Comprehensive Adjoint Sensitivity Analysis Methodology Research Articles

First-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Neural Ordinary Differential Equations: Mathematical Framework and Illustrative Application to the Nordheim–Fuchs Reactor Safety Model

First-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Neural Ordinary Differential Equations: Mathematical Framework and Illustrative Application to the Nordheim–Fuchs Reactor Safety Model

4th-Order-SENS: A Software Module for Efficient and Exact 4th-Order Sensitivity Analysis of Neutron Transport

This work presents a software module called 4th-Order-SENS, which enables the efficient computation of exactly obtained expressions for all sensitivities, up to and including the 4th order, of a functional of the particle flux (e.g., the leakage of particles out of a body) with respect to nuclear parameters (total, scattering, and fission cross sections; nu, chi, sources; and number densities) for systems modeled by the neutron transport equation. The 4th-Order-SENS module implements the nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Linear Systems (nth-CASAM-L), which is the only practically implementable methodology for obtaining the exact expressions of arbitrarily high-order sensitivities of model responses to model parameters, for response-coupled forward/adjoint large-scale linear systems. In addition to presenting the equations that are solved to obtain the 1st-order through 4th-order sensitivities, this work also describes the components of the module 4th-Order-SENS, including the user interface, input file, output files, and several independent code verification capabilities using symmetries and/or finite-difference formulas. The 4th-Order-SENS module is written in Python and Fortran and runs on Linux platforms. Several illustrative applications involving fixed-source problems in one-dimensional spherical and slab geometries are also presented.

Overview of Arbitrarily High-Order Adjoint Sensitivity and Uncertainty Quantification Methodology for Large-Scale Systems

This work reviews from a unified viewpoint the concepts underlying the “nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems” (nth-CASAM-L) and the “nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (nth-CASAM-N) methodologies. The practical application of the nth-CASAM-L methodology is illustrated for an OECD/NEA reactor physics benchmark, while the practical application of the nth-CASAM-N methodology is illustrated for a nonlinear model of reactor dynamics that exhibits periodic and chaotic oscillations. As illustrated both by the general theory and by the examples reviewed in this work, both the nth-CASAM-L and nth-CASAM-N methodologies overcome the curse of dimensionality in sensitivity analysis. The availability of efficiently and exactly computed sensitivities of arbitrarily high order can lead to major advances in all areas that need such high-order sensitivities, including data assimilation, model calibration, uncertainty reduction, and predictive modeling.

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

This work presents the nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (nth-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 nth-CASAM-N is proven to be correct by using mathematical induction. The nth-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.

Fifth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (5th-CASAM-N): I. Mathematical Framework

This work presents the mathematical framework of the “Fifth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (5th-CASAM-N),” which generalizes and extends all of the previous works performed to date on this subject. The 5th-CASAM-N enables the exact and efficient computation of all sensitivities, up to and including fifth-order, of model responses to uncertain model parameters and uncertain boundaries of the system’s domain of definition, thus enabling, inter alia, the quantification of uncertainties stemming from manufacturing tolerances. The 5th-CASAM-N provides a fundamental step towards overcoming the curse of dimensionality in sensitivity and uncertainty analysis.

Fifth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (5th-CASAM-N): II. Paradigm Application to a Bernoulli Model Comprising Uncertain Parameters

This work presents the application of the recently developed “Fifth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (5th-CASAM-N)” to a simplified Bernoulli model. The 5th-CASAM-N builds upon and incorporates all of the lower-order (i.e., the first-, second-, third-, and fourth-order) adjoint sensitivities analysis methodologies. The Bernoulli model comprises a nonlinear model response, uncertain model parameters, uncertain model domain boundaries and uncertain model boundary conditions, admitting closed-form explicit expressions for the response sensitivities of all orders. Illustrating the specific mechanisms and advantages of applying the 5th-CASAM-N for the computation of the response sensitivities with respect to the uncertain parameters and boundaries reveals that the 5th-CASAM-N provides a fundamental step towards overcoming the curse of dimensionality in sensitivity and uncertainty analysis.

The nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems (nth-CASAM-L): I. Mathematical Framework

This work presents the mathematical framework of the nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems (abbreviated as “nth-CASAM-L”), which is conceived for obtaining the exact expressions of arbitrarily-high-order (nth-order) sensitivities of a generic system response with respect to all of the parameters (including boundary and initial conditions) underlying the respective forward/adjoint systems. Since many of the most important responses for linear systems involve the solutions of both the forward and the adjoint linear models that correspond to the respective physical system, the sensitivity analysis of such responses makes it necessary to treat linear systems in their own right, rather than treating them as particular cases of nonlinear systems. This is in contradistinction to responses for nonlinear systems, which can depend only on the forward functions, since nonlinear operators do not admit bona-fide adjoint operators (only a linearized form of a nonlinear operator admits an adjoint operator). The nth-CASAM-L determines the exact expression of arbitrarily-high order sensitivities of responses to the parameters underlying both the forward and adjoint models of a nonlinear system, thus enable the most efficient and accurate computation of such sensitivities. The mathematical framework underlying the nth-CASAM is developed in linearly increasing higher-dimensional Hilbert spaces, as opposed to the exponentially increasing “parameter-dimensional” spaces in which response sensitivities are computed by other methods, thus providing the basis for overcoming the “curse of dimensionality” in sensitivity analysis and all other fields (uncertainty quantification, predictive modeling, etc.) which need such sensitivities. In particular, for a scalar-valued valued response associated with a nonlinear model comprising TP parameters, the 1st-−CASAM-L requires 1 additional large-scale adjoint computation (as opposed to TP large-scale computations, as required by other methods) for computing exactly all of the 1st-−order response sensitivities. All of the (mixed) 2nd-order sensitivities are computed exactly by the 2nd-CASAM-L in at most TP computations, as opposed to TP(TP + 1)/2 computations required by all other methods, and so on. For every lower-order sensitivity of interest, the nth-CASAM-L computes the “TP next-higher-order” sensitivities in one adjoint computation performed in a linearly increasing higher-dimensional Hilbert space. Very importantly, the nth-CASAM-L computes the higher-level adjoint functions using the same forward and adjoint solvers (i.e., computer codes) as used for solving the original forward and adjoint systems, thus requiring relatively minor additional software development for computing the various-order sensitivities.

The nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems (nth-CASAM-L): II. Illustrative Application

This work illustrates the application of the nth-order comprehensive adjoint sensitivity analysis methodology for response-coupled forward/adjoint linear systems (abbreviated as “nth-CASAM-L”) to a paradigm model that describes the transmission of particles (neutrons and/or photons) through homogenized materials, as encountered in radiation protection and shielding. The first-, second-, and third-order sensitivities of responses that depend on both the forward and adjoint particle fluxes are obtained exactly, in closed-form, underscoring the principles and methodology underlying the nth-CASAM-L. The results presented in this work underscore the fundamentally important role of the nth-CASAM-L in the quest to overcome the “curse of dimensionality” in sensitivity analysis, uncertainty quantification and predictive modeling.

High-Order Deterministic Sensitivity Analysis and Uncertainty Quantification: Review and New Developments

This work reviews the state-of-the-art methodologies for the deterministic sensitivity analysis of nonlinear systems and deterministic quantification of uncertainties induced in model responses by uncertainties in the model parameters. The need for computing high-order sensitivities is underscored by presenting an analytically solvable model of neutron scattering in a hydrogenous medium, for which all of the response’s relative sensitivities have the same absolute value of unity. It is shown that the wider the distribution of model parameters, the higher the order of sensitivities needed to achieve a desired level of accuracy in representing the response and in computing the response’s expectation, variance, skewness and kurtosis. This work also presents new mathematical expressions that extend to the sixth-order of the current state-of-the-art fourth-order formulas for computing fourth-order correlations among computed model response and model parameters. Another novelty presented in this work is the mathematical framework of the 3rd-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (3rd-CASAM-N), which enables the most efficient computation of the exact expressions of the 1st-, 2nd- and 3rd-order functional derivatives (“sensitivities”) of a model’s response to the underlying model parameters, including imprecisely known initial, boundary and/or interface conditions. The 2nd- and 3rd-level adjoint functions are computed using the same forward and adjoint computer solvers as used for solving the original forward and adjoint systems. Comparisons between the CPU times are also presented for an OECD/NEA reactor physics benchmark, highlighting the fact that finite-difference schemes would not only provide approximate values for the respective sensitivities (in contradistinction to the 3rd-CASAM-N, which provides exact expressions for the sensitivities) but would simply be unfeasible for computing sensitivities of an order higher than first-order. Ongoing work will generalize the 3rd-CASAM-N to a higher order while aiming to overcome the curse of dimensionality.

On the Need to Determine Accurately the Impact of Higher-Order Sensitivities on Model Sensitivity Analysis, Uncertainty Quantification and Best-Estimate Predictions

This work aims at underscoring the need for the accurate quantification of the sensitivities (i.e., functional derivatives) of the results (a.k.a. “responses”) produced by large-scale computational models with respect to the models’ parameters, which are seldom known perfectly in practice. The large impact that can arise from sensitivities of order higher than first has been highlighted by the results of a third-order sensitivity and uncertainty analysis of an OECD/NEA reactor physics benchmark, which will be briefly reviewed in this work to underscore that neglecting the higher-order sensitivities causes substantial errors in predicting the expectation and variance of model responses. The importance of accurately computing the higher-order sensitivities is further highlighted in this work by presenting a text-book analytical example from the field of neutron transport, which impresses the need for the accurate quantification of higher-order response sensitivities by demonstrating that their neglect would lead to substantial errors in predicting the moments (expectation, variance, skewness, kurtosis) of the model response’s distribution in the phase space of model parameters. The incorporation of response sensitivities in methodologies for uncertainty quantification, data adjustment and predictive modeling currently available for nuclear engineering systems is also reviewed. The fundamental conclusion highlighted by this work is that confidence intervals and tolerance limits on results predicted by models that only employ first-order sensitivities are likely to provide a false sense of confidence, unless such models also demonstrate quantitatively that the second- and higher-order sensitivities provide negligibly small contributions to the respective tolerance limits and confidence intervals. The high-order response sensitivities to parameters underlying large-scale models can be computed most accurately and most efficiently by employing the high-order comprehensive adjoint sensitivity analysis methodology, which overcomes the curse of dimensionality that hampers other methods when applied to large-scale models involving many parameters.

Fourth-Order Adjoint Sensitivity and Uncertainty Analysis of an OECD/NEA Reactor Physics Benchmark: I. Computed Sensitivities

This work extends the investigation of higher-order sensitivity and uncertainty analysis from 3rd-order to 4th-order for a polyethylene-reflected plutonium (PERP) OECD/NEA reactor physics benchmark. Specifically, by applying the 4th-order comprehensive adjoint sensitivity analysis methodology (4th-CASAM) to the PERP benchmark, this work presents the numerical results of the most important 4th-order sensitivities of the benchmark’s total leakage response with respect to the benchmark’s 180 microscopic total cross sections, which includes 180 4th-order unmixed sensitivities and 360 4th-order mixed sensitivities corresponding to the largest 3rd-order ones. The numerical results obtained in this work reveal that the number of 4th-order relative sensitivities that have large values (e.g., greater than 1.0) is far greater than the number of important 1st-, 2nd- and 3rd-order sensitivities. The majority of those large sensitivities involve isotopes 1H and 239Pu contained in the PERP benchmark. Furthermore, it is found that for most groups of isotopes 1H and 239Pu of the PERP benchmark, the values of the 4th-order relative sensitivities are significantly larger than the corresponding 1st-, 2nd- and 3rd-order sensitivities. The overall largest 4th-order relative sensitivity S(4)σt,6g=30,σt,6g=30,σt,6g=30,σt,6g=30=2.720×106 is around 291,000 times, 6350 times and 90 times larger than the corresponding largest 1st-order, 2nd-order and 3rd-order sensitivities, respectively, and the overall largest mixed 4th-order relative sensitivity S(4)σt,630,σt,630,σt,630,σt,530=2.279×105 is also much larger than the largest 2nd-order and 3rd-order mixed sensitivities. The results of the 4th-order sensitivities presented in this work have been independently verified with the results obtained using the well-known finite difference method, as well as with the values of the corresponding symmetric 4th-order sensitivities. The 4th-order sensitivity results obtained in this work will be subsequently used on the 4th-order uncertainty analysis to evaluate their impact on the uncertainties they induce in the PERP leakage response.

Fourth-Order Comprehensive Adjoint Sensitivity Analysis (4th-CASAM) of Response-Coupled Linear Forward/Adjoint Systems: I. Theoretical Framework

The most general quantities of interest (called “responses”) produced by the computational model of a linear physical system can depend on both the forward and adjoint state functions that describe the respective system. This work presents the Fourth-Order Comprehensive Adjoint Sensitivity Analysis Methodology (4th-CASAM) for linear systems, which enables the efficient computation of the exact expressions of the 1st-, 2nd-, 3rd- and 4th-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 underlying the respective forward/adjoint systems. Among the best known such system responses are various Lagrangians, including the Schwinger and Roussopoulos functionals, for analyzing ratios of reaction rates, the Rayleigh quotient for analyzing eigenvalues and/or separation constants, etc., which require the simultaneous consideration of both the forward and adjoint systems when computing them and/or their sensitivities (i.e., functional derivatives) with respect to the model parameters. Evidently, such responses encompass, as particular cases, responses that may depend just on the forward or just on the adjoint state functions pertaining to the linear system under consideration. This work also compares the CPU-times needed by the 4th-CASAM versus other deterministic methods (e.g., finite-difference schemes) for computing response sensitivities These comparisons underscore the fact that the 4th-CASAM is the only practically implementable methodology for obtaining and subsequently computing the exact expressions (i.e., free of methodologically-introduced approximations) of the 1st-, 2nd, 3rd- and 4th-order sensitivities (i.e., functional derivatives) of responses to system parameters, for coupled forward/adjoint linear systems. By enabling the practical computation of any and all of the 1st-, 2nd, 3rd- and 4th-order response sensitivities to model parameters, the 4th-CASAM makes it possible to compare the relative values of the sensitivities of various order, in order to assess which sensitivities are important and which may actually be neglected, thus enabling future investigations of the convergence of the (multivariate) Taylor series expansion of the response in terms of parameter variations, as well as investigating the range of validity of other important quantities (e.g., response variances/covariance, skewness, kurtosis, etc.) that are derived from Taylor-expansion of the response as a function of the model’s parameters. The 4th-CASAM presented in this work provides the basis for significant future advances towards overcoming the “curse of dimensionality” in sensitivity analysis, uncertainty quantification and predictive modeling.

First-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Critical Points in Coupled Nonlinear Systems. I: Mathematical Framework

This work presents the novel first-order comprehensive adjoint sensitivity analysis methodology for critical points (1st-CASAM-CP), which enables the exact and efficient computation of the first-order sensitivities of responses defined at critical points (maxima, minima, saddle points) of coupled nonlinear models of physical systems characterized by imprecisely known parameters underlying the models, boundaries, and interfaces between the coupled systems. Responses defined at critical points are important in many applications, including system optimization, safety analyses and licensing. For the design and licensing of nuclear reactors, such essentially important responses include the maximum temperatures of the fuel and cladding in hot channels. The 1st-CASAM-CP presented in this work makes it possible to determine, using a single large-scale “adjoint” computation, the first-order sensitivities of the magnitude of a response defined at a critical point of a function in the phase-space of the systems’ independent variables. In addition, the 1st-CASAM-CP enables the computation of the sensitivities of the location in phase-space of the critical point at which the respective response is located: one “adjoint” computation is required for each component of the respective critical point in the phase-space of independent variables. By enabling the exact and efficient computation of the sensitivities of responses and of their critical locations to imprecisely known model parameters, boundaries, and interfaces, the 1st-CASAM-CP significantly extends the practicality of analyzing crucially important responses for large-scale systems involving many uncertain parameters, interfaces, and boundaries.

First-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Critical Points in Coupled Nonlinear Systems. II: Application to a Nuclear Reactor Thermal-Hydraulics Safety Benchmark

Responses defined at critical points are particularly important for reactor safety analyses and licensing (e.g., the maximum fuel and/or clad temperature). The novel mathematical framework of the first-order comprehensive adjoint sensitivity analysis methodology for critical points (1st-CASAM-CP) is applied in this work to develop a reactor safety thermal-hydraulics benchmark model which admits exact closed-form expressions for the adjoint functions and for the first-order sensitivities of responses defined at critical points (maxima, minima, saddle points) in physical systems characterized by imprecisely known parameters, external and internal boundaries. This benchmark model is designed for verifying the capabilities and accuracies of computational tools for modeling numerically thermal-hydraulics systems. The unique and extensive capabilities of the 1st-CASAM-CP methodology are demonstrated in this work by considering two responses of paramount importance in reactor safety, namely, (i) the maximum rod surface temperature, which occurs at the imprecisely known interface between the subsystem that models the heat conduction inside the heated rod and the subsystem modeling the heat convection process surrounding the rod; and (ii) the maximum temperature inside the heated rod, which has a critical point with two components, one located at a precisely known boundary of the subsystem that models the heat conduction inside the heated rod, while the other component depends on an imprecisely known boundary (i.e., the rod length). The exact analytical expressions developed in this work for the sensitivities of the maximum internal rod temperature and maximum rod surface temperature, as well as for the sensitivities of the locations where these respective maxima occur, provide exact benchmarks for verifying the accuracy of thermal-hydraulics computational tools. The sensitivities of such responses and of their critical points with respect to model parameters enable the quantification of uncertainties induced by uncertainties stemming from the system’s parameters and boundaries in the respective responses and their underlying critical points.

First order comprehensive adjoint sensitivity analysis methodology (1st-casam) for computing efficiently the exact response sensitivities for physical systems with imprecisely known boundaries and parameters: General theory and illustrative paradigm applications

This work presents the novel first-order comprehensive adjoint sensitivity analysis methodology (1st-CASAM) for computing efficiently, exactly, and exhaustively, the first-order response sensitivities for nonlinear physical systems characterized by imprecisely known (uncertain) parameters model and domain boundary parameters. This novel methodology is designed for generic operator-valued model responses, treating scalar-valued responses as particular cases. The 1st-CASAM highlights the conclusion that response sensitivities to the imprecisely known domain boundaries can arise both from the definition of the system’s response as well as from the equations and boundary conditions defining the model on its imprecisely known domain. The application of this new methodology is illustrated on two paradigm models, each admitting exact closed-form expressions for the sensitivities of typical model responses to the model’s imprecisely known (uncertain) model and boundary parameters, as follows: (a) a nonlinear heat conduction equation; and (b) a neutron diffusion equation modeling a subcritical reactor with an external source. Very importantly, the diffusion equation is subject to vacuum boundary conditions on the extrapolated system boundary, thus illustrating the computation of response sensitivities to boundary parameters involving not only imprecisely known geometrical dimensions but also imprecisely known nuclear data. For the nonlinear heat conduction equation, the value of the temperature that would be measured at a point in phase-space is chosen as a representative scalar-valued response, while the temperature distribution in the entire phase-space, having uncertain boundaries, of the model’s independent variable has been chosen as a representative function-valued response. For the neutron diffusion equation, a reaction-rate within the imprecisely known domain boundaries has been chosen as a representative scalar-valued response, and the neutron flux distribution within the imprecisely known domain boundaries has been chosen as a representative function-valued response. The application of the 1st-CASAM for the operator-valued responses is illustrated using: (a) a pseudo-spectral collocation representation for the sensitivities of the temperature distribution response; and (b) a spectral Fourier representation for the sensitivities of the neutron flux distribution response. These paradigm examples highlight the details of applying the general 1st-CASAM methodology, underscoring the effects of imprecisely known domain boundaries on both the system’s state functions and responses. By enabling, in premiere, the exact computations of function-valued response sensitivities to boundary parameters and conditions, the novel adjoint sensitivity analysis methodology presented in this work enables the quantification of the effects of manufacturing tolerances on the responses of physical and engineering systems.

First-Order Comprehensive Adjoint Method for Computing Operator-Valued Response Sensitivities to Imprecisely Known Parameters, Internal Interfaces and Boundaries of Coupled Nonlinear Systems: I. Mathematical Framework

This work presents the first-order comprehensive adjoint sensitivity analysis methodology (1st-CASAM) for computing efficiently the first-order sensitivities (i.e., functional derivatives) of operator-valued responses (i.e., model results) of general models of coupled nonlinear physical systems characterized by imprecisely known or and/or uncertain parameters, external boundaries, and internal interfaces between the coupled systems. The explicit mathematical formalism developed within the 1st-CASAM for computing the first-order sensitivities of operator-valued response to uncertain internal interfaces and external boundaries in the models’ phase–space enables this methodology to generalize all of the previously published methodologies for computing first-order response sensitivities. The computational resources needed for using forward versus adjoint operators in conjunction with spectral versus collocation methods for computing the response sensitivities are analyzed in detail. By enabling the exact computations of operator-valued response sensitivities to internal interfaces and external boundary parameters and conditions, the 1st-CASAM presented in this work makes it possible, inter alia, to quantify the effects of manufacturing tolerances on operator-valued responses of physical and engineering systems.

Illustrative Application of the 2<sup>nd</sup>-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 1st- and 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.

Second-Order Adjoint Sensitivity Analysis Methodology for Computing Exactly Response Sensitivities to Uncertain Parameters and Boundaries of Linear Systems: Mathematical Framework

This work presents the “Second-Order Comprehensive Adjoint Sensitivity Analysis Methodology (2nd-CASAM)” for the efficient and exact computation of 1st- and 2nd-order response sensitivities to uncertain parameters and domain boundaries of linear systems. The model’s response (i.e., model result of interest) is a generic nonlinear function of the model’s forward and adjoint state functions, and also depends on the imprecisely known boundaries and model parameters. In the practically important particular case when the response is a scalar-valued functional of the forward and adjoint state functions characterizing a model comprising N parameters, the 2nd-CASAM requires a single large-scale computation using the First-Level Adjoint Sensitivity System (1st-LASS) for obtaining all of the first-order response sensitivities, and at most N large-scale computations using the Second-Level Adjoint Sensitivity System (2nd-LASS) for obtaining exactly all of the second-order response sensitivities. In contradistinction, forward other methods would require (N2/2 + 3 N/2) large-scale computations for obtaining all of the first- and second-order sensitivities. This work also shows that constructing and solving the 2nd-LASS requires very little additional effort beyond the construction of the 1st-LASS needed for computing the first-order sensitivities. Solving the equations underlying the 1st-LASS and 2nd-LASS requires the same computational solvers as needed for solving (i.e., “inverting”) either the forward or the adjoint linear operators underlying the initial model. Therefore, the same computer software and “solvers” used for solving the original system of equations can also be used for solving the 1st-LASS and the 2nd-LASS. Since neither the 1st-LASS nor the 2nd-LASS involves any differentials of the operators underlying the original system, the 1st-LASS is designated as a “first-level” (as opposed to a “first-order”) adjoint sensitivity system, while the 2nd-LASS is designated as a “second-level” (rather than a “second-order”) adjoint sensitivity system. Mixed second-order response sensitivities involving boundary parameters may arise from all source terms of the 2nd-LASS that involve the imprecisely known boundary parameters. Notably, the 2nd-LASS encompasses an automatic, inherent, and independent “solution verification” mechanism of the correctness and accuracy of the 2nd-level adjoint functions needed for the efficient and exact computation of the second-order sensitivities.

Illustrative Application of the 2<sup>nd</sup>-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” (2nd-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 1st- and 2nd-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 inter-comparing the performances of all deterministic and statistical sensitivity analysis methods, including the 2nd-CASAM.

The First-Order Comprehensive Sensitivity Analysis Methodology (1st-CASAM) for Scalar-Valued Responses: I. Theory

This work presents the first-order comprehensive adjoint sensitivity analysis methodology (1st-CASAM) for computing efficiently, exactly, and exhaustively, the first-order sensitivities of scalar-valued responses (results of interest) of coupled nonlinear physical systems characterized by imprecisely known model parameters, boundaries and interfaces between the coupled systems. The 1st-CASAM highlights the conclusion that response sensitivities to the imprecisely known domain boundaries and interfaces can arise both from the definition of the system’s response as well as from the equations, interfaces and boundary conditions defining the model and its imprecisely known domain. By enabling, in premiere, the exact computations of sensitivities to interface and boundary parameters and conditions, the 1st-CASAM enables the quantification of the effects of manufacturing tolerances on the responses of physical and engineering systems. Ongoing research will generalize the methodology presented in this work, aiming at computing exactly and efficiently higher-order response sensitivities for coupled systems involving imprecisely known interfaces, parameters, and boundaries.