University of Sussex
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Inverse problems for hyperbolic conservation laws: a Bayesian approach

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posted on 2023-06-09, 23:56 authored by Duc Lam Duong
This thesis contributes to the development of the Bayesian approach to inverse problems for hyperbolic conservation laws. Inverse problems in the context of hyperbolic conservation laws are challenging due to the presence of shock waves and their irreversibility. The first main contribution of this work is the study of the forward problems where we develop a theory of particle trajectories for scalar conservation laws. Motivated from a model of traffic ow, we consider some ordinary differential equations with their right-hand side depending on solutions of scalar conservation laws. Despite the presence of discontinuities in the entropy solutions, it is showed that the trajectories are well-posed by using Filippov theory. Moreover, we prove that the approximate trajectory generated by either the front tracking approximation method or the vanishing viscosity method converges uniformly to the trajectory corresponding to the entropy solution of the scalar conservation law. For certain flux functions, illustrated by traffic flow, we are able to obtain the convergence rate for the approximate trajectory with respect to changes in the initial field or the flux function by combining the front tracking method with Filippov theory. As the second main contribution of the thesis, we study some Bayesian inverse problems for scalar conservation laws and establish several well-posedness and approximation results. Specifically, we consider two types of inverse problems: the inverse problem of recovering the upstream field and the inverse problem of finding the flux function, both from observations of appropriate functions of the entropy solutions of scalar conservation laws. Based on the theory of trajectories developed in the first part of the thesis and the Bayesian inversion theory developed by Stuart, we prove that the statistical solutions to these inverse problems are well-posed and stable with respect to changes in the forward model. Rates of convergence of the approximate posteriors are also given for certain inverse problems.


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University of Sussex

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