The desire to form an image of a region of space from externally collected data arises in applications ranging from detecting and characterizing cancers in the body, to quantifying the distribution of water, oil, or subsurface pollutants, and to the timely accurate identification of explosives in crowded venues. The physics associated with signal propagation and sensing in these problems creates substantial computational challenges for transforming raw data into useful information. Across all applications, these problems share a common structure: the availability of limited measurements collected on the boundary or outside of the medium and a physical, or forward, model mapping the unknown properties of the medium onto the data. Given the (often nonlinear) forward model, noisy data, and available computing resources, we need algorithms for the inverse problem that characterize the medium quickly and efficiently.           

In addition to recovering parameters of interest from noisy measurements, in these applications it is also important to quantify the uncertainties associated with the reconstruction of the parameters. To this end, we adopt a probabilistic (i.e. Bayesian) framework that enables this uncertainty quantification. Another important aspect of this project is that in many scenarios---such as identifying a tumor in a patient, a crack in a wing, or a pool of contaminant in the subsurface---detailed pixelwise images are not always needed for addressing relevant, practical questions. This motivated the use of shape-based approaches for estimation of location, size, and shape of objects of interest without the need to construct detailed images. However, computational algorithms for uncertainty quantification in inverse problems suffer from high computational costs that encumber their applicability to practical scenarios. The research in this project builds on recent methods that exploit randomization to compute accurate estimates of solutions at greatly reduced computational cost, and on the efficient construction of smaller, approximate, reduced order numerical models that are accurate for relevant sets of parameters, and thus reduce the cost of full simulation of the sensing physics.

In this project, we developed several efficient algorithms for uncertainty quantification in Bayesian inverse problems, with a particular emphasis on shape-based inverse problems. Our major contributions can be summarized as follows:

 1. Development of randomized methods to reduce the computational cost of likelihood evaluations in Markov Chain Monte Carlo methods (MCMC) for Partial Differential Equation-based Bayesian Inverse Problems.

2. Analysis of randomized algorithms for low-rank approximations. Development of algorithms for nonlinear model reduction in the Discrete Empirical Interpolation Method framework. These two developments are of interest to scientific computing beyond inverse problems. 

3. Development of low-order parametric approaches for shape-based inverse problems.

4. Development of a Bayesian approach for quantifying reconstruction uncertainty in shape-based inverse problems.

5. Development of Krylov subspace methods for uncertainty quantification in linear Bayesian inverse problems.

Additionally, we validated our approaches on model problems from a suite of applications that exemplify the breadth of the proposed approaches: diffuse optical tomography, photoacoustic tomography, atmospheric CO2 monitoring, etc. 

This project also supported several outreach and educational activities such as accessible talks on imaging and its applications to undergraduates and high school teachers, teaching material for a special topics course at North Carolina State University, tutorial on randomized methods at the SIAM CAIMS Joint Annual Meeting 2020. 

Last Modified: 12/29/2021
Modified by: Arvind K Saibaba