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，数值方法可以显著受益于自适应细化算法，特别是在随机伽辽金方法和随机配点方法中计算多项式函数逼近时。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平稳扩散方程的解。已知该细化过程是可靠的，但此类无界系数格式的理论收敛性长期以来一直是一个开放问题。本文填补了这一空白，特别为对数正态平稳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陈述。