An efficient and robust algorithm for source reconstruction in the Helmholtz equation

We consider the problem of reconstruction of an unknown characteristic interval and block transient thermal source inside a domain. By exploring the definition of an Extended Dirichlet to Neumann map in the time space cylinder that has been introduced in Roberty and Rainha (2010a), we can treat the problem with methods similar to that used in the analysis of the stationary source reconstruction problem. Further, the finite differenceθ-scheme applied to the transient heat conduction equation leads to a model based on a sequence of modified Helmholtz equation solutions. For each modified Helmholtz equation the characteristic interval and parallelepiped source function may be reconstructed uniquely from the Cauchy boundary data. Using representation formula we establish reciprocity functional mapping functions that are solutions of the modified Helmholtz equation to their integral in the unknown characteristic support. Numerical experiment for capture of an interval and an rectangular parallelepiped characteristic source inside a cubic box domain from boundary data are presented in threedimensional and one-dimensional implementations. The problem of centroid determination is addressed and questions are discussed from an computational points of view.

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Integral and Variational Formulations for the Helmholtz Equation Inverse Source Problem

Mathematical Problems in Engineering ◽

10.1155/2012/808913 ◽

2012 ◽

Vol 2012 ◽

pp. 1-28 ◽

Cited By ~ 1

Author(s):

Marcelo L. Rainha ◽

Nilson C. Roberty

Keyword(s):

Functional Equation ◽

Hilbert Space ◽

Integral Equation ◽

Helmholtz Equation ◽

Green's Function ◽

Equivalence Theorem ◽

Functional Structure ◽

Inverse Source Problem ◽

Source Reconstruction ◽

Variational Formulations

The purpose of this paper is to explore the Hilbert space functional structure of the Helmholtz equation inverse source problem. An integral equation for the sources reconstruction based on the composition of the trace and Green's function operators is introduced and compared with the reciprocity source reconstruction methodologies. An equivalence theorem comparing the integral inverse source equation with the variational weak reciprocity gap functional equation is then demonstrated. Some examples on applications to the unitary disk are presented.

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Moving Heat Source Reconstruction from the Cauchy Boundary Data

Mathematical Problems in Engineering ◽

10.1155/2010/987545 ◽

2010 ◽

Vol 2010 ◽

pp. 1-22 ◽

Cited By ~ 2

Author(s):

Nilson C. Roberty ◽

Marcelo L. S. Rainha

Keyword(s):

Helmholtz Equation ◽

Boundary Data ◽

Representation Formula ◽

Transient Heat Conduction ◽

Thermal Source ◽

Source Reconstruction ◽

Modified Helmholtz Equation ◽

Time Space ◽

Definition Of ◽

Transient Thermal

We consider the problem of reconstruction of an unknown characteristic transient thermal source inside a domain. By introducing the definition of an extended dirichlet-to-Neumann map in the time-space cylinder and the adoption of the anisotropic Sobolev-Hilbert spaces, we can treat the problem with methods similar to those used in the analysis of the stationary source reconstruction problem. Further, the finite differenceθscheme applied to the transient heat conduction equation leads to a model based on a sequence of modified Helmholtz equation solutions. For each modified Helmholtz equation the characteristic star-shape source function may be reconstructed uniquely from the Cauchy boundary data. Using representation formula, we establish reciprocity functional mapping functions that are solutions of the modified Helmholtz equation to their integral in the unknown characteristic support.

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