Sparse, adaptive Smolyak quadratures for Bayesian inverse problems

Abstract

Based on the parametric deterministic formulation of Bayesian inverse problems with unknown input parameter from infinite-dimensional, separable Banach spaces proposed in Schwab and Stuart (2012 Inverse Problems 28 045003), we develop a practical computational algorithm whose convergence rates are provably higher than those of Monte Carlo (MC) and Markov chain Monte Carlo methods, in terms of the number of solutions of the forward problem. In the formulation of Schwab and Stuart, the forward problems are parametric, deterministic elliptic partial differential equations, and the inverse problem is to determine the unknown diffusion coefficients from noisy observations comprising linear functionals of the system's response. The sparsity of the generalized polynomial chaos representation of the posterior density being implied by sparsity assumptions on the class of the prior (Schwab and Stuart 2012), we design, analyze and implement a class of adaptive, deterministic sparse tensor Smolyak quadrature schemes for the efficient approximate numerical evaluation of expectations under the posterior, given data. The proposed, deterministic quadrature algorithm is based on a greedy, iterative identification of finite sets of most significant, 'active' chaos polynomials in the posterior density analogous to recently proposed algorithms for adaptive interpolation (Chkifa et al 2012 Report 2012-NN, 2013. Convergence rates for the quadrature approximation are shown, both theoretically and computationally, to depend only on the sparsity class of the unknown, but are bounded independently of the number of random variables activated by the adaptive algorithm. Numerical results for a model problem of coefficient identification with point measurements in a diffusion problem confirm the theoretical results.

Approximation of high-dimensional parametric PDEs

“…In such a case, we have indeed 78) which shows that ρ * j > 1. Note that the values ρ * j increase as t decrease, which in principle results in a better bound for t ν V .…”

“…These results establish for the first time convergence rates immune to the curse of dimensionality, in the sense that they hold with infinitely many variables, see also [44] for a survey dealing in particular with these issues. In a similar infinite dimensional framework, and not covered in our paper, let us mention the following related works: (i) similar holomorphy and approximation results are established in [47,48,58] for specific type of PDEs and control problems, (ii) approximation of integrals by quadratures is discussed in [60,61], (iii) inverse problems are discussed in [78,79,82], following the Bayesian perspective from [87], and (iv) diffusion problems with lognormal coefficients are treated in [50,43,45].…”

“…In this splitting, we have used the fact that if ν / ∈Λ and ν j = 0, we have either ν − e j / ∈Λ or ν − e j ∈ M. Using (7.67), we control the first term A by 78) and by the same argument we obtain…”

Shape Holomorphy of the Stationary Navier--Stokes Equations

“…If V = V (x, t) is a vector field and T t is the corresponding flow, then if u Ω is the solution to some given PDE for the domain Ω, then we definė 21) where the limit needs to be defined in a given topology, for example in the strong sense for a Sobolev space W m,p (Ω). Then, a simple computation shows that if F T is the Fréchet derivative at T of the map T →û T for this topology, we havė…”

“…In relation with such results, the concept of (b, ε)-holomorphy has been exploited in the context of Bayesian inverse problems [21] and for the construction of low-parametric, reduced basis surrogates [3,4]. We next explain items (a) and (b), detailing in particular computational approximation strategies for the efficient computation of sparse approximations of countably-parametric solution families.…”