Adaptive stochastic Galerkin finite element methods: Optimality and non-affine coefficients
Markus Bachmayr, Henrik Eisenmann, Igor Voulis
TL;DR
The paper presents an adaptive stochastic Galerkin finite element framework for parametric elliptic PDEs with infinite-dimensional randomness, formulated through the weak form of $-\nabla\cdot(a(y)\nabla u(y))=f$ on polygonal domains. It extends prior affine-coefficient results to nonlinear parametrizations via an operator-compression scheme and error estimation using finite element frames, achieving near-optimal degrees of freedom and near-optimal costs when a multilevel coefficient structure is available. Key contributions include a compressible operator treatment for both log-affine and general nonlinear diffusion, a polynomial-based diffusion coefficient approximation, and a rigorous complexity and optimality analysis that yields quasi-optimality for $u\in \mathcal{A}^s$ with rates governed by $s$ and the problem dimension. The approach has potential impact for high-dimensional uncertainty quantification in elliptic problems, enabling efficient, robust adaptive discretization with flexible spatial refinements per stochastic mode and applicability to nontrivial geometries.
Abstract
Near-optimal computational complexity of an adaptive stochastic Galerkin method with independently refined spatial meshes for elliptic partial differential equations is shown. The method takes advantage of multilevel structure in expansions of random diffusion coefficients and combines operator compression in the stochastic variables with error estimation using finite element frames in space. A new operator compression strategy is introduced for nonlinear coefficient expansions, such as diffusion coefficients with log-affine structure.