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 remains a challenging open question. This paper advances the theoretical results by providing a quasi-error reduction 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，在随机Galerkin方法和随机配点方法中计算多项式函数逼近时，自适应细化算法能显著提升数值方法的性能。本文研究了一种基于残差的自适应算法，用于求解含对数正态系数的稳态扩散方程。尽管已知该细化过程是可靠的，但对于这类无界系数问题，该方案的理论收敛性仍是一个悬而未决的难题。本文通过给出对数正态稳态扩散问题自适应求解的拟误差缩减结果，推进了理论进展。数值算例验证了该理论结论。