Randomized Residual-Based Error Estimators for Parametrized Equations
Name
18m120364x.pdf
Description
Published version
Size
3.32 MB
Format
Adobe PDF
Checksum (MD5)
37698361f4ce143c616f0f527738b2e3
Author(s)
Patera, Anthony T.
Date Issued
January 2019
Journal
SIAM journal on scientific computing
Publisher
Society for Industrial & Applied Mathematics (SIAM)
Citation
Smetana, Kathrin, Oliver Zahm and Anthony T. Patera. "Randomized Residual-Based Error Estimators for Parametrized Equations." SIAM journal on scientific computing 41 (2019):A900-A926 © 2019 The Author(s)
Version
Final published version
Abstract
We propose a randomized a posteriori error estimator for reduced order approximations of parametrized (partial) differential equations. The error estimator has several important properties: the effectivity is close to unity with prescribed lower and upper bounds at specified high probability; the estimator does not require the calculation of stability (coercivity, or inf-sup) constants; the online cost to evaluate the a posteriori error estimator is commensurate with the cost to find the reduced order approximation; and the probabilistic bounds extend to many queries with only modest increase in cost. To build this estimator, we first estimate the norm of the error with a Monte Carlo estimator using Gaussian random vectors whose covariance is chosen according to the desired error measure, e.g., user-defined norms or quantity of interest. Then, we introduce a dual problem with random right-hand side the solution of which allows us to rewrite the error estimator in terms of the residual of the original equation. In order to have a fast-to-evaluate estimator, model order reduction methods can be used to approximate the random dual solutions. Here, we propose a greedy algorithm that is guided by a scalar quantity of interest depending on the error estimator. Numerical experiments on a multiparametric Helmholtz problem demonstrate that this strategy yields rather low-dimensional reduced dual spaces.
Subjects
Applied Mathematics
Computational Mathematics
MIT Department
Massachusetts Institute of Technology. Department of Mechanical Engineering
Terms of Use
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1137/18m120364x
