We apply the tensor train (TT) decomposition to construct the tensor product polynomial chaos expansion (PCE) of a random field, to solve the stochastic elliptic diffusion PDE with the stochastic Galerkin discretization, and to compute some quantities of interest (mean, variance, and exceedance probabilities). We assume that the random diffusion coefficient is given as a smooth transformation of a Gaussian random field. In this case, the PCE is delivered by a complicated formula, which lacks an analytic TT representation. To construct its TT approximation numerically, we develop the new block TT cross algorithm, a method that computes the whole TT decomposition from a few evaluations of the PCE formula. The new method is conceptually similar to the adaptive cross approximation in the TT format but is more efficient when several tensors must be stored in the same TT representation, which is the case for the PCE. In addition, we demonstrate how to assemble the stochastic Galerkin matrix and to compute the s...

Realistic mathematical models of physical processes contain uncertainties. These models are often described by stochastic differential equations (SDEs) or stochastic partial differential equations (SPDEs) with multiplicative noise. The uncertainties in the right-hand side or the coefficients are represented as random fields. To solve a given SPDE numerically one has to discretise the deterministic operator as well as the stochastic fields. The total dimension of the SPDE is the product of the dimensions of the deterministic part and the stochastic part. To approximate random fields with as few random variables as possible, but still retaining the essential information, the Karhunen–Loève expansion (KLE) becomes important. The KLE of a random field requires the solution of a large eigenvalue problem. Usually it is solved by a Krylov subspace method with a sparse matrix approximation. We demonstrate the use of sparse hierarchical matrix techniques for this. A log-linear computational cost of the matrix-vector product and a log-linear storage requirement yield an efficient and fast discretisation of the random fields presented.