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Carleman Estimates For Globally Convergent Numerical Methods for Coefficient Inverse Problems 

Mikhail V. Klibanov (U of North Carolina Charlotte)
Friday, November 15, 2019 - 4:00pm to 5:00pm
PDL C-38

 Abstract: A Coefficient Inverse Problem (CIP) is a problem of finding an unknown coefficient of a Partial Differential Equation (PDE) from boundary measurements of the solution of this PDE. That coefficient describes a physical property of an unknown medium, e.g. acoustic speed, dielectric constant, electric conductivity, heat conduction, etc. The latter clearly indicates a large volume of applications of CIPs.
However, there is a fundamental challenge in the goal of the numerical solution of any CIP. Indeed, any CIP is both nonlinear and ill-posed. These two factors cause the well known phenomenon of multiple local minima and ravines of any conventional Tikhonov least squares functional. Since, on the other hand, any gradient-like method of the minimization of this functional stops at any point of a local minimum, which is not necessary close to the correct solution, then conventional optimization methods deliver unstable and unreliable solutions for CIPs.
To overcome the above stumbling block, we have invented a radically new numerical method, called "convexification". The convexification constructs globally strictly convex Tikhonov-like functionals for CIPs. The key point of such a functional is the presence of the so-called Carleman Weight Function. This is the function which is involved in the so-called Carleman estimate for the original PDE operator.
The global strict convexity guarantees all nice properties which numerics wants:
1. The absence of local minima of that new functional.
2. The existence of the unique global minimum.
3. The global convergence to the correct solution of the gradient projection method of the optimization of that functional.
4. A good first guess for the solution is no longer necessary.
The convexification is a universal approach which works for a broad variety of CIP.
First, we will present the theory of the convexification for just one example of a CIP for the Helmholtz equation in 3D. Next, we will present numerical results for this CIP. In particular, we will present results for microwave experimental data. The data were collected in our microwave laboratory. Next, we will give an overview of all other convexification-like results for many CIPs which we have obtained so far.

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