The Detection of the Source of Acoustical Noise in Two Dimensions

DeLillo, Thomas K.; Isakov, Victor; Valdivia, Nicolas; Wang, Lianju

doi:10.1137/s0036139900367152

Cited by 62 publications

(28 citation statements)

References 12 publications

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“…The indirect formulations are based on representations to the solution of Equation 1 like the single source or double source representation found in Augusztinovicz (1999); Delillo et al (2000); DeLillo et al (2001;; Raveendra et al (1998);Schuhmacher et al (2003); Tekatlian et al (1996); Vlahopoulos & Raveerdra (1998);Williams et al (2000); Zhang et al (2001;2000). Using the notation of the previous subsection the single source representation is defined as…”

Section: Integral Formulationsmentioning

confidence: 99%

Numerical Methods for Near-Field Acoustic Holography over Arbitrarily Shaped Surfaces

Valdivia¹

2011

Holography - Different Fields of Application

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Section: Integral Formulationsmentioning

confidence: 99%

Numerical Methods for Near-Field Acoustic Holography over Arbitrarily Shaped Surfaces

Valdivia¹

2011

Holography - Different Fields of Application

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“…DeLillo et al [10] detected the source of acoustical noise inside the cabin of a midsize aircraft from measurements of the acoustical pressure field inside the cabin by solving a linear Fredholm integral equation of the first kind and they extended this study to three-dimensional problems, see DeLillo et al [11]. The alternating iterative algorithm of Kozlov et al [12], which reduces the Cauchy problem to solving a sequence of well-posed boundary value problems, was implemented numerically using the boundary element method (BEM) for the two-dimensional modified Helmholtz equation by Marin et al [13].…”

Section: Introductionmentioning

confidence: 99%

A relaxation method of an alternating iterative MFS algorithm for the Cauchy problem associated with the two‐dimensional modified Helmholtz equation

Marin

2011

Numerical Methods Partial

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We propose two algorithms involving the relaxation of either the given Dirichlet data or the prescribed Neumann data on the over-specified boundary in the case of the alternating iterative algorithm of Kozlov et al. (USSR Comput Math Math Phys 31 (1991), 45-52) applied to the Cauchy problem for the two-dimensional modified Helmholtz equation. The two mixed, well-posed and direct problems corresponding to every iteration of the numerical procedure are solved using the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method. For each direct problem considered, the optimal value of the regularization parameter is selected according to the generalized cross-validation criterion. The iterative MFS algorithms with relaxation are tested for Cauchy problems associated with the modified Helmholtz equation in two-dimensional geometries to confirm the numerical convergence, stability, accuracy and computational efficiency of the method.

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“…Depending on the type of the boundary conditions, it is involved in the determination of acoustic cavities [14], the detection of the source of acoustical noise [17], the description of underwater waves [18], the determination of the radiation field surrounding a source of radiation [15], the localization of a tumor in a human body [16], the identification and location of vibratory sources [23], the detection of surface vibrations from interior acoustical pressure [20], etc.…”

Section: Introductionmentioning

confidence: 99%

Iterative Methods for Solving the Cauchy Problem for the Helmholtz Equation

Mpinganzima

2014

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The inverse problem of reconstructing the acoustic, or electromagnetic, field from inexact measurements on a part of the boundary of a domain is important in applications, for instance for detecting the source of acoustic noise. The governing equation for the applications we consider is the Helmholtz equation. More precisely, in this thesis we study the case where Cauchy data is available on a part of the boundary and we seek to recover the solution in the whole domain. The problem is ill-posed in the sense that small errors in the Cauchy data may lead to large errors in the recovered solution. Thus special regularization methods that restore the stability with respect to measurements errors are used.In the thesis, we focus on iterative methods for solving the Cauchy problem. The methods are based on solving a sequence of well-posed boundary value problems. The specific choices for the boundary conditions used are selected in such a way that the sequence of solutions converges to the solution for the original Cauchy problem. For the iterative methods to converge, it is important that a certain bilinear form, associated with the boundary value problem, is positive definite. This is sometimes not the case for problems with a high wave number.The main focus of our research is to study certain modifications to the problem that restore positive definiteness to the associated bilinear form. First we add an artificial interior boundary inside the domain together with a jump condition that includes a parameter µ. We have shown by selecting an appropriate interior boundary and sufficiently large value for µ, we get a convergent iterative regularization method. We have proved the convergence of this method. This method converges slowly. We have therefore developed two conjugate gradient type methods and achieved much faster convergence. Finally, we have attempted to reduce the size of the computational domain by solving well-posed problems only in a strip between the outer and inner boundaries. We demonstrate that by alternating between Robin and Dirichlet conditions on the interior boundary, we can get a convergent iterative regularization method. Numerical experiments are used to illustrate the performance of the methods suggested. Populärvetenskaplig sammanfattningDet inversa problemet att rekonstruera akustiska eller elektromagnetiska fält i ett område från inexakta mätningar på en del av områdets randär viktigt för att t.ex. detektera källan för akustiskt brus. Sådana problem beskrivs av Helmholtz ekvation.I avhandlingen studerar vi sådana problem där Cauchydataär givna endast på en del av områdets rand och man vill finna lösningen i hela området. Vi vill alltså lösa Cauchyproblem för Helmholtz ekvation. Sådana problemär illa ställda, vilket innebär att små fel i Cauchydata kan medföra stora fel i lösningen. Särskilda regulariseringsmetoder måste därför användas för att ge stabilitet med avseende på mätfel.Avhandlingens syfteär att utveckla iterativa lösningsmetoder för dessa Cauchyproblem. Metoderna går ut ...

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The Detection of the Source of Acoustical Noise in Two Dimensions

Cited by 62 publications

References 12 publications

Numerical Methods for Near-Field Acoustic Holography over Arbitrarily Shaped Surfaces

Numerical Methods for Near-Field Acoustic Holography over Arbitrarily Shaped Surfaces

A relaxation method of an alternating iterative MFS algorithm for the Cauchy problem associated with the two‐dimensional modified Helmholtz equation

Iterative Methods for Solving the Cauchy Problem for the Helmholtz Equation

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