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Nonlinearity helps the convergence of the inverse Born series

Nicholas Defilippis, Shari Moskow, John C. Schotland

TL;DR

This work analyzes inverse problems for nonlinear PDEs with Kerr-type cubic nonlinearity, showing that when the linear coefficient is known, the inverse Born series (IBS) converges for sufficiently small data and can recover arbitrarily strong nonlinear terms; the results extend to general polynomial nonlinearities. The authors derive explicit bounds for forward operators, establish convergence radii for both forward and inverse series, and provide 2D numerical reconstructions demonstrating accurate recovery of high-contrast nonlinear coefficients. They argue that exploiting nonlinear structure can ease reconstruction and suggest fast convergence of Newton-type methods for small data, with the first-order inverse Born approximation offering a competitive initial estimate. These findings have implications for optical imaging and seismology where nonlinear media are present, and they highlight the practical viability of IBS in recovering nonlinear coefficients from boundary measurements.

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 [8]. 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.

Nonlinearity helps the convergence of the inverse Born series

TL;DR

This work analyzes inverse problems for nonlinear PDEs with Kerr-type cubic nonlinearity, showing that when the linear coefficient is known, the inverse Born series (IBS) converges for sufficiently small data and can recover arbitrarily strong nonlinear terms; the results extend to general polynomial nonlinearities. The authors derive explicit bounds for forward operators, establish convergence radii for both forward and inverse series, and provide 2D numerical reconstructions demonstrating accurate recovery of high-contrast nonlinear coefficients. They argue that exploiting nonlinear structure can ease reconstruction and suggest fast convergence of Newton-type methods for small data, with the first-order inverse Born approximation offering a competitive initial estimate. These findings have implications for optical imaging and seismology where nonlinear media are present, and they highlight the practical viability of IBS in recovering nonlinear coefficients from boundary measurements.

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 [8]. 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.
Paper Structure (8 sections, 5 theorems, 60 equations, 3 figures)

This paper contains 8 sections, 5 theorems, 60 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Forward problem
  3. Born series
  4. Inverse Born Series
  5. General polynomial nonlinearities
  6. Numerical Reconstructions
  7. Discussion
  8. Acknowledgments

Key Result

Proposition 1

Let $T: C(\overline{\Omega})\rightarrow C(\overline{\Omega})$ be defined by (Tdef) and define $\mu$ by If then $T$ has a unique fixed point on the ball of radius ${\| u_0\|_{{\color{black} C(\overline{\Omega})}}/{2}}$ about $u_0$ in $C(\overline{\Omega})$, and fixed point iteration starting with $u_0$ converges in $C(\Omega)$ to the unique fixed point $u$.

Figures (3)

  • Figure 1: Reconstruction of a high contrast $\beta$. Sources were scaled down with $g_0=0.01$ to ensure convergence of the IBS. The projection of the true $\beta$ onto the regularization space, $\mathcal{K}_1 K_1 \beta$, indicates an expected best case scenario.
  • Figure 2: Reconstruction of $\beta$ with a discontinuous disk shaped inclusion. Sources were scaled moderately with $g_0=.1$. The series captures the shape better than the projection $\mathcal{K}_1 K_1 \beta$.
  • Figure 3: Reconstruction of $\beta$ with a discontinuous disk shaped inclusion and a Gaussian. Sources were scaled moderately with $g_0=.1$. The series differentiates the two inhomogeneities better than the projection $\mathcal{K}_1 K_1 \beta$.

Theorems & Definitions (8)

  • Remark
  • Proposition 1
  • Proposition 2
  • proof
  • Corollary
  • Theorem 1: Convergence of the inverse Born series
  • Theorem 2: Approximation error
  • Remark