相关论文: Theory and implementation of $\mathcal{H}$-matrix …
This manuscript presents an efficient solver for the linear system that arises from the Hierarchical Poincar\'e-Steklov (HPS) discretization of three dimensional variable coefficient Helmholtz problems. Previous work on the HPS method has…
Standard discretization techniques for boundary integral equations, e.g., the Galerkin boundary element method, lead to large densely populated matrices that require fast and efficient compression techniques like the fast multipole method…
We consider the efficient solution of strongly elliptic partial differential equations with random load based on the finite element method. The solution's two-point correlation can efficiently be approximated by means of an…
Boundary element methods for the Helmholtz equation lead to large dense matrices that can only be handled if efficient compression techniques are used. Directional compression techniques can reach good compression rates even for…
Boundary integral equations lead to dense system matrices when discretized, yet they are data-sparse. Using the $\mathcal{H}$-matrix format, this sparsity is exploited to achieve $\mathcal{O}(N\log N)$ complexity for storage and…
Hierarchical Matrix (H-matrix) is an approximation technique which splits a target dense matrix into multiple submatrices, and where a selected portion of submatrices are low-rank approximated. The technique substantially reduces both time…
Standard numerical algorithms like the fast multipole method or $\mathcal{H}$-matrix schemes rely on low-rank approximations of the underlying kernel function. For high-frequency problems, the ranks grow rapidly as the mesh is refined, and…
We consider a fast, data-sparse directional method to realize matrix-vector products related to point evaluations of the Helmholtz kernel. The method is based on a hierarchical partitioning of the point sets and the matrix. The considered…
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…
The discretisation of boundary integral equations for the scalar Helmholtz equation leads to large dense linear systems. Efficient boundary element methods (BEM), such as the fast multipole method (FMM) and $\Hmat$ based methods, focus on…
We present a fast and memory-efficient algorithm for transient, space-time-domain, and elastodynamic boundary-integral analysis. Associated data-sparse approximations and operations are named fast domain partitioning hierarchical matrices…
In this work, we develop an optimization framework for problems whose solutions are well-approximated by Hierarchical Tucker (HT) tensors, an efficient structured tensor format based on recursive subspace factorizations. By exploiting the…
We propose an iterative solution method for the 3D high-frequency Helmholtz equation that exploits a contour integral formulation of spectral projectors. In this framework, the solution in certain invariant subspaces is approximated by…
We present a fast algorithm for linear least squares problems governed by hierarchically block separable (HBS) matrices. Such matrices are generally dense but data-sparse and can describe many important operators including those derived…
In this paper, we study numerical homogenization methods based on integral equations. Our work is motivated by materials such as concrete, modeled as composites structured as randomly distributed inclusions imbedded in a matrix. We…
The purpose of this research is to describe an efficient iterative method suitable for obtaining high accuracy solutions to high frequency time-harmonic scattering problems. The method allows for both refinement of local polynomial degree…
Several researchers have developed a rich toolbox of matrix compression techniques that exploit structure and redundancy in large matrices. Classical methods such as the block low-rank format and the Fast Multipole Method make it possible…
We consider the discretization of time-space diffusion equations with fractional derivatives in space and either 1D or 2D spatial domains. The use of implicit Euler scheme in time and finite differences or finite elements in space, leads to…
In this paper, we propose and analyze the numerical algorithms for fast solution of periodic elliptic problems in random media in $\mathbb{R}^d$, $d=2,3$. We consider the stochastic realizations using checkerboard configuration of the…
This paper introduces a dynamic, error-bounded hierarchical matrix (H-matrix) compression method tailored for Physics-Informed Neural Networks (PINNs). The proposed approach reduces the computational complexity and memory demands of…