Bayesian Approach to Inverse Problems

Abstract

These lecture notes highlight the mathematical and computational structure relating to the formulation of, and development of algorithms for, the Bayesian approach to inverse problems in differential equations. This approach is fundamental in the quantification of uncertainty within applications involving the blending of mathematical models with data. The finite dimensional situation is described first, along with some motivational examples. Then the development of probability measures on separable Banach space is undertaken, using a random series over an infinite set of functions to construct draws; these probability measures are used as priors in the Bayesian approach to inverse problems. Regularity of draws from the priors is studied in the natural Sobolev or Besov spaces implied by the choice of functions in the random series construction, and the Kolmogorov continuity theorem is used to extend regularity considerations to the space of Hölder continuous functions. Bayes' theorem is derived in this prior setting, and here interpreted as finding conditions under which the posterior is absolutely continuous with respect to the prior, and determining a formula for the Radon-Nikodym derivative in terms of the likelihood of the data. Having established the form of the posterior, we then describe various properties common to it in the infinite dimensional setting. These properties include well-posedness, approximation theory, and the existence of maximum a posteriori estimators. We then describe measure-preserving dynamics, again on the infinite dimensional space, including Markov chain-Monte Carlo and sequential Monte Carlo methods, and measure-preserving reversible stochastic differential equations. By formulating the theory and algorithms on the underlying infinite dimensional space, we obtain a framework suitable for rigorous analysis of the accuracy of reconstructions, of computational complexity, as well as naturally constructing algorithms which perform well under mesh refinement, since they are inherently well-defined in infinite dimensions.

Towards quantitative evaluation of wall shear stress from 4D flow imaging

“…Sobel filter [3] On the other hand, regularization is a general principle that aims at incorporating additional structural information to counterbalance the unstable character of inverse problem solutions [22].…”

High-dimensional Gaussian sampling: a review and a unifying approach based on a stochastic proximal point algorithm

“…Note that this method is even simpler when using Σ: then one uses C = chol(Σ) such that Σ = C T C and θ = Cz where z ∼ N (0 d , I d ). Band matrices naturally appear in specific applications, e.g., when the latter involve finite impulse response linear filters [43]. Problems with such structured (sparse or band) matrices have been extensively studied in the literature and as such this paper will not cover them explicitly.…”

“…where, contrary to the MS schemes discussed in the previous section, the two matrices Q 1 and Q 2 are not chosen by the user but directly result from the statistical model under consideration. In particular, such situations arise when deriving hierarchical Bayesian models (see, e.g., [43,72,87]). By capitalizing on possible specific structures of {Q i } i∈ [2] , it may be desirable to separate Q 1 and Q 2 in two different hopefully simpler steps of a Gibbs sampler.…”

A two-step mixed inpainting method with curvature-based anisotropy and spatial adaptivity

“…The CDD inpainting model (5) can take noise into account, for instance, by using TV regularization outside the occluded region. Thus, the resulting diffusion PDE takes the form (6) ∂u…”