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This thesis focuses on iterative methods for the treatment of the steady state neutron transport equation in slab geometry, bounded convex domain of Rn (n = 2,3) and in 1-D spherical geometry. We introduce a generic Alternate Direction Implicit (ADI)-like iterative method based on positive definite and m-accretive splitting (PAS) for linear operator equations with operators admitting such splitting. This method converges unconditionally and its SOR acceleration yields convergence results similar to those obtained in presence of finite dimensional systems with matrices possessing the Young property A. The proposed methods are illustrated by a numerical example in which an integro-differential problem of transport theory is considered. In the particular case where the positive definite part of the linear equation operator is self-adjoint, an upper bound for the contraction factor of the iterative method, which depends solely on the spectrum of the self-adjoint part is derived. As such, this method has been successfully applied to the neutron transport equation in slab and 2-D cartesian geometry and in 1-D spherical geometry. The self-adjoint and m-accretive splitting leads to a fixed point problem where the operator is a 2 by 2 matrix of operators. An infinite dimensional adaptation of minimal residual and preconditioned minimal residual algorithms using Gauss-Seidel, symmetric Gauss-Seidel and polynomial preconditioning are then applied to solve the matrix operator equation. Theoretical analysis shows that the methods converge unconditionally and upper bounds of the rate of residual decreasing which depend solely on the spectrum of the self-adjoint part of the operator are derived. The convergence of theses solvers is illustrated numerically on a sample neutron transport problem in 2-D geometry. Various test cases, including pure scattering and optically thick domains are considered.
The authors consider dynamic types of inverse problems in which the additional information is given by the trace of the direct problem on a (usually time-like) surface of the domain. They discuss theoretical and numerical background of the finite-difference scheme inversion, the linearization method, the method of Gel'fand-Levitan-Krein, the boundary control method, and the projection method and prove theorems of convergence, conditional stability, and other properties of the mentioned methods.
Applied Iterative Methods
This book consists of 20 review articles dedicated to Prof. Philip Roe on the occasion of his 60th birthday and in appreciation of his original contributions to computational fluid dynamics. The articles, written by leading researchers in the field, cover many topics, including theory and applications, algorithm developments and modern computational techniques for industry. Contents: OC A One-Sided ViewOCO: The Real Story (B van Leer); Collocated Upwind Schemes for Ideal MHD (K G Powell); The Penultimate Scheme for Systems of Conservation Laws: Finite Difference ENO with Marquina's Flux Splitting (R P Fedkiw et al.); A Finite Element Based Level-Set Method for Multiphase Flows (B Engquist & A-K Tornberg); The GHOST Fluid Method for Viscous Flows (R P Fedkiw & X-D Liu); Factorizable Schemes for the Equations of Fluid Flow (D Sidilkover); Evolution Galerkin Methods as Finite Difference Schemes (K W Morton); Fluctuation Distribution Schemes on Adjustable Meshes for Scalar Hyperbolic Equations (M J Baines); Superconvergent Lift Estimates Through Adjoint Error Analysis (M B Giles & N A Pierce); Somewhere between the LaxOCoWendroff and Roe Schemes for Calculating Multidimensional Compressible Flows (A Lerat et al.); Flux Schemes for Solving Nonlinear Systems of Conservation Laws (J M Ghidaglia); A LaxOCoWendroff Type Theorem for Residual Schemes (R Abgrall et al.); Kinetic Schemes for Solving SaintOCoVenant Equations on Unstructured Grids (M O Bristeau & B Perthame); Nonlinear Projection Methods for Multi-Entropies NavierOCoStokes Systems (C Berthon & F Coquel); A Hybrid Fluctuation Splitting Scheme for Two-Dimensional Compressible Steady Flows (P De Palma et al.); Some Recent Developments in Kinetic Schemes Based on Least Squares and Entropy Variables (S M Deshpande); Difference Approximation for Scalar Conservation Law. Consistency with Entropy Condition from the Viewpoint of Oleinik's E-Condition (H Aiso); Lessons Learned from the Blast Wave Computation Using Overset Moving Grids: Grid Motion Improves the Resolution (K Fujii). Readership: Researchers and graduate students in numerical and computational mathematics in engineering."
Includes following subjects: Solution of equations in Rn, Finite difference methods, Finite element methods, Techniques of scientific computing, Optimization theory and systems science, Numerical methods for fluids, Numerical methods for solids, Specific applications
This edited review book on Godunov methods contains 97 articles, all of which were presented at the international conference on Godunov Methods: Theory and Applications, held at Oxford, in October 1999, to commemorate the 70th birthday of the Russian mathematician Sergei K. Godunov. The central theme of this book is numerical methods for hyperbolic conservation laws following Godunov's key ideas contained in his celebrated paper of 1959. Hyperbolic conservation laws play a central role in mathematical modelling in several distinct disciplines of science and technology. Application areas include compressible, single (and multiple) fluid dynamics, shock waves, meteorology, elasticity, magnetohydrodynamics, relativity, and many others. The successes in the design and application of new and improved numerical methods of the Godunov type for hyperbolic conservation laws in the last twenty years have made a dramatic impact in these application areas. The 97 papers cover a very wide range of topics, such as design and analysis of numerical schemes, applications to compressible and incompressible fluid dynamics, multi-phase flows, combustion problems, astrophysics, environmental fluid dynamics, and detonation waves. This book will be a reference book on the subject of numerical methods for hyperbolic partial differential equations for many years to come.All contributions are self-contained but do contain a review element. There is a key paper by Peter Sweby in which a general overview of Godunov methods is given. This contribution is particularly suitable for beginners on the subject. This book is unique: it contains virtually everything concerned with Godunov-type methods for conservation laws. As such it will be of particular interest to academics (applied mathematicians, numerical analysts, engineers, environmental scientists, physicists, and astrophysicists) involved in research on numerical methods for partial differential equations; scientists and engineers concerned with new numerical methods and applications to scientific and engineering problems e.g., mechanical engineers, aeronautical engineers, meteorologists; and academics involved in teaching numerical methods for partial differential equations at the postgraduate level.
The goal of Optimal Transport (OT) is to define geometric tools that are useful to compare probability distributions. Their use dates back to 1781. Recent years have witnessed a new revolution in the spread of OT, thanks to the emergence of approximate solvers that can scale to sizes and dimensions that are relevant to data sciences. Thanks to this newfound scalability, OT is being increasingly used to unlock various problems in imaging sciences (such as color or texture processing), computer vision and graphics (for shape manipulation) or machine learning (for regression, classification and density fitting). This monograph reviews OT with a bias toward numerical methods and their applications in data sciences, and sheds lights on the theoretical properties of OT that make it particularly useful for some of these applications. Computational Optimal Transport presents an overview of the main theoretical insights that support the practical effectiveness of OT before explaining how to turn these insights into fast computational schemes. Written for readers at all levels, the authors provide descriptions of foundational theory at two-levels. Generally accessible to all readers, more advanced readers can read the specially identified more general mathematical expositions of optimal transport tailored for discrete measures. Furthermore, several chapters deal with the interplay between continuous and discrete measures, and are thus targeting a more mathematically-inclined audience. This monograph will be a valuable reference for researchers and students wishing to get a thorough understanding of Computational Optimal Transport, a mathematical gem at the interface of probability, analysis and optimization.
Handbook on Numerical Methods for Hyperbolic Problems: Applied and Modern Issues details the large amount of literature in the design, analysis, and application of various numerical algorithms for solving hyperbolic equations that has been produced in the last several decades. This volume provides concise summaries from experts in different types of algorithms, so that readers can find a variety of algorithms under different situations and become familiar with their relative advantages and limitations. - Provides detailed, cutting-edge background explanations of existing algorithms and their analysis - Presents a method of different algorithms for specific applications and the relative advantages and limitations of different algorithms for engineers or those involved in applications - Written by leading subject experts in each field, the volumes provide breadth and depth of content coverage