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Multilevel approximation of parametric and stochastic PDEs

by J. Zech and D. Dung and Ch. Schwab

(Report number 2018-05)

Abstract
We analyze the complexity of the sparse-grid interpolation and sparse-grid quadrature of countably-parametric functions which take values in separable Banach spaces with unconditional bases. Under the provision of a suitably quantified holomorphic dependence on the parameters, we establish dimension-independent convergence rate bounds for sparse-grid approximation schemes. Analogous results are shown in the case that the parametric solutions are obtained as solutions of corresponding parametric-holomorphic, nonlinear operator equations as considered in [A. Cohen and A. Chkifa and Ch. Schwab: Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs, Journ. Math. Pures et Appliquees 103(2) 400-428 (2015)] by means of stable, finite dimensional approximations, for example nonlinear Petrov-Galerkin projections. Error and convergence rate bounds for constructive and explicit multilevel, sparse tensor approximation schemes combining sparse-grid interpolation in the parameter space and general, multilevel discretization schemes in the physical domain are proved. The results considerably generalize several earlier works in terms of the admissible multilevel approximations in the physical domain (comprising general stable Petrov-Galerkin and discrete Petrov-Galerkin schemes, collocation and stable domain approximations) and in terms of the admissible operator equations (comprising smooth, nonlinear locally well-posed operator equations). Additionally, a novel, general computational strategy to localize sequences of nested index sets is given for the anisotropic Smolyak scheme realizing best n-term benchmark convergence rates. We also consider Smolyak-type quadratures in this general setting, for which we establish improved convergence rates based on cancellations in gpc expansions due to symmetries of the probability measure [J. Zech and Ch. Schwab: Convergence rates of high dimensional Smolyak quadrature, Report 2017-27, SAM ETH Zürich]. Several examples illustrating the abstract theory include domain uncertainty quantification (“UQ” for short) for general, linear, second order, elliptic advection-reaction-diffusion equations on polygonal domains, where optimal convergence rates of FEM are known to require local mesh refinement near corners. For these equations, we also consider a combined sparse-grid scheme in physical and parameter space, affording complexity similar to the recent multiindex stochastic collocation approach. Further applications of the presently developed theory comprise evaluations of posterior expectations in Bayesian inverse problems.

Keywords: Uncertainty Quantification, Multilevel Approximation, Smolyak Quadrature, Smolyak Interpolation

@Techreport{ZDS18_759,
  author = {J. Zech and D. Dung and Ch. Schwab},
  title = {Multilevel approximation of parametric and stochastic PDEs},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2018-05},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2018/2018-05.pdf },
  year = {2018}
}

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