English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

We propose and analyze deterministic multilevel approximations for Bayesianinversion of operator equations with uncertain distributed parameters, subjectto additive Gaussian measurement data. The algorithms use a multilevel (ML)approach based on deterministic, higher order quasi-Monte Carlo (HoQMC)quadrature for approximating the high-dimensional expectations, which arise inthe Bayesian estimators, and a Petrov-Galerkin (PG) method for approximatingthe solution to the underlying partial differential equation (PDE). Thisextends the previous single-level approach from [J. Dick, R. N. Gantner, Q. T.Le Gia and Ch. Schwab, Higher order Quasi-Monte Carlo integration for BayesianEstimation. Report 2016-13, Seminar for Applied Mathematics, ETH Z\"urich (inreview)].  We obtain sufficient conditions which allow us to achieve arbitrarily high,algebraic convergence rates in terms of work, which are independent of thedimension of the parameter space. The convergence rates are limited only by thespatial regularity of the forward problem,the discretization order achieved bythe Petrov Galerkin discretization, and by the sparsity of the uncertaintyparametrization. We provide detailed numerical experiments for linear ellipticproblems in two space dimensions, with $s=1024$ parameters characterizing theuncertain input, confirming the theory and showing that the ML HoQMC algorithmsoutperform, in terms of error vs.~computational work, both multilevel MonteCarlo (MLMC) methods and single-level (SL) HoQMC methods.

Multilevel higher order Quasi-Monte Carlo Bayesian Estimation