Related papers: Uniform Error Estimates for the Lanczos Method
We prove sharp, computable error estimates for the propagation of errors in the numerical solution of ordinary differential equations. The new estimates extend previous estimates of the influence of data errors and discretisation errors…
Global and block Krylov subspace methods are efficient iterative solvers for large sparse linear systems with multiple right-hand sides. However, global or block Lanczos-type solvers often exhibit large oscillations in the residual norms…
We study the problem of approximating the eigenspectrum of a symmetric matrix $\mathbf A \in \mathbb{R}^{n \times n}$ with bounded entries (i.e., $\|\mathbf A\|_{\infty} \leq 1$). We present a simple sublinear time algorithm that…
This paper investigates a class of non-autonomous highly oscillatory ordinary differential equations characterized by a linear component inversely proportional to a small parameter $\varepsilon$, with purely imaginary eigenvalues, and an…
In this paper, we develop an optimization-based framework for solving coupled forward-backward stochastic differential equations. We introduce an integral-form objective function and prove its equivalence to the error between consecutive…
Variational procedure is developed that yields lowest frequencies of small-amplitude oscillations of classical Hamiltonian systems. Genuine Lanczos recursion is generalized to treat related non-Hermitian eigenvalue problems.
The Levenberg-Marquardt (LM) method is commonly used for inverting models used to describe geothermal, groundwater, or oil and gas reservoirs. In previous studies LM parameter updates have been made tractable for highly parameterized…
We develop a uniform inference theory for high-dimensional slope parameters in threshold regression models, allowing for either cross-sectional or time series data. We first establish oracle inequalities for prediction errors, and L1…
This paper establishes a comparison theorem for the maximum eigenvalue of a sum of independent random symmetric matrices. The theorem states that the maximum eigenvalue of the matrix sum is dominated by the maximum eigenvalue of a Gaussian…
Lanczos-type algorithms are well known for their inherent instability. They typically breakdown when relevant orthogonal polynomials do not exist. Current approaches to avoiding breakdown rely on jumping over the non-existent polynomials to…
In this paper we present a convergence rate analysis of inexact variants of several randomized iterative methods. Among the methods studied are: stochastic gradient descent, stochastic Newton, stochastic proximal point and stochastic…
We present a comparative study of the application of modern eigenvalue algorithms to an eigenvalue problem arising in quantum physics, namely, the computation of a few interior eigenvalues and their associated eigenvectors for the large,…
We provide tools to help automate the error analysis of algorithms that evaluate simple functions over the floating-point numbers. The aim is to obtain tight relative error bounds for these algorithms, expressed as a function of the unit…
We consider computing the $k$-th eigenvalue and its corresponding eigenvector of a generalized Hermitian eigenvalue problem of $n\times n$ large sparse matrices. In electronic structure calculations, several properties of materials, such as…
In this note, we concentrate on the backward error of the equality constrained indefinite least squares problem. For the normwise backward error of the equality constrained indefinite least square problem, we adopt the linearization method…
Let $u$ be an eigenfunction of the Laplacian on a compact manifold with boundary, with Dirichlet or Neumann boundary conditions, and let $-\lambda^2$ be the corresponding eigenvalue. We consider the problem of estimating the maximum of $u$…
In this paper, we give estimates for both upper and lower bounds of eigenvalues of a simple matrix. The estimates are shaper than the known results.
Every sufficiently big matrix with small spectral norm has a nearby low-rank matrix if the distance is measured in the maximum norm (Udell & Townsend, SIAM J Math Data Sci, 2019). We use the Hanson--Wright inequality to improve the estimate…
Coherent lower previsions are general probabilistic models allowing incompletely specified probability distributions. However, for complete description of a coherent lower prevision -- even on finite underlying sample spaces -- an infinite…
Umbrella sampling efficiently yields equilibrium averages that depend on exploring rare states of a model by biasing simulations to windows of coordinate values and then combining the resulting data with physical weighting. Here, we…