Bayesian Level Set Approach for Inverse Problems with Piecewise Constant Reconstructions

Abstract

There are several challenges associated with inverse problems in which we seek to reconstruct a piecewise constant field, and which we model using multiple level sets. Adopting a Bayesian viewpoint, we impose prior distributions on both the level set functions that determine the piecewise constant regions as well as the parameters that determine their magnitudes. We develop a Gauss-Newton approach with a backtracking line search to efficiently compute the maximum a priori (MAP) estimate as a solution to the inverse problem. We use the Gauss-Newton Laplace approximation to construct a Gaussian approximation of the posterior distribution and use preconditioned Krylov subspace methods to sample from the resulting approximation. To visualize the uncertainty associated with the parameter reconstructions we compute the approximate posterior variance using a matrix-free Monte Carlo diagonal estimator, which we develop in this paper. We will demonstrate the benefits of our approach and solvers on synthetic test problems (photoacoustic and hydraulic tomography, respectively a linear and nonlinear inverse problem) as well as an application to X-ray imaging with real data.

A Bayesian approach for consistent reconstruction of inclusions

“…The level set method has found success in optimization-based approaches, in for example [70], where a descent step is taken in each iteration of an iterative algorithm. A Bayesian maximum a posteriori approach [21] has also been shown to find success for a smoothened level set. We expect that using gradient information in gradient-based MCMC methods would improve the performance significantly.…”

“…Suppose conditions 1 and 2 are satisfied. Then condition A.3 is satisfied for A n as in (20) and a as in (21). (20), for which we note…”

“…Lemma 3.4. Let Π satisfy condition 3 and let γ 0 = Φ(θ 0 ) for some θ 0 ∈ H. If condition 1 and 2 are satisfied, then Condition A.1 is satisfied for a as in (21).…”

“…We could even consider a smooth approximation for H ϵ , as in [21], but this is not necessary for our case. To construct the continuous level set parametrization, take constants c…”

“…In this paper, we consider a Bayesian approach that captures this idea for two parametrizations used in detection of inclusions for nonlinear inverse problems: the star-shaped set and level set parametrizations. These parametrizations are studied rigorously in [9][10][11] and remain popular to Bayesian practitioners: we mention [1,[12][13][14][15][16] in the case of the star-shaped inclusions and [12,[17][18][19][20][21] for the level set inclusions, see also references therein.…”