The nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology, Volume III, Cacuci
Автор: Schm?dgen Название: Unbounded Self-Adjoint Operators on Hilbert Space ISBN: 9400747527 ISBN-13(EAN): 9789400747524 Издательство: Springer Рейтинг: Цена: 10480.00 р. Наличие на складе: Есть у поставщика Поставка под заказ.
Описание: The book is a graduate text on unbounded self-adjoint operators on Hilbert space and their spectral theory with the emphasis on applications in mathematical physics (especially, Schroedinger operators) and analysis (Dirichlet and Neumann Laplacians, Sturm-Liouville operators, Hamburger moment problem) .
Автор: Konrad Schm?dgen Название: Unbounded Self-adjoint Operators on Hilbert Space ISBN: 9400797419 ISBN-13(EAN): 9789400797413 Издательство: Springer Рейтинг: Цена: 9083.00 р. Наличие на складе: Есть у поставщика Поставка под заказ.
Описание: This book explores unbounded self-adjoint operators on Hilbert space and their spectral theory, placing emphasis on applications in mathematical physics and analysis. Addresses advanced topics, and includes many examples and exercises.
Описание: This monograph presents the newly developed method of rigged Hilbert spaces as a modern approach in singular perturbation theory. A key notion of this approach is the Lax-Berezansky triple of Hilbert spaces embedded one into another, which specifies the well-known Gelfand topological triple. All kinds of singular interactions described by potentials supported on small sets (like the Dirac ?-potentials, fractals, singular measures, high degree super-singular expressions) admit a rigorous treatment only in terms of the equipped spaces and their scales. The main idea of the method is to use singular perturbations to change inner products in the starting rigged space, and the construction of the perturbed operator by the Berezansky canonical isomorphism (which connects the positive and negative spaces from a new rigged triplet). The approach combines three powerful tools of functional analysis based on the Birman-Krein-Vishik theory of self-adjoint extensions of symmetric operators, the theory of singular quadratic forms, and the theory of rigged Hilbert spaces. The book will appeal to researchers in mathematics and mathematical physics studying the scales of densely embedded Hilbert spaces, the singular perturbations phenomenon, and singular interaction problems.
Описание: Building from the investigation of the spectrum and spectral singularities and construction of the spectral expansion for the non-self-adjoint Schroedinger operator, the book features a complete spectral analysis of the Mathieu-Schroedinger operator and the Schroedinger operator with a parity-time (PT)-symmetric periodic optical potential.
Описание: This book introduces and discusses the self-adjoint extension problem for symmetric operators on Hilbert space. It presents the classical von Neumann and Krein–Vishik–Birman extension schemes both in their modern form and from a historical perspective, and provides a detailed analysis of a range of applications beyond the standard pedagogical examples (the latter are indexed in a final appendix for the reader’s convenience). Self-adjointness of operators on Hilbert space representing quantum observables, in particular quantum Hamiltonians, is required to ensure real-valued energy levels, unitary evolution and, more generally, a self-consistent theory. Physical heuristics often produce candidate Hamiltonians that are only symmetric: their extension to suitably larger domains of self-adjointness, when possible, amounts to declaring additional physical states the operator must act on in order to have a consistent physics, and distinct self-adjoint extensions describe different physics. Realising observables self-adjointly is the first fundamental problem of quantum-mechanical modelling. The discussed applications concern models of topical relevance in modern mathematical physics currently receiving new or renewed interest, in particular from the point of view of classifying self-adjoint realisations of certain Hamiltonians and studying their spectral and scattering properties. The analysis also addresses intermediate technical questions such as characterising the corresponding operator closures and adjoints. Applications include hydrogenoid Hamiltonians, Dirac–Coulomb Hamiltonians, models of geometric quantum confinement and transmission on degenerate Riemannian manifolds of Grushin type, and models of few-body quantum particles with zero-range interaction. Graduate students and non-expert readers will benefit from a preliminary mathematical chapter collecting all the necessary pre-requisites on symmetric and self-adjoint operators on Hilbert space (including the spectral theorem), and from a further appendix presenting the emergence from physical principles of the requirement of self-adjointness for observables in quantum mechanics.
Описание: 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 computatio
Описание: 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.
Автор: Bertrand Iooss, Clementine Prieur, Fabrice Gamboa, Sebastien Da Veiga Название: Basics and Trends in Sensitivity Analysis: Theory and Practice in R ISBN: 1611976685 ISBN-13(EAN): 9781611976687 Издательство: Mare Nostrum (Eurospan) Рейтинг: Цена: 11161.00 р. Наличие на складе: Нет в наличии.
Описание: This book provides an overview of global sensitivity analysis methods and algorithms, including their theoretical basis and mathematical properties. The authors use a practical point of view and real case studies as well as numerous examples, and applications of the different approaches are illustrated throughout using R code to explain their usage and usefulness in practice. Basics and Trends in Sensitivity Analysis: Theory and Practice in R covers a lot of material, including theoretical aspects of Sobol’ indices as well as sampling-based formulas, spectral methods, and metamodel-based approaches for estimation purposes; screening techniques devoted to identifying influential and noninfluential inputs; variance-based measures when model inputs are statistically dependent (and several other approaches that go beyond variance-based sensitivity measures); and a case study in R related to a COVID-19 epidemic model where the full workflow of sensitivity analysis combining several techniques is presented.This book is intended for engineers, researchers, and undergraduate students who use complex numerical models and have an interest in sensitivity analysis techniques and is appropriate for anyone with a solid mathematical background in basic statistical and probability theories who develops and uses numerical models in all scientific and engineering domains.
Автор: Petropoulos, George Название: Sensitivity Analysis In Earth Observation Modelling ISBN: 0128030119 ISBN-13(EAN): 9780128030110 Издательство: Elsevier Science Рейтинг: Цена: 14654.00 р. Наличие на складе: Есть у поставщика Поставка под заказ.
Описание:
Sensitivity Analysis in Earth Observation Modeling highlights the state-of-the-art in ongoing research investigations and new applications of sensitivity analysis in earth observation modeling. In this framework, original works concerned with the development or exploitation of diverse methods applied to different types of earth observation data or earth observation-based modeling approaches are included. An overview of sensitivity analysis methods and principles is provided first, followed by examples of applications and case studies of different sensitivity/uncertainty analysis implementation methods, covering the full spectrum of sensitivity analysis techniques, including operational products. Finally, the book outlines challenges and future prospects for implementation in earth observation modeling.
Information provided in this book is of practical value to readers looking to understand the principles of sensitivity analysis in earth observation modeling, the level of scientific maturity in the field, and where the main limitations or challenges are in terms of improving our ability to implement such approaches in a wide range of applications. Readers will also be informed on the implementation of sensitivity/uncertainty analysis on operational products available at present, on global and continental scales. All of this information is vital in the selection process of the most appropriate sensitivity analysis method to implement.
Описание: Building from the investigation of the spectrum and spectral singularities and construction of the spectral expansion for the non-self-adjoint Schroedinger operator, the book features a complete spectral analysis of the Mathieu-Schroedinger operator and the Schroedinger operator with a parity-time (PT)-symmetric periodic optical potential.
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