## Near-optimal approximation methods for elliptic PDEs with lognormal coefficients

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Albert Cohen and Giovanni Migliorati
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## Abstract:

This paper studies numerical methods for the approximation of elliptic PDEs with lognormal coefficients of the form $-\operatorname {div}(a\nabla u)=f$ where $a=\exp (b)$ and $b$ is a Gaussian random field. The approximant of the solution $u$ is an $n$-term polynomial expansion in the scalar Gaussian random variables that parametrize $b$. We present a general convergence analysis of weighted least-squares approximants for smooth and arbitrarily rough random field, using a suitable random design, for which we prove optimality in the following sense: their convergence rate matches exactly or closely the rate that has been established in Bachmayr, Cohen, DeVore, and Migliorati [ESAIM Math. Model. Numer. Anal. 51 (2017), pp. 341–363] for best $n$-term approximation by Hermite polynomials, under the same minimial assumptions on the Gaussian random field. This is in contrast with the current state of the art results for the stochastic Galerkin method that suffers the lack of coercivity due to the lognormal nature of the diffusion field. Numerical tests with $b$ as the Brownian bridge confirm our theoretical findings.## References

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## Additional Information

**Albert Cohen**- Affiliation: Sorbonne Université, UPMC Univ Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 4 place Jussieu, 75005 Paris, France
- MR Author ID: 308419
- Email: cohen@ann.jussieu.fr
**Giovanni Migliorati**- Affiliation: Sorbonne Université, UPMC Univ Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 4 place Jussieu, 75005 Paris, France
- MR Author ID: 941196
- ORCID: 0000-0002-6317-5663
- Email: migliorati@ljll.math.upmc.fr
- Received by editor(s): March 25, 2021
- Received by editor(s) in revised form: January 28, 2022, July 25, 2022, September 22, 2022, and January 2, 2023
- Published electronically: February 9, 2023
- © Copyright 2023 American Mathematical Society
- Journal: Math. Comp.
**92**(2023), 1665-1691 - MSC (2020): Primary 41A10, 41A63, 42B08, 42B37, 60G60, 65C30
- DOI: https://doi.org/10.1090/mcom/3825
- MathSciNet review: 4570337