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Monte-Carlo Finite-Volume Methods inUncertainty Quantification for Hyperbolic Conservation Laws

by S. Mishra and Ch. Schwab

(Report number 2017-38)

Abstract
We consider hyperbolic systems of conservation laws and review developments in the general area of computational uncertainty quantification (UQ) for these equations. We focus on non-intrusive sampling methods of the Monte-Carlo (MC) and Multi-level Monte-Carlo (MLMC) type. The modeling of uncertainty, within the framework of random fields and random entropy solutions, is discussed. We also describe (ML)MC finite volume methods and present the underlying error bounds and complexity estimates. Based on these bounds, and numerical experiments, we illustrate the gain in efficiency resulting from the use of MLMC methods in this context. Recent progress in the mathematical UQ frameworks of measure-valued and statistical solutions is briefly presented, with comprehensive literature survey.

Keywords: UQ, MLMC, Finite Volumes, Monte-Carlo

@Techreport{MS17_734,
  author = {S. Mishra and Ch. Schwab},
  title = {Monte-Carlo Finite-Volume Methods inUncertainty Quantification for Hyperbolic Conservation Laws },
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2017-38},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2017/2017-38.pdf },
  year = {2017}
}

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