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

This work continues the illustrative application of the “Second Order Comprehensive Adjoint Sensitivity Analysis Methodology” (2nd-CASAM) to a benchmark mathematical model that can simulate the evolution and/or transmission of particles in a heterogeneous medium. The model response considered in this work is a reaction-rate detector response, which provides the average interactions of particles with the respective detector or, alternatively, the time-average of the concentration of a mixture of substances in a medium. The definition of this model response includes both uncertain boundary points of the benchmark, thereby providing both direct and indirect contributions to the response sensitivities stemming from the boundaries. The exact expressions for the 1stand 2nd-order response sensitivities to the boundary and model parameters obtained in this work can serve as stringent benchmarks for inter-comparing the performances of all (deterministic and statistical) sensitivity analysis methods.


Introduction
This work continues to illustrate the application of the general second-order adjoint sensitivity analysis methodology (2 nd -CASAM) presented in [1] by us-ing the evolution/transmission mathematical benchmark model introduced in [2], but considering a "reaction-rate" detector response, as opposed to the "point-detector" response considered in [2]. As in [2], the mathematical model considered in this work could represent [3] [4] the time-evolution of the concentration of a substance in a homogeneous mixture of materials or, alternatively, it could represent [4] [5] [6] the transmission/attenuation of the flux of uncollided particles (e.g., photons) travelling through a one-dimensional homogenized multi-material slab of imprecisely known thickness.
Although simple, the model 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 application of the 2 nd -CASAM [1] yields the exact expressions of the 1 st -and 2 nd -order sensitivities of the reaction-rate response to the uncertain model and boundary parameters. These exact expressions can be used to benchmark any other statistical or deterministic software used for computing sensitivities.
This work is organized as follows: Section 2 presents the mathematical formulation and expression of a reaction-rate detector for the paradigm evolution/transmission model. Section 3 illustrates the application of the 2 nd -CASAM [1] for obtaining efficiently the exact closed-form expressions of the first-and second-order sensitivities of the reaction-rate detector response to both model and boundary parameters. Section 4 offers concluding remarks.

Mathematical Modeling of a Paradigm Evolution/Transmission Benchmark Problem
The general methodology presented in Part I  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 ( ) t ρ , satisfying the following linear conservation equation: As has been discussed in [2], Equations (1) and (2)  An important typical response of interest for the physical problem modeled by Equations (1) and (2) is a "reaction rate" detector response, which will be denoted as ( ) 2 ; , R ρ α β , and which is represented mathematically by the following functional: where d Σ represents the detector's reaction cross section. 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: Throughout this work, the symbol "  " is used to denote "is defined as" or "is by definition," while the "dagger" ( ) † superscript is used to denote "transposition." For subsequent verification of the expressions that will be obtained for various response sensitivities, the closed-form solution of Equations (1) and (2) is provided below: Although the model parameters in Σ , together with the boundary parameters β  and u β are considered to be imperfectly known and 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: 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: In practice, the nominal solution, denoted as The closed-form expression of ( ) 2 ; , R ρ α β is readily obtained by inserting into Equation (3) the expression of ( ) t ρ provided in Equation (6) and performing the respective integration, which yields the following expression: As indicated be the expression in Equation (10) R ρ α β evidently depends on u β , so its sensitivities with respect to u β will not vanish.

Application of the 2 nd -CASAM for Computing Exactly and
Efficiently the 1 st -and 2 nd -Order Response Sensitivities of ( ) ρ α β with Respect to the 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: R ρ α β defined in Equation (3) is provided by the first-order G-differential of this response, which is obtained by G-differentiating Equation (3) as shown in Part I , to obtain the following expression: The direct effect term, ( ) 2 dir R δ , can be computed directly by using in Equation (15) the nominal solution provided in Equation (9) to obtain: The indirect effect term, ( ) The need for performing the many large-scale computations for obtaining for all possible variations in the model and boundary parameters can be avoided by applying the 2 nd -CASAM presented in [1]. In order to apply the 2 nd -CASAM, the function ( ) t δρ is considered to be an element of a Hilbert space , endowed with the following inner product, denoted as The indirect effect term, ( ) 2 ind R δ , is computed by applying the general 2 nd -CASAM presented in [1] to Equation (17), which commences by using the definition of the inner product provided in Equation (19) to form the inner product of Equation (17) with a square-integrable function ( ) ( ) ( ) ( ) and integrating the left-side of the resulting equation by parts once, so as to transfer the differential operation from The following sequence of operations is performed next using Equation (20) 4) Insert the boundary condition provided in Equation (18) into the resulting expression.
The result of the above sequence of operations is the following expression for where the first-level adjoint function ( ) ( ) 1 t θ appearing in Equation (21) is the solution of the following First-Level Adjoint Sensitivity System (1 st -LASS):  (22) and (23) yields the following expression for the first-level adjoint function ( ) ( ) Adding Equations (21) and (15), and identifying the quantities that multiply the respective parameter variations yields the following expressions for the first-order sensitivities of ( ) 2 ; , R ρ α β in terms of the first-level adjoint func- The results in Equations (26) and (27) As indicated in Equations (28) and (29), the first-order sensitivities of ( )  (1) and (2)]. Thus, inserting the expression of ( ) t ρ and performing the operations indicated in Equations (28) and (29), respectively, yields the following expressions: On the other hand, as indicated in Equations (25) Finally, the sensitivity of ( ) 2 ; , R ρ α β with respect to the boundary parameter β  is computed by using in Equation (30)  .
Notably for the result obtained in Equation (37) Omitting, for notational simplicity, the superscript "zero" (which denotes "nominal values" in this work), the 1 st -order G-differential of the expression provided in Equation (26) is obtained as follows: The direct-effect term defined by Equation (39)  Therefore, the need for solving these equations 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 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) t Ω H will be denoted as Integrating by parts the left-side of Equation (44) so as to transfer the differential operations on The last two terms on the right-side of Equation (45) are now required to represent the indirect-effect term defined in Equation (40) by imposing the following relations: The boundary conditions for Equations (46) Using the conditions given in Equations (18) and (42) in the last terms on the right side of Equation (49) yields the following expression for the indirect-effect term: Adding the direct-effect term defined in Equation (39) The 2 nd -order sensitivities shown in Equations (51)-(56) can be computed after having determined the 2 nd -level adjoint function ( ) ( ) ( ) ( ) ( ) ( ) by solving the 2 nd -LASS comprising Equations (46)-(48) using the nominal parameter values (the superscript "zero," which indicates "nominal values," has been omitted, for simplicity). Since the model parameters i n depend on the index 1, , i N =  , it follows that the right-sides of Equations (46)-(48) also depend on this index. Strictly speaking, therefore, the 2 nd -level adjoint sensitiv- Hence, in the most unfavorable situation, the 2 nd -LASS, comprising Equations u t : The components of the 2 nd -level adjoint function ( ) ( ) Using Equations (63) and (64) in Equations (51)-(56) and performing the respective operations yields the following results for the partial 2 nd -order sensitivities: As before, the right-sides of expressions shown in Equations (65)-(70) 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.
The expressions of the remaining 2 nd -order sensitivities of ( ) 2 ; , R ρ α β can be derived following the same procedure as outlined above, and as also outlined in [2] since these derivations would not illustrate any new concepts, they will not be presented in this work.

Conclusions
The results obtained by applying the general 2 nd -CASAM presented in [1] to the paradigm evolution/transmission benchmark analyzed in this work indicate the following major characteristics of this powerful methodology for computing exactly and efficiently the 1 st -and 2 nd -order sensitivities of model responses with respect to model and boundary parameters: 1) For a model comprising N α distinct but uncertain model parameters and N β distinct but uncertain distinct boundary parameters, a single adjoint computation, to solve the 1 st -LASS, is necessary for computing exactly all of the 2) By considering each 1 st -order sensitivity as a response and developing a corresponding 2 nd -level adjoint sensitivity system (2 nd -LASS) for computing the respective 2 nd -order sensitivities, the application of the 2 nd -CASAM yields exact expressions/values for all (mixed and unmixed) 2 nd -order sensitivities.
For each 1 st -order sensitivity, the solution of each of the 2 nd -LASS is at most a two-component (vector) 2 nd -level adjoint sensitivity function of the form . Hence, although these 2 nd -level adjoint sensitivity functions are independent of any parameter variations, the 2 nd -LASS would need to be solved for N N α β + distinct right-sides (i.e., "source terms"), in the most unfavorable situation. Even in this "worse-case scenario," only the right sides of the 2 nd -LASS would need to be modified in computational computer codes, which is a relatively easy programming task. The left-sides of the 2 nd -LASS equations (which contain differential operators, and which would therefore involve "solvers" that would be much more difficult to modify computationally) remain unchanged.
3) In many practical situations, it is possible to reduce drastically the number of computations involving the 2 nd -LASS. Occasionally, the solutions of some of the 2 nd -LASS can be written down by inspection, without actually solving the American Journal of Computational Mathematics corresponding 2 nd -LASS. For the paradigm evolution problem analyzed in this chapter, for example, the 2 nd -LASS would need to be solved only four times, to compute the 2 nd -level adjoint functions ( ) ( ) ( ) ( ) ( ) ( ) Of course, such a very large reduction in the number of large-scale computation cannot be expected in every practical problem, but in most cases, the number of computations required for computing the complete set of 2 nd -order response sensitivities if far less than the number, ( ) N N α β + , of model parameters. 4) The specific 2 nd -order sensitivities of interest can be selected "a priori," based on the magnitude/importance of the 1 st -order sensitivities. 5) As has been generally shown by Cacuci [1] [2] [3], the mixed 2 nd -order sensitivities are obtained twice, stemming from distinct 2 nd -LASS. This fact enables the 2 nd -CASAM to provide an inherent independent verification of the correctness and accuracy of the 2 nd -level adjoint sensitivity functions that are used to compute the respective mixed 2 nd -order sensitivities.
6) The un-mixed 2 nd -order sensitivities of the form ( ) are obtained only once. Therefore, they can be independently verified either by solving the 2 nd -LFSS, which would yield their exact values, or they can be computed approximately (rather than exactly) by using finite difference formulas in conjunction with re-computations, e.g.,

(
) ( ) ( ) Notably, contributions to the response sensitivities with respect to the uncertain boundary parameters can arise from either the model's boundary conditions, from the definition of the model's response or from both. It has been shown that in some cases, the contributions from the model's boundary conditions may cancel partially the "direct-effect" contributions stemming from the response's definition.
Ongoing work aims at extending the 2 nd -CASAM to include the consideration of additional responses (e.g., ratios of functionals), as well as consideration of coupled physical systems having common imprecisely known internal interfaces in phase-space.

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