scholarly journals The Eigenspace Spectral Regularization Method for Solving Discrete Ill-Posed Systems

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Fredrick Asenso Wireko ◽  
Benedict Barnes ◽  
Charles Sebil ◽  
Joseph Ackora-Prah
Keyword(s):  
Regularization Method ◽  
Sparse Matrix ◽  
Circulant Matrix ◽  
Least Square ◽  
Matrix Operator ◽  
Qr Factorization ◽  
Spectral Matrix ◽  
Spectral Regularization ◽  
Ill Posed ◽  
Hilbert Matrix

This paper shows that discrete linear equations with Hilbert matrix operator, circulant matrix operator, conference matrix operator, banded matrix operator, TST matrix operator, and sparse matrix operator are ill-posed in the sense of Hadamard. Gauss least square method (GLSM), QR factorization method (QRFM), Cholesky decomposition method (CDM), and singular value decomposition (SVDM) failed to regularize these ill-posed problems. This paper introduces the eigenspace spectral regularization method (ESRM), which solves ill-posed discrete equations with Hilbert matrix operator, circulant matrix operator, conference matrix operator, and banded and sparse matrix operator. Unlike GLSM, QRFM, CDM, and SVDM, the ESRM regularizes such a system. In addition, the ESRM has a unique property, the norm of the eigenspace spectral matrix operator κ K = K − 1 K = 1 . Thus, the condition number of ESRM is bounded by unity, unlike the other regularization methods such as SVDM, GLSM, CDM, and QRFM.

scholarly journals The Eigenspace Spectral Regularization Method for solving Discrete Ill-Posed Systems

2021 ◽  
Author(s):  
BENEDICT BARNES ◽  
FREDRICK WIREKO ◽  
Charles Sebil; ◽  
Joseph Ackora-Prah
Keyword(s):  
Regularization Method ◽  
Sparse Matrix ◽  
Circulant Matrix ◽  
Matrix Operator ◽  
Qr Factorization ◽  
Spectral Matrix ◽  
Discrete Equations ◽  
Banded Matrix ◽  
Spectral Regularization ◽  
Ill Posed

In this paper, it is shown that discrete equations with Hilb ert matrix operator, circulant matrix operator, conference matrix operator, banded matrix operator, and sparse matrix operator are ill-posed in the sense of Hadamard. These ill-posed problems cannot be regularized by Gauss Least Square Method (GLSM), QR Factorization Method (QRFM), Cholesky Decomposition Method (CDM) and Singular Value Decomposition (SVDM). To overcome the limitations of these methods of regularization, an Eigenspace Spectral Regularization Method (ESRM) is introduced which solves ill-p os ed discrete equations with Hilb ert matrix operator, circulant matrix operator, conference matrix operator, banded matrix operator, and sparse matrix operator. Unlike GLSM, QRFM, CDM, and SVDM, the ESRM regularize such a system. In addition, the ESRM has a unique property, the norm of the eigenspace spectral matrix operator κ (K) = ||K − 1K|| = 1. Thus, the condition number of ESRM is bounded by unity unlike the other regularization methods such as SVDM, GLSM, CDM, and QRFM.

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.

Research on Compressed Sensing Matrix Based on Data Compressing of Insulator Leakage Current

2013 ◽  
Vol 475-476 ◽  
pp. 451-454
Author(s):  
Xue Ming Zhai ◽  
Xiao Bo You ◽  
Ruo Chen Li ◽  
Yu Jia Zhai ◽  
De Wen Wang
Keyword(s):  
Compressed Sensing ◽  
Data Compression ◽  
Leakage Current ◽  
Sparse Matrix ◽  
Circulant Matrix ◽  
Measurement Matrix ◽  
Sensing Matrix ◽  
Significant Step ◽  
On Line

Insulator fault may lead to the accident of power network,thus the on-line monitoring of insulator is very significant. Low rates wireless network is used for data transmission of leakage current. Making data compression and reconstruction of leakage current with the compressed sensing theory can achieve pretty good results. Determination of measurement matrix is the significant step for realizing the compressed sensing theory. This paper compares multiple measurement matrix of their effect via experiments, putting forward to make data compression and reconstruction of leakage current using Toeplitz matrix, circulant matrix and sparse matrix as measurement matrix, of which the reconstitution effect is almost the same as classical measurement matrix and depletes computational complexity and workload.

scholarly journals Heating source localization in a reduced time

2016 ◽  
Vol 26 (3) ◽  
pp. 623-640 ◽  
Author(s):  
Sara Beddiaf ◽  
Laurent Autrique ◽  
Laetitia Perez ◽  
Jean-Claude Jolly
Keyword(s):  
Heat Conduction ◽  
Source Localization ◽  
Conjugate Gradient ◽  
Regularization Method ◽  
Three Dimensional ◽  
Gradient Algorithm ◽  
Iterative Regularization ◽  
Heating Source ◽  
Ill Posed ◽  
Reduced Time

Abstract Inverse three-dimensional heat conduction problems devoted to heating source localization are ill posed. Identification can be performed using an iterative regularization method based on the conjugate gradient algorithm. Such a method is usually implemented off-line, taking into account observations (temperature measurements, for example). However, in a practical context, if the source has to be located as fast as possible (e.g., for diagnosis), the observation horizon has to be reduced. To this end, several configurations are detailed and effects of noisy observations are investigated.

scholarly journals Inventário de Emissões com Alta Resolução para a Região da Grande Vitória Utilizando o Sistema de Modelagem Integrada WRF-SMOKE-CMAQ

2018 ◽  
Vol 33 (3) ◽  
pp. 521-536 ◽  
Author(s):  
Ayres Geraldo Loriato ◽  
Nadir Salvador ◽  
Ayran Ayres Barbosa Loriato ◽  
Anton Sokolov ◽  
Antonio Paula Nascimento ◽  
...  
Keyword(s):  
Air Quality ◽  
Sparse Matrix ◽  
Matrix Operator ◽  
Espírito Santo ◽  
Multi Scale ◽  
Espirito Santo ◽  
Operator Kernel

Resumo A poluição atmosférica proveniente das atividades antropogênicas tem provocado incômodo à população da Região Metropolitana da Grande Vitória (RGV), Espirito Santo, Brasil. Muitos pacientes como asmáticos, alérgicos, químico sensíveis, cardiacos, pessoas com acidente vascular cerebral (AVC), diabetes, grávidas, idosos e crianças são especialmente suscetíveis a uma pobre qualidade do ar. As complexas interações dos poluentes atmosféricos podem ser entendidas usando modelos de transporte químico, os quais necessitam de inventários de emissões para prover a alocação espacial e temporal das emissões. Na RGV, o Instituto Estadual do Meio Ambiente (IEMA) disponibilizou o inventário de emissões da região para o ano de 2010. Nesse artigo foi realizada uma adaptação do inventário de fontes regional para o Sparse Matrix Operator Kernel Emissions (SMOKE) de modo a ser utilizado em modelos de qualidade do ar (MQA) como o Community Multi-scale Air Quality (CMAQ) Modeling System e outros modelos fotoquímicos. Foram obtidos valores simulados elevados referentes às emissões de ressuspensão em vias, apontando uma possível falha do inventário neste tipo de fonte. Utilizando os fatores de emissão médios de Abu-Allaban et al. (2003) para a ressuspensão das vias locais, foram obtidos resultados mais condizentes aos atuais níveis de concentrações de partículas monitorados na RGV.

A conjugate gradient least square based method for sound field control

2021 ◽  
Vol 263 (5) ◽  
pp. 1029-1040
Author(s):  
Pierangelo Libianchi ◽  
Finn T. Agerkvist ◽  
Elena Shabalina
Keyword(s):  
Conjugate Gradient ◽  
Regularization Parameter ◽  
Sound Field ◽  
Least Square ◽  
Active Set ◽  
Noise Sources ◽  
Field Control ◽  
Ill Posed ◽  
The Right ◽  
Number Of Iterations

In sound field control, a set of control sources is used to match the pressure field generated by noise sources but with opposite phase to reduce the total sound pressure level in a defined area commonly referred to as dark zone. This is usually an ill-posed problem. The approach presented here employs a subspace iterative method where the number of iterations acts as the regularization parameter and controls unwanted side radiation, i.e. side lobes. More iterations lead to less regularization and more side lobes. The number of iterations is controlled by problem-specific stopping criteria. Simulations show the increase of lobing with increased number of iterations. The solutions are analysed through projections on the basis provided by the source strength modes corresponding to the right singular vector of the transfer function matrix. These projections show how higher order pressure modes (left singular vectors) become dominant with larger number of iterations. Furthermore, an active-set type method provides the constraints on the amplitude of the solution which is not possible with the conjugate gradient least square algorithm alone.

Identification of Viscoelastic Properties of Finite Element Structures With Joints

1997 ◽  
Author(s):  
Torsten Herrmann ◽  
Valdas Chaika
Keyword(s):  
Finite Element ◽  
Viscoelastic Properties ◽  
Sparse Matrix ◽  
Maxwell Model ◽  
Least Square ◽  
Identification Algorithm ◽  
Domain Identification ◽  
Generalized Maxwell Model ◽  
Maxwell Models ◽  
Complex Modal Analysis

Abstract Identification of the damping and stiffness parameters of the composite joints in finite element structures is analyzed. For the modeling of the viscoelastic properties of the joints the classical Voigt-Kelvin and generalized Maxwell model (three parameter solid) are used. A time domain identification algorithm for classically and non-classically damped dynamic systems is developed. It is based on the application of an extended Kalman filter and least square technique. The algorithm uses complex modal analysis and sparse matrix technology. Both force and base excited systems are considered. Experimental verification of the identification results is carried out on a test structure. The accuracy of the modeling of damping in the joint using the Voigt-Kelvin and generalized Maxwell models is investigated.

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