## Abstract

In previous works we considered the Helmholtz equation with fixed frequency k outside a discrete set of resonant frequencies, where it is implied that, given a source region _{D}_{a}⊂R^{d}(d=2,3¯) and u_{0}, a solution of the homogeneous scalar Helmholtz equation in a set containing the control region D_{c}⊂R^{d} , there exists an infinite class of boundary data on ∂D_{a} so that the radiating solution to the corresponding exterior scalar Helmholtz problem in R^{d}\_{Da} will closely approximate u_{0} in D_{c}. Moreover, it will have vanishingly small values beyond a certain large enough "far-field" radius R. In this paper we study the minimal energy solution of the above problem (e.g. the solution obtained by using Tikhonov regularization with the Morozov discrepancy principle) and perform a detailed sensitivity analysis. In this regard we discuss the stability of the minimal energy solution with respect to measurement errors as well as the feasibility of the active scheme (power budget and accuracy) depending on: The mutual distances between the antenna, control region and far field radius R; value of the regularization parameter; frequency; location of the source.

Original language | English (US) |
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Pages (from-to) | 1-23 |

Number of pages | 23 |

Journal | Applied Numerical Mathematics |

Volume | 106 |

DOIs | |

State | Published - Aug 1 2016 |

### Bibliographical note

Funding Information:M. Hubenthal was partially supported and D. Onofrei was partially supported by the AFOSR under the 2013 YIP Award FA9550-13-1-0078 and by ONR under the award N00014-15-1-2462 .

Publisher Copyright:

© 2016 IMACS. Published by Elsevier B.V. Allrightsreserved.

Copyright:

Copyright 2018 Elsevier B.V., All rights reserved.

## Keywords

- Active exterior cloaking
- Helmholtz equation
- Integral equations
- Regularization