Convergence and Stability of the Inverse Scattering Series for Diffuse Waves

扩散波与标量波逆散射级数的范数依赖收敛性与稳定性

Norm-dependent convergence and stability of the inverse scattering series for diffuse and scalar waves

本文在Sobolev $H^s$范数下分析标量波与扩散波的正逆散射级数的收敛性与稳定性，揭示频率偏置效应，并为实际反演中的频率加权技术提供理论依据。

不同表面上的相分离与扩散界面近似的收敛性

Phase separation on varying surfaces and convergence of diffuse interface approximations

本文研究欧氏空间广义超曲面上的相分离问题，基于流上广义BV函数证明Modica-Mortola型扩散表面能在锐界面极限下的紧性与下界估计，并拓展至含Willmore能的相分离生物膜场景。

Semiclassical limit to the Vlasov equation with inverse power law potentials

We consider mixed quasi-free states describing $N$ fermions in the mean-field limit. In this regime, the time evolution is governed by the nonlinear Hartree equation. In the large $N$ limit, we study the convergence towards the classical Vlasov equation. Under integrability and regularity assumptions on the initial state, we prove strong convergence in trace and Hilbert-Schmidt norm and provide explicit bounds on the convergence rate for a class of singular potentials of the form $V(x)=|x|^{-α}$, for $α\in(0,1/2)$.

"Gradient-free" diffuse approximations of the Willmore functional and Willmore flow

We introduce new diffuse approximations of the Willmore functional and the Willmore flow. They are based on a corresponding approximation of the perimeter that has been studied by Amstutz-van Goethem [{\em Interfaces Free Bound. 14 (2012)}]. We identify the candidate for the $Γ$--convergence, prove the $Γ$--limsup statement and justify the convergence to the Willmore flow by an asymptotic expansion. Furthermore, we present numerical simulations that are based on the new approximation.

Convergence of general inverse $σ_k$-flow on Kähler manifolds with Calabi Ansatz

We study the convergence behavior of the general inverse $σ_k$-flow on Kähler manifolds with initial metrics satisfying the Calabi Ansatz. The limiting metrics can be either smooth or singular. In the latter case, interesting conic singularities along negatively self-intersected sub-varieties are formed as a result of partial blow-up.

Convergence Rates of Tikhonov Regularizations for Elliptic and Parabolic Inverse Radiativity Problems

We shall study in this paper the Lipschitz type stabilities and convergence rates of Tikhonov regularization for the recovery of the radiativities in elliptic and parabolic systems with Dirichlet boundary conditions. The Lipschitz type stability estimates are derived. Due to the difficulty of the verification of the existing source conditions or nonlinearity conditions for the considered inverse radiativity problems in high dimensional spaces, some new variational source conditions are proposed. The conditions are rigorously verified in general dimensional spaces under the Lipschitz type stabi…

Insufficient convergence of inverse mean curvature flow on asymptotically hyperbolic manifolds

We construct a solution to inverse mean curvature flow on an asymptotically hyperbolic 3-manifold which does not have the convergence properties needed in order to prove a Penrose--type inequality. This contrasts sharply with the asymptotically flat case. The main idea consists in combining inverse mean curvature flow with work done by Shi--Tam regarding boundary behavior of compact manifolds. Assuming the Penrose inequality holds, we also derive a nontrivial inequality for functions on $S^2$.

非齐次 Allen-Cahn 方程向加权平均曲率流的收敛性

Convergence of a heterogeneous Allen-Cahn equation to weighted mean curvature flow

本文研究带移动势阱的非齐次相场模型，证明扩散界面消失时 Allen-Cahn 方程弱解收敛到加权平均曲率流的 BV 解，并建立其弱-强唯一性。

Stability of an inverse problem for the discrete wave equation and convergence results

Using uniform global Carleman estimates for discrete elliptic and semi-discrete hyperbolic equations, we study Lipschitz and logarithmic stability for the inverse problem of recovering a potential in a semi-discrete wave equation, discretized by finite differences in a 2-d uniform mesh, from boundary or internal measurements. The discrete stability results, when compared with their continuous counterparts, include new terms depending on the discretization parameter h. From these stability results, we design a numerical method to compute convergent approximations of the continuous potential.

Discrete regularization and convergence of the inverse problem for 1+1 dimensional wave equation

An inverse boundary value problem for the 1+1 dimensional wave equation $(\partial_t^2 - c(x)^2 \partial_x^2)u(x,t)=0,\quad x\in\mathbb{R}_+$ is considered. We give a discrete regularization strategy to recover wave speed $c(x)$ when we are given the boundary value of the wave, $u(0,t)$, that is produced by a single pulse-like source. The regularization strategy gives an approximative wave speed $\widetilde c$, satisfying a Hölder type estimate $\| \widetilde c-c\|\leq C ε^γ$, where $ε$ is the noise level.

Lipschitz stability at the boundary for time-harmonic diffuse optical tomography

We study the inverse problem in Optical Tomography of determining the optical properties of a medium $Ω\subset\mathbb{R}^n$, with $n\geq 3$, under the so-called diffusion approximation. We consider the time-harmonic case where $Ω$ is probed with an input field that is modulated with a fixed harmonic frequency $ω=\frac{k}{c}$, where $c$ is the speed of light and $k$ is the wave number. We prove a result of Lipschitz stability of the absorption coefficient $μ_a$ at the boundary $\partialΩ$ in terms of the measurements in the case when the scattering coefficient $μ_s$ is assumed to be known and $…

双曲方程的逆源问题与拟可逆性方法的类Lipschitz收敛性

An inverse source problem for hyperbolic equations and the Lipschitz-like convergence of the quasi-reversibility method

针对一般双曲方程逆源问题，提出含Volterra积分方程结合拟可逆性方法的数值解法，证明其类Lipschitz收敛性并通过数值测试验证良好重建精度。

Born and inverse Born series for scattering problems with Kerr nonlinearities

We consider the Born and inverse Born series for scalar waves with a cubic nonlinearity of Kerr type. We find a recursive formula for the operators in the Born series and prove their boundedness. This result gives conditions which guarantee convergence of the Born series, and subsequently yields conditions which guarantee convergence of the inverse Born series. We also use fixed point theory to give alternate explicit conditions for convergence of the Born series. We illustrate our results with numerical experiments.

Uniqueness, stability and global convergence for a discrete inverse elliptic Robin transmission problem

We derive a simple criterion that ensures uniqueness, Lipschitz stability and global convergence of Newton's method for the finite dimensional zero-finding problem of a continuously differentiable, pointwise convex and monotonic function. Our criterion merely requires to evaluate the directional derivative of the forward function at finitely many evaluation points and for finitely many directions. We then demonstrate that this result can be used to prove uniqueness, stability and global convergence for an inverse coefficient problem with finitely many measurements. We consider the problem of d…

Convergence of an inverse problem for discrete wave equations

It is by now well-known that one can recover a potential in the wave equation from the knowledge of the initial waves, the boundary data and the flux on a part of the boundary satisfying the Gamma-conditions of J.-L. Lions. We are interested in proving that trying to fit the discrete fluxes, given by discrete approximations of the wave equation, with the continuous one, one recovers, at the limit, the potential of the continuous model. In order to do that, we shall develop a Lax-type argument, usually used for convergence results of numerical schemes, which states that consistency and uniform …

Boundary regularity for the Ricci equation, geometric convergence, and Gel'fand's inverse boundary problem

This paper explores and ties together three themes. The first is to establish regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The second is to establish geometric convergence of a (sub)sequence of manifolds with boundary with such geometrical bounds and also an upper bound on the diameter and a lower bound on injectivity and boundary injectivity radius, making use of the first part. The third theme involves the uniqueness and conditional st…

Hölder-Logarithmic 稳定性与收敛速率研究反随机源问题

Hölder-Logarithmic Stability and Convergence Rates for an Inverse Random Source Problem

本文研究了反随机源问题的Hölder-对数稳定性与收敛速率，通过复几何光学解和变分源条件，推导了Sobolev正则性下的稳定性估计和收敛速率，并通过数值实验验证了理论结果。