Computational Methods for Sparse Solution of Linear Inverse Problems

Computational and Variational Methods for Inverse Problems

Sparse inverse solution methods for signal and image processing applications

This paper addresses image and signal processing problems where the result most consistent with prior knowledge is the minimum order, or “maximally sparse” solution. These problems arise in such diverse areas as astronomical star image deblurring, neuromagnetic image reconstruction, seismic deconvolution, and thinned array beamformer design. An optimization theoretic formulation for sparse solu...

Inverse Linear Multistep Methods for the Numerical Solution of Initial Value Problems of Ordinary Differential Equations

The well-known explicit linear multistep methods for the numerical solution of ordinary differential equations advance the numerical solution from xn+k_x to xn+k ky comPut'ng some numerical approximation from back values and then evaluating the problem defining function to obtain an approximation of the derivative. In this paper similar methods are proposed that first compute an approximation t...

Stochastic spectral methods for efficient Bayesian solution of inverse problems

We present a reformulation of the Bayesian approach to inverse problems, that seeks to accelerate Bayesian inference by using polynomial chaos (PC) expansions to represent random variables. Evaluation of integrals over the unknown parameter space is recast, more efficiently, as Monte Carlo sampling of the random variables underlying the PC expansion. We evaluate the utility of this technique on...

Consistent Discretization of Linear Inverse Problems using Sparse Stochastic Processes

We introduce a novel discretization paradigm and specify MAP estimators for linear inverse problems by using the theory of continuous-domain sparse stochastic processes. We characterize the complete class of admissible priors for the discretized version of the signal and show that the said class is restricted to the family of infinitely divisible distributions. We also explain the connections b...