Fast direct solvers for elliptic PDEs /

Fast direct solvers for elliptic PDEs /

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Bibliographic Details
Author / Creator:	Martinsson, Per-Gunnar, author.
Imprint:	Philadelphia, PA : Society for Industrial and Applied Mathematics, [2020] ©2020
Description:	xv, 315 pages : illustrations (some color), charts ; 25 cm.
Language:	English
Series:	CBMS-NSF regional conference series in applied mathematics ; 96 CBMS-NSF regional conference series in applied mathematics ; 96.
Subject:	Differential equations, Elliptic -- Numerical solutions. Differential equations, Partial -- Numerical solutions. Differential equations, Elliptic -- Numerical solutions. Differential equations, Partial -- Numerical solutions.
Format:	Print Book
URL for this record:	http://pi.lib.uchicago.edu/1001/cat/bib/12355426

Hidden Bibliographic Details
Varying Form of Title:	Fast direct solvers for elliptic partial differential equations
ISBN:	9781611976038 1611976030 9781611976045
Notes:	Includes bibliographical references (pages 301-311) and index.
Summary:	"This is a book about how to efficiently solve elliptic partial differential equations numerically"-- 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 such 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 the dense matrices that arise in the discretization of integral operators, such as the fast multipole method and fast direct solvers. The book also describes modern linear algebraic techniques that accelerate computations, such as randomized algorithms, interpolative decompositions, and data-sparse and rank-structured hierarchical matrix representations.

Table of Contents:

Scope and aims
How global operators rise in solvers for elliptic PDEs
I. Linear algebra
Matrix factorizations and low-rank approximation
Randomized methods for low-rank approximation
Fast algorithms for rank-structured matrices
II. The fast multipole method
Fast summation and multipole expansions
The fast multipole method
Extensions and improvements to the basic FMM
The potential evaluation map
III. Integral equation methods
Integral equation formulations
Extensions of integral equation-based methods
Discretization of integral equations
IV. Fast direct solvers for integral equations
A simple direct solver for integral equations
A multilevel scheme
Additional topics on HBS matrices
Interpolative decompositions and skeletonization
Constructing a rank-structured representation of a matrix
Direct solvers based on discrete scattering matrices
V. Fast direct solvers for sparse matrices
An introduction to fast solvers for linear elliptic PDEs
Direct sparse solvers
Fast direct sparse solvers
Linear complexity "sweeping" schemes
A geometry-based view of nested dissection
Spectral collocation methods
The hierarchical Poincaré-Steklove (HPS) method
Extensions of the HPS method
Fast solvers for elliptic problems on lattices.