Low-rank tensor methods for solving PDEs with uncertain coefficients and Bayesian Update surrogate

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 will start with discretization, applying 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 will obtain a large linear system (stochastic Galerkin matrix), which has nice tensor properties and allows a low-rank tensor representation. I will 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, histogram in a low-rank tensor format. In the second part we will develop a Bayesian Update surrogate. It will allow 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.