BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Elisabeth Ullmann (TU Munich) DTSTART:20200916T140000Z DTEND:20200916T150000Z DTSTAMP:20260423T041509Z UID:E-NLA/16 DESCRIPTION:Title: A pproximation of parametric covariance matrices\nby Elisabeth Ullmann ( TU Munich) as part of E-NLA - Online seminar series on numerical linear al gebra\n\n\nAbstract\nCovariance operators model the spatial\, temporal or other correlation between collections of random variables. In modern appli cations these random variables are often associated with an infinite-dimen sional or high-dimensional function space. Examples are the solution of a partial differential equation with random coefficients in uncertainty quan tification (UQ)\, or Gaussian process regression in machine learning. When a suitable discretization of the function space has been applied\, the di scretized covariance operator becomes a very large matrix - the covariance matrix - with a size that is of the order of the dimension of the discret e space squared.\n\nCovariance matrices are naturally symmetric and positi ve semi-definite\, but in the applications we are interested in\, they are typically dense. To avoid the enormous cost of creating and handling thes e dense matrices\, efficient low-rank approximations such as the pivoted C holesky decomposition\, or the adaptive cross approximation (ACA) have bee n developed during the last decade.\n\nBut the story does not end here sin ce recently\, the attention has shifted to parameterized covariance operat ors. This is due to their increased modeling capacity\, e.g.\, in Bayesian inverse problems or Gaussian process regression with hyperparameters in m achine learning. Now we are faced with the task to approximate a parametri c covariance matrix where the parameter itself is a random process. Simply repeating the ACA or pivoted Cholesky decomposition for different paramet er values is inefficient and most certainly too expensive in practise.\n\n We introduce and study two algorithms for the approximation of parametric families of covariance matrices. The first approach is a (non-certified) a pproximation\, and employs a reduced basis associated with a collection of eigenvectors for specific parameter values. The second approach is a cert ified extension of the ACA where the approximation error is controlled in the Wasserstein-2 distance of two Gaussian measures. Both approaches rely on an affine linear expansion of the covariance operator with respect to t he parameter. This keeps the computational cost under control. Notably\, b oth algorithms do not require regular meshes in the covariance operator di scretization and can be used on irregular domains.\n\nThis talk describes joint work with Daniel Kressner (EPFL)\, Jonas Latz (University of Cambrid ge)\, Stefano Massei (TU/e) and Marvin Eisenberger (TUM).\n LOCATION:https://researchseminars.org/talk/E-NLA/16/ END:VEVENT END:VCALENDAR