The heterogeneous Helmholtz problem with spherical symmetry: Green’s operator and stability estimates

Inverse Boundary Value Problem For The Helmholtz Equation: Quantitative Conditional Lipschitz Stability Estimates

We study the inverse boundary value problem for the Helmholtz equation using the Dirichlet-toNeumann map at selected frequencies as the data. A conditional Lipschitz stability estimate for the inverse problem holds in the case of wavespeeds that are a linear combination of piecewise constant functions (following a domain partition) and gives a framework in which the scheme converges. The stabil...

A Cauchy Problem for Helmholtz Equation : Regularization and Error Estimates

In this paper, the Cauchy problem for the Helmholtz equation is investigated. It is known that such problem is severely ill-posed. We propose a new regularization method to solve it based on the solution given by the method of separation of variables. Error estimation and convergence analysis have been given. Finally, we present numerical results for several examples and show the effectiveness ...

Stability Estimates for the Lowest Eigenvalue of a Schrödinger Operator

Abstract. There is a family of potentials that minimize the lowest eigenvalue of a Schrödinger eigenvalue under the constraint of a given L norm of the potential. We give effective estimates for the amount by which the eigenvalue increases when the potential is not one of these optimal potentials. Our results are analogous to those for the isoperimetric problem and the Sobolev inequality. We al...

Vekua theory for the Helmholtz operator

A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator

The Helmholtz equation governing wave propagation and scattering phenomena is difficult to solve numerically. Its discretization with piecewise linear finite elements results in typically large linear systems of equations. The inherently parallel domain decomposition methods constitute hence a promising class of preconditioners. An essential element of these methods is a good coarse space. Here...