The concept of convexification-based numerical methods was proposed recently by Alexandre Timonov and the Speaker. Both analytical and computational results will be presented in this talk.
Convexification-based numerical methods are constructed for a broad class of multidimensional coefficient inverse problems. These methods address rigorously the well known problem of multiple local minima of conventional least squares objective functions. An inverse problem is reformulated as a Cauchy-like problem for a non-linear integro-differential PDE, in which the unknown coefficient is not present. The latter problem is approximately solved via sequential minimization of a finite sequence of weighted least squares strictly convex objective functions. Weights in those objective functionals are the so-called Carleman Weight Functions, which are involved in the Carleman estimate for a certain differential operator.
The main advantage of the convexification approach is that convergence of the resulting algorithm to the exact solution is guaranteed regardless on the distance between the starting vector and this solution. The only condition is that the starting vector should belong to a compact set of ones a priori choice, i.e., correctness set. It is important, however that conditions of smallness are not imposed on the correctness set.
Thus, numerical methods with guaranteed global convergence are constructed. In particular, this means that inverse problems concerning with imaging of unknown targets embedded in unknown strongly scattering heterogeneous backgrounds can, in principle, be addressed.
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