scholarly journals A New Method for Determining Optimal Regularization Parameter in Near-Field Acoustic Holography

2018 ◽  
Vol 2018 ◽  
pp. 1-13
Author(s):  
Yue Xiao
Keyword(s):  
Point Source ◽  
Regularization Parameter ◽  
Near Field ◽  
New Method ◽  
Tikhonov Regularization Method ◽  
Acoustic Holography ◽  
Reconstruction Process ◽  
Curve Method ◽  
Univariate Optimization ◽  
Equivalent Sources

Tikhonov regularization method is effective in stabilizing reconstruction process of the near-field acoustic holography (NAH) based on the equivalent source method (ESM), and the selection of the optimal regularization parameter is a key problem that determines the regularization effect. In this work, a new method for determining the optimal regularization parameter is proposed. The transfer matrix relating the source strengths of the equivalent sources to the measured pressures on the hologram surface is augmented by adding a fictitious point source with zero strength. The minimization of the norm of this fictitious point source strength is as the criterion for choosing the optimal regularization parameter since the reconstructed value should tend to zero. The original inverse problem in calculating the source strengths is converted into a univariate optimization problem which is solved by a one-dimensional search technique. Two numerical simulations with a point driven simply supported plate and a pulsating sphere are investigated to validate the performance of the proposed method by comparison with the L-curve method. The results demonstrate that the proposed method can determine the regularization parameter correctly and effectively for the reconstruction in NAH.

Identifying an Unknown Source in the Poisson Equation with a Super Order Regularization Method

2019 ◽  
Vol 17 (07) ◽  
pp. 1950030 ◽  
Author(s):  
Zhi Li ◽  
Zhenyu Zhao ◽  
Zehong Meng ◽  
Baoqin Chen ◽  
Duan Mei
Keyword(s):  
Poisson Equation ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Solution Process ◽  
New Method ◽  
Tikhonov Regularization Method ◽  
Optimal Error ◽  
Penalty Term ◽  
The Poisson Equation ◽  
Unknown Source

This paper develops a new method to deal with the problem of identifying the unknown source in the Poisson equation. We obtain the regularization solution by the Tikhonov regularization method with a super-order penalty term. The order optimal error bounds can be obtained for various smooth conditions when we choose the regularization parameter by a discrepancy principle and the solution process of the new method is uniform. Numerical examples show that the proposed method is effective and stable.

scholarly journals A New Method for Optimal Regularization Parameter Determination in the Inverse Problem of Load Identification

2016 ◽  
Vol 2016 ◽  
pp. 1-16 ◽  
Author(s):  
Wei Gao ◽  
Kaiping Yu ◽  
Ying Wu
Keyword(s):  
Inverse Problem ◽  
Regularization Method ◽  
Cross Validation ◽  
Regularization Parameter ◽  
New Method ◽  
Parameter Determination ◽  
Load Identification ◽  
Noise Levels ◽  
Regularization Parameters ◽  
Curve Method

According to the regularization method in the inverse problem of load identification, a new method for determining the optimal regularization parameter is proposed. Firstly, quotient function (QF) is defined by utilizing the regularization parameter as a variable based on the least squares solution of the minimization problem. Secondly, the quotient function method (QFM) is proposed to select the optimal regularization parameter based on the quadratic programming theory. For employing the QFM, the characteristics of the values of QF with respect to the different regularization parameters are taken into consideration. Finally, numerical and experimental examples are utilized to validate the performance of the QFM. Furthermore, the Generalized Cross-Validation (GCV) method and theL-curve method are taken as the comparison methods. The results indicate that the proposed QFM is adaptive to different measuring points, noise levels, and types of dynamic load.

scholarly journals Determination of a space-dependent force function in the one-dimensional wave equation

2014 ◽  
Vol 12 (1) ◽  
Author(s):  
S. O. Hussein ◽  
D. Lesnic
Keyword(s):  
Input Data ◽  
Regularization Parameter ◽  
Stable Solution ◽  
Force Function ◽  
Tikhonov Regularization Method ◽  
Vibrating String ◽  
Ill Posed ◽  
Curve Method ◽  
The One

<p class="p1">The determination of an unknown spacewice dependent force function acting on a vibrating string from over-specied Cauchy boundary data is investigated numerically using the boundary element method (BEM) combined with a regularized method of separating variables. This linear inverse problem is ill-posed since small errors in the input data cause large errors in the output force solution. Consequently, when the input data is contaminated with noise we use the Tikhonov regularization method in order to obtain a stable solution. The choice of the regularization parameter is based on the L-curve method. Numerical results show that the solution is accurate for exact data and stable for noisy data.</p>

STUDY OF SOUND SOURCE IDENTIFICATION BASED ON BEAMFORMING AND PATCH NEAR-FIELD ACOUSTIC HOLOGRAPHY

2009 ◽  
Vol 17 (03) ◽  
pp. 219-245 ◽  
Author(s):  
WENQIANG JIA ◽  
JIN CHEN ◽  
CHAO YANG ◽  
JIAQING LI
Keyword(s):  
Regularization Parameter ◽  
Near Field ◽  
Sound Field ◽  
Reconstruction Error ◽  
Tikhonov Regularization Method ◽  
Superposition Method ◽  
Sound Sources ◽  
Extrapolation Technique ◽  
Measurement Surface ◽  
Data Extrapolation

A study for applying data extrapolation technique based on wave superposition method (WSM) is proposed to overcome the disadvantages of near-field acoustical holography (NAH). Unlike conformal NAH, where the measurement surface surrounds the entire structure, in the patch holography the measurement surface needs only be approximately as large as the patch on the structure surface where the reconstruction is required. At first, a microphone array is used to acquire the sound pressure field radiated from a vibrator; then a beamforming method is adopted to locate sound sources; after that, a serial of equivalent sources are collocated around these sound sources; at last, a data extrapolation technique based on WSM is applied to extend the measurement aperture and reconstruct the sound field. Since the data extrapolation algorithm requires the inversion of Green's function matrix which may be ill-conditioned, Tikhonov regularization method is used to invert it, and the value of the regularization parameter is determined by the L-curve criteria. The effectiveness of this method is demonstrated by numerical simulation with two pulse ball model, and also experiment is carried out in a semi-anechoic chamber using two sound boxes. The results confirm that the exterior sound field can be accurately reconstructed with few iterative times and the reconstruction error is sufficiently suppressed by the proposed method.

DISTRIBUTED SOURCE BOUNDARY POINT METHOD-BASED NEARFIELD ACOUSTIC HOLOGRAPHY FOR RECONSTRUCTING THE ACOUSTIC RADIATION FROM ARBITRARILY SHAPED OBJECTS

2006 ◽  
Vol 14 (04) ◽  
pp. 379-395
Author(s):  
C. X. BI ◽  
X. Z. CHEN ◽  
J. CHEN
Keyword(s):  
Boundary Point ◽  
Measurement Errors ◽  
Regularization Parameter ◽  
Point Method ◽  
Tikhonov Regularization Method ◽  
Acoustic Holography ◽  
Reconstruction Procedure ◽  
Distributed Source ◽  
Boundary Point Method ◽  
Nearfield Acoustic Holography

Nearfield acoustic holography (NAH) is an indirect technique for identifying noise sources and visualizing acoustic field. Recently, several different methods, such as the spatial Fourier transform method, the boundary element method (BEM) and the Helmholtz equation-least squares (HELS) method, have been used to realize the NAH successfully. In the paper, a novel numerical method, the distributed source boundary point method (DSBPM), is proposed to realize the NAH. In the method, the transfer matrices from the reconstructed surface to the hologram surface are constructed indirectly by a set of particular solution sources located inside the vibrating structure, and their inverses are carried out by singular value decomposition (SVD) technique. Additionally, considering the high sensitivity of the reconstructed solution to measurement errors, the Tikhonov regularization method is implemented to stabilize the reconstruction procedure and the regularization parameter is determined by L-curve criterion. Compared with the BEM-based NAH, the variable interpolation, the numerical quadrature, and the treatments of singular integral and nonuniqueness of solution are all avoided in the proposed method. Two numerical examples and an experiment are investigated to validate the feasibility and correctness of the proposed method.

scholarly journals A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation

2020 ◽  
Vol 18 (1) ◽  
pp. 1685-1697
Author(s):  
Zhenyu Zhao ◽  
Lei You ◽  
Zehong Meng
Keyword(s):  
Cauchy Problem ◽  
Laplace Equation ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Solution Process ◽  
Tikhonov Regularization Method ◽  
Hermite Expansion ◽  
The Cauchy Problem ◽  
Ill Posed

Abstract In this paper, a Cauchy problem for the Laplace equation is considered. We develop a modified Tikhonov regularization method based on Hermite expansion to deal with the ill posed-ness of the problem. The regularization parameter is determined by a discrepancy principle. For various smoothness conditions, the solution process of the method is uniform and the convergence rate can be obtained self-adaptively. Numerical tests are also carried out to verify the effectiveness of the method.

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