \hspace{0.17em}{\displaystyle \sum _{i=1}^{2^n}{〈{\psi }^{(n+1)}\left(i;x\right),{\phi }^{(n+1)}\left(i;x\right)〉}_1}.$$

(51)

The (n + 1)^th-Level Adjoint Sensitivity System [(n + 1)^th-LASS ] for the (n + 1)^th-level adjoint function $A^{(n+1)}\left(2^n;j_n;\dots ;j_1;x\right)$ is obtained by using the inner product defined in Equation (51) as follows:

$$\begin{array}{l}{\left\{{〈A^{(n+1)}\left(2^n;x\right),V{M}^{\left(n+1\right)}\left(2^n;x\right)〉}_2\right\}}_{{\alpha }^0}={\left\{{\left[P^{\left(n+1\right)}\left(U^{\left(n+1\right)};A^{(n+1)};V^{\left(n+1\right)};\alpha \right)\right]}_{\partial {\mathsf{\Omega }}_x}\right\}}_{{\alpha }^0}\\ +{\left\{{〈V^{\left(n+1\right)}\left(2^n;x\right),A{M}^{\left(n+1\right)}\left[2^n\times 2^n;U^{\left(n+1\right)}\left(2^n;x\right);\alpha \right]A^{(n+1)}\left(2^n;x\right)〉}_n\right\}}_{{\alpha }^0},\end{array}$$

(52)

where the quantity ${\left\{{\left[P^{\left(n+1\right)}\left(U^{\left(n+1\right)};A^{(n+1)};V^{\left(n+1\right)};\alpha \right)\right]}_{\partial {\mathsf{\Omega }}_x}\right\}}_{{\alpha }^0}$ denotes the corresponding bilinear concomitant on the domain’s boundary, evaluated at the nominal values for the parameters and respective state functions.

In terms of the (n + 1)^th-level adjoint function $A^{(n+1)}\left(2^n;j_n;\dots ;j_1;x\right)$, the indirect-effect term defined by Equation (39) will have the following expression:

$$\begin{array}{l}{\left\{\mathsf{\delta }{R}^{\left(n\right)}\left[j_n;\dots ;j_1;U^{\left(n\right)};A^{(n)};\alpha ;V^{\left(n\right)};\mathsf{\delta }{A}^{(n)}\right]\right\}}_{ind}=-{\left\{{\left[{\widehat{P}}^{\left(n+1\right)}\left(U^{\left(n+1\right)};A^{(n+1)};\mathsf{\delta }\alpha \right)\right]}_{\partial {\mathsf{\Omega }}_x}\right\}}_{{\alpha }^0}\\ +{\left\{{〈A^{(n+1)}\left(2^n;j_n;\dots ;j_1;x\right),Q_V^{\left(n+1\right)}\left[2^n;U^{\left(n+1\right)}\left(2^n;j_{n-1};\dots ;j_1;x\right);\alpha ;\mathsf{\delta }\alpha \right]〉}_n\right\}}_{{\alpha }^0}.\end{array}$$

(53)

where $A^{(n+1)}\left(2^n;j_n;\dots ;j_1;x\right)$ is the solution of the following (n + 1)^th-LASS:

$$\begin{array}{l}A{M}^{\left(n+1\right)}\left[2^n\times 2^n;U^{\left(n+1\right)}\left(2^n;x\right);\alpha \right]A^{(n+1)}\left(2^n;j_n;\dots ;j_1;x\right)\\ =Q_A^{\left(n+1\right)}\left[2^n;j_n;\dots ;j_1;U^{\left(n+1\right)}\left(2^n;j_{n-1};\dots ;j_1;x\right);\alpha \right]\hspace{0.17em},\end{array}$$

(54)

subject to boundary conditions represented in operator form as follows:

$$\begin{array}{l}{\left\{B_A^{\left(n+1\right)}\left[2^n;U^{\left(n\right)}\left(2^n;j_{n-1};\dots ;j_1;x\right);A^{(n)}\left(2^n;j_n;\dots ;j_1;x\right);\alpha \right]\right\}}_{{\alpha }^0}=0\left[2^n\right],\\ x\in \partial {\mathsf{\Omega }}_x\left({\alpha }^0\right);\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{j}_1=1,\dots ,TP;\hspace{0.17em}\hspace{0.17em}{j}_2=1,\dots ,j_1;\hspace{0.17em}\dots ;\hspace{0.17em}{j}_n=1,\dots ,j_{n-1}.\end{array}$$

(55)

In Equation (53), the quantity ${\left\{{\left[{\widehat{P}}^{\left(n+1\right)}\left(U^{\left(n+1\right)};A^{(n+1)};\mathsf{\delta }\alpha \right)\right]}_{\partial {\mathsf{\Omega }}_x}\right\}}_{{\alpha }^0}$ denotes residual boundary terms that may have not vanished automatically after having used the boundary conditions provided in Equations (41) and (55) to eliminate from Equation (52) all unknown values of the (n + 1)^th-level variational function $V^{\left(n+1\right)}\left(2^n;j_{n-1};\dots ;j_1;x\right)$.

The quantities that appear in the definition of the (n + 1)^th-LASS, cf. Equations (54) and (55), are defined as follows:

$$\begin{array}{l}{\mathrm{AM}}^{\left(n+1\right)}\left[2^n\times 2^n;U^{\left(n+1\right)}\left(2^n;j_{n-1};\dots ;j_1;x\right);\alpha \right]\triangleq {\left[{\mathrm{VM}}^{\left(n+1\right)}\left(2^n\times 2^n;U^{\left(n+1\right)};\alpha \right)\right]}^{*}\\ =\left(\begin{array}{cc}{\left\{{\left[{\mathrm{VM}}^{\left(n\right)}\left(2^{n-1}\times 2^{n-1}\right)\right]}^{*}\right\}}^{†}& {\left\{{\left[{\mathrm{VM}}_{21}^{\left(n+1\right)}\left(2^{n-1}\times 2^{n-1}\right)\right]}^{*}\right\}}^{†}\\ 0\left[2^{n-1}\times 2^{n-1}\right]& {\left\{{\left[{\mathrm{VM}}_{22}^{\left(n+1\right)}\left(2^{n-1}\times 2^{n-1}\right)\right]}^{*}\right\}}^{†}\end{array}\right),\end{array}$$

(56)

$$\begin{array}{l}{Q}_A^{\left(n+1\right)}\left[2^n;j_n;\dots ;j_1;U^{\left(n+1\right)}\left(2^n;x\right);\alpha \right]\triangleq \left\{q_A^{\left(n+1\right)}\left[1;j_n;\dots ;j_1;U^{\left(n+1\right)}\left(2^n;x\right);\alpha \right]\right.,\dots \\ \dots ,{\left.q_A^{\left(n+1\right)}\left[2^n;j_n;\dots ;j_1;U^{\left(n+1\right)}\left(2^n;x\right);\alpha \right]\right\}}^{†},\hspace{0.17em}\hspace{0.17em}{j}_1=1,\dots ,TP;\hspace{0.17em}\hspace{0.17em}\dots ;\hspace{0.17em}{j}_n=1,\dots ,j_{n-1};\end{array}$$

(57)

$$q_A^{\left(n+1\right)}\left[1;j_n;\dots ;j_1;U^{\left(n+1\right)}\left(x\right);\alpha \right]\triangleq \partial S^{\left(n\right)}\left[j_n;\dots ;j_1;U^{\left(n+1\right)};\alpha \right]/\partial u\left(x\right);$$

(58)

$$q_A^{\left(n+1\right)}\left[2;j_n;\dots ;j_1;U^{\left(n+1\right)}\left(x\right);\alpha \right]\triangleq \partial S^{\left(n\right)}\left[j_n;\dots ;j_1;U^{\left(n+1\right)};\alpha \right]/\partial a^{\left(1\right)}\left(x\right);$$

(59)

$$\begin{array}{l}for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}n\ge 3\hspace{0.17em}:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{q}_A^{\left(n+1\right)}\left[2^k+i;j_n;\dots ;j_1;U^{\left(n+1\right)}\left(x\right);\alpha \right]\\ \triangleq \frac{\partial S^{\left(n\right)}\left[j_n;\dots ;j_1;U^{\left(n+1\right)};\alpha \right]}{\partial a^{(k+1)}\left(i;j_k;\dots ;j_1;x\right)};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}k=1,\dots ,n-1;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1,\dots ,2^k.\end{array}$$

(60)

The final expression of the total differential expressed by Equation (17) is obtained by inserting into Equation (38) the expression for the indirect-effect term obtained in Equation (53), which yields the following expression for the (n + 1)^th-order sensitivities $R^{\left(n+1\right)}\left[j_{n+1};\dots ;j_1;U^{\left(n+1\right)};A^{(n+1)};\alpha \right]\equiv {\partial }^{n+1}R\left[u\left(x\right);\alpha \right]/\partial {\mathsf{\alpha }}_{j_{n+1}}\dots \partial {\mathsf{\alpha }}_{j_1}$ of the response $R\left[u\left(x\right);\alpha \right]$ with respect to the parameters ${\mathsf{\alpha }}_{j_1},\dots ,{\mathsf{\alpha }}_{j_{n+1}}$, for $j_1=1,\dots ,TP;\hspace{0.17em}\hspace{0.17em}\dots ;\hspace{0.17em}{j}_n=1,\dots ,j_{n-1}$:

$$\begin{array}{l}{R}^{\left(n+1\right)}\left[j_{n+1};\dots ;j_1;U^{\left(n+1\right)};A^{\left(n+1\right)};\alpha \right]\equiv {\partial }^{n+1}R\left[u\left(x\right);\alpha \right]/\partial {\alpha }_{j_{n+1}}\dots \partial {\alpha }_{j_1}=\\ \frac{\partial R^{\left(n\right)}\left[j_n;\dots ;j_1;U^{\left(n\right)}\left(x\right);A^{\left(n\right)}\left(x\right);\alpha \right]}{\partial {\alpha }_{j_{n+1}}}-\frac{{\left[\partial {\widehat{P}}^{\left(n+1\right)}\left(U^{\left(n+1\right)};A^{\left(n+1\right)};\mathsf{\delta }\alpha \right)\right]}_{\partial {\mathsf{\Omega }}_x}}{\partial {\alpha }_{j_{n+1}}}\\ +{\displaystyle {\displaystyle \sum }_{i=1}^{2^n}}\langle a^{\left(n+1\right)}\left(i;j_n;\dots ;j_1;x\right),s_V^{\left(n+1\right)}{\left[i;j_{n+1};\dots ;j_1;U^{\left(n+1\right)}\left(x\right);\alpha \right]\rangle }_1.\\ \triangleq {\displaystyle \underset{{\mathsf{\lambda }}_1\left(\alpha \right)}{\overset{{\mathsf{\omega }}_1\left(\alpha \right)}{{\displaystyle \int }}}}\dots {\displaystyle \underset{{\mathsf{\lambda }}_{TI}\left(\alpha \right)}{\overset{{\mathsf{\omega }}_{TI}\left(\alpha \right)}{{\displaystyle \int }}}}{S}^{\left(n+1\right)}\left[j_{n+1};\dots ;j_1;U^{\left(n+1\right)}\left(2^n;x\right);A^{\left(n+1\right)}\left(2^n;x\right);\alpha \right]d{x}_1\dots d{x}_{TI}.\end{array}$$

(61)

The expression obtained in Equation (61) is identical to the expression that would be obtained by replacing the index n with (n + 1) in the expression obtained in Equation (25), thus completing the proof, by mathematical induction, of the validity/correctness of the conjectured general expressions underlying the n^th-CASAM-N.

3. Discussion

This work has presented the n^th-Order Comprehensive Sensitivity Analysis Methodology for Nonlinear Systems (abbreviated as “n^th-CASAM-N”), which enables the hitherto very difficult, if not intractable, exact computation of arbitrarily high-order response sensitivities with respect to uncertain model and boundary parameters. The mathematical framework underlying the n^th-CASAM-N is proven to be correct by using mathematical induction. The n^th-CASAM-N is formulated in linearly increasing higher-dimensional Hilbert spaces—as opposed to exponentially increasing parameter-dimensional spaces—thus overcoming the curse of dimensionality in sensitivity analysis of nonlinear systems. For very large and complex models, the n^th-CASAM-N remains the only practical method for computing response sensitivities comprehensively and accurately.

The question of when to stop computing progressively higher-order sensitivities has been addressed by Cacuci [19] in conjunction with the question of convergence of the Taylor-series expansion of the response in terms of the uncertain model parameters, since this Taylor-series expansion is the fundamental premise for the expressions provided by the “propagation of errors” methodology for the cumulants of the model response distribution in the phase space of model parameters. The convergence of this Taylor-series, which depends on both the response sensitivities to parameters and the uncertainties associated with the parameter distribution, must be ensured. This can be done by ensuring that the combination of parameter uncertainties and response sensitivities is sufficiently small to fall inside the radius of convergence of the Taylor-series expansion. If the Taylor-series fails to converge, targeted experiments must be performed in order to reduce the largest sensitivities as well as the largest uncertainties (particularly standard deviations) that affect the most important parameters, by applying the principles of the BERRU-PM [20] methodology to obtain best-estimate parameter values with reduced uncertainties (model calibration).

Taken together, the n^th-CASAM-N and n^th-CASAM-L are the only practical methods for computing response sensitivities comprehensively and accurately for large and complex models, as has been illustrated by the application [9,11] of the n^th-CASAM-L to the OECD/NEA reactor physics benchmark. Illustrative paradigm applications of the n^th-CASAM-N are currently being finalized.