On the use of conjugate gradient-type methods for boundary integral equations

1993 ◽  
Vol 12 (4) ◽  
pp. 214-232 ◽  
Author(s):  
J. E. Romate
Keyword(s):  
Integral Equations ◽  
Conjugate Gradient ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Gradient Type ◽  
Conjugate Gradient Type Methods

Fast direct solvers for integral equations in complex three-dimensional domains

Acta Numerica ◽  
2009 ◽  
Vol 18 ◽  
pp. 243-275 ◽  
Author(s):  
Leslie Greengard ◽  
Denis Gueyffier ◽  
Per-Gunnar Martinsson ◽  
Vladimir Rokhlin
Keyword(s):  
Integral Equations ◽  
Degrees Of Freedom ◽  
Boundary Integral Equations ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Three Dimensions ◽  
Computer Hardware ◽  
Matrix Vector Multiplication ◽  
Gradient Type ◽  
Mathematical Foundations

Methods for the solution of boundary integral equations have changed significantly during the last two decades. This is due, in part, to improvements in computer hardware, but more importantly, to the development of fast algorithms which scale linearly or nearly linearly with the number of degrees of freedom required. These methods are typically iterative, based on coupling fast matrix-vector multiplication routines with conjugate-gradient-type schemes. Here, we discuss methods that are currently under development for the fast, direct solution of boundary integral equations in three dimensions. After reviewing the mathematical foundations of such schemes, we illustrate their performance with some numerical examples, and discuss the potential impact of the overall approach in a variety of settings.

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