Tensor-structured methods for parameter dependent and stochastic elliptic PDEs

Modern methods of tensor-product decomposition allow an efficient data-sparse approximation of functions and operators in higher dimensions [5]. The recent quantics-TT (QTT) tensor method allows to represent the multidimensional data with log-volume complexity [1, 2, 3]. We discuss the convergence rate of the Tucker, canonical and QTT stochastic collocation tensor approximations to the solution of multi-parametric elliptic PDEs, and describe efficient iterative methods for solving arising highdimensional parameter-dependent algebraic systems of equations [4, 6]. Such PDEs arise, in particular, in the parametric, deterministic reformulation of elliptic PDEs with random field input, based on the M -term truncated Karhunen-Loève expansion. We consider both the case of additive and log-additive dependence on the multivariate parameter in RM . The local-global versions of the canonical and QTTrank estimates for the system matrix in terms of the parameter space dimension is presented. We discuss tensor-truncated iteration based on the construction of solution-adaptive preconditioner providing linear complexity in M . Various numerical tests presented indicate that the computational complexity of the canonical and QTT tensor methods scales almost linearly in the dimension of parametric space M , and the adaptive preconditioner provides robust convergence in both additive and log-additive cases.