A fast direct solver for boundary integral equations in two dimensions
Journal of Computational Physics
A Fast $ULV$ Decomposition Solver for Hierarchically Semiseparable Representations
SIAM Journal on Matrix Analysis and Applications
Superfast Multifrontal Method for Large Structured Linear Systems of Equations
SIAM Journal on Matrix Analysis and Applications
The effective conductivity of random checkerboards
Journal of Computational Physics
Fast construction of hierarchical matrix representation from matrix-vector multiplication
Journal of Computational Physics
A composite preconditioner for the electromagnetic scattering from a large cavity
Journal of Computational Physics
Robust Approximate Cholesky Factorization of Rank-Structured Symmetric Positive Definite Matrices
SIAM Journal on Matrix Analysis and Applications
A fast direct solver for elliptic problems on general meshes in 2D
Journal of Computational Physics
A Fast Randomized Algorithm for Computing a Hierarchically Semiseparable Representation of a Matrix
SIAM Journal on Matrix Analysis and Applications
Journal of Scientific Computing
A fast nested dissection solver for Cartesian 3D elliptic problems using hierarchical matrices
Journal of Computational Physics
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We describe a fast and robust method for solving the large sparse linear systems that arise upon the discretization of elliptic partial differential equations such as Laplace's equation and the Helmholtz equation at low frequencies. While most existing fast schemes for this task rely on so called "iterative" solvers, the method described here solves the linear system directly (to within an arbitrary predefined accuracy). The method is described for the particular case of an operator defined on a square uniform grid, but can be generalized other geometries. For a grid containing N points, a single solve requires O(Nlog驴2 N) arithmetic operations and $O(\sqrt{N}\log N)$ storage. Storing the information required to perform additional solves rapidly requires O(Nlog驴N) storage. The scheme is particularly efficient in situations involving domains that are loaded on the boundary only and where the solution is sought only on the boundary. In this environment, subsequent solves (after the first) can be performed in $O(\sqrt{N}\log N)$ operations. The efficiency of the scheme is illustrated with numerical examples. For instance, a system of size 106脳106 is directly solved to seven digits accuracy in four minutes on a 2.8 GHz P4 desktop PC.