An Interpolation Method for Karhunen-Loève Expansion of Random Field Discretization

In the context of global mean square error concerning the number of random variables in the representation, the Karhunen-Loeve (KL) expansion is the optimal series expansion method for random field discretization. The computational efficiency and accuracy of the KL expansion are contingent upon the accurate resolution of the Fredholm integral eigenvalue problem (IEVP). The paper proposes an interpolation method based on different interpolation basis functions such as moving least squares (MLS), least squares (LS), and finite element method (FEM) to solve the IEVP. Compared with the Galerkin method based on finite element or Legendre polynomials, the main advantage of the interpolation method is that, in the calculation of eigenvalues and eigenfunctions in one-dimensional random fields, the integral matrix containing covariance function only requires a single integral, which is less than a two-folded integral by the Galerkin method. The effectiveness and computational efficiency of the proposed interpolation method are verified through various one-dimensional examples. Furthermore, based on the KL expansion and polynomial chaos expansion, the stochastic analysis of two-dimensional regular and irregular domains is conducted, and the basis function of the extended finite element method (XFEM) is introduced as the interpolation basis function in two-dimensional irregular domains to solve the IEVP.