The Second-Order Adjoint Sensitivity Analysis Methodology generalizes the First-Order Theory presented in the author’s previous books published by CRC Press. This breakthrough has many applications in sensitivity and uncertainty analysis, optimization, data assimilation, model calibration, and reducing uncertainties in model predictions. The book has many illustrative examples that will help readers understand the complexity of the subject and will enable them to apply this methodology to problems in their own fields. Highlights: • Covers a wide range of needs, from graduate students to advanced researchers • Provides a text positioned to be the primary reference for high-order sensitivity and uncertainty analysis • Applies to all fields involving numerical modeling, optimization, quantification of sensitivities in direct and inverse problems in the presence of uncertainties. About the Author: Dan Gabriel Cacuci is a South Carolina SmartState Endowed Chair Professor and the Director of the Center for Nuclear Science and Energy, Department of Mechanical Engineering at the University of South Carolina. He has a Ph.D. in Applied Physics, Mechanical and Nuclear Engineering from Columbia University. He is also the recipient of many awards including four honorary doctorates, the Ernest Orlando Lawrence Memorial award from the U.S. Dept. of Energy and the Arthur Holly Compton, Eugene P. Wigner and the Glenn Seaborg Awards from the American Nuclear Society.

As computer-assisted modeling and analysis of physical processes have continued to grow and diversify, sensitivity and uncertainty analyses have become indispensable investigative scientific tools in their own right. While most techniques used for these analyses are well documented, there has yet to appear a systematic treatment of the method based

This text describes a comprehensive adjoint sensitivity analysis methodology (nth-CASAM), developed by the author, which enablesthe efficient and exact computation of arbitrarily high-order functional derivatives of model responses to model parameters in large-scale systems. The nth-CASAM framework is set in linearly increasing Hilbert spaces, each of state-function-dimensionality, as opposed to exponentially increasing parameter-dimensional spaces, thereby overcoming the so-called “curse of dimensionality” in sensitivity and uncertainty analysis. The nth-CASAM is applicable to any model; the larger the number of model parameters, the more efficient the nth-CASAM becomes for computing arbitrarily high-order response sensitivities. The book will be helpful to those working in the fields of sensitivity analysis, uncertainty quantification, model validation, optimization, data assimilation, model calibration, sensor fusion, reduced-order modelling, inverse problems and predictive modelling. This Volume Two, the second of three, presents the large-scale application of the nth-CASAM to perform a representative fourth-order sensitivity analysis of the Polyethylene-Reflected Plutonium benchmark described in the Nuclear Energy Agency (NEA) International Criticality Safety Benchmark Evaluation Project (ICSBEP) Handbook. This benchmark is modeled mathematically by the Boltzmann particle transport equation, involving 21,976 imprecisely-known parameters, the numerical solution of which requires representative large-scale computations. The sensitivity analysis presented in this volume is the most comprehensive ever performed in the field of reactor physics and the results presented in this book prove, perhaps counter-intuitively, that many of the 4th-order sensitivities are much larger than the corresponding 3rd-order ones, which are, in turn, much larger than the 2nd-order ones, all of which are much larger than the 1st-order sensitivities. Currently, the nth-CASAM is the only known methodology which enables such large-scale computations of exactly obtained expressions of arbitrarily-high-order response sensitivities.

The computational models of physical systems comprise parameters, independent and dependent variables. Since the physical processes themselves are seldom known precisely and since most of the model parameters stem from experimental procedures which are also subject to imprecisions, the results predicted by these models are also imprecise, being affected by the uncertainties underlying the respective model. The functional derivatives (also called “sensitivities”) of results (also called “responses”) produced by mathematical/computational models are needed for many purposes, including: (i) understanding the model by ranking the importance of the various model parameters; (ii) performing “reduced-order modeling” by eliminating unimportant parameters and/or processes; (iii) quantifying the uncertainties induced in a model response due to model parameter uncertainties; (iv) performing “model validation,” by comparing computations to experiments to address the question “does the model represent reality?” (v) prioritizing improvements in the model; (vi) performing data assimilation and model calibration as part of forward “predictive modeling” to obtain best-estimate predicted results with reduced predicted uncertainties; (vii) performing inverse “predictive modeling”; (viii) designing and optimizing the system. This 3-Volume monograph describes a comprehensive adjoint sensitivity analysis methodology, developed by the author, which enables the efficient and exact computation of arbitrarily high-order sensitivities of model responses in large-scale systems comprising many model parameters. The qualifier “comprehensive” is employed to highlight that the model parameters considered within the framework of this methodology also include the system’s uncertain boundaries and internal interfaces in phase-space. The model’s responses can be either scalar-valued functionals of the model’s parameters and state variables (e.g., as customarily encountered in optimization problems) or general function-valued responses. Since linear operators admit bona-fide adjoint operators, responses of models that are linear in the state functions (i.e., dependent variables) can depend simultaneously on both the forward and the adjoint state functions. Hence, the sensitivity analysis of such responses warrants the treatment of 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 state functions, since nonlinear operators do not admit bona-fide adjoint operators (only a linearized form of a nonlinear operator may admit an adjoint operator). Thus, Volume 1 of this book 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 the most efficient computation of exactly obtained mathematical expressions of arbitrarily-high-order (nth-order) sensitivities of a generic system response with respect to all of the parameters underlying the respective forward/adjoint systems. Volume 2 of this book presents the application of the nth-CASAM-L to perform a fourth-order sensitivity and uncertainty analysis of an OECD/NEA reactor physics benchmark which is representative of a large-scale model comprises many (21,976) uncertain parameters, thereby amply illustrating the unique potential of the nth-CASAM-L to enable the exact and efficient computation of chosen high-order response sensitivities to model parameters. Volume 3 of this book presents the “nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (abbreviation: nth-CASAM-N) for the practical, efficient, and exact computation of arbitrarily-high order sensitivities of responses to model parameters for systems that are also nonlinear in their underlying state functions. Such computations are not feasible with any other methodology. The application of the nth-CASAM-L and the nth-CASAM-N overcomes the so-called “curse of dimensionality” in sensitivity and uncertainty analysis, thus revolutionizing all of the fields of activities which require accurate computation of response sensitivities. Since this monograph includes many illustrative, fully worked-out, paradigm problems, it can serve as a textbook or as supplementary reading for graduate courses in academic departments in the natural sciences and engineering.

This text describes a comprehensive adjoint sensitivity analysis methodology (C-ASAM), developed by the author, enabling the efficient and exact computation of arbitrarily high-order functional derivatives of model responses to model parameters in large-scale systems. The model’s responses can be either scalar-valued functionals of the model’s parameters and state variables (as customarily encountered, e.g., in optimization problems) or general function-valued responses, which are often of interest but are currently not amenable to efficient sensitivity analysis. The C-ASAM framework is set in linearly increasing Hilbert spaces, each of state-function-dimensionality, as opposed to exponentially increasing parameter-dimensional spaces, thereby breaking the so-called “curse of dimensionality” in sensitivity and uncertainty analysis. The C-ASAM applies to any model; the larger the number of model parameters, the more efficient the C-ASAM becomes for computing arbitrarily high-order response sensitivities. The text includes illustrative paradigm problems which are fully worked-out to enable the thorough understanding of the C-ASAM’s principles and their practical application. The book will be helpful to those working in the fields of sensitivity analysis, uncertainty quantification, model validation, optimization, data assimilation, model calibration, sensor fusion, reduced-order modelling, inverse problems and predictive modelling. It serves as a textbook or as supplementary reading for graduate course on these topics, in academic departments in the natural, biological, and physical sciences and engineering. This Volume Three, the third of three, covers systems that are nonlinear in the state variables, model parameters and associated responses. The selected illustrative paradigm problems share these general characteristics. A separate Volume One covers systems that are linear in the state variables.

This unique reference is the first to cover the theory of adjoint sensitivity analysis and explains how it can be applied to different types of electromagnetic structures. It is an invaluable book for anyone looking for an in-depth understanding of this useful theory for application in high-frequency electromagnetic problems. It uses the popular FDTD method to show how wideband sensitivities can be efficiently estimated for different types of materials and structures, and includes plenty of well-explained MATLAB(R) examples to help readers absorb the content more easily. Topics covered include a review of FDTD and an introduction to adjoint sensitivity analysis; sensitivity of the fields to changes in material parameters; sensitivity of S parameters; extension to dispersive material parameters, where the underlying FDTD algorithm must be modified; second-order sensitivity analysis; time-domain responses; and applications to nonlinear and anisotropic materials. This book will make the theory more understandable to the broadest possible audience. It will be useful for researchers and advanced students involved in computational techniques for electromagnetics, and other disciplines such as microwave, optics, acoustics, and semiconductor modelling.