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On the well-posedness of Bayesian inversion for PDEs with ill-posed forward problems
(2021)SAM Research ReportWe study the well-posedness of Bayesian inverse problems for PDEs, for which the underlying forward problem may be ill-posed. Such PDEs, which include the fundamental equations of fluid dynamics, are characterized by the lack of rigorous global existence and stability results as well as possible non-convergence of numerical approximations. Under very general hypotheses on approximations to these PDEs, we prove that the posterior measure, ...Report -
Well-posedness of Bayesian inverse problems for hyperbolic conservation laws
(2021)SAM Research ReportWe study the well-posedness of the Bayesian inverse problem for scalar hyperbolic conservation laws where the statistical information about inputs such as the initial datum and (possibly discontinuous) flux function are inferred from noisy measurements. In particular, the Lipschitz continuity of the measurement to posterior map as well as the stability of the posterior to approximations, are established with respect to the Wasserstein ...Report -
Statistical solutions of hyperbolic systems of conservation laws: numerical approximation
(2019)SAM Research ReportReport -
On the vanishing viscosity limit of statistical solutions of the incompressible Navier-Stokes equations
(2021)SAM Research ReportWe study statistical solutions of the incompressible Navier--Stokes equation and their vanishing viscosity limit. We show that a formulation using correlation measures, which are probability measures accounting for spatial correlations, and moment equations is equivalent to statistical solutions in the Foiac{s}--Prodi sense. Under the assumption of weak scaling, a weaker version of Kolmogorov's self-similarity at small scales hypothesis ...Report