Globally strictly convex cost functional for an inverse parabolic problem

Recovering of dielectric constants of explosives via a globally strictly convex cost functional

The inverse problem of estimating dielectric constants of explosives using boundary measurements of one component of the scattered electric field is addressed. It is formulated as a coefficient inverse problem for a hyperbolic differential equation. After applying the Laplace transform, a new cost functional is constructed and a variational problem is formulated. The key feature of this functional is the presence of the Carleman Weight Function for the Laplacian. The strict convexity of this functional on a bounded set in a Hilbert space of an arbitrary size is proven. This allows for establis…

关于全局双曲时空中的完备陷俘子流形

On complete trapped submanifolds in globally hyperbolic spacetimes

本文研究全局双曲时空中抛物型类空子流形的刚性与不存在性，通过因果平均曲率向量场条件，并应用于广义相对论中的陷俘曲面，恢复并推广了标准静态及广义罗伯逊-沃克时空中的相关结果。

Numerical solution of an ill-posed Cauchy problem for a quasilinear parabolic equation using a Carleman weight function

This is the first publication in which an ill-posed Cauchy problem for a quasi- linear PDE is solved numerically by a rigorous method. More precisely, we solve the side Cauchy problem for a 1-d quasilinear parabolc equation. The key idea is to minimize a strictly convex cost functional with the Carleman Weight Function in it. Previous publications about numerical solutions of ill-posed Cauchy problems were considering only linear equations.

Carleman weight functions for a globally convergent numerical method for ill-posed Cauchy problems for some quasilinear PDEs

In a series of publications of the second author, including some with coauthors, globally strictly convex Tikhonov-like functionals were constructed for some nonlinear ill-posed problems. The main element of such a functional is the presence of the Carleman Weight Function. Compared with previous publications, the main novelty of this paper is that the existence of the regularized solution (i.e. the minimizer) is proved rather than assumed. The method works for both ill-posed Cauchy problems for some quasilinear PDEs of the second order and for some Coefficient Inverse Problems. However, to si…