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Abstract In previous work of the authors, we investigated the Born and inverse Born series for a scalar wave equation with linear and nonlinear terms, the nonlinearity being cubic of Kerr type (DeFilippiset al2023Inverse Problems39125015). We reported conditions which guarantee convergence of the inverse Born series, enabling recovery of the coefficients of the linear and nonlinear terms. In this work, we show that if the coefficient of the linear term is known, an arbitrarily strong Kerr nonlinearity can be reconstructed, for sufficiently small data. Additionally, we show that similar convergence results hold for general polynomial nonlinearities. Our results are illustrated with numerical examples.
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On unifying randomized methods for inverse problems
Abstract This work unifies the analysis of various randomized methods for solving linear and nonlinear inverse problems with Gaussian priors by framing the problem in a stochastic optimization setting. By doing so, we show that many randomized methods are variants of a sample average approximation (SAA). More importantly, we are able to prove a single theoretical result that guarantees the asymptotic convergence for a variety of randomized methods. Additionally, viewing randomized methods as an SAA enables us to prove, for the first time, a single non-asymptotic error result that holds for randomized methods under consideration. Another important consequence of our unified framework is that it allows us to discover new randomization methods. We present various numerical results for linear, nonlinear, algebraic, and PDE-constrained inverse problems that verify the theoretical convergence results and provide a discussion on the apparently different convergence rates and the behavior for various randomized methods.
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- PAR ID:
- 10482675
- Publisher / Repository:
- IOPScience: IOP Publishing Ltd
- Date Published:
- Journal Name:
- Inverse Problems
- Volume:
- 39
- Issue:
- 7
- ISSN:
- 0266-5611
- Page Range / eLocation ID:
- 075010
- Subject(s) / Keyword(s):
- randomization, Bayesian inversion, ensemble Kalman filter, randomized maximum a posteriori
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1088/1361-6420/acd36e
