Convexification for a 3D inverse scattering problem with the moving point source

Abstract

For the first time, we develop in this paper the globally convergent convexification numerical method for a Coefficient Inverse Problem for the 3D Helmholtz equation for the case when the backscattering data are generated by a point source running along an interval of a straight line and the wavenumber is fixed. Thus, by varying the wavenumber, one can reconstruct the dielectric constant depending not only on spatial variables but the wavenumber (i.e. frequency) as well. Our approach relies on a new derivation of a boundary value problem for a system of coupled quasilinear elliptic partial differential equations. This is done via an application of a special truncated Fourier-like method. First, we prove the Lipschitz stability estimate for this problem via a Carleman estimate. Next, using the Carleman Weight Function generated by that estimate, we construct a globally strictly convex cost functional and prove the global convergence to the exact solution of the gradient projection method. Finally, our theoretical finding is verified via several numerical tests with computationally simulated data. These tests demonstrate that we can accurately recover all three important components of targets of interest: locations, shapes and dielectric constants. In particular, large target/background contrasts in dielectric constants (up to 10:1) can be accurately calculated.

32 pages; 11 figures; 2 tables

References in corpus (2)

• The Cauchy problem of coupled elliptic sine-Gordon equations with noise: Analysis of a general kernel-based regularization and reliable tools of computing
• Numerical Solution of a Coefficient Inverse Problem with Multi-Frequency Experimental Raw Data by a Globally Convergent Algorithm

Cited by in corpus (7)

• A convergent numerical method to recover the initial condition of nonlinear parabolic equations from lateral Cauchy data
• The gradient descent method for the convexification to solve boundary value problems of quasi-linear PDEs and a coefficient inverse problem
• Carleman estimates and the contraction principle for an inverse source problem for nonlinear hyperbolic equation
• Numerical viscosity solutions to Hamilton-Jacobi equations via a Carleman estimate and the convexification method
• A Carleman-based numerical method for quasilinear elliptic equations with over-determined boundary data and applications
• Carleman contraction mapping for a 1D inverse scattering problem with experimental time-dependent data
• A globally convergent numerical method for a 3D coefficient inverse problem for a wave-like equation