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.

Continuing the author’s previous work on modeling, this book presents the most recent advances in high-order predictive modeling. The author begins with the mathematical framework of the 2nd-BERRU-PM methodology, an acronym that designates the “second-order best-estimate with reduced uncertainties (2nd-BERRU) predictive modeling (PM).” The 2nd-BERRU-PM methodology is fundamentally anchored in physics-based principles stemming from thermodynamics (maximum entropy principle) and information theory, being formulated in the most inclusive possible phase-space, namely the combined phase-space of computed and measured parameters and responses. The 2nd-BERRU-PM methodology provides second-order output (means and variances) but can incorporate, as input, arbitrarily high-order sensitivities of responses with respect to model parameters, as well as arbitrarily high-order moments of the initial distribution of uncertain model parameters, in order to predict best-estimate mean values for the model responses (i.e., results of interest) and calibrated model parameters, along with reduced predicted variances and covariances for these predicted responses and parameters.

A comprehensive reference on data assimilation and inverse problems, and their applications across a broad range of geophysical disciplines, ideal for researchers and graduate students. It highlights the importance of data assimilation for understanding dynamical processes of the Earth and its space environment, and summarises recent advances.