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Abstract We consider the Born and inverse Born series for scalar waves with a cubic nonlinearity of Kerr type. We find a recursive formula for the operators in the Born series and prove their boundedness. This result gives conditions which guarantee convergence of the Born series, and subsequently yields conditions which guarantee convergence of the inverse Born series. We also use fixed point theory to give alternate explicit conditions for convergence of the Born series. We illustrate our results with numerical experiments.
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This content will become publicly available on November 26, 2025
Nonlinearity helps the convergence of the inverse Born series
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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- PAR ID:
- 10617614
- Publisher / Repository:
- Inverse Problems
- Date Published:
- Journal Name:
- Inverse Problems
- Volume:
- 40
- Issue:
- 12
- ISSN:
- 0266-5611
- Page Range / eLocation ID:
- 125020
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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