Related papers: Randomized sketching of nonlinear eigenvalue probl…
The adaptive Antoulas-Anderson (AAA) algorithm for rational approximation is a widely used method for the efficient construction of highly accurate rational approximations to given data. While AAA can often produce rational approximations…
We present a method for solving nonlinear eigenvalue problems using rational approximation. The method uses the AAA method by Nakatsukasa, S\`{e}te, and Trefethen to approximate the nonlinear eigenvalue problem by a rational eigenvalue…
We introduce a new algorithm for approximation by rational functions on a real or complex set of points, implementable in 40 lines of Matlab and requiring no user input parameters. Even on a disk or interval the algorithm may outperform…
A selection of algorithms for the rational approximation of matrix-valued functions are discussed, including variants of the interpolatory AAA method, the RKFIT method based on approximate least squares fitting, vector fitting, and a method…
Many algorithms for approximating data with rational functions are built on interpolation or least-squares approximation. Inspired by the adaptive Antoulas-Anderson (AAA) algorithm for the univariate case, the parametric adaptive…
We describe a strategy for solving nonlinear eigenproblems numerically. Our approach is based on the approximation of a vector-valued function, defined as solution of a non-homogeneous version of the eigenproblem. This approximation step is…
This paper presents a method for computing eigenvalues and eigenvectors for some types of nonlinear eigenvalue problems. The main idea is to approximate the functions involved in the eigenvalue problem by rational functions and then apply a…
Approximations based on rational functions are widely used in various applications across computational science and engineering. For univariate functions, the adaptive Antoulas-Anderson algorithm (AAA), which uses the barycentric form of a…
Rational approximation appears in many contexts throughout science and engineering, playing a central role in linear systems theory, special function approximation, and many others. There are many existing methods for solving the rational…
We propose AAA rational approximation as a method for interpolating or approximating smooth functions from equispaced data samples. Although it is always better to approximate from large numbers of samples if they are available, whether…
We propose new approximate alternating projection methods, based on randomized sketching, for the low-rank nonnegative matrix approximation problem: find a low-rank approximation of a nonnegative matrix that is nonnegative, but whose…
Given a symmetric matrix $A$, we show from the simple sketch $GAG^T$, where $G$ is a Gaussian matrix with $k = O(1/\epsilon^2)$ rows, that there is a procedure for approximating all eigenvalues of $A$ simultaneously to within $\epsilon…
High-dimensional representations, such as radial basis function networks or tile coding, are common choices for policy evaluation in reinforcement learning. Learning with such high-dimensional representations, however, can be expensive,…
In this work, we propose an extensive numerical study on approximating the absolute value function. The methods presented in this paper compute approximants in the form of rational functions and have been proposed relatively recently, e.g.,…
We introduce a theoretical framework for the rational approximation of optical response functions in resonant photonic systems. The framework is based on the AAA algorithm and further allows to solve the underlying nonlinear eigenproblems…
We consider the Adaptive Antoulas-Anderson (AAA) rational interpolation algorithm recently developed by Trefethen and co-authors, which can be viewed as a type of moment-matching technique for system realization and approximation. We…
This paper proposes a new framework for computing low-rank solutions to nonlinear matrix equations arising from spatial discretization of nonlinear partial differential equations: low-rank Anderson acceleration (lrAA). lrAA is an adaptation…
The power method is one of the most fundamental tools for extracting top principal components from data through low-rank matrix approximation. Yet, when the target rank is large, the cost of matrix multiplication associated with this…
Recently, Bessa et al. (PODS 2023) showed that sketches based on coordinated weighted sampling theoretically and empirically outperform popular linear sketching methods like Johnson-Lindentrauss projection and CountSketch for the ubiquitous…
This survey highlights the recent advances in algorithms for numerical linear algebra that have come from the technique of linear sketching, whereby given a matrix, one first compresses it to a much smaller matrix by multiplying it by a…