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Iterative inversion schemes for the Born series and the reduced inverse Born series

Akari Ishida, Manabu Machida

TL;DR

The paper develops fast Newton-type iterative schemes for inverting the Born series in nonlinear inverse problems and establishes their connection to reduced inverse Born series. It proves contraction-based convergence for both the standard and reduced variants under explicit operator bounds and shows how the fast scheme effectively implements a reduced IBS without exponential growth in complexity. Numerical tests in a 2D diffusion/radial setting demonstrate dramatically reduced computation times with reconstructions that closely match those produced by the full inverse Born series. The results offer a scalable, PDE-ready approach to nonlinear imaging that bypasses the prohibitive cost of higher-order Born terms while preserving accuracy under suitable regularization.

Abstract

Nonlinear inverse problems have complicated landscapes. Hence the calculation with naive iterative schemes (e.g., Gauss-Newton or conjugate gradients) is trapped in local minima. The (first) Born approximation can avoid this trapping but linearization is required. Nonlinear inverse problems can be solved without linearization by means of the inverse Born series. However, the computational cost of its standard recursive implementation grows exponentially when nonlinear terms are taken into account. In this work we revisit a Newton-type iterative scheme to invert the Born series and develop a fast variant. The relation between this fast scheme and the reduced inverse Born series is shown.

Iterative inversion schemes for the Born series and the reduced inverse Born series

TL;DR

The paper develops fast Newton-type iterative schemes for inverting the Born series in nonlinear inverse problems and establishes their connection to reduced inverse Born series. It proves contraction-based convergence for both the standard and reduced variants under explicit operator bounds and shows how the fast scheme effectively implements a reduced IBS without exponential growth in complexity. Numerical tests in a 2D diffusion/radial setting demonstrate dramatically reduced computation times with reconstructions that closely match those produced by the full inverse Born series. The results offer a scalable, PDE-ready approach to nonlinear imaging that bypasses the prohibitive cost of higher-order Born terms while preserving accuracy under suitable regularization.

Abstract

Nonlinear inverse problems have complicated landscapes. Hence the calculation with naive iterative schemes (e.g., Gauss-Newton or conjugate gradients) is trapped in local minima. The (first) Born approximation can avoid this trapping but linearization is required. Nonlinear inverse problems can be solved without linearization by means of the inverse Born series. However, the computational cost of its standard recursive implementation grows exponentially when nonlinear terms are taken into account. In this work we revisit a Newton-type iterative scheme to invert the Born series and develop a fast variant. The relation between this fast scheme and the reduced inverse Born series is shown.

Paper Structure

This paper contains 15 sections, 10 theorems, 136 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Born series
  3. Iterative schemes
  4. Proofs of Theorems \ref{['mainthm1']} and \ref{['mainthm2']}
  5. Proof of Therem \ref{['mainthm1']}
  6. Proof of Theorem \ref{['mainthm2']}
  7. Inverse Born series
  8. Reduced inverse Born series
  9. Revisiting inverse operators
  10. Numerical tests
  11. A simple invertible case
  12. Radial problem: Setup
  13. Radial problem: Forward data
  14. Radial problem: Reconstruction
  15. Concluding remarks

Key Result

Theorem 3.1

We assume that there exists a constant $b\in(1,\infty)$ such that $\|I-\mathcal{K}_1K\|\le1-1/b$. Let $(\eta^{(n)})_{n=0}^{\infty}$ be the sequence constructed by (iterative1) with $\eta^{(0)}\equiv0$. Then the iteration in (iterative1) admits a unique fixed-point $\eta^*$. Furthermore,

Figures (3)

  • Figure 1: In the case of $\eta_a\equiv0.2$. (Left) The reconstructed $\eta$ by the fast iterative scheme (\ref{['iterativeRed1']}). (Right) The reconstructed $\eta$ by the inverse Born series (\ref{['invBorn']}). In both panels, the black line shows the true shape of the target and the red line is the best reconstruction given in (\ref{['etaproj']}); furthermore, the purple, green, light blue, ocher, and dark blue lines show 1st (linear), 2nd, 3rd, 4th, and 5th reconstructions, respectively.
  • Figure 2: Same as Fig. \ref{['fig1_02']} but $\eta_a\equiv0.4$.
  • Figure 3: Comparison of the reconstructed $\eta$ by the fast iterative scheme (\ref{['iterativeRed1']}) and the inverse Born series (\ref{['invBorn']}). Results for (Left) $\eta_a=0.2$ and (Right) $\eta_a=0.4$ are shown. The purple and dark blue lines show reconstructions by the 1st- and 5th-order fast iterations, and green and ocher dots show reconstructions by the 1st- and 5th-order inverse Born series.

Theorems & Definitions (22)

  • Remark 2.1
  • Remark 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Definition 5.1
  • Example 5.2
  • Example 5.3
  • Lemma 5.4
  • proof
  • Lemma 6.1
  • ...and 12 more