Second-Order Adjoint Sensitivity Analysis Methodology Research Articles

Computation of High-Order Sensitivities of Model Responses to Model Parameters—II: Introducing the Second-Order Adjoint Sensitivity Analysis Methodology for Computing Response Sensitivities to Functions/Features of Parameters

This work introduces a new methodology, which generalizes the extant second-order adjoint sensitivity analysis methodology for computing sensitivities of model responses to primary model parameters. This new methodology enables the computation, with unparalleled efficiency, of second-order sensitivities of responses to functions of uncertain model parameters, including uncertain boundaries and internal interfaces, for linear and/or nonlinear models. Such functions of primary model parameters customarily describe characteristic “features” of the system under consideration, including correlations modeling material properties, flow regimes, etc. The number of such “feature” functions is considerably smaller than the total number of primary model parameters. By enabling the computations of exact expressions of second-order sensitivities of model responses to model “features”, the number of required large-scale adjoint computations is greatly reduced. As shown in this work, obtaining the first- and second-order sensitivities to the primary model parameters from the corresponding response sensitivities to the feature functions can be performed analytically, thereby involving just the respective function/feature of parameters rather than the entire model. By replacing large-scale computations involving the model with relatively trivial computations involving just the feature functions, this new second-order adjoint sensitivity analysis methodology reaches unsurpassed efficiency. The applicability and unparalleled efficiency of this “2nd-Order Function/Feature Adjoint Sensitivity Analysis Methodology” (2nd-FASAM) is illustrated using a paradigm particle transport model that involves feature functions of many parameters, while admitting closed-form analytic solutions. Ongoing work will generalize the mathematical framework of the 2nd-FASAM to enable the computation of arbitrarily high-order sensitivities of model responses to functions/features of model parameters.

Illustrating Important Effects of Second-Order Sensitivities on Response Uncertainties in Reactor Physics

This paper illustrates the relative importance of the largest first- and second-order sensitivities of the leakage response of an OECD/NEA reactor physics benchmark (a polyethylene-reflected plutonium sphere) to the benchmark’s underlying total cross sections. It will be shown that numerous 2nd-order sensitivities of the leakage response with respect to the total cross sections are significantly larger than the largest corresponding 1st-order sensitivities. In particular, the contributions of the 2nd-order sensitivities cause the mean (expected) value of the response to differ appreciably from its computed value and also cause the response distribution to be skewed towards positive values relative to the mean. Neglecting these large 2nd-order sensitivities would cause very large non-conservative errors by under-reporting the response’s variance and expected value. The results presented in this paper also underscore the need for obtaining reliable cross section covariance data, which are currently unavailable. Finally, comparing the CPU-times needed for computations, this paper demonstrates that the Second-Order Adjoint Sensitivity Analysis Methodology is the only practical method for computing 2nd-order sensitivities exactly, without introducing methodological errors, for large-scale systems characterized by many uncertain parameters.

ON THE IMPORTANCE OF SECOND-ORDER RESPONSE SENSITIVITIES TO NUCLEAR DATA IN REACTOR PHYSICS UNCERTAINTY ANALYSIS

This invited keynote presentation compares the relative importance of 1st-order versus 2nd-order sensitivities of the leakage response of an OECD/NEA benchmark (polyethylene-reflected plutonium sphere) to the nuclear data characterizing this benchmark. The imprecisely known parameters underlying the neutron transport computational model for this benchmark include 180 group-averaged total microscopic cross sections, 21600 group-averaged scattering microscopic cross sections, 60 parameters describing the fission process, 30 parameters describing the fission spectrum, 10 parameters describing the system’s sources, and 6 isotopic number densities. Thus, this benchmark comprises 21886 1st-order sensitivities of the leakage response with respect to the model parameters, and 478,996,996 2nd-order sensitivities, of which 239,509,441 are distinct. The exact deterministic computation of all of these 1st- and 2nd-order sensitivities was made possible by the application of the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) developed by Cacuci. Thousands (out of the 32 400 elements) of the 2nd-order sensitivities of the leakage response with respect to the total cross sections turned out to be significantly larger than the largest corresponding 1st-order sensitivities, contrary to some previously held beliefs in the reactor physics community. Hence, it will be shown that neglecting the 2nd-order sensitivities to total cross sections would cause very large non-conservative errors by under-reporting the response’s variance and expected value. The 2nd-order sensitivities also cause the response distribution to be skewed towards positive values relative to the expected value, which, in turn, is significantly larger than the computed value of the leakage response. The result presented in this paper also underscore the need for obtaining reliable cross section covariance data, which are not available at this time.

Comprehensive Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) Applied to a Subcritical Experimental Reactor Physics Benchmark: V. Computation of Mixed 2nd-Order Sensitivities Involving Isotopic Number Densities

This work applies the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) to compute the mixed 2nd-order sensitivities of a polyethylene-reflected plutonium (PERP) benchmark’s leakage response with respect to the benchmark’s imprecisely known isotopic number densities and the other benchmark imprecisely known parameters, including: (i) the 6 × 180 mixed 2nd-order sensitivities involving the total microscopic cross sections; (ii) the 6 × 21,600 mixed 2nd-order sensitivities involving the scattering microscopic cross sections; (iii) the 6 × 60 mixed 2nd-order sensitivities involving the fission microscopic cross sections; and (iv) the 6 × 60 mixed 2nd-order sensitivities involving the average number of neutrons produced per fission. It is shown that many of these mixed 2nd-order sensitivities involving the isotopic number densities have very large values. Most of the large sensitivities involve the isotopic number density of 239Pu, and the microscopic total, scattering or fission cross sections for the 12th or 30th energy groups of 239Pu or 1H, respectively. The 2nd-order mixed sensitivity of the PERP leakage response with respect to the isotopic number density of 239Pu and the microscopic total cross section for the 30th energy group of 1H is the largest of the above mentioned sensitivities, attaining the value −94.91.

Comprehensive Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) Applied to a Subcritical Experimental Reactor Physics Benchmark. VI: Overall Impact of 1st- and 2nd-Order Sensitivities on Response Uncertainties

This work applies the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) to compute the 1st-order and unmixed 2nd-order sensitivities of a polyethylene-reflected plutonium (PERP) benchmark’s leakage response with respect to the benchmark’s imprecisely known isotopic number densities. The numerical results obtained for these sensitivities indicate that the 1st-order relative sensitivity to the isotopic number densities for the two fissionable isotopes have large values, which are comparable to, or larger than, the corresponding sensitivities for the total cross sections. Furthermore, several 2nd-order unmixed sensitivities for the isotopic number densities are significantly larger than the corresponding 1st-order ones. This work also presents results for the first-order sensitivities of the PERP benchmark’s leakage response with respect to the fission spectrum parameters of the two fissionable isotopes, which have very small values. Finally, this work presents the overall summary and conclusions stemming from the research findings for the total of 21,976 first-order sensitivities and 482,944,576 second-order sensitivities with respect to all model parameters of the PERP benchmark, as presented in the sequence of publications in the Special Issue of Energies dedicated to “Sensitivity Analysis, Uncertainty Quantification and Predictive Modeling of Nuclear Energy Systems”.

Comprehensive Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) Applied to a Subcritical Experimental Reactor Physics Benchmark: IV. Effects of Imprecisely Known Source Parameters

By applying the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) to the polyethylene-reflected plutonium (PERP) benchmark, this work presents results for the first- and second-order sensitivities of this benchmark’s leakage response with respect to the spontaneous fission source parameters. The numerical results obtained for these sensitivities indicate that the 1st-order relative sensitivity of the leakage response to the source parameters for the two fissionable isotopes in the benchmark are all positive, signifying that an increase in the source parameters will cause an increase in the total neutron leakage from the PERP sphere. The 1st- and 2nd-order relative sensitivities with respect to the source parameters for 239Pu are very small (10−4 or less). In contradistinction, the 1st-order and several 2nd-order relative sensitivities of the leakage response with respect to the source parameters of 240Pu are large. Large values (e.g., greater than 1.0) are also displayed by several mixed 2nd-order relative sensitivities of the leakage response with respect to parameters involving the source and: (i) the total cross sections; (ii) the average neutrons per fission; and (iii) the isotopic number densities. On the other hand, the values of the mixed 2nd-order relative sensitivities with respect to parameters involving the source and: (iv) the scattering cross sections; and (v) and the fission cross sections are smaller than 1.0. It is also shown that the effects of the 1st- and 2nd-order sensitivities of the PERP benchmark’s leakage response with respect to the benchmark’s source parameters on the moments (expected value, variance and skewness) of the PERP benchmark’s leakage response distribution are negligibly smaller than the corresponding effects (on the response distribution) stemming from uncertainties in the total, fission and scattering cross sections.

Comprehensive Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) Applied to a Subcritical Experimental Reactor Physics Benchmark: I. Effects of Imprecisely Known Microscopic Total and Capture Cross Sections

The subcritical polyethylene-reflected plutonium (PERP) metal fundamental physics benchmark, which is included in the Nuclear Energy Agency (NEA) International Criticality Safety Benchmark Evaluation Project (ICSBEP) Handbook, has been selected to serve as a paradigm illustrative reactor physics system for the application of the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) that was developed by Cacuci. The 2nd-ASAM enables the exhaustive deterministic computation of the exact values of the 1st-order and 2nd-order sensitivities of a system response to the parameters underlying the respective system. The PERP benchmark is numerically modeled in this work by using the deterministic multigroup neutron transport equation discretized in the spatial and angular independent variables. Thus, the numerical model of the PERP benchmark developed includes the following imprecisely known uncertain parameters: 180 group-averaged total microscopic cross sections, 21,600 group-averaged scattering microscopic cross sections, 120 fission process parameters, 60 fission spectrum parameters, 10 parameters describing the experiment’s nuclear sources, and six isotopic number densities. Thus, the numerical simulation model for the PERP benchmark comprises 21,976 uncertain parameters, which implies that, for any response of interest, there are a total of 21,976 first-order sensitivities and 482,944,576 second-order sensitivities with respect to the model parameters. Computing these sensitivities exactly represents the largest sensitivity analysis endeavor ever carried out in the field of reactor physics. Only 241,483,276 are distinct from each other, and some of these turned out to be zero due to the symmetry of the 2nd-order sensitivities. The numerical results for all of these sensitivities, together with discussions of their major impacts, will be presented in a sequence of publications in the Special Issue of Energies dedicated to “Sensitivity Analysis, Uncertainty Quantification and Predictive Modeling of Nuclear Energy Systems”. This work is the first in this sequence, presenting formulas of general use for neutron transport problems, along with the numerical results that were produced by these formulas for the 180 first-order and 32,400 second-order sensitivities of the PERP leakage response with respect to the neutron transport model’s group-averaged isotopic total cross sections. For comparison, this work also presents formulas of general use and numerical results for the 180 first-order and 32,400 second-order sensitivities of the PERP leakage response with respect to the neutron transport model’s group-averaged isotopic capture cross sections. It has been widely believed hitherto that, for reactor physics systems modeled by the neutron transport or diffusion equations, the second-order sensitivities are all much smaller than the first-order ones. However, contrary to this widely held belief, the numerical results that were obtained in this work prove, for the first time ever, that many of the 2nd-order sensitivities are much larger than the corresponding 1st-order ones, so their effects can become much larger than the corresponding effects stemming from the 1st-order sensitivities. For example, the 2nd-order sensitivities of the PERP leakage response cause the expected value of this response to be significantly larger than the corresponding computed value. The importance of the 2nd-order sensitivities increases as the relative standard deviations for the cross sections increase. For the extreme case of fully correlated cross sections, for example, neglecting the 2nd-order sensitivities would cause an error as large as 2000% in the expected value of the leakage response and up to 6000% in the variance of the leakage response. The significant effects of the mixed 2nd-order sensitivities underscore the need for reliable values for the correlations that might exist among the total cross sections, which are unavailable at this time. The 2nd-order sensitivities with respect to the total cross sections also cause the response distribution to be skewed towards positive values relative to the expected value. Hence, neglecting the 2nd-order sensitivities could potentially cause very large non-conservative errors by under-reporting of the response variance and expected value.

Comprehensive Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) Applied to a Subcritical Experimental Reactor Physics Benchmark: II. Effects of Imprecisely Known Microscopic Scattering Cross Sections

This work continues the presentation commenced in Part I of the second-order sensitivity analysis of nuclear data of a polyethylene-reflected plutonium (PERP) benchmark using the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM). This work reports the results of the computations of the first- and second-order sensitivities of this benchmark’s computed leakage response with respect to the benchmark’s 21,600 parameters underlying the computed group-averaged isotopic scattering cross sections. The numerical results obtained for the 21,600 first-order relative sensitivities indicate that the majority of these were small, the largest having relative values of O (10−2). Furthermore, the vast majority of the (21600)2 second-order sensitivities with respect to the scattering cross sections were much smaller than the corresponding first-order ones. Consequently, this work shows that the effects of variances in the scattering cross sections on the expected value, variance, and skewness of the response distribution were negligible in comparison to the corresponding effects stemming from uncertainties in the total cross sections, which were presented in Part I. On the other hand, it was found that 52 of the 21600 × 180 mixed second-order sensitivities of the leakage response with respect to the scattering and total microscopic cross sections had values that were significantly larger than the unmixed second-order sensitivities of the leakage response with respect to the group-averaged scattering microscopic cross sections. The first- and second-order mixed sensitivities of the PERP benchmark’s leakage response with respect to the scattering cross sections and the other benchmark parameters (fission cross sections, average number of neutrons per fission, fission spectrum, isotopic atomic number densities, and source parameters) have also been computed and will be reported in subsequent works.

Comprehensive Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) Applied to a Subcritical Experimental Reactor Physics Benchmark: III. Effects of Imprecisely Known Microscopic Fission Cross Sections and Average Number of Neutrons per Fission

The Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) is applied to compute the first-order and second-order sensitivities of the leakage response of a polyethylene-reflected plutonium (PERP) experimental system with respect to the following nuclear data: Group-averaged isotopic microscopic fission cross sections, mixed fission/total, fission/scattering cross sections, average number of neutrons per fission (), mixed /total cross sections, /scattering cross sections, and /fission cross sections. The numerical results obtained indicate that the 1st-order relative sensitivities for these nuclear data are smaller than the 1st-order sensitivities of the PERP leakage response with respect to the total cross sections but are larger than those with respect to the scattering cross sections. The vast majority of the 2nd-order unmixed sensitivities are smaller than the corresponding 1st-order ones, but several 2nd-order mixed relative sensitivities are larger than the 1st-order ones. In particular, several 2nd-order sensitivities for 239Pu are significantly larger than the corresponding 1st-order ones. It is also shown that the effects of the 2nd-order sensitivities of the PERP benchmark’s leakage response with respect to the benchmark’s parameters underlying the average number of neutrons per fission, , on the moments (expected value, variance, and skewness) of the PERP benchmark’s leakage response distribution are negligible by comparison to the corresponding effects (on the response distribution) stemming from uncertainties in the total cross sections, but are larger than the corresponding effects (on the response distribution) stemming from uncertainties in the fission and scattering cross sections.

Second-order adjoint sensitivity analysis of a general ratio of functionals of the forward and adjoint fluxes in a multiplying nuclear system with source

In the present work, the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) is applied to a generic multiplying subcritical nuclear system comprising a non-fission neutron source in order to compute efficiently and exactly all of the 1st- and 2nd-order functional derivatives (“sensitivities”) of a system response with respect to the system’s many parameters. The system response is defined to be a ratio of two general nonlinear functionals of the forward and adjoint particle fluxes, while the system parameters include isotopic number densities, microscopic cross sections, fission spectrum parameters, neutron sources, and detector response functions. Such ratios of functionals represent the most general models for quantities of fundamental importance in reactor physics, including eigenvalues, ratios of reaction rates and forward/adjoint flux-weighted multi-group cross sections. The exact expressions obtained in this work will be used in subsequent works to perform a large-scale sensitivity analysis of a polyethylene-reflected plutonium metal sphere experimental benchmark to the nuclear data characterizing this benchmark, comprising 21,976 first-order sensitivities and 482,944,576 second-order sensitivities.

Application of the Second-Order Comprehensive Adjoint Sensitivity Analysis Methodology to Compute First- and Second-Order Sensitivities of Flux Functionals in a Multiplying System with Source

This work presents an application of the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) to the neutron transport Boltzmann equation that models a multiplying subcritical system comprising a nonfission neutron source to compute efficiently and exactly all of the first- and second-order functional derivatives (sensitivities) of a detector’s response to all of the model’s parameters, including isotopic number densities, microscopic cross sections, fission spectrum, sources, and detector response function. As indicated by the general theoretical considerations underlying the 2nd-ASAM, the number of computations required to obtain the first and second orders increases linearly in augmented Hilbert spaces as opposed to increasing exponentially in the original Hilbert space. The results presented in this work are currently being implemented in several production-oriented three-dimensional neutron transport code systems for analyzing specific subcritical systems.

Second-order sensitivities of a general functional of the forward and adjoint fluxes in a multiplying nuclear system with source

This work presents an application of the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) to compute efficiently and exactly all of the 1st- and 2nd-order functional derivatives (“sensitivities”) of a generic scalar-valued response to parameters for a multiplying subcritical system comprising a non-fission neutron source. The response is defined to be a nonlinear functional of the forward and adjoint particle fluxes while the system parameters include isotopic number densities, microscopic cross sections, fission spectrum, sources, and detector response functions. As indicated by the general theoretical considerations underlying the 2nd-ASAM, the number of computations required to obtain the 1st- and 2nd-order increases linearly in augmented Hilbert spaces as opposed to increasing exponentially in the original Hilbert space. This unique feature provides the fundamental reason for the unmatched efficiency of the 2nd-ASAM for computing exactly all of the 1st- and 2nd-order responses sensitivities to model parameters. The results presented in this work are currently being implemented in several production-oriented three dimensional neutron transport code systems for analyzing specific subcritical systems.

Second-order sensitivity and uncertainty analysis of the Roussopoulos-functional for a subcritical multiplying nuclear system with source

The Roussopoulos-functional has long been used for selecting trial functions for the forward and adjoint fluxes when computing reaction rates in nuclear systems and/or particle detector responses, because this functional was considered in previous works and textbooks to be accurate to second-order in variations of the model’s forward and adjoint fluxes. However, by applying the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM), this work shows that the first-order sensitivities of the Roussopoulos-functional to the system’s parameters (including isotopic number densities, microscopic cross sections, fission spectrum, forward and adjoint sources) are not identically zero. This means that the Roussopoulos-functional is not accurate to second-order in the flux variations if the system parameters are imprecisely (rather than perfectly well) known, as is invariably the case in practice. This work also illustrates the application of the 2nd-ASAM for computing efficiently the correct expressions of the second-order sensitivities of the Roussopoulos-functional to model parameters. The results presented in this work indicate the correct path for future possible uses of the Roussopoulos-functional for performing sensitivity and uncertainty analyses of both forward and inverse problems in nuclear systems.

Second-Order Sensitivity Analysis of Uncollided Particle Contributions to Radiation Detector Responses

This work presents an application of Cacuci’s Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) to the simplified Boltzmann equation that models the transport of uncollided particles through a medium to compute efficiently and exactly all of the first- and second-order derivatives (sensitivities) of a detector’s response with respect to the system’s isotopic number densities, microscopic cross sections, source emission rates, and detector response function. The off-the-shelf PARTISN multigroup discrete ordinates code is employed to solve the equations underlying the 2nd-ASAM. The accuracy of the results produced using PARTISN is verified by using the results of three test configurations: (1) a homogeneous sphere, for which the response is the exactly known total uncollided leakage, (2) a multiregion two-dimensional (r-z) cylinder, and (3) a two-region sphere for which the response is a reaction rate. For the homogeneous sphere, results for the total leakage as well as for the respective first- and second-order sensitivities are in excellent agreement with the exact benchmark values. For the nonanalytic problems, the results obtained by applying the 2nd-ASAM to compute sensitivities are in excellent agreement with central-difference estimates. The efficiency of the 2nd-ASAM is underscored by the fact that, for the cylinder, only 12 adjoint PARTISN computations were required by the 2nd-ASAM to compute all of the benchmark’s 18 first-order sensitivities and 224 second-order sensitivities, in contrast to the 877 PARTISN calculations needed to compute the respective sensitivities using central finite differences, and this number does not include the additional calculations that were required to find appropriate values of the perturbations to use for the central differences.

The Second-Order Adjoint Sensitivity Analysis Methodology for Nonlinear Systems—II: Illustrative Application to a Nonlinear Heat Conduction Problem

This work presents an illustrative application of the second-order adjoint sensitivity analysis methodology (2nd-ASAM) to a paradigm nonlinear heat conduction benchmark, which models a conceptual experimental test section containing heated rods immersed in liquid lead—bismuth eutectic (LBE). This benchmark admits an exact solution, thereby making transparent the underlying mathematical derivations. The general theory underlying 2nd-ASAM indicates that for a physical system comprising Nα parameters, the computation of all of the first- and second-order response sensitivities requires (per response) at most Nα large-scale computations using the first-level adjoint sensitivity system (1st-LASS) and second-level adjoint sensitivity system (2nd-LASS), respectively. For this illustrative problem, six large-scale adjoint computations sufficed to compute exactly all 5 first-order and 15 distinct second-order derivatives of the temperature response to the five model parameters. The construction and solution of the 2nd-LASS require very little additional effort beyond the construction of the adjoint sensitivity system needed for computing the first-order sensitivities. Very significantly, only the sources on the right sides of the heat conduction differential operator needed to be modified; the left side of the differential equations (and hence the solver in large-scale practical applications) remains unchanged.The second-order sensitivities play the following roles: (1) They cause the expected value of the response to differ from the computed nominal value of the response; for the nonlinear heat conduction benchmark, however, these differences were insignificant over the range of temperatures (400 to 900 K) considered. (2) They contribute to increasing the response variances and modifying the response covariances, but for the nonlinear heat conduction benchmark, their contribution was smaller than that stemming from the first-order response sensitivities, over the range of temperatures (400 to 900 K) considered. (3) They provide the leading contributions to causing asymmetries in the response distribution. For the benchmark test section considered in this work, the heat source, the boundary heat flux, and the temperature at the bottom boundary of the test section would cause the temperature distribution in the test section to be skewed significantly toward values lower than the mean temperature. On the other hand, the model parameters entering the nonlinear, temperature-dependent expression of the LBE conductivity would cause the temperature distribution in the test section to be skewed significantly toward values higher than the mean temperature. These opposite effects partially cancel each other. Consequently, the cumulative effects of model parameter uncertainties on the skewness of the temperature distribution in the test section are such that the temperature distribution in the LBE is skewed slightly toward higher temperatures in the cooler part of the test section but becomes increasingly skewed toward temperatures lower than the mean temperature in the hotter part of the test section. Notably, the influence of the model parameter that controls the strength of the nonlinearity in the heat conduction coefficient for this LBE test section benchmark would be strong if it were the only uncertain model parameter. However, if all of the other model parameters are also uncertain, all having equal relative standard deviations, the uncertainties in the heat source and boundary heat flux diminish the impact of uncertainties in the nonlinear heat conduction coefficient for the range of temperatures (400 to 900 K) considered for this LBE test section benchmark.Ongoing work aims at generalizing the 2nd-ASAM to enable the exact and efficient computation of higher-order response sensitivities. The availability of such higher-order sensitivities is expected to affect significantly the fields of optimization and predictive modeling, including uncertainty quantification, data assimilation, model calibration, and extrapolation.

The Second-Order Adjoint Sensitivity Analysis Methodology for Nonlinear Systems—I: Theory

The use of adjoint methods for computing first-order sensitivities (i.e., functional derivatives) of results (responses) produced by a computational model to the model’s parameters was initiated in the nuclear engineering sciences in the 1940s. The field of nuclear science and engineering also provided pioneering works, during the 1970s, for computing second-order response sensitivities of responses associated with the adjoint neutron and radiation transport and/or diffusion equations. These works generally indicated that the second-order sensitivities of responses such as reaction rates and the system’s effective multiplication factor to cross sections were computationally intensive to obtain, requiring O(Nα2) large-scale computations per response, for a system comprising Nα model parameters, and were considerably smaller than the corresponding first-order sensitivities. These results likely gave rise to the generally held opinion that second-order sensitivities are generally insignificant in reactor physics, which may, in turn, have led to diminishing interest in developing efficient methods for computing second-order sensitivities for nuclear engineering systems.This work presents the second-order adjoint sensitivity analysis methodology (2nd-ASAM) for nonlinear systems, which yields exactly and efficiently the second-order functional derivatives of physical system responses (i.e., system performance parameters) to the system’s model parameters. For a physical system comprising Nα parameters, forward methods require a total of (Nα2/2+3Nα/2) large-scale computations for obtaining all of the first- and second-order sensitivities, for all system responses. In contradistinction, the 2nd-ASAM requires one large-scale computation using the first-level adjoint sensitivity system (1st-LASS) for obtaining all of the first-order sensitivities, followed by at most Nα large-scale computations using the second-level adjoint sensitivity systems, for obtaining exactly all of the second-order sensitivities of a functional-type response. The construction, implementation, and solution of the 2nd-ASAM require very little additional effort beyond the construction of the 1st-LASS needed for computing the first-order sensitivities. Furthermore, because of the symmetry properties of the second-order sensitivities, the 2nd-ASAM comprises the inherent automatic solution verification of the correctness and accuracy of the second-level adjoint functions used for the efficient and exact computation of the second-order sensitivities. The use of the 2nd-ASAM to compute exactly all of the second-order response sensitivities to model input parameters is expected to enable significant advances in related scientific disciplines, particularly the areas of uncertainty quantification and predictive modeling, including model validation, reduced-order modeling, data assimilation, model calibration, and extrapolation.

Second-Order Adjoint Sensitivity and Uncertainty Analysis of a Heat Transport Benchmark Problem—I: Analytical Results

This work presents an illustrative application of the second-order adjoint sensitivity analysis methodology (2nd-ASAM) to a heat transport benchmark problem that simulates the radial heat conduction in a heated fuel rod together with the axial heat transport along the coolant channel surrounding the rod. This paradigm benchmark is sufficiently simple to admit an exact solution, thereby making transparent the mathematical derivations underlying the 2nd-ASAM, and also for performing solution verification of computational fluid dynamics simulation tools such as the FLUENT Adjoint Solver, a code that has been used for computing thermal-hydraulic processes within the G4M Reactor. The G4M Reactor is an innovative small modular fast reactor cooled by lead-bismuth eutectic.The benchmark problem has six thermal-hydraulic parameters representative of the G4M Reactor, as well as of a test section that is under design for analyzing thermal-hydraulic phenomena expected within the reactor’s core. Thus, the benchmark exhibits 6 first-order and 21 second-order sensitivities for the temperature distribution at any location within the heated rod (and/or on its surface) and just as many sensitivities for the temperature distribution at any location within the coolant. Locations of particular importance are those where the rod temperature attains its maximum along the rod’s axis or on the rod’s surface, as well as the coolant’s exit, where the coolant temperature attains its maximum.The general theory underlying the 2nd-ASAM indicates that for a physical system comprising Nα parameters, the computation of all of the first- and second-order response sensitivities requires Nα large-scale computations involving correspondingly constructed adjoint systems, which we called second-level adjoint sensitivity systems (2nd-LASS). In practice, however, the actual number of large-scale adjoint computations can be significantly smaller; in particular, all 54 first- and second-order rod and coolant temperature sensitivities for the benchmark presented in this work were obtained using only seven independent adjoint computations. Furthermore, the construction and solution of the 2nd-LASS require very little additional effort beyond the construction of the first-level adjoint sensitivity system (1st-LASS) needed for computing the first-order sensitivities. Very significantly, only the sources on the right side of the differential operator needed to be modified; the left side of the respective differential equations (and hence the solver in large-scale practical applications) remained unchanged from the 1st-LASS and from the original equations when these are linear in the unknown state function.This work also shows that the second-order sensitivities have the following impacts on the computed moments of the response distribution: (a) they cause the expected value of the response to differ from the computed nominal value of the response, (b) they contribute to the response variances and covariances, and (c) they contribute decisively to causing asymmetries in the response distribution. Indeed, neglecting the second-order sensitivities would nullify the third-order response correlations stemming from normally distributed parameters and hence would nullify the skewness of the response; consequently, events occurring in a response’s long and/or short tails could be missed.

Second-order adjoint sensitivity analysis methodology (2nd-ASAM) for computing exactly and efficiently first- and second-order sensitivities in large-scale linear systems: I. Computational methodology

This work presents the second-order forward and adjoint sensitivity analysis methodologies (2nd-FSAM and 2nd-ASAM) for computing exactly and efficiently the second-order functional derivatives of physical (engineering, biological, etc.) system responses (i.e., “system performance parameters”) to the system's model parameters. The definition of “system parameters” used in this work includes all computational input data, correlations, initial and/or boundary conditions, etc. For a physical system comprising Nα parameters and Nr responses, we note that the 2nd-FSAM requires a total of (Nα2/2+3Nα/2) large-scale computations for obtaining all of the first- and second-order sensitivities, for all Nr system responses. On the other hand, for one functional-type system response, the 2nd-ASAM requires one large-scale computation using the first-level adjoint sensitivity system for obtaining all of the first-order sensitivities, followed by at most Nα large-scale computations using the second-level adjoint sensitivity systems for obtaining exactly all of the second-order sensitivities. Therefore, the 2nd-FSAM should be used when Nr≫Nα, while the 2nd-ASAM should be used when Nα≫Nr. The original 2nd-ASAM presented in this work should enable the hitherto very difficult, if not intractable, exact computation of all of the second-order response sensitivities (i.e., functional Gateaux-derivatives) for large-systems involving many parameters, as usually encountered in practice. Very importantly, the implementation of the 2nd-ASAM requires very little additional effort beyond the construction of the adjoint sensitivity system needed for computing the first-order sensitivities.

Second-order adjoint sensitivity analysis methodology (2nd-ASAM) for computing exactly and efficiently first- and second-order sensitivities in large-scale linear systems: II. Illustrative application to a paradigm particle diffusion problem

This work presents an illustrative application of the second-order adjoint sensitivity analysis methodology (2nd-ASAM) to a paradigm neutron diffusion problem, which is sufficiently simple to admit an exact solution, thereby making transparent the underlying mathematical derivations. The general theory underlying 2nd-ASAM indicates that, for a physical system comprising Nα parameters, the computation of all of the first- and second-order response sensitivities requires (per response) at most (2Nα+1) “large-scale” computations using the first-level and, respectively, second-level adjoint sensitivity systems (1st-LASS and 2nd-LASS). Very importantly, however, the illustrative application presented in this work shows that the actual number of adjoint computations needed for computing all of the first- and second-order response sensitivities may be significantly less than (2Nα+1) per response. For this illustrative problem, four “large-scale” adjoint computations sufficed for the complete and exact computations of all 4 first- and 10 distinct second-order derivatives. Furthermore, the construction and solution of the 2nd-LASS requires very little additional effort beyond the construction of the adjoint sensitivity system needed for computing the first-order sensitivities. Very significantly, only the sources on the right-sides of the diffusion (differential) operator needed to be modified; the left-side of the differential equations (and hence the “solver” in large-scale practical applications) remained unchanged.All of the first-order relative response sensitivities to the model parameters have significantly large values, of order unity. Also importantly, most of the second-order relative sensitivities are just as large, and some even up to twice as large as the first-order sensitivities. In the illustrative example presented in this work, the second-order sensitivities contribute little to the response variances and covariances. However, they have the following major impacts on the computed moments of the response distribution: (a) they cause the “expected value of the response” to differ from the “computed nominal value of the response”; and (b) they contribute decisively to causing asymmetries in the response distribution. Indeed, neglecting the second-order sensitivities would nullify the third-order response correlations, and hence would nullify the skewness of the response. Consequently, any events occurring in a response's long and/or short tails, which are characteristic of rare but decisive events (e.g., major accidents, catastrophes), would likely be missed. The 2nd-ASAM is expected to affect significantly other fields that need efficiently computed second-order response sensitivities, e.g., optimization, data assimilation/adjustment, model calibration, and predictive modeling.