# Numerical stochastic homogenization by quasilocal effective diffusion tensors

## TL;DR

This paper introduces a numerical method for homogenizing elliptic problems with random small-scale diffusion, producing a deterministic quasilocal model with error estimates to quantify uncertainty effects.

## Contribution

It develops a novel quasilocal effective diffusion tensor approach for stochastic homogenization with rigorous error analysis.

## Key findings

- Effective deterministic model derived from random diffusion tensors
- Error estimates quantify uncertainty impact
- Method enables efficient numerical simulations

## Abstract

This paper proposes a numerical upscaling procedure for elliptic boundary value problems with diffusion tensors that vary randomly on small scales. The resulting effective deterministic model is given through a quasilocal discrete integral operator, which can be further compressed to an effective partial differential operator. Error estimates consisting of a priori and a posteriori terms are provided that allow one to quantify the impact of uncertainty in the diffusion coefficient on the expected effective response of the process.