2023/02/06 by Martin Eigel, Eigel, Martin, Hegemann, Nando
Computer Science · Decision Sciences · Engineering · #65N12 #65N15 #65N50 #65Y20 #68Q25 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design
paper · pdf · doi:10.48550/arxiv.2302.02839
openalex publication_date 2023/02/06 · openalex created_date 2023/02/09 · openalex updated_date 2026/08/03
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.