Logarithmic Stability Estimate Research Articles

On the Logarithmic Stability Estimates of Non-autonomous Systems: Applications to Control Systems

On the Logarithmic Stability Estimates of Non-autonomous Systems: Applications to Control Systems

Stability estimates for an inverse boundary value problem for biharmonic operators with first order perturbation from partial data

In this paper we study an inverse boundary value problem for the biharmonic operator with first order perturbation. Our geometric setting is that of a bounded simply connected domain in the Euclidean space of dimension three or higher. Assuming that the inaccessible portion of the boundary is flat, and we have knowledge of the Dirichlet-to-Neumann map on the complement, we prove logarithmic type stability estimates for both the first and the zeroth order perturbation of the biharmonic operator.

Regularization of the inverse Laplace transform by mollification

In this paper we study the inverse Laplace transform. We first derive a new global logarithmic stability estimate that shows that the inversion is severely ill-posed. Then we propose a regularization method to compute the inverse Laplace transform using the concept of mollification. Taking into account the exponential instability we derive a criterion for selection of the regularization parameter. We show that by taking the optimal value of this parameter we improve significantly the convergence of the method. Finally, making use of the holomorphic extension of the Laplace transform, we suggest a new PDEs based numerical method for the computation of the solution. The effectiveness of the proposed regularization method is demonstrated through several numerical examples.

Inverse problems for general parabolic systems and application to Ornstein-Uhlenbeck equation

We investigate the link between inverse problems and final state observability for a general class of parabolic systems. We generalize a stability result for initial data known for the case of self-adjoint dissipative operators. More precisely, we consider a system governed by an analytic semigroup. Under the assumption of final state observability, we prove a logarithmic stability estimate depending on the analyticity angle of the semigroup. This is done by using a general logarithmic convexity result. The abstract result is illustrated by considering the Ornstein–Uhlenbeck equation.

Sharp uniqueness and stability of solution for an inverse source problem for the Schrödinger equation

This article is concerned with the uniqueness and the stability for an inverse source problem of determining a spatially varying factor f(x) of a source term given by with suitably given R(t) of the Schrödinger equation with time independent coefficients. In order to establish these results, for the Schrödinger equation, we prove (i) a logarithmic conditional stability estimate for a Cauchy problem attached with the zero Dirichlet boundary conditions on the whole boundary, (ii) the unique continuation for the Cauchy problem from an arbitrary small part of a lateral boundary. We do not assume any constraints on the geometry of the subboundary and an observation time length. The key is an integral transform with a kernel solving a null controllability problem for the one-dimensional Schrödinger equation, which transforms a solution of the Schrödinger equation to a solution of an elliptic equation.

Hölder stability for a semilinear elliptic inverse problem

We are concerned with the problem of determining the nonlinear term in a semilinear elliptic equation by boundary measurements. Precisely, we improve [5, Theorem 1.3], where a logarithmic type stability estimate was proved. We show actually that we have a Hölder stability estimate with less boundary measurements and less regular nonlinearities. We establish our stability inequality by following the same method as in [4]. This method consists in constructing special solutions vanishing on a subboundary of the domain.

Discrete Calderón problem with partial data

In this work, we are interested in analyzing the well-known Calderón problem, which is an inverse boundary value problem of determining a coefficient function of an elliptic partial differential equation from the knowledge of the associated Dirichlet-to-Neumann map on the boundary of a domain. We consider the discrete version of the Calderon inverse problem with partial boundary data; in particular, we establish logarithmic stability estimates for the discrete Calderón problem, in dimension , for the discrete H −r -norm on the boundary under suitable a priori bounds. The proof of our main result is based on a new discrete Carleman estimate for the discrete Laplacian operator with boundary observations.

Stability estimates for the inverse boundary value problem for the first order perturbation of the biharmonic operator

In this paper, we prove logarithmic stability estimates for the magnetic and electric potentials associated to the first order perturbation of the biharmonic operator, when the Dirichlet-to-Neumann maps are made on the whole boundary.

Stable determination of an anisotropic inclusion in the Schrödinger equation from local Cauchy data

We consider the inverse problem of determining an inclusion contained in a body for a Schrödinger type equation by means of local Cauchy data. Both the body and the inclusion are made by inhomogeneous and anisotropic materials. Under mild a priori assumptions on the unknown inclusion, we establish a logarithmic stability estimate in terms of the local Cauchy data. In view of possible applications, we also provide a stability estimate in terms of an ad-hoc misfit functional.

Stable determination of a second order perturbation of the polyharmonic operator by boundary measurements

In this paper, we consider the inverse boundary value problem for the polyharmonic operator. We prove that the second order perturbations are uniquely determined by the corresponding Dirichlet to Neumann map. More precisely, we show in dimension n≥3, a logarithmic type stability estimate for the inverse problem under consideration.

Recovering the initial condition in the one-phase Stefan problem

<p style='text-indent:20px;'>We consider the problem of recovering the initial condition in the one-dimensional one-phase Stefan problem for the heat equation from the knowledge of the position of the melting point. We first recall some properties of the free boundary solution. Then we study the uniqueness and stability of the inversion. The principal contribution of the paper is a new logarithmic type stability estimate that shows that the inversion may be severely ill-posed. The proof is based on integral equations representation techniques, and the unique continuation property for parabolic type solutions. We also present few numerical examples operating with noisy synthetic data.</p>

Local logarithmic stability of an inverse coefficient problem for a singular heat equation with an inverse-square potential

An inverse problem of the determination of the coefficient in the equation: is considered. The main difficulty here as compared with the existing result is that there is a singular potential in the equation. A local logarithmic stability estimate is obtained using the method of Carleman estimates. Our proof relies on the Bukhgeim–Klibanov method which was originated in [Bukhgeim AL, Klibanov MV. Uniqueness in the large of a class of multidiimensional inverse problems. Dokl Akad Nauk SSSR. 1981;17:244–247] to prove inverse source or coefficient results.

Identification of a boundary influx condition in a one-phase Stefan problem

We consider a one-dimensional one-phase inverse Stefan problem for the heat equation. It consists in recovering a boundary influx condition from the knowledge of the position of the moving front and the initial state. We derived a logarithmic stability estimate that shows that the inversion may be severely ill-posed. The proof is based on integral equations and unique continuation of holomorphic functions. We also proposed a direct algorithm with a regularization term to solve the nonlinear inverse problem. Several numerical tests using noisy data are provided with relative errors.

Stably determining time-dependent convection–diffusion coefficients from a partial Dirichlet-to-Neumann map

We study in this paper the inverse problem for the dynamical convection–diffusion equation. More precisely, we set logarithmic stability estimates in the determination of the two time-dependent first-order convection term and the scalar potential appearing in the heat equation. The observations here are taken only on an arbitrary open subset of the boundary and are given by a partial Dirichlet-to-Neumann map. For this end, we will reduce our initial problem into an auxiliary one then we will construct particular solutions and apply a special parabolic Carleman estimate.

On Single Measurement Stability for the Fractional Calderón Problem

In this short article we complement the known single measurement uniqueness result for the fractional Calderon problem by a single measurement logarithmic stability estimate. To this end, we combin...

An improved quasi-reversibility method for a terminal-boundary value multi-species model with white Gaussian noise

Upon the recent development of the quasi-reversibility method for terminal value parabolic problems in Nguyen et al. (2019), it is imperative to investigate the convergence analysis of this regularization method in the stochastic setting. In this paper, we positively unravel this open question by focusing on a coupled system of Dirichlet reaction–diffusion equations with additive white Gaussian noise on the terminal data. In this regard, the approximate problem is designed by adding the so-called perturbing operator to the original problem and by exploiting the Fourier reconstructed terminal data. By this way, Gevrey-type source conditions are included, while we successfully maintain the logarithmic stability estimate of the corresponding stabilized operator, which is necessary for the error analysis. As the main theme of this work, we prove the error bounds for the concentrations and for the concentration gradients, driven by a large amount of weighted energy-like controls involving the expectation operator. Compared to the classical error bounds in L2 and H1 that we obtained in the previous studies, our analysis here needs a higher smoothness of the true terminal data to ensure their reconstructions from the stochastic fashion. Two numerical examples are provided to corroborate the theoretical results.

Stable determination of polygonal inclusions in Calderón’s problem by a single partial boundary measurement

We are concerned with the Calderón problem of determining the unknown conductivity of a body from the associated boundary measurement. We establish a logarithmic type stability estimate in terms of the Hausdorff distance in determining the support of a convex polygonal inclusion by a single partial boundary measurement. We also derive the uniqueness result in a more general scenario where the conductivities are piecewise constants supported in a nested polygonal geometry. Our methods in establishing the stability and uniqueness results have a significant technical initiative and a strong potential to apply to other inverse boundary value problems.

The Calder\xf3n problem for the fractional Schr\xf6dinger equation with drift

We investigate the Calderón problem for the fractional Schrödinger equation with drift, proving that the unknown drift and potential in a bounded domain can be determined simultaneously and uniquely by an infinite number of exterior measurements. In particular, in contrast to its local analogue, this nonlocal problem does not enjoy a gauge invariance. The uniqueness result is complemented by an associated logarithmic stability estimate under suitable apriori assumptions. Also uniqueness under finitely many generic measurements is discussed. Here the genericity is obtained through singularity theory which might also be interesting in the context of hybrid inverse problems. Combined with the results from Ghosh et al. (Uniqueness and reconstruction for the fractional Calderón problem with a single easurement, 2018. arXiv:1801.04449), this yields a finite measurements constructive reconstruction algorithm for the fractional Calderón problem with drift. The inverse problem is formulated as a partial data type nonlocal problem and it is considered in any dimension nge 1.

A logarithmic estimate for the inverse source scattering problem with attenuation in a two-layered medium

Abstract The paper aims a logarithmic stability estimate for the inverse source problem of the one-dimensional Helmholtz equation with attenuation factor in a two layer medium. We establish a stability by using multiple frequencies at the two end points of the domain which contains the compact support of the source functions.

Logarithmic stability of an inverse problem for Biot’s consolidation system in poro-elasticity

In this paper, we consider a coupled system of mixed hyperbolic-parabolic type, which describes Biot’s consolidation model in poro-elasticity. We study an inverse problem of determining five spatially varying coefficients in the model, i.e. two Lamé coefficients, the secondary consolidation effects and two densities, by three measurements of displacement in an arbitrary subboundary and temperature in an arbitrary neighborhood of the boundary over a time interval. By assuming that, in a neighborhood of the boundary of the spatial domain, the densities, secondary consolidation effects and the Lamé coefficients are known, we prove a logarithmic stability estimate for the inverse problem.