Illustrative Application of the 2 nd -Order Adjoint Sensitivity Analysis Methodology to a Paradigm Linear Evolution/Transmission Model: Point-Detector Response

This work illustrates the application of the “Second Order Comprehensive Adjoint Sensitivity Analysis Methodology” (2 nd -CASAM) to a mathematical model that can simulate the evolution and/or transmission of particles in a heterogeneous medium. The model response is the value of the model’s state function (particle concentration or particle flux) at a point in phase-space, which would simulate a pointwise measurement of the respective state function. This paradigm model admits exact closed-form expressions for all of the 1 st - and 2 nd -order response sensitivities to the model’s uncertain parameters and domain boundaries. These closed-form expressions can be used to verify the numerical results of production and/or commercial software, e.g., particle transport codes. Furthermore, this paradigm model comprises many uncertain parameters which have relative sensitivities of identical magnitudes. Therefore, this paradigm model could serve as a stringent benchmark for in-ter-comparing the performances of all deterministic and statistical sensitivity analysis methods, including the 2 nd -CASAM. Illustrative Application of the 2 nd -Order Adjoint Sensitivity Analysis Methodology to a Paradigm Linear Evolution/Transmission Model: Point-Detector Response.


Introduction
The application of the general second-order adjoint sensitivity analysis metho-dology presented in [1] is illustrated in this work by means of a simple mathematical model which expresses a conservation law of the model's state function. This paradigm model is representative of transmission of particles and/or radiation through materials [2] [3], chemical kinetics processes [4] [5], radioactive decay modeled by the Bateman equation, etc.
Although the model is simple, it comprises a large number of model parameters, thereby involving a correspondingly large number of sensitivities (i.e., functional derivatives) of the model's responses to the model parameters. Furthermore, the model has been deliberately designed so that a large number of relative response sensitivities display identical values. The fact that the model has a large number of parameters and the fact that all but a few relative sensitivities have identical values would make it very difficult, if not impossible, to use statistical methods to compute the first-and second-order sensitivities of the responses to all of the parameters of this model, since the computational costs would be prohibitive. Of course, statistical methods would not be able to compute the exact values of these first-and second-order sensitivities. For such models, involving many parameters but relatively few responses, the Second-Order Comprehensive Adjoint Sensitivity Analysis Methodology (2 nd -CASAM) for Linear Systems, presented in Part I [1], is best suited for computing exactly and efficiently the first-and second-order response sensitivities.
This work is organized as follows: Section 2 presents the paradigm evolution model. Section 3 presents the application of the 2 nd -CASAM [1] for efficiently computing the exact closed-form expressions of the first-and second-order sensitivities of a "point-type" response to both model and boundary parameters. The concluding remarks offered in Section 4 highlight the comprehensive verification mechanism which is inherently built into the 2 nd -CASAM [1] to ensure that the second-level adjoint functions are derived and computed correctly. All in all, the exact expressions of the 1 st -and 2 nd -order sensitivities presented in this work provide stringent benchmarks for the verification of the accuracies of any other methods, deterministic and/or statistical, for performing sensitivity analysis.

Mathematical Modeling of a Paradigm Evolution/Transmission Benchmark Problem
The general 2 nd -CASAM methodology presented in [1] is applied in this work to a simple paradigm model, admitting a closed-form analytic solution for convenient verification of all results to be obtained, which simulates a typical evolution or attenuation of a quantity that will be denoted as ( ) The simple evolution system represented by Equations (1) and (2) occurs in the mathematical modeling of many physical systems.
For example, the dependent variable ( ) t ρ could represent [2] [3] the evolution of the concentration of a substance in a homogeneous mixture of N materials, from an imprecisely known initial quantity, denoted as in ρ , measured at an initial-time value t β =  towards an imprecisely known final-time value u t β = .
The quantities i n and i σ would represent various imprecisely known material (e.g., chemical) properties of the i th -material ( ) Alternatively, ( ) The following functional, denoted as ( ) 1 ; , R ρ α β , can represent mathematically such a measurement: δ − denotes the well-known Dirac-delta (impulse) functional. In Equation (3), the vector α denotes the "vector of model parameters" and defined as follows: Similarly, the vector β denotes the "vector of boundary parameters" and is defined as follows: ( ) subject to uncertainties, the actual probability distributions of these parameters are not known in practice. Usually, only the "nominal" (or "mean") values and the respective variations from the nominal values (e.g., standard deviations) of the respective components are known. The nominal values will be denoted using the superscript "zero" so that the vector comprising the nominal values of the model parameters, denoted as 0 α , will be defined for the system under consideration as follows: †  0  0  0  0  0  0  0  0  0  1  1  1 , , , , , , , , , Similarly, the vector comprising the nominal values of the boundary parameters is denoted as 0 β and is defined for the system under consideration as follows: Altogether, the physical system modeled by Equations (1) through (7)  For subsequent verification of the expressions that will be obtained for various response sensitivities, the closed-form solution of Equations (1) and (2) Using Equation (9) in Equation (3)

Application of the 2 nd -CASAM for Computing Exactly and Efficiently the 1 st -and 2 nd -Order Response Sensitivities of a "Point Detector" Response to Uncertain Model and Boundary Parameters
The variations between the true and the nominal values of the model and boundary parameters will be considered to constitute the components of the vectors δα and δβ , respectively, defined as follows: Since the state function is related to the model and boundary parameters α and β through Equations (1) and (2), it follows that the variations and δβ in the model and boundary parameters will cause a corresponding variation in The total first-order sensitivity of the response Equation (3) is provided [6] by the 1 st -order total sensitivity (G-differential) ( ) The variation Since the closed-form solution represented by Equation (9) is not available in practice, the direct effect term, ( )  . This sequence of steps yields the following relation: The following sequence of operations is performed next using Equation (

4) Insert the boundary condition provided in Equation (17) into Equation (19).
The result of the above sequence of operations is the following expression for where the first-level adjoint function ( ) ( ) In terms of the first-level adjoint function ( ) ( ) 1 t ψ , the partial sensitivities of ( ) 1 ; , R ρ α β with respect to the variations in the model parameters are the quantities in Equation (20) that multiply the respective parameter variations, namely: Recalling the expression of the direct effect term, ( ) 1 dir R δ , defined in Equation (15), yields the following additional first-order sensitivity: Since neither the direct-effect nor the indirect-effect terms depend on the variation It is evident from Equations (23) through (27) that the sensitivities of the response ( ) 1 ; , R ρ α β can be computed by fast quadrature methods applied to the integrals appearing in these expressions, after the 1 st -level adjoint function has been obtained by solving once the 1 st -LASS, which comprises Equa-tions (21) and (22). Notably, the 1 st -LASS needs to be solved once only since the 1 st -LASS does not depend on any variations in the model parameters or state functions. Particularly important is the response sensitivity to the "initial condition" in ρ since, as Equation (25) indicates, the value of the 1 st -level adjoint  is proportional to the response sensitivity to the "initial condition". Since the value of the 1 st -level adjoint  can be obtained only after computing the entire evolution of ( ) ( ) to the "initial-time" 0 t β =  , it becomes apparent that response sensitivities to initial conditions provide a stringent verification procedure for assessing the accuracy of the solution of the 1 st -LASS.
Solving the 1 st -LASS, cf. Equations (21) and (22), yields the following expression for the 1 st -level adjoint function ( ) ( ) is the customary Heaviside unit-step functional, defined as Inserting the result from Equation (29) into Equations (23)-(26), respectively, yields the following expressions: The magnitudes of the 1 st -order relative sensitivities provide a quantitative measure for ranking the importance of the respective parameters in affecting the response (e.g., the importance of the respective parameter's uncertainty in contributing to the overall uncertainty in the response). For the paradigm illustrative evolution problem considered in this work, Equations (23) and (24) indicate the important fact that the relative sensitivities of the response to the parameters i σ , ( )( ) , and the relative sensitivities of the response to the parameters i n , ( )( ) , respectively, happen to be identical, for all of these 2N model parameters, since Therefore, statistical methods that use a priori screening techniques to reduce the number of model parameters that are actually considered in the respective statistical uncertainty/sensitivity analysis will very likely fail to achieve their goal for problems that have many parameters with identical relative sensitivities, as is the case shown in Equation (36). Hence, this illustrative paradigm problem, which has many model parameters that have identical relative sensitivities, would be a prime candidate for testing the various statistical methods for sensitivity and uncertainty analysis. In contrast, a single large-scale computation for obtaining the adjoint function ( ) ( ) The results for the 1 st -order response sensitivities obtained in Section 2.1 can also be verified by noting that the solution of the 1 st -LFSS, comprising Equations (16) and (17), has the following expression: The starting point for obtaining expressions of the 2 nd -order response sensitivities is provided by the G-differentials of the expressions shown in Equations (23)-(27). To keep the notation as simple as possible, the superscript "zero" will henceforth be omitted (except where stringently needed) when denoting "nominal values," since it will be clear from the derivations to follow that all 1 st -and 2 nd -order sensitivities are to be evaluated at the nominal values of parameters.

Results for the 2nd-Order Response Sensitivities Corresponding to
The first-order G-differential of Equation (23) yields: d , The direct-effect term defined by Equation (39)   Therefore, the need for solving these equations (which depend on parameter variations) will be circumvented by expressing the indirect-effect term defined in Equation (40) in an alternative way so as to eliminate the appearance of . The inner product between two elements will be denoted as and is defined as follows: Writing Equations (16) and (41) in matrix form, as follows: and using the definition given in Equation (43), we now construct the inner product of Equation (44) with a square integrable two-component function H to obtain the following relation: Integrating by parts the left-side of Equation (45) so as to transfer the differential operations on The last two terms on the right-side of Equation (46) The boundary conditions for Equations (47) and (48) Using the conditions given in Equations (17) The 2 nd -order sensitivities shown in Equations (52)-(57) can be computed after having determined the 2 nd -level adjoint function , it follows that the right-sides of Equations (47) and (48) also depend on this index. Strictly speaking, therefore, the 2 nd -level adjoint sensitivity function Hence, in the most unfavorable situation, the 2 nd -LASS, comprising Equations (47)-(49) would need to be solved numerically for each distinct value i n , for a total of N-times. Even in such a "worse-case scenario," however, only the right sides (i.e., "sources") of Equations (47) and (48) would need to be modified, which is relatively easy to implement computationally. The left-sides of these equations remain unchanged, since they are independent of the index The components of the 2 nd -level adjoint function ( ) ( )   (58), to obtain the following expressions for the components of the 2 nd -level adjoint function Using Equations (64) and (65) in Equations (52)-(57) and performing the respective operations yields the following results for the respective partial 2 nd -order sensitivities: As before, the right-sides of expressions shown in Equations (66)-(71) are to be evaluated at the nominal values for the parameters, but the superscript "zero," which indicates "nominal values," has been omitted, for notational simplicity.

Results for the 2nd-Order Response Sensitivities Corresponding to
Computing the first-order G-differential of Equation (24)     , satisfies the following 2 nd -LASS: The sources on the right-sides of the 2 nd -LASS defined by Equations (76)-(78) are to be evaluated at the nominal values for the parameters, but the superscript "zero," which indicates "nominal values," has been omitted, for notational simplicity.
Comparing Equations (76)-(78) to Equations (47)-(49) and recalling Equations (59)-(61) indicates that the components of the 2 nd -level adjoint function have the following expressions: Adding the direct-effect term defined in Equation (73) Inserting the expressions obtained in Equations (79) and (80) for the components of the 2 nd -level adjoint function

Results for the 2nd-Order Response Sensitivities Corresponding to
The 2 nd -order response sensitivities corresponding to ( ) 1 ; , in R ρ ρ ∂ ∂ α β will be calculated in this Section by taking the G-differential of Equation (25). Since the model responses need to be written in the form of an inner product in order to apply the adjoint sensitivity analysis methodology, Equation (25) is re-written in the following form: Taking the G-differential of Equation (93) yields and The direct-effect defined in Equation (95)  in the forward function, as in Sections 2.2.1 and 2.2.2. Therefore, the 2 nd -level adjoint function that would be needed to recast the indirect-effect term defined in Equation (96), by following the same general procedure as used in Sections 2.2.1 and 2.2.2, would be a one-component (as opposed to a "two-component" vector) function. Thus, the 2 nd -LASS needed to recast the indirect-effect term defined in Equation (96) is constructed by following a procedure similar to the one that was used in Section 2.1, by applying the definition provided in Equation (18) to construct the inner product of a square-integrable function ( ) ( ) ( ) ( ) with Equation (41) and integrating the left-side of the resulting equation by parts once, so as to transfer the differential operation from This sequence of steps yields the following relation: The last term on the right-side of Equation (97) is now required to represent the indirect-effect term defined in Equation (96). This is accomplished by requiring that Adding the direct-effect term defined in Equation (95) The closed-form solution of the 2 nd -LASS provided in Equations (98) and (99) has the following expression: Replacing the result for the 2 nd -level adjoint function obtained in Equation (106) The following sequence of operations is now performed using Equation (114) The solution of Equations (116) and (117) is: In terms of the 2 nd -level adjoint function ( ) ( ) where ( ) ( ) ( ) ( The last two terms on the right-side of Equation (138) will represent the indirect-effect term defined in Equation (136) by requiring that Using Equations (137)- (141) and (17) in Equation (136) yields the following expression for the indirect-effect term defined in Equation (136):