73 citations · 398 across the 44 of their papers we have counts for
40 papers · 1 filter
A Complete Resolution of Forsythe's Conjecture for Restarted Conjugate Gradients
Matthew J. Colbrook, George Stepaniants, Alex Townsend
Forsythe's conjecture, published in 1968, asserts that for each restart length , every exact-arithmetic restarted conjugate-gradient iteration on a real symmetric positive defin…
A Proof of the Forsythe Conjecture for the Two-Step Restarted Conjugate Gradient Method
Matthew J. Colbrook, George Stepaniants, Alex Townsend
Forsythe conjectured in 1968 that the normalized residual directions of restarted conjugate gradients approach a two-cycle. We prove the conjecture for restart length two: unless t…
A zero-one law for one-shot system identification
Nicolas Boullé, Diana Halikias, Samuel E. Otto +1
Can a model be identified from one experiment? We study analytic systems that are linearly parameterized by a combination of prescribed dictionary terms, such as partial differenti…
Avoiding discretization issues for nonlinear eigenvalue problems
Matthew J. Colbrook, Alex Townsend
The first step when solving an infinite-dimensional eigenvalue problem is often to discretize it. We show that one must be extremely careful when discretizing nonlinear eigenvalue…
Leveraging the Hankel norm approximation and block-AAA algorithms in reduced order modeling
Annan Yu, Alex Townsend
Large-scale linear, time-invariant (LTI) dynamical systems are widely used to characterize complicated physical phenomena. We propose a two-stage algorithm to reduce the order of a…
Are sketch-and-precondition least squares solvers numerically stable?
Maike Meier, Yuji Nakatsukasa, Alex Townsend +1
Sketch-and-precondition techniques are efficient and popular for solving large least squares (LS) problems of the form with and . This i…
Learning Green's functions associated with time-dependent partial differential equations
Nicolas Boullé, Seick Kim, Tianyi Shi +1
Neural operators are a popular technique in scientific machine learning to learn a mathematical model of the behavior of unknown physical systems from data. Neural operators are es…
On the stability of unevenly spaced samples for interpolation and quadrature
Annan Yu, Alex Townsend
Unevenly spaced samples from a periodic function are common in signal processing and can often be viewed as a perturbed equally spaced grid. In this paper, we analyze how the uneve…
Parallel algorithms for computing the tensor-train decomposition
Tianyi Shi, Maximilian Ruth, Alex Townsend
The tensor-train (TT) decomposition expresses a tensor in a data-sparse format used in molecular simulations, high-order correlation functions, and optimization. In this paper, we…
Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems
Matthew J. Colbrook, Alex Townsend
Koopman operators are infinite-dimensional operators that globally linearize nonlinear dynamical systems, making their spectral information valuable for understanding dynamics. How…
A generalization of the randomized singular value decomposition
Nicolas Boullé, Alex Townsend
The randomized singular value decomposition (SVD) is a popular and effective algorithm for computing a near-best rank approximation of a matrix using matrix-vector products…
Data-driven Algorithms for signal processing with trigonometric rational functions
Heather Wilber, Anil Damle, Alex Townsend
Rational approximation schemes for reconstructing periodic signals from samples with poorly separated spectral content are described. These methods are automatic and adaptive, requ…
An optimal complexity spectral method for Navier--Stokes simulations in the ball
Nicolas Boullé, Jonasz Słomka, Alex Townsend
We develop a spectral method for solving the incompressible generalized Navier--Stokes equations in the ball with no-flux and prescribed slip boundary conditions. The algorithm ach…
Learning elliptic partial differential equations with randomized linear algebra
Nicolas Boullé, Alex Townsend
Given input-output pairs of an elliptic partial differential equation (PDE) in three dimensions, we derive the first theoretically-rigorous scheme for learning the associated Green…
Over-parametrized neural networks as under-determined linear systems
Austin R. Benson, Anil Damle, Alex Townsend
We draw connections between simple neural networks and under-determined linear systems to comprehensively explore several interesting theoretical questions in the study of neural n…
The ultraspherical spectral element method
Daniel Fortunato, Nicholas Hale, Alex Townsend
We introduce a novel spectral element method based on the ultraspherical spectral method and the hierarchical Poincaré-Steklov scheme for solving second-order linear partial differ…
Computing spectral measures of self-adjoint operators
Matthew J. Colbrook, Andrew Horning, Alex Townsend
Using the resolvent operator, we develop an algorithm for computing smoothed approximations of spectral measures associated with self-adjoint operators. The algorithm can achieve a…
Anderson Acceleration Based on the Sobolev Norm for Contractive and Noncontractive Fixed-Point Operators
Yunan Yang, Alex Townsend, Daniel Appelö
Anderson acceleration (AA) is a technique for accelerating the convergence of fixed-point iterations. In this paper, we apply AA to a sequence of functions and modify the norm in i…
Computing with functions in the ball
Nicolas Boullé, Alex Townsend
A collection of algorithms in object-oriented MATLAB is described for numerically computing with smooth functions defined on the unit ball in the Chebfun software. Functions are nu…
A sparse spectral method on triangles
Sheehan Olver, Alex Townsend, Geoff Vasil
In this paper, we demonstrate that many of the computational tools for univariate orthogonal polynomials have analogues for a family of bivariate orthogonal polynomials on the tria…
Error localization of best L1 polynomial approximants
Yuji Nakatsukasa, Alex Townsend
An important observation in compressed sensing is that the minimizer of an underdetermined linear system is equal to the minimizer when there exists a sparse solu…
FEAST for differential eigenvalue problems
Andrew Horning, Alex Townsend
An operator analogue of the FEAST matrix eigensolver is developed to compute the discrete part of the spectrum of a differential operator in a region of interest in the complex pla…
On the compressibility of tensors
Tianyi Shi, Alex Townsend
Tensors are often compressed by expressing them in low rank tensor formats. In this paper, we develop three methodologies that bound the compressibility of a tensor: (1) Algebraic…
On the singular values of matrices with high displacement rank
Alex Townsend, Heather Wilber
We introduce a new ADI-based low rank solver for , where has rapidly decaying singular values. Our approach results in both theoretical and practical gains, including…