Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients

By Andrea Barth, Christoph Schwab, Nathaniel Zollinger. Numerische Mathematik, 2011

370 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Numerical Analysis, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/890deac0615a775d394d439de074a06b26b0f345

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Citations in the corpus

Adversarial Uncertainty Quantification in Physics-Informed Neural Networks (2018, background): “However, despite the fact that much progress has been made over the last two decades, the most popular methods for scientific computing under uncertainty, such as polynomial chaos expansions [17, 18, 19], sparse grid quadratures [20, 21], multi-level/multi-fidelity Monte Carlo sampling [22, 23], pro”
N-term Wiener chaos approximation rates for elliptic PDEs with lognormal Gaussian random inputs (2014, methodology): “One principal aim of this work is to prove that, indeed, adaptive truncations of of Wiener polynomial chaos expansions can afford higher rates of convergence in terms of the number N of PDE solves than the commonly used Monte Carlo sampling methods or even their more efficient variants, the multi-le”
Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs* (2011, methodology): “(4.109) In terms of NL = O(2Ld), the computational cost is L∑ =0 M N (logN ) k−1 NL(logNL) k+2 −1 if 2s < d, NL(logNL) k+3 −1 if 2s = d, N 2s/d L (logNL) k+2 −1 if 2s > d. (4.110) We refer to Barth et al. (2010) for a proof and further details.”
First order second moment analysis for stochastic interface problems based on low-rank approximation (2013, methodology): “Recently, the efficient multilevel MC method has been proposed to solve elliptic PDEs with stochastic coefficients in asymptotically optimal complexity [5].”
Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients (2011, methodology): “[Barth, Schwab, Zollinger, 2010] is concerned with the application of multilevel Monte Carlo methods to elliptic PDEs with random coefficients, though they assume smoother coefficients fields than we consider in this paper (see section 5).”