Probabilistic Interpretation of the Calderón Problem

具有外部数据的分数阶各向异性Calderón问题

Fractional anisotropic Calderón problem with external data

本文在欧几里得空间中解决了二维及以上的分数阶各向异性Calderón问题，证明了部分外部Dirichlet--to--Neumann映射可确定黎曼度量至微分同胚，并给出了两种证明方法。

利用Caffarelli-Silvestre延拓重访各向异性分数阶Calderón问题

Revisiting the Anisotropic Fractional Calderón Problem Using the Caffarelli-Silvestre Extension

本文利用Caffarelli-Silvestre延拓方法重访各向异性分数阶Calderón问题，证明热核与波核可互相恢复，并建立Poisson核与波核的直接联系，从而简化恢复过程并联系局部与非局部问题。

Probabilistic interpretation of electrical impedance tomography

In this paper, we give probabilistic interpretations of both, the forward and the inverse problem of electrical impedance tomography with possibly anisotropic, merely measurable conductivities: Using the theory of symmetric Dirichlet spaces, Feynman-Kac type formulae corresponding to different electrode models on bounded Lipschitz domains are derived. Moreover, we give a probabilistic interpretation of the Calderón inverse conductivity problem in terms of reflecting diffusion processes and their corresponding boundary trace processes.

Symmetrization of exterior parabolic problems and probabilistic interpretation

We prove a comparison theorem for the averages of the solutions of two exterior parabolic problems, the second being the "symmetrization" of the first one, by using approximation of the Schwarz symmetrization by polarizations, as it was introduced in [4]. This comparison provides an alternative proof, based on PDEs, of the isoperimetric inequality for the Wiener sausage, which was proved in [14].

Beltrami场的Calderón问题

A Calderón Problem for Beltrami Fields

本文研究3维黎曼流形上Beltrami场的法向到切向映射，证明该零阶伪微分算子的全符号决定边界度量Taylor级数，并实现实解析单连通流形的重建。

On statistical Calderón problems

For $D$ a bounded domain in $\mathbb R^d, d \ge 2,$ with smooth boundary $\partial D$, the non-linear inverse problem of recovering the unknown conductivity $γ$ determining solutions $u=u_{γ, f}$ of the partial differential equation \begin{equation*} \begin{split} \nabla \cdot(γ\nabla u)&=0 \quad \text{ in }D, \\ u&=f \quad \text { on } \partial D, \end{split} \end{equation*} from noisy observations $Y$ of the Dirichlet-to-Neumann map \[f \mapsto Λ_γ(f) = {γ\frac{\partial u_{γ,f}}{\partial ν}}\Big|_{\partial D},\] with $\partial/\partial ν$ denoting the outward normal derivative, is considered…