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Model Order Reduction Methods in Computational Uncertainty Quantification

by P. Chen and Ch. Schwab

(Report number 2015-28)

Abstract
This work surveys mathematical foundations of Model Order Reduction (MOR for short) techniques in accelerating computational forward and inverse UQ. Operator equations (comprising elliptic and parabolic Partial Differential Equations (PDEs for short) and Boundary Integral Equations (BIEs for short)) with distributed uncertain input, being an element of an infinite-dimensional, separable Banach space $X$, are admitted. Using an unconditional basis of $X$, computational UQ for these equations is reduced to numerical solution of countably parametric operator equations with smooth parameter dependence. In computational forward UQ, efficiency of MOR is based on recent sparsity results for countably-parametric solutions which imply upper bounds on Kolmogorov $N$-widths of the manifold of (countably-)parametric solutions and Quantities of Interest (QoI for short) with dimension-independent convergence rates. Subspace sequences which realize the $N$-width convergence rates are obtained by greedy search algorithms in the solution manifold. Heuristic search strategies in parameter space based on finite searches over anisotropic sparse grids in parameter space render greedy searches in reduced basis construction feasible. Instances of the parametric forward problems which arise in the greedy searches are assumed to be discretized by abstract classes of Petrov-Galerkin (PG for short) discretizations of the parametric operator equation, covering most conforming primal, dual and mixed Finite Element Methods (FEM), as well as certain space-time Galerkin schemes for the application problem of interest. Based on the PG discretization, MOR for both linear and nonlinear, affine and nonaffine parametric problems are presented. Computational inverse UQ for the mentioned operator equations is considered in the Bayesian setting of [M. Dashti and A.M. Stuart: Inverse problems a Bayesian perspective, arXiv:1302.6989v3, this Handbook]. The (countably-)parametric Bayesian posterior density inherits, in the absence of concentration effects for small observation noise covariance, the sparsity and $N$-width bounds of the (countably-)parametric manifolds of solution and QoI. This allows, in turn, for the deployment of MOR techniques for the parsimonious approximation of the parametric Bayesian posterior density, with convergence rates which are only limited by the sparsity of the uncertain inputs in the forward model.

Keywords: uncertainty quantification, sparse grid, reduced basis, empirical interpolation, greedy algorithm, high-fidelity, Petrov-Galerkin, a posteriori error estimate, a priori error estimate, Bayesian inversion

@Techreport{CS15_618,
  author = {P. Chen and Ch. Schwab},
  title = {Model Order Reduction Methods in Computational Uncertainty Quantification},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2015-28},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2015/2015-28.pdf },
  year = {2015}
}

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