Methods and Applications of Analysis

Volume 18 (2011)

Number 1

Subspaces of stability in the Cauchy Problem for the Helmholtz equation

Pages: 1 – 30

DOI: http://dx.doi.org/10.4310/MAA.2011.v18.n1.a1

Authors

Victor Isakov (Department of Mathematics and Statistics, Wichita State University, Wichita, Kansas, U.S.A.)

Stefan Kindermann (Industrial Mathematics Institute, Johannes Kepler University, Linz, Austria)

Abstract

We study the stability in the Cauchy Problem for the Helmholtz equation in dependence of the wave number $k$. For simple geometries, we show analytically that this problem is getting more stable with increasing $k$. In more detail, there is a subspace of the data space on which the Cauchy Problem is well posed, and this subspace grows with larger $k$. We call this a subspace of stability. For more general geometries, we study the ill-posedness by computing the singular values of some operators associated with the corresponding well-posed (direct) boundary value problems. Numerical computations of the singular values show that an increasing subspace of stability exists for more general domains and boundary data. Assuming the existence of such subspaces we develop a theory of (Tikhonov type) regularization for the Cauchy Problem, giving convergence rates and regularization parameter choice which improve with growing $k$.

Keywords

Cauchy Problem, Helmholtz equation, subspace of stability, conditional stability

2010 Mathematics Subject Classification

35B60, 35R25, 35R30, 65F15

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