Inverse transport with isotropic sources and angularly averaged measurements
Guillaume Bal Ian Langmore François Monard
Inverse Problems & Imaging 2008, 2(1): 23-42 doi: 10.3934/ipi.2008.2.23
We consider the reconstruction of a spatially-dependent scattering coefficient in a linear transport equation from diffusion-type measurements. In this setup, the contribution to the measurement is an integral of the scattering kernel against a product of harmonic functions, plus an additional term that is small when absorption and scattering are small. The linearized problem is severely ill-posed. We construct a regularized inverse that allows for reconstruction of the low frequency content of the scattering kernel, up to quadratic error, from the nonlinear map. An iterative scheme is used to improve this error so that it is small when the high frequency content of the scattering kernel is small.
keywords: Inverse Transport Theory Averaged Measurements Optical Tomography Complex Geometrical Optics Solutions.
Bayesian inverse problems with Monte Carlo forward models
Guillaume Bal Ian Langmore Youssef Marzouk
Inverse Problems & Imaging 2013, 7(1): 81-105 doi: 10.3934/ipi.2013.7.81
The full application of Bayesian inference to inverse problems requires exploration of a posterior distribution that typically does not possess a standard form. In this context, Markov chain Monte Carlo (MCMC) methods are often used. These methods require many evaluations of a computationally intensive forward model to produce the equivalent of one independent sample from the posterior. We consider applications in which approximate forward models at multiple resolution levels are available, each endowed with a probabilistic error estimate. These situations occur, for example, when the forward model involves Monte Carlo integration. We present a novel MCMC method called $MC^3$ that uses low-resolution forward models to approximate draws from a posterior distribution built with the high-resolution forward model. The acceptance ratio is estimated with some statistical error; then a confidence interval for the true acceptance ratio is found, and acceptance is performed correctly with some confidence. The high-resolution models are rarely run and a significant speed up is achieved.
    Our multiple-resolution forward models themselves are built around a new importance sampling scheme that allows Monte Carlo forward models to be used efficiently in inverse problems. The method is used to solve an inverse transport problem that finds applications in atmospheric remote sensing. We present a path-recycling methodology to efficiently vary parameters in the transport equation. The forward transport equation is solved by a Monte Carlo method that is amenable to the use of $MC^3$ to solve the inverse transport problem using a Bayesian formalism.
keywords: inverse problems importance sampling Linear transport Markov chain Monte Carlo. perturbation Monte Carlo Bayesian

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