# A multiscale method for semi-linear elliptic equations with localized uncertainties and non-linearities

## TL;DR

This paper introduces a multiscale numerical method combining domain decomposition, adaptive sampling, and iterative algorithms to efficiently solve semi-linear elliptic stochastic PDEs with localized uncertainties and non-linearities.

## Contribution

It presents a novel multiscale approach that integrates local polynomial approximations and convergence acceleration for semi-linear elliptic equations with uncertainties.

## Key findings

- Method achieves convergence and robustness under general assumptions.
- Numerical experiments demonstrate efficiency on nonlinear diffusion-reaction problems.
- Aitken's relaxation accelerates the iterative convergence.

## Abstract

A multiscale numerical method is proposed for the solution of semi-linear elliptic stochastic partial differential equations with localized uncertainties and non-linearities, the uncertainties being modeled by a set of random parameters. It relies on a domain decomposition method which introduces several subdomains of interest (called patches) containing the different sources of uncertainties and non-linearities. An iterative algorithm is then introduced, which requires the solution of a sequence of linear global problems (with deterministic operators and uncertain right-hand sides), and non-linear local problems (with uncertain operators and/or right-hand sides) over the patches. Non-linear local problems are solved using an adaptive sampling-based least-squares method for the construction of sparse polynomial approximations of local solutions as functions of the random parameters. Consistency, convergence and robustness of the algorithm are proved under general assumptions on the semi-linear elliptic operator. A convergence acceleration technique (Aitken's dynamic relaxation) is also introduced to speed up the convergence of the algorithm. The performances of the proposed method are illustrated through numerical experiments carried out on a stationary non-linear diffusion-reaction problem.

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Source: https://tomesphere.com/paper/1704.05331