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We investigate the convergence rate of the quantics-TT (QTT) stochas-tic collocation tensor approximations to solutions of multiparametric elliptic PDEs andconstruct efficient iterative methods for solving arising high-dimensional parameter-dependent algebraic systems of equations. Such PDEs arise, for example, in the para-metric, deterministic reformulation of elliptic PDEs with random field inputs, based,for example, on the M-term truncated Karhunen-Loève expansion. We consider boththe case of additive and log-additive dependence on the multivariate parameter. Thelocal-global versions of the QTT-rank estimates for the system matrix in terms of theparameter space dimension is proven. Similar rank bounds are observed in numerics forthe solutions of the discrete linear system. We propose QTT-truncated iteration basedon the construction of solution-adaptive preconditioner that provides robust conver-gence in both additive and log-additive cases. Various numerical tests indicate that thenumerical complexity scales almost linearly in the dimension of parametric space M.

Quantics-TT Collocation Approximation of Parameter-Dependent and Stochastic Elliptic PDEs