Abstract
We develop hybrid projection methods for computing solutions to large-scale inverse problems, where the solution represents a sum of different stochastic components. Such scenarios arise in many imaging applications (e.g., anomaly detection in atmospheric emissions tomography) where the reconstructed solution can be represented as a combination of two or more components and each component contains different smoothness or stochastic properties. In a deterministic inversion or inverse modeling framework, these assumptions correspond to different regularization terms for each solution in the sum. Although various prior assumptions can be included in our framework, we focus on the scenario where the solution is a sum of a sparse solution and a smooth solution. For computing solution estimates, we develop hybrid projection methods for solution decomposition that are based on a combined flexible and generalized Golub–Kahan process. This approach integrates techniques from the generalized Golub–Kahan bidiagonalization and the flexible Krylov methods. The benefits of the proposed methods are that the decomposition of the solution can be done iteratively, and the regularization terms and regularization parameters are adaptively chosen at each iteration. Numerical results from photoacoustic tomography and atmospheric inverse modeling demonstrate the potential for these methods to be used for anomaly detection.
Original language | English |
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Pages (from-to) | S97-S119 |
Number of pages | 23 |
Journal | SIAM Journal on Scientific Computing |
Early online date | 18 Jul 2023 |
DOIs | |
Publication status | E-pub ahead of print - 18 Jul 2023 |
Bibliographical note
Funding:This work was partially supported by the National Science Foundation ATD program under grants DMS-2026841, 2026830, and 2026835. This work was partially supported by National Natural Science Foundation of China under grant 12101406, and Shanghai Science and Technology Innovation Program under grant 21YF1429100. This material was also based upon work partially supported by the National Science Foundation under Grant DMS-1638521 to the Statistical and Ap-plied Mathematical Sciences Institute. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation
Keywords
- inverse problems
- hybrid methods
- generalized Golub--Kahan
- flexible methods
- Tikhonov regularization
- Bayesian inverse problems