A phase-field approach for detecting cavities via a Kohn-Vogelius type functional

一种利用拓扑灵敏度分析和狄利克雷-Neumann成本函数方法的快速重建算法

A fast reconstruction algorithm for geometric inverse problems using topological sensitivity analysis and Dirichlet-Neumann cost functional approach

本文提出了一种基于Kohn-Vogelius公式和拓扑灵敏度分析的快速算法，用于从边界测量中重建浸入各向异性介质中的物体，通过拓扑优化方法最小化能量型函数，利用拓扑渐近展开重构未知物体。

On a two-phase Serrin-type problem and its numerical computation

We consider an overdetermined problem of Serrin-type with respect to an operator in divergence form with piecewise constant coefficients. We give sufficient condition for unique solvability near radially symmetric configurations by means of a perturbation argument relying on shape derivatives and the implicit function theorem. This problem is also treated numerically, by means of a steepest descent algorithm based on a Kohn-Vogelius functional.

On Runge approximation and Lipschitz stability for a finite-dimensional Schrödinger inverse problem

In this note we reprove the Lipschitz stability for the inverse problem for the Schrödinger operator with finite-dimensional potentials by using quantitative Runge approximation results. This provides a quantification of the Schrödinger version of the argument from Kohn and Vogelius in Comm. Pure Appl. Math. (1985) and presents a slight variant of the strategy considered by Alessandrini, de Hoop, Gaburro and Sincich in Asymptotic Analysis (2018) which may prove useful also in the context of more general operators.

Uniqueness, Lipschitz stability and reconstruction for the inverse optical tomography problem

In this paper, we consider the inverse problem of recovering a diffusion and absorption coefficients in steady-state optical tomography problem from the Neumann-to-Dirichlet map. We first prove a Global uniqueness and Lipschitz stability estimate for the absorption parameter provided that the diffusion is known. Then, we prove a Lipschitz stability result for simultaneous recovery of diffusion and absorption. In both cases the parameters belong to a known finite subspace with a priori known bounds. The proofs relies on a monotonicity result combined with the techniques of localized potentials.…

Stability estimates for Navier-Stokes equations and application to inverse problems

In this work, we present some new Carleman inequalities for Stokes and Oseen equations with non-homogeneous boundary conditions. These estimates lead to log type stability inequalities for the problem of recovering the solution of the Stokes and Navier-Stokes equations from both boundary and distributed observations. These inequalities fit the well-known unique continuation result of Fabre and Lebeau [18]: the distributed observation only depends on interior measurement of the velocity, and the boundary observation only depends on the trace of the velocity and of the Cauchy stress tensor measu…

A regularization approach for an inverse source problem in elliptic systems from single Cauchy data

In this paper we investigate the problem of identifying the source term in an elliptic system from a single noisy measurement couple of the Neumann and Dirichlet data. A variational method of Tikhonov-type regularization with specific misfit term of Kohn-Vogelius-type and quadratic stabilizing penalty term is suggested to tackle this linear inverse problem. The method also appears as a variant of the Lavrentiev regularization. For the occurring linear inverse problem in infinite dimensional Hilbert spaces, convergence and rate results can be found from the general theory of classical Tikhonov …