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IPI

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.

IPI

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.

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.

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