A version of the convexification numerical method for a Coefficient Inverse Problem for a 1D hyperbolic PDE is presented. The data for this problem are generated by a single measurement event. This method converges globally. The most important element of the construction is the presence of the Carleman Weight Function in a weighted Tikhonovlike functional. This functional is strictly convex on a certain bounded set in a Hilbert space, and the diameter of this set is an arbitrary positive number. The global convergence of the gradient projection method is established. Computational results demonstrate a good performance of the numerical method for noisy data.

A new numerical method to solve an inverse source problem for the radiative transfer equation involving the absorption and scattering terms, with incomplete data, is proposed. No restrictive assumption on those absorption and scattering coefficients is imposed. The original inverse source problem is reduced to boundary value problem for a system of coupled partial differential equations of the first order. The unknown source function is not a part of this system. Next, we write this system in the fully discrete form of finite differences. That discrete problem is solved via the quasi-reversibility method. We prove the existence and uniqueness of the regularized solution. Especially, we prove the convergence of regularized solutions to the exact one as the noise level in the data tends to zero via a new discrete Carleman estimate. Numerical simulations demonstrate good performance of this method even when the data is highly noisy.

The forward problem here is the Cauchy problem for a 1D hyperbolic PDE with a variable coefficient in the principal part of the operator. That coefficient is the spatially distributed dielectric constant. The inverse problem consists of the recovery of that dielectric constant from backscattering boundary measurements. The data depend on one variable, which is time. To address this problem, a new version of the convexification method is analytically developed. The theory guarantees the global convergence of this method. Numerical testing is conducted for both computationally simulated and experimental data. Experimental data, which are collected in the field, mimic the problem of the recovery of the spatially distributed dielectric constants of antipersonnel land mines and improvised explosive devices.

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