For the inverse problem in physical models, one measures the solution and infers the model parameters using information from the collected data. Oftentimes, these data are inadequate and render the inverse problem illposed. We study the illposedness in the context of optical imaging, which is a medical imaging technique that uses light to probe (bio)tissue structure. Depending on the intensity of the light, the forward problem can be described by different types of equations. Highenergy light scatters very little, and one uses the radiative transfer equation (RTE) as the model; lowenergy light scatters frequently, so the diffusion equation (DE) suffices to be a good approximation. A multiscale approximation links the hyperbolictype RTE with the parabolictype DE. The inverse problems for the two equations have a multiscale passage as well, so one expects that as the energy of the photons diminishes, the inverse problem changes from well to illposed. We study this stability deterioration using the Bayesian inference. In particular, we use the Kullback–Leibler divergence between the prior distribution and the posterior distribution based on the RTE to prove that the information gain from the measurement vanishes as the energy of the photons decreases, so that the inverse problem is illposed in the diffusive regime. In the linearized setting, we also show that the mean square error of the posterior distribution increases as we approach the diffusive regime.
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On Bayesian data assimilation for PDEs with illposed forward problems
Abstract We study Bayesian data assimilation (filtering) for timeevolution Partial differential equations (PDEs), for which the underlying forward problem may be very unstable or illposed. Such PDEs, which include the Navier–Stokes equations of fluid dynamics, are characterized by a high sensitivity of solutions to perturbations of the initial data, a lack of rigorous global wellposedness results as well as possible nonconvergence of numerical approximations. Under very mild and readily verifiable general hypotheses on the forward solution operator of such PDEs, we prove that the posterior measure expressing the solution of the Bayesian filtering problem is stable with respect to perturbations of the noisy measurements, and we provide quantitative estimates on the convergence of approximate Bayesian filtering distributions computed from numerical approximations. For the Navier–Stokes equations, our results imply uniform stability of the filtering problem even at arbitrarily small viscosity, when the underlying forward problem may become illposed, as well as the compactness of numerical approximants in a suitable metric on timeparametrized probability measures.
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 Award ID(s):
 2042454
 NSFPAR ID:
 10341609
 Date Published:
 Journal Name:
 Inverse Problems
 Volume:
 38
 Issue:
 8
 ISSN:
 02665611
 Page Range / eLocation ID:
 085012
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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