Sarah Vallélian

Kui Ren, Sarah Vallélian

This work is concerned with uncertainty quantification problems for image reconstructions in quantitative photoacoustic imaging (PAT), a recent hybrid imaging modality that utilizes the photoacoustic effect to achieve high-resolution imaging of optical properties of tissue-like heterogeneous media. We quantify mathematically and computationally the impact of uncertainties in various model parameters of PAT on the accuracy of reconstructed optical properties. We derive, via sensitivity analysis, analytical bounds on error in image reconstructions in some simplified settings, and develop a computational procedure, based on the method of polynomial chaos expansion, for such error characterization in more general settings. Numerical simulations based on synthetic data are presented to illustrate the main ideas.

D. Andrew Brown, Arvind Saibaba, Sarah Vallélian

In Bayesian inverse problems, the posterior distribution is used to quantify uncertainty about the reconstructed solution. In practice, Markov chain Monte Carlo algorithms often are used to draw samples from the posterior distribution. However, implementations of such algorithms can be computationally expensive. We present a computationally efficient scheme for sampling high-dimensional Gaussian distributions in ill-posed Bayesian linear inverse problems. Our approach uses Metropolis-Hastings independence sampling with a proposal distribution based on a low-rank approximation of the prior-preconditioned Hessian. We show the dependence of the acceptance rate on the number of eigenvalues retained and discuss conditions under which the acceptance rate is high. We demonstrate our proposed sampler by using it with Metropolis-Hastings-within-Gibbs sampling in numerical experiments in image deblurring, computerized tomography, and NMR relaxometry.