Numerically solving high-dimensional random parametric PDEs poses a challenging computational problem. It is well-known that numerical methods can greatly benefit from adaptive refinement algorithms, in particular when functional approximations in polynomials are computed as in stochastic Galerkin finite element methods. This work investigates a residual based adaptive algorithm, akin to classical adaptive FEM, used to approximate the solution of the stationary diffusion equation with lognormal coefficients, i.e. with a non-affine parameter dependence of the data. It is known that the refinement procedure is reliable but the theoretical convergence of the scheme for this class of unbounded coefficients remains a challenging open question. This paper advances the theoretical state-of-the-art by providing a quasi-error reduction result for the adaptive solution of the lognormal stationary diffusion problem. The presented analysis generalizes previous results in that guaranteed convergence for uniformly bounded coefficients follows directly as a corollary. Moreover, it highlights the fundamental challenges with unbounded coefficients that cannot be overcome with common techniques. A computational benchmark example illustrates the main theoretical statement.

翻译：高维随机参数化偏微分方程的数值求解是一个具有挑战性的计算问题。众所周知，数值方法可以极大地受益于自适应细化算法，特别是在计算多项式函数逼近时，如随机 Galerkin 有限元法。本研究探讨了一种基于残差的自适应算法，类似于经典的自适应有限元法，用于逼近具有对数正态系数（即数据具有非仿射参数依赖性）的稳态扩散方程的解。已知该细化过程是可靠的，但针对此类无界系数，该方案的理论收敛性仍然是一个具有挑战性的开放性问题。本文通过为对数正态稳态扩散问题的自适应求解提供一个拟误差缩减结果，推进了该领域的理论前沿。所提出的分析推广了先前的结果，使得一致有界系数情况下的保证收敛性可直接作为推论得出。此外，它突显了无界系数带来的基本挑战，这些挑战无法通过常用技术克服。一个计算基准示例阐明了主要的理论陈述。