A Quasi Monte Carlo Method for Large-Scale Inverse Problems

We consider large-scale linear inverse problems with a simulation-based algorithm that approximates the solution within a low-dimensional subspace. The algorithm uses Tikhonov regularization, regression, and low-dimensional linear algebra calculations and storage. For sampling efficiency, we implement importance sampling schemes, specially tailored to the structure of inverse problems. We emphasize various alternative methods for approximating the optimal sampling distribution and we demonstrate their impact on the reduction of simulation noise. The performance of our algorithm is tested on a practical inverse problem arising from Fredholm integral equations of the first kind.

[1]  R. A. Leibler,et al.  Matrix inversion by a Monte Carlo method , 1950 .

[2]  J. Halton.  A Retrospective and Prospective Survey of the Monte Carlo Method , 1970 .

[3]  C. W. Groetsch,et al.  The theory of Tikhonov regularization for Fredholm equations of the first kind , 1984 .

[4]  Y. Marzouk,et al.  Large-Scale Inverse Problems and Quantification of Uncertainty , 1994 .

[5]  John N. Tsitsiklis,et al.  Neuro-Dynamic Programming , 1996, Encyclopedia of Machine Learning.

[6]  Mario Bertero,et al.  Introduction to Inverse Problems in Imaging , 1998 .

[7]  Dianne P. O'Leary,et al.  Near-Optimal Parameters for Tikhonov and Other Regularization Methods , 2001, SIAM J. Sci. Comput..

[8]  E. Somersalo,et al.  Statistical and computational inverse problems , 2004 .

[9]  Richard S. Sutton,et al.  Reinforcement Learning: An Introduction , 1998, IEEE Trans. Neural Networks.

[10]  Petros Drineas,et al.  Fast Monte Carlo Algorithms for Matrices II: Computing a Low-Rank Approximation to a Matrix , 2006, SIAM J. Comput..

[11]  Petros Drineas,et al.  Fast Monte Carlo Algorithms for Matrices III: Computing a Compressed Approximate Matrix Decomposition , 2006, SIAM J. Comput..

[12]  Petros Drineas,et al.  Fast Monte Carlo Algorithms for Matrices I: Approximating Matrix Multiplication , 2006, SIAM J. Comput..

[13]  Peter W. Glynn,et al.  Stochastic Simulation: Algorithms and Analysis , 2007 .

[14]  Christiane Lemieux.  Quasi–Monte Carlo Constructions , 2009 .

[15]  C. Lemieux.  Monte Carlo and Quasi-Monte Carlo Sampling , 2009 .

[16]  D. Bertsekas,et al.  Journal of Computational and Applied Mathematics Projected Equation Methods for Approximate Solution of Large Linear Systems , 2022 .

[17]  P. Hansen.  Discrete Inverse Problems: Insight and Algorithms , 2010 .