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 and stochastic collocations methods. This work investigates a residual based adaptive algorithm used to approximate the solution of the stationary diffusion equation with lognormal coefficients. It is known that the refinement procedure is reliable, but the theoretical convergence of the scheme for this class of unbounded coefficients has long been an open question. This paper fills this gap and in particular provides a convergence results for the adaptive solution of the lognormal stationary diffusion problem. A computational example supports the theoretical statement.

翻译：求解高维随机参数化偏微分方程是一个具有挑战性的计算问题。众所周知，数值方法能从自适应细化算法中受益匪浅，特别是在随机伽辽金法和随机配点法中计算多项式函数逼近时尤为如此。本文研究了一种基于残差的自适应算法，用于逼近具有对数正态系数的稳态扩散方程的解。已知细化过程是可靠的，但该方案针对这类无界系数的理论收敛性长期以来一直是一个未解问题。本文填补了这一空白，特别为对数正态稳态扩散问题的自适应求解提供了收敛性结果。一个计算实例支持了理论结论。