Bayesian approach to inverse scattering with topological priors

Abstract

We propose a Bayesian inference framework to estimate uncertainties in inverse scattering problems. Given the observed data, the forward model and their uncertainties, we find the posterior distribution over a finite parameter field representing the objects. To construct the prior distribution we use a topological sensitivity analysis. We demonstrate the approach on the Bayesian solution of 2D inverse problems in light and acoustic holography with synthetic data. Statistical information on objects such as their center location, diameter size, orientation, as well as material properties, are extracted by sampling the posterior distribution. Assuming the number of objects known, comparison of the results obtained by Markov Chain Monte Carlo sampling and by sampling a Gaussian distribution found by linearization about the maximum a posteriori estimate show reasonable agreement. The latter procedure has low computational cost, which makes it an interesting tool for uncertainty studies in 3D. However, MCMC sampling provides a more complete picture of the posterior distribution and yields multi-modal posterior distributions for problems with larger measurement noise. When the number of objects is unknown, we devise a stochastic model selection framework.

Multifrequency Topological Derivative Approach to Inverse Scattering Problems in Attenuating Media

“…When the material properties of the objects are unknown, we could generate initial approximations to their geometry by related topological energy techniques [45,[50][51][52] and to their material parameters by hybrid gradient methods [38]. Uncertainty due to noise can be addressed by combining topological priors with Bayesian techniques [46].…”

“…For our inverse scattering problem in an attenuating half space, we have checked that increasing the level of noise δ defined in (8) from δ = 0.01 to δ = 0.1 results are almost identical. A detailed study of the effect of noise on a topological derivative-based approximation is performed in [46] in a Bayesian framework, though it is exemplified in different emitter/receiver configurations.…”

Uncertainty quantification in Covid-19 spread: Lockdown effects

“…By sampling this posterior distribution, we can visualize the uncertainty in the inference of parameters for a given data set. To do so, we will resort to Markov Chain Monte Carlo Sampling [33] , [34] . Once we have a large collection of samples, we can extract information from the model (2) – (7) with quantified uncertainty, such as the global number of people who have been affected by the virus the last day of the period we are considering.…”

“…In our case, slight asymmetry is caused by discarding negative values. In principle, we could try to improve our estimate of the parameter values that maximize the likelihood by optimization procedures [33] . In practice, enforcing the positivity constraint while doing it may be problematic, and the best samples provide reasonable approximations for our purposes.…”

The interior inverse scattering problem for a two-layered cavity using the Bayesian method

“…Recently as a result of its good interpretation of mathematical models, the Bayesian method has attracted intensive attention for the inverse problems [9,10,24]. The readers are referred to [3,4,8,14,15,19,20,26,29] for the applications of the Bayesian method in different kinds of inverse scattering problems. From the perspective of the Bayesian framework, inverse problems can be rephrased into the form of statistical inferences.…”