Increased stability in the continuation of solutions to the Helmholtz equation

@article{Hrycak2004IncreasedSI,
  title={Increased stability in the continuation of solutions to the Helmholtz equation},
  author={Tomasz Hrycak and Victor Isakov},
  journal={Inverse Problems},
  year={2004},
  volume={20},
  pages={697-712}
}
In this paper we give an analytical derivation and numerical evidence of how stability in the Cauchy problem for the Helmholtz equation grows with frequency. This effect depends on convexity properties of the surface where the Cauchy data are given. Proofs use Carleman estimates and the theory of elliptic boundary value problems in Sobolev spaces. Our theory is illustrated by numerical experiments, including an example in the nearfield acoustical holography. 

Tables from this paper

On increased stability in the continuation of the Helmholtz equation
In this paper, we give analytical and numerical evidence of increasing stability in the Cauchy problem for the Helmholtz equation in the whole domain when frequency is growing. This effect depends
Increased Stability in the Cauchy Problem for Some Elliptic Equations
We derive some bounds which can be viewed as an evidence of increasing stability in the Cauchy Problem for the Helmholtz equation with lower order terms when frequency is growing. These bounds hold
Stability analysis of a continuation problem for the Helmholtz equation
We investigate the continuation problem for the elliptic equation. The continuation problem is formulated in the operator form Aq = f . Singular values of the operator A are presented and analyzed
Increasing stability of the continuation for the Maxwell system
In this paper, we obtain bounds showing increasing stability of the continuation for solutions of the stationary Maxwell system when the wave number k is growing. We reduce this system to a new
An a posteriori truncation method for some Cauchy problems associated with Helmholtz-type equations
In this article, we investigate some Cauchy problems associated with Helmholtz-type equations in rectangle using the method of truncation. Not alike the previous literature, we propose to choose the
An Integral Equations Method for the Cauchy Problem Connected with the Helmholtz Equation
We are concerned with the Cauchy problem connected with the Helmholtz equation. We propose a numerical method, which is based on the Helmholtz representation, for obtaining an approximate solution to
Approximate solution of a Cauchy problem for the Helmholtz equation
A problem of reconstruction of the radiation field in a domain from experimental data given on a part of boundary is considered. For the model problem described by a Cauchy problem for the Helmholtz
A new regularization method for the cauchy problem of the helmholtz equation with nonhomogeneous cauchy data
In this paper, we investigate the Cauchy problem for the Helmholtz equation in the infinite strip {(x, y) | x ∈ R, 0 < y < 1} with nonhomogeneous Cauchy data given at y = 0. The problem is severely
...
...

References

SHOWING 1-10 OF 17 REFERENCES
Increased stability in the continuation of solutions to the Helmholtz equation
In this paper we give an analytical derivation and numerical evidence of how stability in the Cauchy problem for the Helmholtz equation grows with frequency. This effect depends on convexity
Exponential instability in an inverse problem for the Schrodinger equation
We consider the problem of the determination of the potential from the Dirichlet to Neumann map of the Schrodinger operator. We show that this problem is severely ill-posed. The results extend to
Stability in diffraction tomography and a nonlinear “basic theorem”
The stability problem is studied for reconstruction of the refraction coefficient from boundary measurements of solutions of the Helmholtz equation at a fixed time-frequency. An answer is given in
Stable determination of conductivity by boundary measurements
We consider the problem of determining the scalar coefficient γ in the elliptic equation div(γ grad u) = 0 in ω when, for every Dirichlet datum u = ∅ on ∂ω , the Neumann datum γ(∂/ ∂ n)u = ∧.γ∅ is
The linear sampling method in inverse electromagnetic scattering theory
We survey the linear sampling method for solving the inverse scattering problem for time-harmonic electromagnetic waves at fixed frequency. We consider scattering by an obstacle as well as scattering
Inverse/Observability Estimates for Second-Order Hyperbolic Equations with Variable Coefficients
Abstract We consider a general second-order hyperbolic equation defined on an open bounded domain Ω ⊂  R n with variable coefficients in both the elliptic principal part and in the first-order terms
Inverse Problems for Partial Differential Equations
Auxiliary information from functional analysis and theory of differential equations the basic notions and notations inequalities some concepts and theorems of functional analysis linear partial
L
  • Wang, The detection of surface vibrations from interior acoustical pressure Inverse Problems 19
  • 2003
...
...