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The reduced basis method is a model reduction technique yielding substantial savings of computational time when a solution to a parametrized equation has to be computed for many values of the parameter. Certification of the approximation is…
A new O(N) algorithm based on a recursion method, in which the computational effort is proportional to the number of atoms N, is presented for calculating the inverse of an overlap matrix which is needed in electronic structure calculations…
As the demand for AI computation rapidly increases, more hardware is being developed to efficiently perform the low-precision matrix multiplications required by such workloads. However, these operations are generally not directly applicable…
This paper proposes and analyzes an a posteriori error estimator for the finite element multi-scale discretization approximation of the Steklov eigenvalue problem. Based on the a posteriori error estimates, an adaptive algorithm of shifted…
The multiplication of matrices is an important arithmetic operation in computational mathematics. In the context of hierarchical matrices, this operation can be realized by the multiplication of structured block-wise low-rank matrices,…
An efficient algorithm for computing eigenvectors of a matrix of integers by exact computation is proposed. The components of calculated eigenvectors are expressed as polynomials in the eigenvalue to which the eigenvector is associated, as…
In this paper, we propose variants of forward-backward splitting method for solving the system of splitting inclusion problem. We propose a conceptual algorithm containing three variants, each having a different projection steps. The…
The iterative algorithm recently proposed by Waxman for solving eigenvalue problems, which relies on the method of moments, has been modified to improve its convergence considerably without sacrificing its benefits or elegance. The…
We present a new numerical technique to solve large-scale eigenvalue problems. It is based on the projection technique, used in strongly correlated quantum many-body systems, where first an effective approximate model of smaller complexity…
In this paper, we propose a method for emulating double-precision general matrix--matrix multiplication (DGEMM), a fundamental and performance-critical kernel in many high-performance computing applications. Ozaki-I and Ozaki-II are…
This paper provides an accurate method to obtain the bidiagonal factorization of many generalized Pascal matrices, which in turn can be used to compute with high relative accuracy the eigenvalues, singular values and inverses of these…
In this paper we propose an approach to approximate a truncated singular value decomposition of a large structured matrix. By first decomposing the matrix into a sum of Kronecker products, our approach can be used to approximate a large…
Solving symmetric positive definite linear problems is a fundamental computational task in machine learning. The exact solution, famously, is cubicly expensive in the size of the matrix. To alleviate this problem, several linear-time…
The fundamental matrix (FM) describes the geometric relations that exist between two images of the same scene. Different error criteria are used for estimating FMs from an input set of correspondences. In this paper, the accuracy and…
In this paper we propose an efficiently preconditioned Newton method for the computation of the leftmost eigenpairs of large and sparse symmetric positive definite matrices. A sequence of preconditioners based on the BFGS update formula is…
The power method is a basic method for computing the dominant eigenpair of a matrix. In this paper, we propose a structure-preserving power-like method for computing the dominant conjugate pair of purely imaginary eigenvalues and the…
This paper introduces an efficient algorithm for finding the dominant generalized eigenvectors of a pair of symmetric matrices. Combining tools from approximation theory and convex optimization, we develop a simple scalable algorithm with…
In this paper a novel numerical approximation of parametric eigenvalue problems is presented. We motivate our study with the analysis of a POD reduced order model for a simple one dimensional example. In particular, we introduce a new…
In this paper we consider a family of algorithms for approximate implicitization of rational parametric curves and surfaces. The main approximation tool in all of the approaches is the singular value decomposition, and they are therefore…
The purpose of this article is to propose ODE based approaches for the numerical evaluation of matrix functions $f(A)$, a question of major interest in the numerical linear algebra. To this end, we model $f(A)$ as the solution at a finite…