Conditional Stability Estimate Research Articles

Determining the initial distribution in space-fractional diffusion by a negative exponential regularization method

In this paper, we consider the backward problem for diffusion equation with space fractional Laplacian, i.e. determining the initial distribution from the final value measurement data. In order to overcome the ill-posedness of the backward problem, we present a so-called negative exponential regularization method to deal with it. Based on the conditional stability estimate and an a posteriori regularization parameter choice rule, the convergence rate estimate are established under a-priori bound assumption for the exact solution. Finally, several numerical examples are proposed to show that the numerical methods are effective.

Inverse Boundary Value Problem For The Helmholtz Equation: Quantitative Conditional Lipschitz Stability Estimates

We study the inverse boundary value problem for the Helmholtz equation using the Dirichlet-to-Neumann map at selected frequencies as the data. A conditional Lipschitz stability estimate for the inverse problem holds in the case of wavespeeds that are a linear combination of piecewise constant functions (following a domain partition) and gives a framework in which the scheme converges. The stability constant grows exponentially as the number of subdomains in the domain partition increases. We establish an order optimal upper bound for the stability constant. We eventually realize computational experiments to demonstrate the stability constant evolution for three dimensional wavespeed reconstruction.

Stability for Some Inverse Problems for Transport Equations

In this article, we consider inverse problems of determining a source term and a coefficient of a first-order partial differential equation and prove conditional stability estimates with minimum boundary observation data and relaxed condition on the principal part.

A Modified Method for a Cauchy Problem of the Helmholtz Equation

In this paper, a Cauchy problem for the Helmholtz equation is investigated. It is well known that this problem is severely ill-posed in the sense that the solution (if it exists) does not depend continuously on the given Cauchy data. To overcome such difficulties, we propose a modified regularization method to approximate the solution of this problem, and then analyze the stability and convergence of the proposed regularization method based on the conditional stability estimates. Finally, we present two numerical examples to illustrate that the proposed regularization method works well.

Stability and uniqueness for a two-dimensional inverse boundary value problem for less regular potentials

We consider inverse boundary value problems for the Schrodinger equations in two dimensions. Within less regular classes of potentials, we establish a conditional stability estimate of logarithmic order. Moreover we prove the uniqueness within $L^p$-class of potentials with $p > 2$.

Verification of a variational source condition for acoustic inverse medium scattering problems

This paper is concerned with the classical inverse scattering problem to recover the refractive index of a medium given near or far field measurements of scattered time-harmonic acoustic waves. It contains the first rigorous proof of (logarithmic) rates of convergence for Tikhonov regularization under Sobolev smoothness assumptions for the refractive index. This is achieved by combining two lines of research, conditional stability estimates via geometrical optics solutions and variational regularization theory.

Rate-Independent Dynamics and Kramers-Type Phase Transitions in Nonlocal Fokker–Planck Equations with Dynamical Control

The hysteretic behavior of many-particle systems with non-convex free energy can be modeled by nonlocal Fokker–Planck equations that involve two small parameters and are driven by a time-dependent constraint. In this paper we consider the fast reaction regime related to Kramers-type phase transitions and show that the dynamics in the small-parameter limit can be described by a rate-independent evolution equation with hysteresis. For the proof we first derive mass-dissipation estimates by means of Muckenhoupt constants, formulate conditional stability estimates, and characterize the mass flux between the different phases in terms of moment estimates that encode large deviation results. Afterwards we combine all these partial results and establish the dynamical stability of localized peaks as well as sufficiently strong compactness results for the basic macroscopic quantities.

Stability estimate and the modified regularization method for a Cauchy problem of the fractional diffusion equation

In this paper we investigate a non-characteristic Cauchy problem for a fractional diffusion equation. Using the Fourier transformation technique, we give a conditional stability estimate on the solution. Since the problem is highly ill-posed in the Hadamard sense, a modified version of the Tikhonov regularization technique is devised for stable numerical reconstruction of the solution. An error bound with optimal order is proven. For illustration, several numerical experiments are constructed to demonstrate the feasibility and efficiency of the proposed method.

Conditional stability in a backward parabolic system

We consider a parabolic system with components in a bounded spatial domain over a time interval whose principal part is coupled and discuss the backward problem in time of determining initial data , from , . We prove two conditional stability estimates under suitable a priori boundness assumptions on the solution : and , where and . The proof is based on a Carleman estimate with the weight function with some constant and a large parameter .

The index function and Tikhonov regularization for ill-posed problems

In this paper, we study the regularizing properties of the conditional stability estimates in ill-posed problems. First, we analyze how conditional stability estimates occur, and which properties the corresponding index functions must obey. In addition, we adapt the convergence analysis for the Tikhonov regularization in Banach spaces where the difference between the approximated solution and the exact one in metric measure is taken into account. We conclude this study with a comparison of stability estimates and variational inequalities, another emerging tool in Banach space regularization.

Determination of an electromagnetic potential for the Dirac equation

In this paper, we study the inverse problems of determining a spatially varying electromagnetic potential of the Dirac equation by overdetermining data on a boundary or in a subdomain over a sufficiently long time interval. By suitably choosing initial values once or twice, we prove conditional stability estimates for the inverse problems. Our arguments are based on the Carleman estimate.

Conditional stability in determining a zeroth-order coefficient in a half-order fractional diffusion equation by a Carleman estimate

An inverse problem of determining a zeroth-order coefficient in a one-dimensional fractional diffusion equation of half-order in time is investigated. Under some assumptions on the regularity of the solutions and coefficients, we prove a conditional stability estimate by some additional data. The key is a Carleman estimate, but since we have no Carleman estimates for the fractional diffusion equation, we further take the t-derivative of half-order to obtain the equation where the principal term is . The nonhomogeneous term is coupled with derivatives of the difference between two coefficients, and so we need an additional Carleman estimate, which is different from usual coefficient inverse problems for partial differential equations.

Two Tikhonov‐type regularization methods for inverse source problem on the Poisson equation

In this paper, we investigate a problem of the identification of an unknown source on Poisson equation from some fixed location. A conditional stability estimate for an inverse heat source problem is proved. We show that such a problem is mildly ill‐posed and further present two Tikhonov‐type regularization methods (a generalized Tikhonov regularization method and a simplified generalized Tikhonov regularization method) to deal with this problem. Convergence estimates are presented under the a priori choice of the regularization parameter. Numerical results are presented to illustrate the accuracy and efficiency of our methods. Copyright © 2012 John Wiley & Sons, Ltd.

Unique continuation on a line for the Helmholtz equation

In this article, local unique continuation on a line for solutions of the Helmholtz equation is discussed. The fundamental solution of the exterior problem for the Helmholtz equation have a logarithmic singularity which behaves similar to those of the interior problem for the Laplace equation in two dimension. A Hölder-type conditional stability estimate of the proposed exterior problem for the Helmholtz equation is obtained by adopting the complex extension method in Cheng and Yamamoto [J. Cheng and M. Yamamoto, Unique continuation on a line for harmonic functions, Inverse Probl. 14 (1998), pp. 869–882]. Finally, a regularization scheme based on the collocation method is compatible with the Hölder-type stability estimate provided that the line does not intersect the boundary of the domain for both the Laplace and the Helmholtz equations.

Carleman estimate for stochastic parabolic equations and inverse stochastic parabolic problems

In this paper, we establish a global Carleman estimate for stochastic parabolic equations. Based on this estimate, we study two inverse problems for stochastic parabolic equations. One is concerned with a determination problem of the history of a stochastic heat process through the observation at the final time T for which we obtain a conditional stability estimate. The other is an inverse source problem with observation on the lateral boundary. We derive the uniqueness of the source.

An implicit leap-frog discontinuous Galerkin method for the time-domain Maxwell’s equations in metamaterials

Numerical simulation of metamaterials play a very important role in the design of invisibility cloak, and sub-wavelength imaging. In this paper, we propose a leap-frog discontinuous Galerkin method to solve the time-dependent Maxwell’s equations in metamaterials. Conditional stability and error estimates are proved for the scheme. The proposed algorithm is implemented and numerical results supporting the analysis are provided.

Conditional Stability Estimates for Ill-Posed PDE Problems by Using Interpolation

The focus of this article is on conditional stability estimates for ill-posed inverse problems in partial differential equations. Conditional stability estimates have been obtained in related literature by a couple different methods. In this article, we propose a method called interpolation method, which is based on interpolation in variable Hilbert scales. We provide the theoretical background of this method and show that optimal conditional stability estimates are obtained. The capabilities of our method are illustrated by a comprehensive collection of different inverse and ill-posed PDE problems containing elliptic and parabolic problems, one source problem and the problem of analytic continuation.

Inverse problem and null-controllability for parabolic systems

Abstract In this paper, we present some abstract results giving a general connection between null-controllability and several inverse problems for a class of parabolic equations. We obtain some conditional stability estimates for the inverse problems consisting of determining the initial condition and the source term, from interior or boundary measurements. We apply this framework for Stokes system with interior and boundary observations, for a coupling of two Stokes system and a linear fluid-structure system.

Convergence Estimates for Some Regularization Methods to Solve a Cauchy Problem of the Laplace Equation

In this paper, we give a general proof on convergence estimates for some regularization methods to solve a Cauchy problem for the Laplace equation in a rectangular domain. The regularization methods we considered are: a non-local boundary value problem method, a boundary Tikhonov regularization method and a generalized method. Based on the conditional stability estimates, the convergence estimates for various regularization methods are easily obtained under the simple verifications of some conditions. Numerical results for one example show that the proposed numerical methods are effective and stable.

Two numerical methods for a Cauchy problem for modified Helmholtz equation

We consider a Cauchy problem for the modified Helmholtz equation. A conditional stability estimate is proved. Two order optimal regularization methods and error estimates are investigated. Numerical experiment shows that these methods work well.