Alapfogalmak
The paper proposes a novel collocation-type numerical stochastic homogenization method that leverages a super-localized basis function construction to achieve significant computational savings during the sampling phase.
Kivonat
The paper presents a novel numerical stochastic homogenization method for solving random diffusion problems with small correlation lengths in the coefficient field. The method is based on a recently introduced localization technique called the Super-Localized Orthogonal Decomposition (SLOD), which leads to basis functions with super-exponential decay. This allows for a computationally efficient collocation-type formulation that avoids communication between the local basis functions during the sampling phase.
The key highlights of the paper are:
The proposed method is based on the SLOD, which identifies an almost local basis of the solution space by minimizing the conormal derivative of the localized responses to carefully chosen local source terms.
The collocation-type formulation of the method leads to a coarse stiffness-type matrix that can be assembled without any communication between the local basis functions, enabling improved parallelization and significant speed-ups.
Under standard assumptions of quantitative stochastic homogenization, the paper provides an a posteriori error estimate for the coarse-scale approximation, where certain SLOD-specific quantities contribute in an a posteriori manner.
A worst-case a priori error analysis is conducted for one of the SLOD-specific quantities, and several numerical experiments are performed to study the effect of the correlation length and other discretization parameters on the accuracy of the approximation.
Statisztikák
The paper does not contain any explicit numerical data or statistics. It focuses on the theoretical development and analysis of the proposed numerical stochastic homogenization method.
Idézetek
The paper does not contain any striking quotes that support the key logics. The content is presented in a technical, mathematical style.