2017/07/14 by Sergey Dolgov, Robert Scheichl, Dolgov, Sergey +1 · 2 citations
Decision Sciences · Mathematics · #65F10 #65F30 #65N22 #65N30 #65N35 #FOS: Mathematics #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design #Statistical and numerical algorithms #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.1707.04562
openalex publication_date 2017/07/14 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We consider the approximate solution of parametric PDEs using the low-rank\nTensor Train (TT) decomposition. Such parametric PDEs arise for example in\nuncertainty quantification problems in engineering applications. We propose an\nalgorithm that is a hybrid of the alternating least squares and the TT cross\nmethods. It computes a TT approximation of the whole solution, which is\nbeneficial when multiple quantities of interest are sought. This might be\nneeded, for example, for the computation of the probability density function\n(PDF) via the maximum entropy method [Kavehrad and Joseph, IEEE Trans. Comm.,\n1986]. The new algorithm exploits and preserves the block diagonal structure of\nthe discretized operator in stochastic collocation schemes. This disentangles\ncomputations of the spatial and parametric degrees of freedom in the TT\nrepresentation. In particular, it only requires solving independent PDEs at a\nfew parameter values, thus allowing the use of existing high performance PDE\nsolvers. In our numerical experiments, we apply the new algorithm to the\nstochastic diffusion equation and compare it with preconditioned steepest\ndescent in the TT format, as well as with (multilevel) quasi-Monte Carlo and\ndimension-adaptive sparse grids methods. For sufficiently smooth random fields\nthe new approach is orders of magnitude faster.\n