Sampling-free Bayesian inversion with adaptive hierarchical tensor representations. (12th February 2018)

• Record Type: Journal Article
• Authors: Eigel, Martin; Marschall, Manuel; Schneider, Reinhold
• Is Part Of: Inverse problems. Volume 34:Number 3(2018:Mar.)
• Publication Date: 2018-02-12
• Subjects: Bayesian inversion -- partial differential equations with random coefficients -- tensor representation -- tensor train -- uncertainty quantification -- stochastic finite element methods -- low-rank approximation; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/aaa998
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: A sampling-free approach to Bayesian inversion with an explicit polynomial representation of the parameter densities is developed, based on an affine-parametric representation of a linear forward model. This becomes feasible due to the complete treatment in function spaces, which requires an efficient model reduction technique for numerical computations. The advocated perspective yields the crucial benefit that error bounds can be derived for all occuring approximations, leading to provable convergence subject to the discretization parameters. Moreover, it enables a fully adaptive a posteriori control with automatic problem-dependent adjustments of the employed discretizations. The method is discussed in the context of modern hierarchical tensor representations, which are used for the evaluation of a random PDE (the forward model) and the subsequent high-dimensional quadrature of the log-likelihood, alleviating the 'curse of dimensionality'. Numerical experiments demonstrate the performance and confirm the theoretical results.