Tensor train approximation of moment equations for elliptic equations with lognormal coefficient

We study an elliptic equation with stochastic coefficient modeled as a lognormal random field. A perturbation approach is adopted, expanding the solution in Taylor series around the nominal value of the coefficient. The resulting recursive deterministic problem satisfied by the expected value of the stochastic solution, the so called moment equation, is discretized with a full tensor product finite element technique. To overcome the incurred curse of dimensionality the solution is sought in a low-rank tensor format, the so called Tensor Train format. We develop an algorithm for solving the recursive first moment problem approximately in the Tensor Train format and show its effectiveness with numerical examples.