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Quantics-TT collocation approximation of parameter-dependent and stochastic elliptic PDEs
Boris N. Khoromskij and Ivan V. Oseledets
We investigate the convergence rate of QTT stochastic collocation tensor approximations to solutions of multi-parametric elliptic PDEs, and construct efficient iterative methods for solving arising high-dimensional parameter-dependent algebraic systems of equations. Such PDEs arise, for example, in the parametric, deterministic reformulation of elliptic PDEs with random field inputs, based for example, on the $M$-term truncated Karhunen-Loève expansion. We consider both the case of additive and log-additive dependence on the multivariate parameter. The local-global versions of the QTT-rank estimates for the system matrix in terms of the parameter space dimension is proven. Similar rank bounds are observed in numerics for the solutions of the discrete linear system. We propose QTT-truncated iteration based on the construction of solution-adaptive preconditioner. Various numerical tests indicate that the numerical complexity scales almost linearly in the dimension of parametric space, and the adaptive preconditioner provides robust convergence in both additive and log-additive cases.