Convexification of a 3-D coefficient inverse scattering problem

Abstract: A version of the so-called "convexification" numerical method for a coefficient inverse scattering problem for the 3D Hemholtz equation is developed analytically and tested numerically. Backscattering data are used, which result from a single direction of the propagation of the incident plane wave on an interval of frequencies. The method converges globally. The idea is to construct a weighted Tikhonov-like functional. The key element of this functional is the presence of the so-called Carleman Weight Function (CWF). This is the function which is involved in the Carleman estimate for the Laplace operator. This functional is strictly convex on any appropriate ball in a Hilbert space for an appropriate choice of the parameters of the CWF. Thus, both the absence of local minima and convergence of minimizers to the exact solution are guaranteed. Numerical tests demonstrate a good performance of the resulting algorithm. Unlikeprevious the so-called tail functions globally convergent method, we neither do not impose the smallness assumption of the interval of wavenumbers, nor we do not iterate with respect to the so-called tail functions.

• The paper introduces a globally convergent convexification method for 3D coefficient inverse scattering using a Carleman-weighted Tikhonov framework.
• It demonstrates accurate recovery of spatially-varying dielectric coefficients with reconstruction errors below 9% even under 15% noise.
• The method overcomes ill-posedness by constructing a strictly convex cost functional, ensuring robust convergence without strict initialization.

Convexification Method for 3D Coefficient Inverse Scattering Problems

Problem Setting and Methodological Framework

This work addresses the development and analysis of a convexification-based numerical method for the 3D coefficient inverse scattering problem (CISP) associated with the Helmholtz equation. The CISP under consideration involves backscattering data generated by a single incident plane wave, spanning an interval of frequencies. The principal objective is the stable recovery of the unknown spatially-varying dielectric coefficient $c(\mathbf{x})$ in the domain $\Omega$ from multi-frequency boundary measurements.

The approach is a direct extension of previous convexification methodology applied to 1D settings. The core difficulty arises from the severe ill-posedness and nonlinearity of the CISP, precluding unique continuation by classical least squares or standard Tikhonov regularization due to the proliferation of suboptimal local minima. The work builds upon globally convergent methods, which guarantee approximate recovery with error bounds dependent solely on data noise and discretization, without requiring proximity of initialization.

Construction of Strictly Convex Functionals

The methodological centerpiece is the construction of a globally strictly convex cost functional via the inclusion of a Carleman Weight Function (CWF) inspired by Carleman estimates and applied within a Tikhonov-like regularization framework. The CWF, $\varphi_\lambda(z) = \exp(2\lambda(z+s)^{-\nu})$, weights the residual of the associated PDE operator, and can be tuned to enforce strict convexity on balls of arbitrarily large radius in the appropriate Hilbert space.

Key elements include:

• Recasting the CISP as an integro-differential equation for a nonlinear transform of the total field, incorporating asymptotic expansions at high wavenumber.
• Introduction of the tail function approximation to enable decoupling of the nonlinear coefficient recovery and field update steps, without requiring iterative tail updates as in previous "tail function" methods.
• Formulation of the main cost functional $J_{\lambda,\rho}(p)$, designed so that both absence of spurious local minima and convergence of minimizers—under noisy data and discretization—are established analytically.

Central theorems rigorously demonstrate that for suitable $\lambda$, the functional is strictly convex on any bounded set. Global convergence of the gradient projection method is proved, and explicit rates are given for the convergence of minimizers to the true coefficient as the noise level $\delta \to 0$, with all constants independent of initialization.

Numerical Implementation and Results

The convexification algorithm is realized in a fully discrete framework using finite differences. Several nontrivial aspects are addressed:

• Data simulation uses the Lippmann-Schwinger integral equation with FFT-based solvers and simulated target inclusions, emulating physical experiments for detection of mine-like or IED-like targets.
• Preprocessing (data propagation) is employed to move measurements from distant planes to locations proximate to the inclusion, mitigating the impact of noise and enhancing the identifiability of target features.
• Strong numerical smoothing is observed as a consequence of the propagation and Carleman-weighted minimization.
• The main algorithm includes initialization of boundary data via propagation, computation of auxiliary fields, minimization of the weighted functionals for both the tail and main variable (using conjugate gradients), and back-calculation of the coefficient.

Numerical experiments demonstrate:

• Accurate location and dielectric property recovery for inclusions of varying size and contrast, including challenging cases of multiple inclusions with distinct parameters.
• Relative computational errors for the reconstructed coefficients do not exceed $9\%$ even under $15\%$ additive noise, with spatial localization accurate within grid resolution.
• The error in the reconstructed coefficient is consistently less than the noise level in the data, and the algorithm is robust with respect to initialization due to global convexity.

Theoretical and Practical Implications

This work provides a scalable globally convergent scheme for 3D CISPs, avoiding the limitations of previous globally convergent approaches that required small wavenumber intervals or iterative tail updates. By leveraging robust Carleman estimates and convexification, the ill-posedness is controlled at the level of the cost functional rather than post hoc regularization.

Practically, such methodology enables high-fidelity imaging and parameter identification with minimal measurement data, which is critical in applications like landmine and IED detection where full boundary control is unavailable. The method accommodates single measurement events over frequency intervals, further broadening applicability in resource-constrained or hostile environments.

Theoretically, this establishes a template for extending convexification strategies (traditionally limited to simpler PDEs or 1D) to more realistic 3D inverse problems. The flexibility of the CWF and Tikhonov weights in ensuring global convexity across large function balls points toward similar constructions in other classes of inverse PDE problems.

Future Directions

Several topics for further development emerge:

• Extension to fully discrete analogs in more general geometries, including adaptive or nonuniform grids, and alternative discretization such as finite elements.
• Rigorous analysis in the context of Maxwell’s system for applications where the scalar Helmholtz equation is insufficient.
• Investigation of the interplay between data propagation, Carleman weight design, and noise regularization for maximally efficient data utilization.
• Integration with more sophisticated data acquisition (e.g., compressive or phaseless data) and consideration of more severe modeling errors.

Conclusion

This paper addresses a fundamental open problem in coefficient inverse scattering, providing a globally convergent algorithm for 3D CISPs that is analytically justified via strict convexity from Carleman-weighted Tikhonov regularization. The algorithm exhibits strong numerical performance, achieving reconstruction errors well below the noise level and robust localization of inclusions with minimal a priori information. The approach offers a pathway for future rigorous and effective inversion algorithms for complex, high-dimensional nonlinear inverse problems (1801.04404).