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Adaptive Quasi-Monte Carlo Finite Element Methods for parametric elliptic PDEs

by M. Longo

(Report number 2021-03)

Abstract
We introduce novel adaptive methods to approximate moments of solutions of Partial Differential Equations (PDEs) with uncertain parametric inputs. A typical problem in Un- certainty Quantification is the approximation of the expected values of Quantities of Interest of the solution, which requires the efficient numerical approximation of high-dimensional in- tegrals. We perform this task by a class of deterministic Quasi-Monte Carlo integration rules derived from Polynomial lattices, that allows to control a-posteriori the integration error with- out querying the governing PDE and does not incur the curse of dimensionality. Based on an abstract formulation of Adaptive Finite Element methods for deterministic problems, we infer convergence of the combined adaptive algorithms in the parameter and physical space. We propose a selection of examples of PDEs admissible for these algorithms. Finally, we present numerical evidence of convergence for a model diffusion PDE.

Keywords: Uncertainty Quantification, Adaptive Finite Element Methods, High-dimensional Integration, Quasi-Monte Carlo

@Techreport{L21_945,
  author = {M. Longo},
  title = {Adaptive Quasi-Monte Carlo Finite Element Methods for parametric elliptic PDEs},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2021-03},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2021/2021-03.pdf },
  year = {2021}
}

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