Related papers: Error bounds for the approximation of matrix funct…
Let $A$ be a square complex matrix; $z_1$, ..., $z_{N}\in\mathbb C$ be arbitrary (possibly repetitive) points of interpolation; $f$ be an analytic function defined on a neighborhood of the convex hull of the union of the spectrum…
We present a unified framework to efficiently approximate solutions to fractional diffusion problems of stationary and parabolic type. After discretization, we can take the point of view that the solution is obtained by a matrix-vector…
The generalized Lanczos trust-region (GLTR) method is one of the most popular approaches for solving large-scale trust-region subproblem (TRS). Recently, Jia and Wang [Z. Jia and F. Wang, \emph{SIAM J. Optim., 31 (2021), pp. 887--914}]…
Compared to the classical Lanczos algorithm, the $s$-step Lanczos variant has the potential to improve performance by asymptotically decreasing the synchronization cost per iteration. However, this comes at a cost. Despite being…
This note establishes a theoretical framework for finding (potentially overparameterized) approximations of a function on a compact set with a-priori bounds for the generalization error. The approximation method considered is to choose,…
The challenge to measure exposures regularly forces financial institutions into a choice between an overwhelming computational burden or oversimplification of risk. To resolve this unsettling dilemma, we systematically investigate replacing…
We consider the approximation of $B^T (A+sI)^{-1} B$ where $A\in\mathbb{R}^{n\times n}$ is large, symmetric positive definite, and has a dense spectrum, and $B\in\mathbb{R}^{n\times p}$, $p\ll n$. Our target application is the computation…
The quadratic numerical range $W^2(A)$ is a subset of the standard numerical range of a linear operator which still contains its spectrum. It arises naturally in operators which have a $2 \times 2$ block structure, and it consists of at…
In an unnormalized Krylov subspace framework for solving symmetric systems of linear equations, the orthogonal vectors that are generated by a Lanczos process are not necessarily on the form of gradients. Associating each orthogonal vector…
Over the last several decades, improvements in the fields of analytic combinatorics and computer algebra have made determining the asymptotic behaviour of sequences satisfying linear recurrence relations with polynomial coefficients largely…
We consider the problem of approximating the von Neumann entropy of a large, sparse, symmetric positive semidefinite matrix $A$, defined as $\operatorname{tr}(f(A))$ where $f(x)=-x\log x$. After establishing some useful properties of this…
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…
Given a large square matrix $A$ and a sufficiently regular function $f$ so that $f(A)$ is well defined, we are interested in the approximation of the leading singular values and corresponding singular vectors of $f(A)$, and in particular of…
The numerical computation of equilibrium reward gradients for Markov chains appears in many applications for example within the policy improvement step arising in connection with average reward stochastic dynamic programming. When the state…
We consider the task of computing solutions of linear systems that only differ by a shift with the identity matrix as well as linear systems with several different right hand sides. In the past Krylov subspace methods have been developed…
As computational machines become larger and more complex, the probability of hardware failure rises. ``Silent errors'', or bit flips, may not be immediately apparent but can cause detrimental effects to algorithm behavior. In this work, we…
Error bounds, which refer to inequalities that bound the distance of vectors in a test set to a given set by a residual function, have proven to be extremely useful in analyzing the convergence rates of a host of iterative methods for…
The conventional rounding error analysis provides worst-case bounds with an associated failure probability and ignores the statistical property of the rounding errors. In this paper, we develop a new statistical rounding error analysis for…
Mathematical methods provide useful framework for the analysis and design of complex systems. In newer contexts such as biology, however, there is a need to both adapt existing methods as well as to develop new ones. Using a combination of…
Many problems in Science and Engineering give rise to linear integral equations of the first kind with a smooth kernel. Discretization of the integral operator yields a matrix, whose singular values cluster at the origin. We describe the…