# 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.

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## References

SHOWING 1-10 OF 17 REFERENCES

Increased stability in the continuation of solutions to the Helmholtz equation

- Mathematics
- 2004

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

- Mathematics
- 2001

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”

- Mathematics
- 2003

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

- Mathematics
- 1988

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

- Mathematics
- 2003

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

- Mathematics
- 1999

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

- Mathematics
- 2002

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…

Continuous dependence on data for solutions of partial differential equations with a prescribed bound

- Mathematics
- 1960

L

- Wang, The detection of surface vibrations from interior acoustical pressure Inverse Problems 19
- 2003