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A Simple Collocation-Type Approach to Numerical Stochastic Homogenization

Moritz Hauck, Hannah Mohr, Daniel Peterseim

Abstract

This paper proposes a novel collocation-type numerical stochastic homogenization method for prototypical stochastic homogenization problems with random coefficient fields of small correlation lengths. The presented method is based on a recently introduced localization technique that enforces a super-exponential decay of the basis functions relative to the underlying coarse mesh, resulting in considerable computational savings during the sampling phase. More generally, the collocation-type structure offers a particularly simple and computationally efficient construction in the stochastic setting with minimized communication between the patches where the basis functions of the method are computed. An error analysis that bridges numerical homogenization and the quantitative theory of stochastic homogenization is performed. In a series of numerical experiments, we study the effect of the correlation length and the discretization parameters on the approximation quality of the method.

A Simple Collocation-Type Approach to Numerical Stochastic Homogenization

Abstract

This paper proposes a novel collocation-type numerical stochastic homogenization method for prototypical stochastic homogenization problems with random coefficient fields of small correlation lengths. The presented method is based on a recently introduced localization technique that enforces a super-exponential decay of the basis functions relative to the underlying coarse mesh, resulting in considerable computational savings during the sampling phase. More generally, the collocation-type structure offers a particularly simple and computationally efficient construction in the stochastic setting with minimized communication between the patches where the basis functions of the method are computed. An error analysis that bridges numerical homogenization and the quantitative theory of stochastic homogenization is performed. In a series of numerical experiments, we study the effect of the correlation length and the discretization parameters on the approximation quality of the method.
Paper Structure (12 sections, 9 theorems, 96 equations, 6 figures)

This paper contains 12 sections, 9 theorems, 96 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Model problem
  3. Numerical stochastic homogenization method
  4. Error analysis
  5. Error analysis using LOD techniques
  6. Localization error indicator
  7. Riesz stability
  8. Practical implementation
  9. Sampling of the space $\boldsymbol{Y}$
  10. Stable local source terms
  11. Numerical experiments
  12. Auxiliary results

Key Result

Lemma 4.3

Let $\boldsymbol{v} \in L^2(\Omega; H^1_0(D_T))$ for almost all $\omega \in \Omega$ be defined as the weak solution to The $L^2$-representation of the Fréchet derivative of $(\boldsymbol{\varphi}_T^\mathrm{loc}, \mathds{1}_K)_{L^2(K)}$ is then given by where $\otimes\colon {\mathbb R}^d \times {\mathbb R}^d \rightarrow {\mathbb R}^{d\times d}$ denotes the outer product.

Figures (6)

  • Figure 3.1: Illustration of the localized basis functions ${\mathbb E}[\boldsymbol{\varphi}^\mathrm{loc}]$ obtained by the novel stochastic homogenization method on successively refined meshes for a piecewise constant random coefficient with a correlation length of $\varepsilon = 2^{-7}$ in two spatial dimensions. Various values of the oversampling parameter are depicted with $\ell = 1$ (left), $\ell = 2$ (middle), and $\ell = 3$ (right). The corresponding right-hand sides $g$ are shown in green.
  • Figure 6.1: Piecewise constant weighting function $w_T$ for an interior element $T$ with $\ell = 2$ in two spatial dimensions.
  • Figure 7.1: Depiction of $\sigma$ for a $\mathcal{T}_\varepsilon$-piecewise constant random coefficient in two spatial dimensions. Left: in dependence of the coarse mesh size $H$ for $\varepsilon=2^{-8}$; Right: in dependence of the correlation length $\varepsilon$ for $H=2^{-4}$.
  • Figure 7.2: Depiction of the Riesz stability constant $C_\mathrm{rb}$ of the stochastic SLOD as a function of the coarse mesh size $H$ for a $\mathcal{T}_\varepsilon$-piecewise constant random coefficient with $\varepsilon=2^{-8}$ in two spatial dimensions.
  • Figure 7.3: Plot of the relative $L^2$-errors $\|\Pi_H\boldsymbol{u}_h - \bar{u}_{H,h,\ell}\|_{L^2(\Omega;L^2(D))}$ of the proposed SLOD method for a $\mathcal{T}_\varepsilon$-piecewise constant random coefficient in one spatial dimension. Left: errors as functions of the coarse mesh size $H$ for fixed $\varepsilon = 2^{-8}$ and several oversampling parameters $\ell$; Right: errors in dependency of the correlation length $\varepsilon$ for fixed $H=2^{-4}$ and several values of $\ell$.
  • ...and 1 more figures

Theorems & Definitions (19)

  • Lemma 4.3: $L^2$-representation of Fréchet derivative
  • proof
  • Remark 4.4: Tilde notation
  • Lemma 4.5: $L^4$-regularity of localized basis functions
  • proof
  • Theorem 4.6: A posteriori error bound
  • proof
  • Lemma 5.1: Upper bound on $\sigma$
  • proof
  • Corollary 5.2: Combined error bound
  • ...and 9 more