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A. Litvinenko presented his KAUST research work at Nottingham University

3/24/2017

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Abstract:

This talk consists of two parts. In the first part we use low-rank tensor methods to solve elliptic PDE with uncertain coefficients. We start with discretization, applying the Karhunen-Loeve Expansion (KLE) to separate spatial and stochastic variables and applying (generalized ) Polynomial Chaos Expansion (PCE). PCE approximates complicated distributions of random variables in multi-variate Hermite basis with Gaussian random variables, which are very comfortable for further computations. After discretization, we obtain a large linear system (stochastic Galerkin matrix), which has nice tensor properties and allows a low-rank tensor representation. I give some examples of low-rank tensor approximations. After solving this large linear system, we obtain coefficients of the solution in the KLE and PCE basis. Having such low-rank representation for the solution, we will compute its maximum (infinity norm), level sets, histograms in a low-rank tensor format. In the second part, we will develop a Bayesian Update surrogate. It allows us to update KLE-PCE coefficients of the uncertain solution and coefficients, if some additional measurements available. The uniqueness of this approach is that it doesn’t require sampling (like Markov Chain Monte Carlo). It updates prior PCE coefficients direct to posterior PCE coefficients.

Literature

1. M. Espig, W. Hackbusch, A. Litvinenko, H.G. Matthies, E. Zander, Efficient analysis of high dimensional data in tensor formats

Sparse Grids and Applications, 31-56, 2012.

2. M. Espig, W. Hackbusch, A Litvinenko, H.G. Matthies, P. Wähnert Efficient low-rank approximation of the stochastic Galerkin matrix in tensor formats Computers & Mathematics with Applications 67 (4), 818-829

3. S. Dolgov, B.N. Khoromskij, A. Litvinenko, H.G. Matthies, Polynomial Chaos Expansion of Random Coefficients and the Solution of Stochastic Partial Differential Equations in the Tensor Train Format, IAM/ASA J. Uncertainty Quantification 3 (1), 1109-1135, 2015

4. B.V. Rosić, B.V., Litvinenko, A., Pajonk, O., Matthies, H.G.: Sampling-free linear Bayesian update of polynomial chaos representations. Journal of Computational Physics 231, 5761–5787 (2012).

5. H.G. Matthies, A. Litvinenko, B.V. Rosic, E. Zander, Bayesian Parameter Estimation via Filtering and Functional Approximations, arXiv preprint arXiv:1611.09293