Abstract
We consider the approximate solution of parametric PDEs using the lowrank Tensor Train (TT) decomposition. Such parametric PDEs arise for example in uncertainty quantification problems in engineering applications. We propose an algorithm that is a hybrid of the alternating least squares and the TT cross methods. It computes a TT approximation of the whole solution, which is beneficial when multiple quantities of interest are sought. This might be needed, for example, for the computation of the probability density function (PDF) via the maximum entropy method [Kavehrad and Joseph, IEEE Trans. Comm., 1986]. The new algorithm exploits and preserves the block diagonal structure of the discretized operator in stochastic collocation schemes. This disentangles computations of the spatial and parametric degrees of freedom in the TT representation. In particular, it only requires solving independent PDEs at a few parameter values, thus allowing the use of existing high performance PDE solvers. In our numerical experiments, we apply the new algorithm to the stochastic diffusion equation and compare it with preconditioned steepest descent in the TT format, as well as with (multilevel) quasiMonte Carlo and dimensionadaptive sparse grids methods. For sufficiently smooth random fields the new approach is orders of magnitude faster.
Original language  English 

Pages (fromto)  260291 
Number of pages  32 
Journal  SIAM/ASA Journal on Uncertainty Quantification 
Volume  7 
Issue number  1 
Early online date  7 Mar 2019 
DOIs  
Publication status  Published  2019 
Keywords
 math.NA
 65F10, 65F30, 65N22, 65N30, 65N35
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Sergey Dolgov
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