SUBSPACES OF STABILITY IN THE CAUCHY PROBLEM FOR THE HELMHOLTZ EQUATION

We study the stability in the Cauchy Problem for the Helmholtz equation in de- pendence 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 val- ues of some operators associated with 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 when there are trapped rays of geometrical optics. 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.

[1]  F. Natterer,et al.  A propagation-backpropagation method for ultrasound tomography , 1995 .

[2]  Bernd Hofmann,et al.  Modulus of continuity for conditionally stable ill-posed problems in Hilbert space , 2008 .

[3]  Stefan A. Sauter,et al.  Is the Pollution Effect of the FEM Avoidable for the Helmholtz Equation Considering High Wave Numbers? , 1997, SIAM Rev..

[4]  E. Williams.  Regularization methods for near-field acoustical holography. , 2001, The Journal of the Acoustical Society of America.

[5]  Victor Isakov,et al.  Increased stability in the continuation of solutions to the Helmholtz equation , 2004 .

[6]  On increased stability in the continuation of the Helmholtz equation , 2007 .

[7]  V. Isakov.  Increased stability in the continuation for the Helmholtz equation with variable coefficient , 2005 .

[8]  T. Wei,et al.  Modified Tikhonov regularization method for the Cauchy problem of the Helmholtz equation , 2009 .

[9]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[10]  C. Bardos,et al.  Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary , 1992 .

[11]  V. Isakov,et al.  Increasing stability of the continuation for the Maxwell system , 2010 .

[13]  Stefan Kindermann,et al.  On the Degree of Ill-posedness for Linear Problems with Noncompact Operators , 2010 .

[14]  Xiang-Tuan Xiong,et al.  Two approximate methods of a Cauchy problem for the Helmholtz equation , 2007 .

[15]  Luca Rondi,et al.  The stability for the Cauchy problem for elliptic equations , 2009, 0907.2882.

[16]  Stefan Kindermann,et al.  On the convergence of the quasioptimality criterion for (iterated) Tikhonov regularization , 2008 .

[17]  Fritz John,et al.  Continuous dependence on data for solutions of partial differential equations with a prescribed bound , 1960 .

[18]  Nicolas Valdivia,et al.  The detection of surface vibrations from interior acoustical pressure , 2003 .

[19]  Nicolas Valdivia,et al.  Acoustic source identification using multiple frequency information , 2009 .

[20]  Derek B. Ingham,et al.  Conjugate gradient-boundary element solution to the Cauchy problem for Helmholtz-type equations , 2003 .

[21]  J. Hadamard,et al.  Lectures on Cauchy's Problem in Linear Partial Differential Equations , 1924 .

[22]  G. Bao,et al.  Inverse medium scattering for the Helmholtz equation at fixed frequency , 2005 .

[23]  Thorsten Hohage,et al.  Logarithmic convergence rates of the iteratively regularized Gauss - Newton method for an inverse potential and an inverse scattering problem , 1997 .

[24]  Teresa Regińska,et al.  Approximate solution of a Cauchy problem for the Helmholtz equation , 2006 .