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Solving a 1-D inverse medium scattering problem using a new multi-frequency globally strictly convex objective functional

Thanh T. Nguyen, Michael V. Klibanov

TL;DR

The paper addresses the 1-D inverse medium scattering problem using multi-frequency backscatter data to recover the spatially varying dielectric coefficient $c(x)$ in a Helmholtz model $u''(x)+k^2 c(x)u(x)=-\delta(x-x^0)$.A globally strictly convex objective functional is constructed via a Carleman weight, built on a reformulation using $v(x,k)=u_x/(k^2 u)$ and a truncated Fourier expansion, enabling global convergence without a good initial guess.Theoretical contributions include a Carleman estimate, strict convexity for large $\lambda$, Lipschitz continuity of the objective’s gradient, and error estimates linking data noise to reconstruction error; computational work develops discretization in $x$ and $k$ and a quasi-Newton minimization strategy.Numerical experiments on simulated data demonstrate accurate reconstructions with modest basis cardinality and robustness to noise, illustrating practical viability of the globally convergent approach and guiding extensions to higher dimensions.

Abstract

We propose a new approach to constructing globally strictly convex objective functional in a 1-D inverse medium scattering problem using multi-frequency backscattering data. The global convexity of the proposed objective functional is proved using a Carleman estimate. Due to its convexity, no good first guess is required in minimizing this objective functional. We also prove the global convergence of the gradient projection algorithm and derive an error estimate for the reconstructed coefficient. Numerical results show reasonable reconstruction accuracy for simulated data.

Solving a 1-D inverse medium scattering problem using a new multi-frequency globally strictly convex objective functional

TL;DR

The paper addresses the 1-D inverse medium scattering problem using multi-frequency backscatter data to recover the spatially varying dielectric coefficient $c(x)$ in a Helmholtz model $u''(x)+k^2 c(x)u(x)=-\delta(x-x^0)$.A globally strictly convex objective functional is constructed via a Carleman weight, built on a reformulation using $v(x,k)=u_x/(k^2 u)$ and a truncated Fourier expansion, enabling global convergence without a good initial guess.Theoretical contributions include a Carleman estimate, strict convexity for large $\lambda$, Lipschitz continuity of the objective’s gradient, and error estimates linking data noise to reconstruction error; computational work develops discretization in $x$ and $k$ and a quasi-Newton minimization strategy.Numerical experiments on simulated data demonstrate accurate reconstructions with modest basis cardinality and robustness to noise, illustrating practical viability of the globally convergent approach and guiding extensions to higher dimensions.

Abstract

We propose a new approach to constructing globally strictly convex objective functional in a 1-D inverse medium scattering problem using multi-frequency backscattering data. The global convexity of the proposed objective functional is proved using a Carleman estimate. Due to its convexity, no good first guess is required in minimizing this objective functional. We also prove the global convergence of the gradient projection algorithm and derive an error estimate for the reconstructed coefficient. Numerical results show reasonable reconstruction accuracy for simulated data.

Paper Structure

This paper contains 9 sections, 7 theorems, 87 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Globally strictly convex functional
  4. Convexity, global convergence, and accuracy estimate
  5. Discretization and algorithm
  6. Discretization with respect to $x$
  7. Algorithm
  8. Numerical results
  9. Concluding Remarks

Key Result

Lemma 3.1

Let $x^0 < 0$ be the position of the point source and $u$ be the solution of problem (1.1)--(1.2). Under the condition (1.3), we have $u\left( x,k\right) \neq 0$ for all $x\in \left[ 0,b\right]$ and for all $k>0$.

Figures (9)

  • Figure 1: Comparison between the exact coefficient $c(x)$ and the approximate coefficient computed by \ref{['eq:c2']} with 3 basis functions and exact functions $v_n$. The derivative $v^{\prime }_n(x)$ is approximated by a finite difference quotient.
  • Figure 2: Comparison of the exact and reconstructed coefficient $c(x)$ for Example 1 for $5\%$ noisy data and the initial guess $Q_h^{(0)} \equiv 0$.
  • Figure 3: Comparison of the exact and reconstructed functions $V_n$ for Example 1 for $5\%$ noisy data and the initial guess $Q_h^{(0)} \equiv 0$.
  • Figure 4: Comparison of the exact and reconstructed coefficient $c(x)$ for Example 1 for noiseless data and with the initial guess $Q_h^{(0)} = Q^*$.
  • Figure 5: Comparison of the exact and reconstructed functions $V_n$ for Example 1 for noiseless data and with the initial guess $Q_h^{(0)} = Q^*$.
  • ...and 4 more figures

Theorems & Definitions (12)

  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • Remark 3.1
  • Lemma 4.1
  • Lemma 4.2
  • Theorem 4.3: Convexity
  • Theorem 4.4
  • Theorem 4.5: Global convergence
  • Theorem 4.6: Error estimates
  • ...and 2 more