Numerical approximation of elliptic problems with log-normal random coefficients
For researchers in uncertainty quantification, this provides an efficient preconditioning strategy that accelerates existing numerical methods for random elliptic PDEs.
This work proposes using a Wick-type elliptic model as a preconditioner to accelerate Monte Carlo and stochastic Galerkin finite element methods for elliptic equations with log-normal random coefficients, achieving second-order accuracy in standard deviation and exact recovery in the infinite correlation length limit.
In this work, we consider a non-standard preconditioning strategy for the numerical approximation of the classical elliptic equations with log-normal random coefficients. In \cite{Wan_model}, a Wick-type elliptic model was proposed by modeling the random flux through the Wick product. Due to the lower-triangular structure of the uncertainty propagator, this model can be approximated efficiently using the Wiener chaos expansion in the probability space. Such a Wick-type model provides, in general, a second-order approximation of the classical one in terms of the standard deviation of the underlying Gaussian process. Furthermore, when the correlation length of the underlying Gaussian process goes to infinity, the Wick-type model yields the same solution as the classical one. These observations imply that the Wick-type elliptic equation can provide an effective preconditioner for the classical random elliptic equation under appropriate conditions. We use the Wick-type elliptic model to accelerate the Monte Carlo method and the stochastic Galerkin finite element method. Numerical results are presented and discussed.