scholarly journals Fourth-Order Adjoint Sensitivity Analysis of an OECD/NEA Reactor Physics Benchmark: II. Mathematical Expressions and CPU-Time Comparisons for Computing 4<sup>th</sup>-Order Sensitivities

2021 ◽  
Vol 11 (02) ◽  
pp. 133-156
Author(s):  
Dan Gabriel Cacuci ◽  
Ruixian Fang
Keyword(s):  
Sensitivity Analysis ◽  
Fourth Order ◽  
Adjoint Sensitivity Analysis ◽  
Reactor Physics ◽  
Adjoint Sensitivity ◽  
Cpu Time ◽  
Mathematical Expressions

scholarly journals 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

Energies ◽  
2020 ◽  
Vol 13 (10) ◽  
pp. 2580 ◽  
Author(s):  
Ruixian Fang ◽  
Dan G. Cacuci
Keyword(s):  
Sensitivity Analysis ◽  
Cross Section ◽  
Cross Sections ◽  
Second Order ◽  
Adjoint Sensitivity Analysis ◽  
Number Density ◽  
Reactor Physics ◽  
Adjoint Sensitivity ◽  
Energy Group ◽  
Analysis Methodology

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.

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

Energies ◽  
2021 ◽  
Vol 14 (11) ◽  
pp. 3335
Author(s):  
Dan Gabriel Cacuci
Keyword(s):  
Sensitivity Analysis ◽  
Linear Systems ◽  
Fourth Order ◽  
Model Parameters ◽  
Adjoint Sensitivity Analysis ◽  
Adjoint Sensitivity ◽  
Adjoint Systems ◽  
Functional Derivatives ◽  
Adjoint State ◽  
State Functions

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.

scholarly journals 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

Energies ◽  
2020 ◽  
Vol 13 (7) ◽  
pp. 1674 ◽  
Author(s):  
Dan G. Cacuci ◽  
Ruixian Fang ◽  
Jeffrey A. Favorite
Keyword(s):  
Sensitivity Analysis ◽  
Cross Sections ◽  
Relative Sensitivity ◽  
Second Order ◽  
Model Parameters ◽  
Adjoint Sensitivity Analysis ◽  
Reactor Physics ◽  
Adjoint Sensitivity ◽  
First Order ◽  
Analysis Methodology

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”.

scholarly journals A Case Study in Adjoint Sensitivity Analysis of Parameter Calibration

2016 ◽  
Vol 80 ◽  
pp. 201-211
Author(s):  
Johannes Lotz ◽  
Marc Schwalbach ◽  
Uwe Naumann
Keyword(s):  
Sensitivity Analysis ◽  
Adjoint Sensitivity Analysis ◽  
Parameter Calibration ◽  
Adjoint Sensitivity
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