Related papers: MRRR-based Eigensolvers for Multi-core Processors …
This article introduces an algorithm for implicit High Dimensional Model Representation (HDMR) of the Bellman equation. This approximation technique reduces memory demands of the algorithm considerably. Moreover, we show that HDMR enables…
We present Accelerated Cyclic Reduction (ACR), a distributed-memory fast direct solver for rank-compressible block tridiagonal linear systems arising from the discretization of elliptic operators, developed here for three dimensions.…
Most iterative algorithms for eigenpair computation consist of two main steps: a subspace update (SU) step that generates bases for approximate eigenspaces, followed by a Rayleigh-Ritz (RR) projection step that extracts approximate…
Let $n\ge 2$ be an integer. Let $R_n$ denote the $n\times n$ tridiagonal matrix with $-1$'s on the sub-diagonal, $1$'s on the super-diagonal, $-1$ in the $(1,1)$ entry, $1$ in the $(n,n)$ entry and zeros elsewhere. We find the eigen-pairs…
Ridge regression (RR) is an important machine learning technique which introduces a regularization hyperparameter $\alpha$ to ordinary multiple linear regression for analyzing data suffering from multicollinearity. In this paper, we present…
The principal minors of a tridiagonal matrix satisfy two-term and three-term recurrences [1, 2]. Based on these facts, the current article presents a new efficient and reliable hybrid numerical algorithm for evaluating general n-th order…
General purpose optimization techniques can be used to solve many problems in engineering computations, although their cost is often prohibitive when the number of degrees of freedom is very large. We describe a multilevel approach to speed…
This work studies the problem of maximizing a higher degree real homogeneous multivariate polynomial over the unit sphere. This problem is equivalent to finding the leading eigenvalue of the associated symmetric tensor of higher order,…
We are concerned with the fastest possible direct numerical solution algorithm for a thin-banded or tridiagonal linear system of dimension $N$ on a distributed computing network of $N$ nodes that is connected in a binary communication tree.…
An algebraic soliton of the massive Thirring model (MTM) is expressed by the simplest rational solution of the MTM with the spatial decay of $\mathcal{O}(x^{-1})$. The corresponding potential is related to a simple embedded eigenvalue in…
In this short note, we present a novel method for computing exact lower and upper bounds of eigenvalues of a symmetric tridiagonal interval matrix. Compared to the known methods, our approach is fast, simple to present and to implement, and…
Numerical solution of nonlinear eigenvalue problems (NEPs) is frequently encountered in computational science and engineering. The applicability of most existing methods is limited by matrix structures, property of eigen-solutions, size of…
Many of the classic graph problems cannot be solved in the Massively Parallel Computation setting (MPC) with strongly sublinear space per machine and $o(\log n)$ rounds, unless the 1-vs-2 cycles conjecture is false. This is true even on…
Given a family of nearly commuting symmetric matrices, we consider the task of computing an orthogonal matrix that nearly diagonalizes every matrix in the family. In this paper, we propose and analyze randomized joint diagonalization (RJD)…
The Density Matrix Renormalization Group (DMRG) algorithm is a powerful tool for solving eigenvalue problems to model quantum systems. DMRG relies on tensor contractions and dense linear algebra to compute properties of condensed matter…
Inspired by the quantum computing algorithms for Linear Algebra problems [HHL,TaShma] we study how the simulation on a classical computer of this type of "Phase Estimation algorithms" performs when we apply it to solve the Eigen-Problem of…
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…
High resolution magnetic resonance~(MR) imaging~(MRI) is desirable in many clinical applications, however, there is a trade-off between resolution, speed of acquisition, and noise. It is common for MR images to have worse through-plane…
In recent years, contour-based eigensolvers have emerged as a standard approach for the solution of large and sparse eigenvalue problems. Building upon recent performance improvements through non-linear least square optimization of…
A new method of solution to the local spin density approximation to the electronic Schr\"{o}dinger equation is presented. The method is based on an efficient, parallel, adaptive multigrid eigenvalue solver. It is shown that adaptivity is…