Related papers: Hierarchical interpolative factorization for ellip…
This paper introduces the hierarchical interpolative factorization for integral equations (HIF-IE) associated with elliptic problems in two and three dimensions. This factorization takes the form of an approximate generalized LU…
The hierarchical interpolative factorization (HIF) offers an efficient way for solving or preconditioning elliptic partial differential equations. By exploiting locality and low-rank properties of the operators, the HIF achieves…
The hierarchical interpolative factorization for elliptic partial differential equations is a fast algorithm for approximate sparse matrix inversion in linear or quasilinear time. Its accuracy can degrade, however, when applied to strongly…
We present a method for updating certain hierarchical factorizations for solving linear integral equations with elliptic kernels. In particular, given a factorization corresponding to some initial geometry or material parameters, we can…
Boundary value problems involving elliptic PDEs such as the Laplace and the Helmholtz equations are ubiquitous in mathematical physics and engineering. Many such problems can be alternatively formulated as integral equations that are…
The hierarchical matrix framework partitions matrices into subblocks that are either small or of low numerical rank, enabling linear storage complexity and efficient matrix-vector multiplication. This work focuses on the $H^2$-matrix format…
This paper presents a general method for applying hierarchical matrix skeletonization factorizations to the numerical solution of boundary integral equations with possibly rank-deficient integral operators. Rank-deficient operators arise in…
This note proposes an efficient preconditioner for solving linear and semi-linear parabolic equations. With the Crank-Nicholson time stepping method, the algebraic system of equations at each time step is solved with the conjugate gradient…
We present a fast direct solver for structured linear systems based on multilevel matrix compression. Using the recently developed interpolative decomposition of a low-rank matrix in a recursive manner, we embed an approximation of the…
The high cost associated with the evaluation of Hartree-Fock exchange (HFX) makes hybrid functionals computationally challenging for large systems. In this work, we present an efficient way to accelerate HFX calculations with numerical…
This work investigates diagonalization-based methods for efficiently solving linear evolution problems, with a particular focus on the heat equation. The plain diagonalization of the differential operator, though effective for elliptic…
Factorization of large dense matrices are ubiquitous in engineering and data science applications, e.g. preconditioners for iterative boundary integral solvers, frontal matrices in sparse multifrontal solvers, and computing the determinant…
High-order methods gain more and more attention in computational fluid dynamics. However, the potential advantage of these methods depends critically on the availability of efficient elliptic solvers. With spectral-element methods, static…
We present a fast direct solver for boundary integral equations on complex surfaces in three dimensions using an extension of the recently introduced recursive strong skeletonization scheme. For problems that are not highly oscillatory, our…
A new fast multipole formulation for solving elliptic difference equations on unbounded domains and its parallel implementation are presented. These difference equations can arise directly in the description of physical systems, e.g.…
In this paper, we propose a new class of operator factorization methods to discretize the integral fractional Laplacian $(-\Delta)^\frac{\alpha}{2}$ for $\alpha \in (0, 2)$. The main advantage of our method is to easily increase numerical…
Triangular factorizations are an important tool for solving integral equations and partial differential equations with hierarchical matrices ($\mathcal{H}$-matrices). Experiments show that using an $\mathcal{H}$-matrix LR factorization to…
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…
We introduce the strong recursive skeletonization factorization (RS-S), a new approximate matrix factorization based on recursive skeletonization for solving discretizations of linear integral equations associated with elliptic partial…
Integral equations are commonly encountered when solving complex physical problems. Their discretization leads to a dense kernel matrix that is block or hierarchically low-rank. This paper proposes a new way to build a low-rank…