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 ill-posed. We study the ill-posedness 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. High-energy light scatters very little, and one uses the radiative transfer equation (RTE) as the model; low-energy light scatters frequently, so the diffusion equation (DE) suffices to be a good approximation. A multiscale approximation links the hyperbolic-type RTE with the parabolic-type 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 ill-posed. 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 ill-posed 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.
Bibliographical noteFunding Information:
Funding: Q.L. acknowledges support from Vilas Early Career award. The research is supported in part by NSF via grant DMS-1750488 and the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin Madison with funding from the Wisconsin Alumni Research Foundation. K.N. acknowledges support from NSF grant DMS-1750488. L.W. acknowledges support from NSF via grant DMS-1846854.
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- Asymptotic analysis
- Bayesian approach
- Inverse problems
- Multiscale modeling
- Stability deterioration