arXiv subjects

Ilse Ipsen

Publications and source records attributed to Ilse Ipsen.

4 recordsLinked to original sources

arXiv2018

A Bayesian Conjugate Gradient Method

A fundamental task in numerical computation is the solution of large linear systems. The conjugate gradient method is an iterative method which offers rapid convergence to the solution, particularly when an effective preconditioner is employed. However, for more challenging systems a substantial error can be present even after many iterations have been performed. The estimates obtained in this case are of little value unless further information can be provided about the numerical error. In this paper we propose a novel statistical model for this numerical error set in a Bayesian framework. Our approach is a strict generalisation of the conjugate gradient method, which is recovered as the posterior mean for a particular choice of prior. The estimates obtained are analysed with Krylov subspace methods and a contraction result for the posterior is presented. The method is then analysed in a simulation study as well as being applied to a challenging problem in medical imaging.

stat.ME↗

arXiv2018

Eigenvector continuation with subspace learning

A common challenge faced in quantum physics is finding the extremal eigenvalues and eigenvectors of a Hamiltonian matrix in a vector space so large that linear algebra operations on general vectors are not possible. There are numerous efficient methods developed for this task, but they generally fail when some control parameter in the Hamiltonian matrix exceeds some threshold value. In this work we present a new technique called eigenvector continuation that can extend the reach of these methods. The key insight is that while an eigenvector resides in a linear space with enormous dimensions, the eigenvector trajectory generated by smooth changes of the Hamiltonian matrix is well approximated by a very low-dimensional manifold. We prove this statement using analytic function theory and propose an algorithm to solve for the extremal eigenvectors. We benchmark the method using several examples from quantum many-body theory.

nucl-th↗

arXiv2017

Structural Convergence Results for Approximation of Dominant Subspaces from Block Krylov Spaces

This paper is concerned with approximating the dominant left singular vector space of a real matrix $A$ of arbitrary dimension, from block Krylov spaces generated by the matrix $AA^T$ and the block vector $AX$. Two classes of results are presented. First are bounds on the distance, in the two and Frobenius norms, between the Krylov space and the target space. The distance is expressed in terms of principal angles. Second are quality of approximation bounds, relative to the best approximation in the Frobenius norm. For starting guesses $X$ of full column-rank, the bounds depend on the tangent of the principal angles between $X$ and the dominant right singular vector space of $A$. The results presented here form the structural foundation for the analysis of randomized Krylov space methods. The innovative feature is a combination of traditional Lanczos convergence analysis with optimal approximations via least squares problems.

math.NA↗

arXiv2014

kappa_SQ: A Matlab package for randomized sampling of matrices with orthonormal columns

The kappa_SQ software package is designed to assist researchers working on randomized row sampling. The package contains a collection of Matlab functions along with a GUI that ties them all together and provides a platform for the user to perform experiments. In particular, kappa_SQ is designed to do experiments related to the two-norm condition number of a sampled matrix, $κ(SQ)$, where $S$ is a row sampling matrix and $Q$ is a tall and skinny matrix with orthonormal columns. Via a simple GUI, kappa_SQ can generate test matrices, perform various types of row sampling, measure $κ(SQ)$, calculate bounds and produce high quality plots of the results. All of the important codes are written in separate Matlab function files in a standard format which makes it easy for a user to either use the codes by themselves or incorporate their own codes into the kappa_SQ package.

math.NA↗