$\mathcal{H}$-matrix based second moment analysis for rough random fields and finite element discretizations
For computational scientists solving stochastic PDEs, this method reduces computational cost significantly for challenging correlation structures.
The paper presents an H-matrix based method to efficiently approximate the two-point correlation of the solution to strongly elliptic PDEs with random loads, achieving essentially linear time for 3D finite element discretizations even for non-smooth or short-correlation data.
We consider the efficient solution of strongly elliptic partial differential equations with random load based on the finite element method. The solution's two-point correlation can efficiently be approximated by means of an $\mathcal{H}$-matrix, in particular if the correlation length is rather short or the correlation kernel is non-smooth. Since the inverses of the finite element matrices which correspond to the differential operator under consideration can likewise efficiently be approximated in the $\mathcal{H}$-matrix format, we can solve the correspondent $\mathcal{H}$-matrix equation in essentially linear time by using the $\mathcal{H}$-matrix arithmetic. Numerical experiments for three-dimensional finite element discretizations for several correlation lengths and different smoothness are provided. They validate the presented method and demonstrate that the computation times do not increase for non-smooth or shortly correlated data.