Stability of Karhunen–Loève expansion for the simulation of Gaussian stochastic fields using Galerkin scheme

• The positive definiteness of truncated covariance function is examined.
• It is shown that truncated covariance function is not positive definite.
• A guideline to avoid numerical stability problems is proposed.
• Two numerically more efficient modifications of covariance function are proposed.

The paper investigates the problem of numerical stability of the Karhunen–Loève expansion for the simulation of Gaussian stochastic fields using Galerkin scheme. The instability is expressed as loss of positive definiteness of covariance matrix and is the result of modifications of standard exponential covariance functions that are commonly applied to increase the sparsity of the covariance matrix. The loss of positive definiteness of covariance matrix limits the use of efficient eigenvalue solvers that are needed for the solution of the resulting generalized eigenvalue problem. Two modifications of the shape of covariance function to avoid instability problems and at the same time to raise the numerical efficiency of Karhunen–Loève expansion by increasing the sparsity of the covariance matrix are proposed. The effects of the proposed modifications are demonstrated on numerical examples.