Numerical Methods for Inverse Problems of the Linear Boltzmann Transport Equation

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The objective of this project is to develop fast and robust numerical reconstruction methods for linear and nonlinear inverse transport problems appeared in many application areas. The main approach proposed is to incorporate a priori information into optimization-based algorithms that have been developed in the past. The a priori information includes both knowledge on the unknowns to be reconstructed and knowledge on numerical methods for forward transport problems. More precisely, the proposed research include: (1) to develop methods that utilize a priori information to reduce the number of unknowns in the reconstruction, and to reconstruct features in the object of interests; (2) to accelerate the reconstruction process by analyzing the structure of the forward transport problem so that larger set of data can be used in the reconstruction process; and (3) to develop efficient Bayesian computational methods for uncertainty quantification in transport-based imaging problems. The proposed research lies between numerical mathematics and applications. From computational point of view, developing fast, robust and accurate reconstruction algorithms for ill-posed transport-based inverse problems is very challenging, not only because of the advanced numerical optimization, numerical partial differential equations techniques it involves, but also because of the fact that a deep understanding of the theory of ill-posed inverse problems is required. Many ideas developed in this proposal should have straightforward applications in numerical solutions of other model-based inverse problems. From application point of view, inverse transport problems find applications in various areas such as medical imaging (mainly optical tomography and optical molecular imaging), detection and imaging in random media, and atmospheric optics. The proposed research can potentially has long-term impacts in those application areas, improving both the stability and the accuracy of those imaging methods. Some of the ideas and techniques developed in this project will be incorporated into a graduate level class on numerical methods for inverse problems which presumably will benefit graduates who are interested in applying mathematical and computational techniques to solve real world problems.