Observability inequality for the wave equation
In this paper, only Carleman estimates are used, without energy estimates, we derive observability inequality. The main tool consists in the use of a new Carleman estimate.
math.AP↗
arXiv subjects
Suliang Si
Publications and source records attributed to Suliang Si.
Arefinement of the Bukhgeim-Klibanov method
In this article, we improve the classical Bukhgeim-Klibanov method presented in [1],which can be used to prove the conditional stability of inverse source problem for a hyperbolic equation from the measurement on the subboundary. A major ingredient of our proof is a novel Carleman estimate. This inequality eliminates the need to extend the solution in time, therefore simplifies the existing proofs, which is widely applicable to various evolution equations.
math.AP↗
Determination of the potential by a fixed angle scattering data
In this paper, we show that a compactly supported potential is uniquely determined by the far field pattern at a fixed angle. Our method is based on a new Carleman estimate and the ideas introduced by Bukhgeim and Klibanov on the use of Carleman estimates for inverse problems.
math.AP↗
Inverse source problem for a hyperbolic equation by Carleman estimates
In this article, we provide a modified argument for proving the conditional stability of inverse source problem for a hyperbolic equation. Our method does not require any extension of solution with respect to time and therefore simplifies the existing proofs, which is widely applicable to various evolution equations.
math.AP↗
Increasing stability for inverse acoustic source problems
In this paper, we show the increasing stability of the inverse source problems for the acoustic wave equation in the full space R3.The goal is to understand increasing stability for wave equation in the time domain. If the time and spatial variables of the source term can be separated with compact support, the increasing stability estimates of the $L^2$-norm of the acoustic source function can be established. The stability estimates consist of two parts: the Lipschitz type data discrepancy and the high time tail of the source functions. As the time increases, the latter decreases and thus becomes negligible.
math.AP↗
Uniqueness, stability and algorithm for an inverse wave-number-dependent source problems
This paper is concerned with an inverse wavenumber/frequency-dependent source problem for the Helmholtz equation. In two and three dimensions, the unknown source term is supposed to be compactly supported in spatial variables but independent on one spatial variable. The dependence of the source function on wavenumber/frequency is supposed to be unknown. Based on the Dirichlet-Laplacian and Fourier-Transform methods, we develop two effcient non-iterative numerical algorithms to recover the wavenumber-dependent source. Uniqueness proof and increasing stability analysis are carried out in terms of the boundary measurement data of Dirichlet kind. Numerical experiments are conducted to illustrate the effectiveness and efficiency of the proposed methods.
math.NA↗
Increasing stability for inverse acoustic source problems in the time domain
This paper is concerned with inverse source problems for the acoustic wave equation in the full space R^3, where the source term is compactly supported in both time and spatial variables. The main goal is to investigate increasing stability for the wave equation in terms of the interval length of given parameters (e.g., bandwith of the temporal component of the source function). We establish increasing stability estimates of the L^2 -norm of the source function by using only the Dirichlet boundary data. Our method relies on the Huygens principle, the Fourier transform and explicit bounds for the continuation of analytic functions.
math.AP↗
Increasing stability for inverse source problem with limited-aperture far field data at multi-frequencies
We study the increasing stability of an inverse source problem for the Helmholtz equation from limited-aperture far field data at multiple wave numbers. The measurement data are givenby the far field patterns $u^\infity(\hat{x},k)$ for all observation directions in some neighborhood of a fixed direction $\hat{x}$ and for all wave numbers k belonging to a finite interval $(0,K)$. In this paper, we discuss the increasing stability with respect to the width of the wavenumber interval $K>1$. In three dimensions we establish stability estimates of the $L^2$-norm and $H^{-1}$-norm of the source function from the far field data. The ill-posedness of the inverse source problem turns out to be of Hölder type while increasing the wavenumber band K. We also discuss an analytic continuation argument of the far-field data with respect to the wavenumbers at a fixed direction.
math.AP↗
Increasing stability for the inverse source problems in electrodynamics
We are concerned with increasing stability in the inverse source problems for the time-dependent Maxwell equations in R^3 , where the source term is compactly supported in both time and spatial variables. By using the Fourier transform, sharp bounds of the analytic continuation and the Huygens principle, increasing stability estimates of the L^2 -norm of the source function are obtained. The main goal of this paper is to understand increasing stability for the Maxwell equations in the time domain.
math.AP↗
Increasing stability in the n-dimensional inverse source problem with multi-frequencies
In this paper, we show for the first time the increasing stability of the inverse source problem for the n-dimensional Helmholtz equation at multiple wave numbers, which is different from the two-or three-dimensional Helmholtz equation. In addition, we develop a new, unified approach to study increasing stability in any dimension. The method is based on the Fourier transform and explicit bounds for analytic continuation.
math.AP↗