Fast Direct Solvers For Elliptic Pdes

Author : Per-Gunnar Martinsson
Publisher : SIAM
Page : 332 pages
File Size : 40,7 Mb
Release : 2019-12-16
Category : Mathematics
ISBN : 9781611976045

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Fast Direct Solvers for Elliptic PDEs by Per-Gunnar Martinsson Pdf

Fast solvers for elliptic PDEs form a pillar of scientific computing. They enable detailed and accurate simulations of electromagnetic fields, fluid flows, biochemical processes, and much more. This textbook provides an introduction to fast solvers from the point of view of integral equation formulations, which lead to unparalleled accuracy and speed in many applications. The focus is on fast algorithms for handling dense matrices that arise in the discretization of integral operators, such as the fast multipole method and fast direct solvers. While the emphasis is on techniques for dense matrices, the text also describes how similar techniques give rise to linear complexity algorithms for computing the inverse or the LU factorization of a sparse matrix resulting from the direct discretization of an elliptic PDE. This is the first textbook to detail the active field of fast direct solvers, introducing readers to modern linear algebraic techniques for accelerating computations, such as randomized algorithms, interpolative decompositions, and data-sparse hierarchical matrix representations. Written with an emphasis on mathematical intuition rather than theoretical details, it is richly illustrated and provides pseudocode for all key techniques. Fast Direct Solvers for Elliptic PDEs is appropriate for graduate students in applied mathematics and scientific computing, engineers and scientists looking for an accessible introduction to integral equation methods and fast solvers, and researchers in computational mathematics who want to quickly catch up on recent advances in randomized algorithms and techniques for working with data-sparse matrices.

Author : Craig C. Douglas,Gundolf Haase,Ulrich Langer
Publisher : SIAM
Page : 153 pages
File Size : 42,6 Mb
Release : 2003-01-01
Category : Technology & Engineering
ISBN : 0898718171

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A Tutorial on Elliptic PDE Solvers and Their Parallelization by Craig C. Douglas,Gundolf Haase,Ulrich Langer Pdf

This compact yet thorough tutorial is the perfect introduction to the basic concepts of solving partial differential equations (PDEs) using parallel numerical methods. In just eight short chapters, the authors provide readers with enough basic knowledge of PDEs, discretization methods, solution techniques, parallel computers, parallel programming, and the run-time behavior of parallel algorithms to allow them to understand, develop, and implement parallel PDE solvers. Examples throughout the book are intentionally kept simple so that the parallelization strategies are not dominated by technical details.

Author : Marián Vajtersic
Publisher : Springer Science & Business Media
Page : 310 pages
File Size : 50,6 Mb
Release : 2013-03-09
Category : Computers
ISBN : 9789401707015

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Algorithms for Elliptic Problems by Marián Vajtersic Pdf

This volume deals with problems of modern effective algorithms for the numerical solution of the most frequently occurring elliptic partial differential equations. From the point of view of implementation, attention is paid to algorithms for both classical sequential and parallel computer systems. The first two chapters are devoted to fast algorithms for solving the Poisson and biharmonic equation. In the third chapter, parallel algorithms for model parallel computer systems of the SIMD and MIMD types are described. The implementation aspects of parallel algorithms for solving model elliptic boundary value problems are outlined for systems with matrix, pipeline and multiprocessor parallel computer architectures. A modern and popular multigrid computational principle which offers a good opportunity for a parallel realization is described in the next chapter. More parallel variants based in this idea are presented, whereby methods and assignments strategies for hypercube systems are treated in more detail. The last chapter presents VLSI designs for solving special tridiagonal linear systems of equations arising from finite-difference approximations of elliptic problems. For researchers interested in the development and application of fast algorithms for solving elliptic partial differential equations using advanced computer systems.

Author : Martin H. Schultz
Publisher : Unknown
Page : 472 pages
File Size : 46,7 Mb
Release : 1981
Category : Differential equations, Elliptic
ISBN : UCAL:B4405983

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Elliptic Problem Solvers by Martin H. Schultz Pdf

Author : Darren Crowdy
Publisher : SIAM
Page : 456 pages
File Size : 46,7 Mb
Release : 2020-04-20
Category : Mathematics
ISBN : 9781611976151

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Solving Problems in Multiply Connected Domains by Darren Crowdy Pdf

Whenever two or more objects or entities—be they bubbles, vortices, black holes, magnets, colloidal particles, microorganisms, swimming bacteria, Brownian random walkers, airfoils, turbine blades, electrified drops, magnetized particles, dislocations, cracks, or heterogeneities in an elastic solid—interact in some ambient medium, they make holes in that medium. Such holey regions with interacting entities are called multiply connected. This book describes a novel mathematical framework for solving problems in two-dimensional, multiply connected regions. The framework is built on a central theoretical concept: the prime function, whose significance for the applied sciences, especially for solving problems in multiply connected domains, has been missed until recent work by the author. This monograph is a one-of-a-kind treatise on the prime function associated with multiply connected domains and how to use it in applications. The book contains many results familiar in the simply connected, or single-entity, case that are generalized naturally to any number of entities, in many instances for the first time. Solving Problems in Multiply Connected Domains is aimed at applied and pure mathematicians, engineers, physicists, and other natural scientists; the framework it describes finds application in a diverse array of contexts. The book provides a rich source of project material for undergraduate and graduate courses in the applied sciences and could serve as a complement to standard texts on advanced calculus, potential theory, partial differential equations and complex analysis, and as a supplement to texts on applied mathematical methods in engineering and science.

Author : Fioralba Cakoni,David Colton,Houssem Haddar
Publisher : SIAM
Page : 259 pages
File Size : 55,5 Mb
Release : 2022-12-07
Category : Mathematics
ISBN : 9781611977424

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Inverse Scattering Theory and Transmission Eigenvalues by Fioralba Cakoni,David Colton,Houssem Haddar Pdf

Inverse scattering theory is a major theme in applied mathematics, with applications to such diverse areas as medical imaging, geophysical exploration, and nondestructive testing. The inverse scattering problem is both nonlinear and ill-posed, thus presenting challenges in the development of efficient inversion algorithms. A further complication is that anisotropic materials cannot be uniquely determined from given scattering data. In the first edition of Inverse Scattering Theory and Transmission Eigenvalues, the authors discussed methods for determining the support of inhomogeneous media from measured far field data and the role of transmission eigenvalue problems in the mathematical development of these methods. In this second edition, three new chapters describe recent developments in inverse scattering theory. In particular, the authors explore the use of modified background media in the nondestructive testing of materials and methods for determining the modified transmission eigenvalues that arise in such applications from measured far field data. They also examine nonscattering wave numbers—a subset of transmission eigenvalues—using techniques taken from the theory of free boundary value problems for elliptic partial differential equations and discuss the dualism of scattering poles and transmission eigenvalues that has led to new methods for the numerical computation of scattering poles. This book will be of interest to research mathematicians and engineers and physicists working on problems in target identification. It will also be useful to advanced graduate students in many areas of applied mathematics.

Author : Jens M. Melenk,Ilaria Perugia,Joachim Schöberl,Christoph Schwab
Publisher : Springer Nature
Page : 571 pages
File Size : 41,6 Mb
Release : 2023-06-30
Category : Mathematics
ISBN : 9783031204326

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Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2020+1 by Jens M. Melenk,Ilaria Perugia,Joachim Schöberl,Christoph Schwab Pdf

The volume features high-quality papers based on the presentations at the ICOSAHOM 2020+1 on spectral and high order methods. The carefully reviewed articles cover state of the art topics in high order discretizations of partial differential equations. The volume presents a wide range of topics including the design and analysis of high order methods, the development of fast solvers on modern computer architecture, and the application of these methods in fluid and structural mechanics computations.

Author : Anonim
Publisher : Unknown
Page : 776 pages
File Size : 40,9 Mb
Release : 2006
Category : Mathematical statistics
ISBN : UOM:39015060948372

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SIAM Journal on Scientific Computing by Anonim Pdf

Author : Garrett Birkhoff,Arthur Schoenstadt
Publisher : Academic Press
Page : 588 pages
File Size : 46,9 Mb
Release : 2014-05-10
Category : Mathematics
ISBN : 9781483263397

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Elliptic Problem Solvers by Garrett Birkhoff,Arthur Schoenstadt Pdf

Elliptic Problem Solvers, II covers the proceedings of the Elliptic Problem Solvers Conference, held at the Naval Postgraduate School in Monterey, California from January 10 to 12, 1983. The book focuses on various aspects of the numerical solution of elliptic boundary value problems. The selection first offers information on building elliptic problem solvers with ELLPACK; presentation and evolution of the club module; and a fourth order accurate fast direct method for the Helmholtz equation. The text then examines the ITPACK project, CMMPAK, solving elliptic problems on an array processor system, and parallel architectures for iterative methods on adaptive, block structured grids. Topics include adaptive solution algorithm, data structure, elliptic problem solvers, input data, and vector ITPACK. The publication ponders on conjugate gradient preconditioners for vector and parallel processors; an algebra for systolic computation; and an incomplete-Cholesky factorization by a matrix partition algorithm. The book also tackles the numerical solution of a model equation near the onset of the Rayleigh-Benard instability; numerical methods for solving coupled semiconductor equations on a minicomputer; and analysis of nonlinear elliptic systems arising in reaction/diffusion modeling. The selection is highly recommended for researchers interested in elliptic problem solvers.

Author : Anonim
Publisher : Unknown
Page : 28 pages
File Size : 55,7 Mb
Release : 1986
Category : Science
ISBN : STANFORD:36105024802576

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NASA Technical Paper by Anonim Pdf

Author : United States. National Aeronautics and Space Administration
Publisher : Unknown
Page : 610 pages
File Size : 55,7 Mb
Release : 1975
Category : Aeronautics
ISBN : UIUC:30112008548510

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NASA Technical Note by United States. National Aeronautics and Space Administration Pdf

Author : Garrett Birkhoff,Arthur L. Schoenstadt
Publisher : Unknown
Page : 600 pages
File Size : 52,5 Mb
Release : 1984
Category : Education
ISBN : UOM:39015015704581

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Elliptic Problem Solvers II by Garrett Birkhoff,Arthur L. Schoenstadt Pdf

Author : Ed Bueler
Publisher : SIAM
Page : 407 pages
File Size : 45,7 Mb
Release : 2020-10-22
Category : Mathematics
ISBN : 9781611976311

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PETSc for Partial Differential Equations: Numerical Solutions in C and Python by Ed Bueler Pdf

The Portable, Extensible Toolkit for Scientific Computation (PETSc) is an open-source library of advanced data structures and methods for solving linear and nonlinear equations and for managing discretizations. This book uses these modern numerical tools to demonstrate how to solve nonlinear partial differential equations (PDEs) in parallel. It starts from key mathematical concepts, such as Krylov space methods, preconditioning, multigrid, and Newton’s method. In PETSc these components are composed at run time into fast solvers. Discretizations are introduced from the beginning, with an emphasis on finite difference and finite element methodologies. The example C programs of the first 12 chapters, listed on the inside front cover, solve (mostly) elliptic and parabolic PDE problems. Discretization leads to large, sparse, and generally nonlinear systems of algebraic equations. For such problems, mathematical solver concepts are explained and illustrated through the examples, with sufficient context to speed further development. PETSc for Partial Differential Equations addresses both discretizations and fast solvers for PDEs, emphasizing practice more than theory. Well-structured examples lead to run-time choices that result in high solver performance and parallel scalability. The last two chapters build on the reader’s understanding of fast solver concepts when applying the Firedrake Python finite element solver library. This textbook, the first to cover PETSc programming for nonlinear PDEs, provides an on-ramp for graduate students and researchers to a major area of high-performance computing for science and engineering. It is suitable as a supplement for courses in scientific computing or numerical methods for differential equations.

Author : Sin-Chung Chang
Publisher : Unknown
Page : 28 pages
File Size : 45,9 Mb
Release : 1986
Category : Differential equations, Elliptic
ISBN : UIUC:30112106614321

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Solution of Elliptic Partial Differential Equations by Fast Poisson Solvers Using a Local Relaxation Factor by Sin-Chung Chang Pdf

Author : Patrick J. Roache
Publisher : CRC Press
Page : 212 pages
File Size : 44,8 Mb
Release : 1995-06-29
Category : Mathematics
ISBN : 0849373786

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Elliptic Marching Methods and Domain Decomposition by Patrick J. Roache Pdf

One of the first things a student of partial differential equations learns is that it is impossible to solve elliptic equations by spatial marching. This new book describes how to do exactly that, providing a powerful tool for solving problems in fluid dynamics, heat transfer, electrostatics, and other fields characterized by discretized partial differential equations. Elliptic Marching Methods and Domain Decomposition demonstrates how to handle numerical instabilities (i.e., limitations on the size of the problem) that appear when one tries to solve these discretized equations with marching methods. The book also shows how marching methods can be superior to multigrid and pre-conditioned conjugate gradient (PCG) methods, particularly when used in the context of multiprocessor parallel computers. Techniques for using domain decomposition together with marching methods are detailed, clearly illustrating the benefits of these techniques for applications in engineering, applied mathematics, and the physical sciences.