2015/03/11 by Sergey Dolgov, Boris N. Khoromskij, Dolgov, Sergey +5 · 2 citations
Decision Sciences · Environmental Science · Mathematics · Physics and Astronomy · #15A69 #60H15 #60H35 #65C30 #65F10 #FOS: Mathematics #Hydrology and Drought Analysis #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design #Tensor decomposition and applications #Wind and Air Flow Studies
paper · pdf · doi:10.48550/arxiv.1503.03210
openalex publication_date 2015/03/11 · openalex created_date 2022/09/20 · openalex updated_date 2026/07/28
We apply the Tensor Train (TT) decomposition to construct the tensor product\nPolynomial Chaos Expansion (PCE) of a random field, to solve the stochastic\nelliptic diffusion PDE with the stochastic Galerkin discretization, and to\ncompute some quantities of interest (mean, variance, exceedance probabilities).\nWe assume that the random diffusion coefficient is given as a smooth\ntransformation of a Gaussian random field. In this case, the PCE is delivered\nby a complicated formula, which lacks an analytic TT representation. To\nconstruct its TT approximation numerically, we develop the new block TT cross\nalgorithm, a method that computes the whole TT decomposition from a few\nevaluations of the PCE formula. The new method is conceptually similar to the\nadaptive cross approximation in the TT format, but is more efficient when\nseveral tensors must be stored in the same TT representation, which is the case\nfor the PCE. Besides, we demonstrate how to assemble the stochastic Galerkin\nmatrix and to compute the solution of the elliptic equation and its\npost-processing, staying in the TT format.\n We compare our technique with the traditional sparse polynomial chaos and the\nMonte Carlo approaches. In the tensor product polynomial chaos, the polynomial\ndegree is bounded for each random variable independently. This provides higher\naccuracy than the sparse polynomial set or the Monte Carlo method, but the\ncardinality of the tensor product set grows exponentially with the number of\nrandom variables. However, when the PCE coefficients are implicitly\napproximated in the TT format, the computations with the full tensor product\npolynomial set become possible. In the numerical experiments, we confirm that\nthe new methodology is competitive in a wide range of parameters, especially\nwhere high accuracy and high polynomial degrees are required.\n