[Paper Review] Convexification method for a coefficient inverse problem and its performance for experimental backscatter data for buried targets

Michael V. Klibanov, Aleksandr E. Kolesov+1 more|arXiv (Cornell University)|May 19, 2018

Numerical methods in inverse problems34 references6 citations

TL;DR

This paper presents a convexification method for solving a 3D coefficient inverse problem (CIP) with experimental backscatter data from buried targets, using a Carleman weight function to construct a globally strictly convex cost functional. The method ensures global convergence without local minima, achieving accurate reconstruction of dielectric constants and locations even with high noise levels, as validated on real experimental data from buried objects with errors below 10% in most cases.

ABSTRACT

We present in this paper a novel numerical reconstruction method for solving a 3D coefficient inverse problem with scattering data generated by a single direction of the incident plane wave. This inverse problem is well-known to be a highly nonlinear and ill-posed problem. Therefore, optimization-based reconstruction methods for solving this problem would typically suffer from the local-minima trapping and require strong a priori information of the solution. To avoid these problems, in our numerical method, we aim to construct a cost functional with a globally strictly convex property, whose minimizer can provide a good approximation for the exact solution of the inverse problem. The key ingredients for the construction of such functional are an integro-differential formulation of the inverse problem and a Carleman weight function. Under a (partial) finite difference approximation, the global strict convexity is proven using the tool of Carleman estimates. The global convergence of the gradient projection method to the exact solution is proven as well. We demonstrate the efficiency of our reconstruction method via a numerical study of experimental backscatter data for buried objects.

Motivation & Objective

• To develop a globally convergent numerical method for 3D coefficient inverse problems (CIPs) with scattering data from a single incident plane wave direction.
• To overcome the limitations of traditional optimization-based methods, such as local minima trapping and dependence on strong a priori information.
• To extend the convexification method to experimental backscatter data for buried targets, a significantly more challenging case than simulated data.
• To demonstrate global convergence and high accuracy in reconstructing both dielectric constants and spatial locations of buried objects from real-world noisy measurements.

Proposed method

• The method constructs a weighted cost functional using a Carleman weight function (CWF), ensuring global strict convexity of the functional.
• The CWF is derived from Carleman estimates for the Helmholtz operator, enabling global convergence of the gradient projection method.
• A semidiscrete finite difference approximation is used for two spatial variables, while the third remains continuous, balancing computational feasibility and theoretical rigor.
• The method avoids iterative tail function refinement and does not require a small wavenumber interval, improving robustness over prior globally convergent methods.
• Data preprocessing includes reference subtraction and backpropagation to improve signal-to-noise ratio and estimate target locations.
• The reconstruction is performed by minimizing the convex cost functional using a gradient projection method, proven to converge globally to the true solution within a neighborhood dependent only on noise and discretization errors.

Experimental results

Research questions

• RQ1 Can a convexification-based method achieve global convergence for a 3D coefficient inverse problem using experimental backscatter data from buried targets?
• RQ2 How does the use of a Carleman weight function ensure global strict convexity in the cost functional, enabling convergence to the true solution without local minima?
• RQ3 What is the performance of the method in reconstructing both dielectric constants and spatial locations of buried objects under high noise and limited data?
• RQ4 How does the method compare to prior globally convergent approaches in terms of assumptions, such as requiring a small wavenumber interval or iterative tail function refinement?

Key findings

• The method successfully reconstructed dielectric constants for six buried targets with computational errors ranging from 1.23% to 9.63%, all below or comparable to measurement errors.
• The location of the maximum reconstructed coefficient was within 0.25 m of the true target location in all cases, demonstrating high spatial accuracy.
• For object 3 (rock), the reconstructed dielectric constant was 5.07 with a 9.63% error, which is lower than its 21.3% measurement error, indicating robustness.
• The method achieved global convergence without requiring prior knowledge of the solution’s neighborhood, relying only on noise and discretization levels.
• The reconstructed images for objects 2 and 4 (geode and sycamore) closely matched the exact shapes, as visualized using Paraview contour filters.
• The optimal frequency and wavenumber intervals were determined for each target, with the method showing consistent performance across different materials and sizes.