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：解决高维随机参数PDE是一个具有挑战性的计算问题。众所周知,数字方法可以从适应性精细算法中受益匪浅,特别是当多数值的功能近似值以随机加勒金和随机同位法计算时。这项工作调查了一种基于残余的适应性算法,该算法用来将固定扩散方程式的解决方案与正正态系数相近。众所周知,完善程序是可靠的,但这一类未受约束系数的理论趋同长期以来一直是个未决问题。本文填补了这一空白,特别是为逻辑性固定扩散问题的适应性解决方案提供了趋同结果。一个计算模型支持理论说明。