Related papers: A Direct Elliptic Solver Based on Hierarchically L…
In this work, we propose a parallel-in-time solver for linear and nonlinear ordinary differential equations. The approach is based on an efficient multilevel solver of the Schur complement related to a multilevel time partition. For linear…
The objective of this study is to present a novel, efficient, and fast direct method for solving linear systems of equations whose coefficient matrix is a tridiagonal Quasi-Toeplitz matrix. Such matrices are frequently encountered in the…
This paper presents an efficient parallel direct algorithm with near-optimal complexity for the compact fourth and sixth-order approximation of the three-dimensional Helmholtz equations [1] with the problem coefficient depending on only one…
Hierarchical matrices provide a highly memory-efficient way of storing dense linear operators arising, for example, from boundary element methods, particularly when stored in the H^2 format. In such data-sparse representations, iterative…
We present a parallel hierarchical solver for general sparse linear systems on distributed-memory machines. For large-scale problems, this fully algebraic algorithm is faster and more memory-efficient than sparse direct solvers because it…
The objective of this work is to present a novel approach for the solution of Pentadiagonal Toeplitz systems of equations that is both faster and more effective than existing classical direct methods. The distinctive structure of…
A direct solver is introduced for solving overdetermined linear systems involving nonuniform discrete Fourier transform matrices. Such matrices can be transformed into a Cauchy-like form that has hierarchical low rank structure. The rank…
A numerical method for solving elliptic PDEs with variable coefficients on two-dimensional domains is presented. The method is based on high-order composite spectral approximations and is designed for problems with smooth solutions. The…
A numerical method for variable coefficient elliptic problems on two dimensional domains is described. The method is based on high-order spectral approximations and is designed for problems with smooth solutions. The resulting system of…
When solving partial differential equations (PDEs) using finite difference or finite element methods, efficient solvers are required for handling large sparse linear systems. In this paper, a recursive sparse LU decomposition for matrices…
The paper describes a sparse direct solver for the linear systems that arise from the discretization of an elliptic PDE on a two dimensional domain. The scheme decomposes the domain into thin subdomains, or ``slabs'' and uses a two-level…
Efficient solutions of large-scale, ill-conditioned and indefinite algebraic equations are ubiquitously needed in numerous computational fields, including multiphysics simulations, machine learning, and data science. Because of their…
We describe a fast, direct solver for elliptic partial differential equations on a two-dimensional hierarchy of adaptively refined, Cartesian meshes. Our solver, inspired by the Hierarchical Poincar\'e-Steklov (HPS) method introduced by…
We consider the numerical solution of large scale time-harmonic Maxwell equations. To this day, this problem remains difficult, in particular because the equations are neither Hermitian nor semi-definite. Our approach is to compare…
Structured dense matrices result from boundary integral problems in electrostatics and geostatistics, and also Schur complements in sparse preconditioners such as multi-frontal methods. Exploiting the structure of such matrices can reduce…
This paper presents a GPU-accelerated framework for solving block tridiagonal linear systems that arise naturally in numerous real-time applications across engineering and scientific computing. Through a multi-stage permutation strategy…
In this paper we accomplish the development of the fast rank-adaptive solver for tensor-structured symmetric positive definite linear systems in higher dimensions. In [arXiv:1301.6068] this problem is approached by alternating minimization…
Symmetric linear solves are fundamental to a wide range of scientific and engineering applications, from climate modeling and structural analysis to machine learning and optimization. These workloads often rely on Cholesky (POTRF)…
Cyclic reduction is a method for the solution of (block-)tridiagonal linear systems. In this note we review the method tailored to hermitian positive definite banded linear systems. The reviewed method has the following advantages: It is…
In this article, we introduce a fast and memory efficient solver for sparse matrices arising from the finite element discretization of elliptic partial differential equations (PDEs). We use a fast direct (but approximate) multifrontal…