☆ 4.4 Article

Determination of a coefficient in the wave equation with a single measurement

APPLICABLE ANALYSIS (2008)

Journal

APPLICABLE ANALYSIS

Volume 87, Issue 8, Pages 901-920

Publisher

TAYLOR & FRANCIS LTD

DOI: 10.1080/00036810802369249

Keywords

Carleman estimates; inverse problems

Categories

Mathematics, Applied

Funding

Japan Society [15340027]
Ministry of Education, Cultures, Sports and Technology [15654015]

Ask authors/readers for more resources

Protocol

Community support

Reagent

Community support

Abstract

We consider an inverse problem of finding the coefficient of the second-order derivatives in a second-order hyperbolic equation with variable coefficients. Under a weak regularity assumption and a geometrical condition on the metric, we prove the uniqueness in a multidimensional hyperbolic inverse problem with a single measurement of Neumann data on a suitable sub-boundary. Moreover we show that our uniqueness yields the Lipschitz stability estimate in L(2) space for solution to the inverse problem. The key is a Carleman estimate for a hyperbolic operator with variable coefficients.

Authors

I am an author on this paper

Click your name to claim this paper and add it to your profile.

Reviews

Primary Rating

4.4

Not enough ratings

Secondary Ratings

Novelty

-

Significance

-

Scientific rigor

-

Rate this paper

Recommended

Article Mathematics, Applied

Carleman estimates and some inverse problems for the coupled quantitative thermoacoustic equations by partial boundary layer data. Part II: Some inverse problems

Michel Cristofol, Shumin Li, Yunxia Shang

Summary: This paper focuses on determining coefficients and source term in a strongly coupled quantitative thermoacoustic system of equations. By adapting a Carleman estimate established in the first part of this paper series, stability estimates of Holder type involving the observation of only one component - either temperature or pressure - are proven.

MATHEMATICAL METHODS IN THE APPLIED SCIENCES (2023)

Add to Collection

Article Mathematics, Applied

Carleman estimates and some inverse problems for the coupled quantitative thermoacoustic equations by boundary data. Part I: Carleman estimates

Shumin Li, Yunxia Shang

Summary: In this paper, Carleman estimates and inverse problems for the coupled quantitative thermoacoustic equations are considered. In part I, Carleman estimates for the coupled quantitative thermoacoustic equations are established under suitable conditions, and the applications of these estimates to inverse problems are discussed in the succeeding part II paper.

JOURNAL OF INVERSE AND ILL-POSED PROBLEMS (2022)

Add to Collection

Article Mathematics, Applied

INVERSE PARABOLIC PROBLEMS BY CARLEMAN ESTIMATES WITH DATA TAKEN AT INITIAL OR FINAL TIME MOMENT OF OBSERVATION

Oleg Imanuvilov, Masahiro Yamamoto

Summary: This paper investigates a parabolic equation in a bounded domain & omega; over a time interval (0, T) with homogeneous Neumann boundary condition. It focuses on an inverse problem of determining a zeroth-order spatially varying coefficient through additional data of the solution u: u|(0,T )x & UGamma; and u(t0, & BULL;) in & omega; with t0 = 0 or t0 = T. The paper establishes a conditional Lipschitz stability estimate and uniqueness for the case t0 = T, and proves the uniqueness for the case t0 = 0 under additional conditions for & UGamma;. The uniqueness result adapts the method of M.V. Klibanov (Inverse Problems 8 (1992) 575-596) to the inverse problem in a bounded domain & omega;, by modifying the inverse parabolic problem to an inverse hyperbolic problem.

INVERSE PROBLEMS AND IMAGING (2023)

Add to Collection

Article Automation & Control Systems

CARLEMAN ESTIMATES OF REFINED STOCHASTIC BEAM EQUATIONS AND APPLICATIONS

Yongyi Yu, Ji-Feng Zhang

Summary: This paper investigates the global Carleman estimates for refined stochastic beam equations. By establishing a fundamental weighted identity and applying Carleman estimates, the exact controllability of the refined system and the uniqueness of the inverse problem are proved.

SIAM JOURNAL ON CONTROL AND OPTIMIZATION (2022)

Add to Collection

Article Mathematics, Applied

CARLEMAN ESTIMATES FOR A MAGNETOHYDRODYNAMICS SYSTEM AND APPLICATION TO INVERSE SOURCE PROBLEMS

Xinchi Huang, Masahiro Yamamoto

Summary: In this article, we examine the linearized magnetohydrodynamics system for incompressible flow in a three-dimensional bounded domain. We establish two types of Carleman estimates by combining the Carleman estimates for parabolic and elliptic equations. We then utilize these estimates to prove the Ho center dot lder type stability results for some inverse source problems.

MATHEMATICAL CONTROL AND RELATED FIELDS (2022)

Add to Collection

Article Mathematics, Applied

INVERSE PROBLEMS FOR A HALF-ORDER TIME-FRACTIONAL DIFFUSION EQUATION IN ARBITRARY DIMENSION BY CARLEMAN ESTIMATES

Xinchi Huang, Atsushi Kawamoto

Summary: In this study, a half-order time-fractional diffusion equation in arbitrary dimension is considered, and inverse problems of determining the source term or the diffusion coefficient from spatial data at an arbitrarily fixed time are investigated under some additional assumptions. The stability estimate of Lipschitz type in the inverse problems is established, with proofs based on the BukhgeimKlibanov method using Carleman estimates.

INVERSE PROBLEMS AND IMAGING (2022)

Add to Collection

Article Mathematics, Applied

Carleman estimates with sharp weights and boundary observability for wave operators with critically singular potentials

Alberto Enciso, Arick Shao, Bruno Vergara

Summary: Researchers established a new family of Carleman inequalities capturing natural boundary conditions and H-1 energy, and applied these estimates to prove a boundary observability property for the associated wave equations.

JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY (2021)

Add to Collection

Article Mathematics

Global Lipschitz stability for inverse problems of wave equations on Lorentzian manifolds

Hiroshi Takase

Summary: We prove global Lipschitz stability for an inverse source problem of a system of wave equations on a Lorentzian manifold. The conventional method is not sufficient for the application to obtain the Lipschitz stability for the hyperbolic partial differential equation with time-dependent coefficients, and further innovations are needed. In this paper, we present an improved global Carleman estimate and an energy estimate to obtain the Lipschitz stability.

JOURNAL OF DIFFERENTIAL EQUATIONS (2023)

Add to Collection

Article Mathematics

Inverse problems for first-order hyperbolic equations with time-dependent coefficients

Giuseppe Floridia, Hiroshi Takase

Summary: The study proves global Lipschitz stability for inverse source and coefficient problems of first-order linear hyperbolic equations, where the key lies in choosing the length of integral curves to construct a weight function for the Carleman estimate. These integral curves correspond to characteristic curves in certain cases.

JOURNAL OF DIFFERENTIAL EQUATIONS (2021)

Add to Collection

Article Mathematics, Applied

Inverse coefficient problem for a magnetohydrodynamics system by Carleman estimates

Xinchi Huang

Summary: This article examines a magnetohydrodynamics system for incompressible flow in a three-dimensional bounded domain, presenting stability results for the inverse coefficient problem and Carleman type inequalities for solutions and unknown coefficients. The proofs of the stability results are completed based on the Carleman estimates provided.

APPLICABLE ANALYSIS (2021)

Add to Collection

Article Mathematics, Applied

Inverse problems for stochastic parabolic equations with additive noise

Ganghua Yuan

Summary: This paper investigates two inverse problems for stochastic parabolic equations, one involving determining the history of a stochastic heat process and the random heat source, and the other involving determining two kinds of sources simultaneously. A new global Carleman estimate for the stochastic parabolic equation is the main tool for solving these inverse problems, leading to a conditional stability result.

JOURNAL OF INVERSE AND ILL-POSED PROBLEMS (2021)

Add to Collection

Article Mathematics, Applied

Inverse parabolic problems of determining functions with one spatial-component independence by Carleman estimate

Oleg Y. Imanuvilov, Yavar Kian, Masahiro Yamamoto

Summary: In this paper, we investigate an inverse problem for a parabolic equation, where we aim to determine a coefficient independent of one spatial component, using lateral boundary data. We utilize a Carleman estimate to provide a conditional stability estimate for the inverse problem. Additionally, we establish similar results for the corresponding inverse source problem.

JOURNAL OF INVERSE AND ILL-POSED PROBLEMS (2022)

Add to Collection

Article Mathematics, Applied

Carleman estimates and boundedness of associated multiplier operators

Eunhee Jeong, Yehyun Kwon, Sanghyuk Lee

Summary: The paper discusses the range of Carleman estimates, particularly focusing on the p, q ranges when dealing with Laplacian Delta, wave operator square, and heat operator. By utilizing uniform Sobolev type estimates and investigating the L-p-L-q boundedness of related multiplier operators, they were able to infer the applicable range of Carleman estimates and obtain some unique continuation results.

COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS (2022)

Add to Collection

Article Mathematics

Carleman estimates for the wave equation in heterogeneous media with non-convex interface

Lucie Baudouin, Pamela Godoy, Alberto Mercado

Summary: This article focuses on a wave equation with a discontinuous main coefficient, which models the evolution of wave amplitude in a medium composed of at least two different materials with different propagation speeds. The article discusses the construction of Carleman weights for this wave operator, allowing for generalizations to interfaces that are not necessarily boundaries of convex sets.

JOURNAL OF DIFFERENTIAL EQUATIONS (2022)

Add to Collection

Article Mathematics, Applied

Holder stability and uniqueness for the mean field games system via Carleman estimates

Michael V. Klibanov, Jingzhi Li, Hongyu Liu

Summary: This paper investigates the mathematical properties of the mean field games system (MFGS). It shows that the uniqueness of solutions to the MFGS can be guaranteed when only two terminal conditions or two initial conditions are given. Furthermore, Holder stability estimates are established. Two new Carleman estimates are introduced as the main mathematical tools, which may have applications in other contexts related to coupled parabolic PDEs.

STUDIES IN APPLIED MATHEMATICS (2023)

Add to Collection

Article Mathematics, Applied

A Carleman estimate for the linear magnetoelastic waves system and an inverse source problem in a bounded conductive medium

Mourad Bellassoued, Chaima Moufid, Masahiro Yamamoto

APPLICABLE ANALYSIS (2020)

Add to Collection

Article Mathematics, Applied

An inverse problem of finding two time-dependent coefficients in second order hyperbolic equations from Dirichlet to Neumann map

Mourad Bellassoued, Ibtissem Ben Aicha

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS (2019)

Add to Collection

Article Mathematics, Applied

An inverse problem for the linear Boltzmann equation with a time-dependent coefficient

Mourad Bellassoued, Yosra Boughanja

INVERSE PROBLEMS (2019)

Add to Collection

Article Mathematics

Simultaneous determination of two coefficients in the Riemannian hyperbolic equation from boundary measurements

Mourad Bellassoued, Zouhour Rezig

ANNALS OF GLOBAL ANALYSIS AND GEOMETRY (2019)

Add to Collection

Article Mathematics

Stability estimate in the inverse scattering for a single quantum particle in an external short-range potential

Mourad Bellassoued, Luc Robbiano

COMPTES RENDUS MATHEMATIQUE (2019)

Add to Collection

Article Mathematics, Applied

Stability estimate for an inverse problem of the convection-diffusion equation

Mourad Bellassoued, Imen Rassas

JOURNAL OF INVERSE AND ILL-POSED PROBLEMS (2020)

Add to Collection

Article Physics, Mathematical

Recovery of coefficients in the linear Boltzmann equation

Mourad Bellassoued, Yosra Boughanja

JOURNAL OF MATHEMATICAL PHYSICS (2019)

Add to Collection

Article Mathematics, Applied

Stable recovery of time dependent coefficient from arbitrary measurements for wave equation

Mourad Bellassoued, Oumaima Ben Fraj

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS (2020)

Add to Collection

Article Mathematics, Applied

A partial data inverse problem for the electro-magnetic wave equation and application to the related Borg-Levinson theorem

Mourad Bellassoued, Yosra Mannoubi

INVERSE PROBLEMS (2020)

Add to Collection

Article Mathematics, Applied

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

Mourad Bellassoued, Oumaima Ben Fraj

Summary: In this paper, we investigate the inverse problem for the dynamical convection-diffusion equation, setting logarithmic stability estimates in determining the time-dependent convection term and scalar potential. Observations are made on an arbitrary open subset of the boundary and given by a partial Dirichlet-to-Neumann map. The initial problem is reduced to an auxiliary one, with particular solutions constructed and a special parabolic Carleman estimate applied.

INVERSE PROBLEMS (2021)

Add to Collection

Article Mathematics, Applied

Analysis of the heart-torso conductivity parameters recovery inverse problem in cardiac electrophysiology ECG modelling

Abir Amri, Mourad Bellassoued, Moncef Mahjoub, Nejib Zemzemi

Summary: In this paper, we prove a stability estimate of the conductivity parameters identification problem in cardiac electrophysiology. The main difficulty that we solve in this paper is related to the transmission conditions between the heart and the torso. Using Carleman estimates and the Bukhgeim and Klibanov approach, we establish a Lipschitz stability estimate of cardiac and torso conductivity parameters.

APPLICABLE ANALYSIS (2023)

Add to Collection

Article Mathematics, Applied

LIPSCHITZ STABILITY IN DETERMINATION OF COEFFICIENTS IN A TWO-DIMENSIONAL BOUSSINESQ SYSTEM BY ARBITRARY BOUNDARY OBSERVATION

Mourad Bellassoued, Chaima Moufid

Summary: This paper considers the inverse problem of determining two spatially varying coefficients in the two-dimensional Boussinesq system from observed data of velocity vector and temperature in a given subboundary. Using Carleman estimates, a Lipschitz stability result is proven.

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S (2021)

Add to Collection

Article Mathematics, Applied

STABLE DETERMINATION OF A VECTOR FIELD IN A NON-SELF-ADJOINT DYNAMICAL SCHRODINGER EQUATION ON RIEMANNIAN MANIFOLDS

Mourad Bellassoued, Ibtissem Ben Aicha, Zouhour Rezig

Summary: This paper addresses an inverse problem for a non-self-adjoint Schrodinger equation on a compact Riemannian manifold, aiming to stably determine a real vector field from the dynamical Dirichlet-to-Neumann map. In dimensions n >= 2, a Holder type stability estimate for the inverse problem is established, primarily relying on reduction to an equivalent problem for an electro-magnetic Schrodinger equation and the use of a Carleman estimate designed for elliptic operators.

MATHEMATICAL CONTROL AND RELATED FIELDS (2021)

Add to Collection

Article Mathematics, Applied

STABLE RECONSTRUCTION OF THE VOLATILITY IN A REGIME-SWITCHING LOCAL-VOLATILITY MODEL

Mourad Bellassoued, Raymond Brummelhuis, Michel Cristofol, Eric Soccorsi

MATHEMATICAL CONTROL AND RELATED FIELDS (2020)

Add to Collection

Article Mathematical & Computational Biology

Ionic parameters identification of an inverse problem of strongly coupled PDE's system in cardiac electrophysiology using Carleman estimates

Yassine Abidi, Mourad Bellassoued, Moncef Mahjoub, Nejib Zemzemi

MATHEMATICAL MODELLING OF NATURAL PHENOMENA (2019)

Add to Collection

No Data Available

© Peeref 2019-2024. All rights reserved.