2 research outputs found
Entropy solutions of a scalar conservation law modeling sedimentation in vessels with varying cross-sectional area
The sedimentation of an ideal suspension in a vessel with variable cross-sectional area can be described by an initial-boundary value problem for a scalar nonlinear hyperbolic conservation law with a nonconvex flux function and a weight function that depends on spatial position. The sought unknown is the local solids' volume fraction. For the most important cases of vessels with downward-decreasing cross-sectional area and flux functions with at most one infection point, entropy solutions of this problem are constructed by the method of characteristics. Solutions exhibit discontinuities that mostly travel at variable speed, i.e., they are curved in the space-time plane. These trajectories are given by ordinary differential equations that arise from the jump condition. It is shown that three qualitatively different solutions may occur in dependence of the initial concentration. The potential application of the findings is a new method of flux identification via settling tests in a suitably shaped vessel. Related models also arise in flows of vehicular traffic, pedestrians, in pipes with varying cross-sectional area, and on curved surfaces
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Inverse problems for hyperbolic conservation laws: a Bayesian approach
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 et.al., 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