A matrix-free isogeometric Galerkin method for Karhunen–Loève approximation of random fields using tensor product splines, tensor contraction and interpolation based quadrature

The Karhunen–Loève series expansion (KLE) decomposes a stochastic process into an infinite of pairwise uncorrelated random variables and L2-orthogonal functions. For any given truncation order the basis is optimal in sense that total mean squared error minimized. orthogonal functions are determined as solution eigenvalue problem corresponding to homogeneous Fredholm integral equation second kind, which computationally challenging for several reasons. Firstly, Galerkin discretization requires numerical integration over 2d dimensional domain, where d, this work, denotes spatial dimension. Secondly, main system matrix discretized weak-form dense. Consequently, computational complexity classical finite element formation assembly procedures well memory requirements direct techniques become quickly intractable with increasing polynomial degree, number elements degrees freedom. objective work significantly reduce bottlenecks associated KLE. We present matrix-free strategy, embarrassingly parallel scales favorably size degree. Our approach based on (1) interpolation quadrature minimizes required points; (2) inexpensive reformulation generalized standard problem; (3) matrix–vector product iterative solvers. Two higher-order three-dimensional C0-conforming multipatch benchmarks illustrate exceptional performance combined high accuracy robustness.