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.

scholarly journals Performance Research on Iterative Methods for Image Deblurring

2019 ◽  
Vol 8 (2S3) ◽  
pp. 1047-1056
Keyword(s):  
Performance Analysis ◽  
Iterative Algorithms ◽  
Motion Blur ◽  
Least Square ◽  
Ringing Artifacts ◽  
Ill Posed ◽  
And Performance ◽  
Restoration Algorithms ◽  
Traditional Approaches ◽  
Number Of Iterations

This paper introduces the iterative image restoration algorithms for the elimination of linearly varying blurs from the images degraded by motion blur and additive noise. Iterative algorithms are very operative for this applications since they include different types of prior knowledge about the class of reasonable solutions. These algorithms are robust in nature to the errors in estimating the blurring operators and can be used to remove the non-stationary blurs. Performance analysis and limitations of traditional approaches such as Inverse, Wiener and Constrained Least Square filters (CLS) are discussed with respect to the iterations. Role and choice of imposing a constraint on the solutions of the algorithms which gives better restoration results are discussed. Regularization methods are debated to Minimis extreme noise amplifications due to ill-posed conditions in the inverse deblurring problems and it is shown that the reduction of noise effects can be achieved by terminating the algorithm after finite number of iterations. It is shown that restoration algorithms with constraints and spatially adaptability reduces the effects of ringing artifacts significantly. The rate of convergence of the algorithms based on the variations in the number of iterations are discussed and performance analysis, limitations and Comparison with the experimental results are presented.

scholarly journals Method of iteration of solving invariant equations with an approximate operator in the case of an arbitrary choice of the regularization parameter

2019 ◽  
Vol 54 (4) ◽  
pp. 408-416
Author(s):  
O. V. Matysik ◽  
V. F. Savchuk
Keyword(s):  
Error Estimate ◽  
Error Estimates ◽  
Regularization Parameter ◽  
A Priori ◽  
Operator Equations ◽  
Mathematical Economics ◽  
Prove Theorem ◽  
A Value ◽  
Ill Posed ◽  
The Right

In the introduction, the object of investigation is indicated – incorrect problems described by first-kind operator equations. The subject of the study is an explicit iterative method for solving first-kind equations. The aim of the paper is to prove the convergence of the proposed method of simple iterations with an alternating step alternately and to obtain error estimates in the original norm of a Hilbert space for the cases of self-conjugated and non self-conjugated problems. The a priori choice of the regularization parameter is studied for a source-like representable solution under the assumption that the operator and the right-hand side of the equation are given approximately. In the main part of the work, the achievement of the stated goal is expressed in four reduced and proved theorems. In Section 1, the first-kind equation is written down and a new explicit method of simple iteration with alternating steps is proposed to solve it. In Section 2, we consider the case of the selfconjugated problem and prove Theorem 1 on the convergence of the method and Theorem 2, in which an error estimate is obtained. To obtain an error estimate, an additional condition is required – the requirement of the source representability of the exact solution. In Section 3, the non-self-conjugated problem is solved, the convergence of the proposed method is proved, which in this case is written differently, and its error estimate is obtained in the case of an a priori choice of the regularization parameter. In sections 2 and 3, the error estimates obtained are optimized, that is, a value is found – the step number of the iteration, in which the error estimate is minimal. Since incorrect problems constantly arise in numerous applications of mathematics, the problem of studying them and constructing methods for their solution is topical. The obtained results can be used in theoretical studies of solution of first-kind operator equations, as well as applied ill-posed problems encountered in dynamics and kinetics, mathematical economics, geophysics, spectroscopy, systems for complete automatic processing and interpretation of experiments, plasma diagnostics, seismic and medicine.

AN ERROR ANALYSIS OF LAVRENTIEV REGULARIZATION IN LEARNING THEORY

2009 ◽  
Vol 02 (01) ◽  
pp. 129-140
Author(s):  
J. K. Sahoo ◽  
Arindama Singh
Keyword(s):  
Inverse Problem ◽  
Learning Theory ◽  
Regularization Parameter ◽  
Least Square ◽  
Support Vector ◽  
Lavrentiev Regularization ◽  
Vector Machines ◽  
Ill Posed ◽  
Regularization Networks ◽  
Context Of Learning

In this paper we study how Lavrentiev regularization can be used in the context of learning theory, especially in regularization networks that are closely related to support vector machines. We briefly discuss formulations of learning from examples in the context of ill-posed inverse problem and regularization. We then study the interplay between the Lavrentiev regularization of the concerned continuous and discretized ill-posed inverse problems. As the main result of this paper, we give an improved probabilistic bound for the regularization networks or least square algorithms, where we can afford to choose the regularization parameter in a larger interval.

scholarly journals Regularized inversion of full tensor magnetic gradient data

2016 ◽  
pp. 13-20
Author(s):  
Я. Ван ◽  
Д.В. Лукьяненко ◽  
А.Г. Ягола
Keyword(s):  
Tikhonov Regularization ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Regularization Parameter ◽  
Three Dimensional ◽  
Fredholm Integral Equations ◽  
Minimization Method ◽  
Magnetic Gradient ◽  
Regularized Inversion ◽  
Ill Posed

Рассматриваются особенности численной реализации решения трехмерной обратной задачи обращения полных тензорных магнитно-градиентных данных, которая моделируется системой двух трехмерных интегральных уравнений Фредгольма 1-го рода. Для решения этой некорректно поставленной задачи применяется алгоритм, основанный на минимизации функционала А.Н. Тихонова. В качестве метода минимизации используется метод сопряженных градиентов. Выбор параметра регуляризации осуществляется в соответствии с версией обобщенного принципа невязки, в которой учитываются ошибки округления, существенные при решении задач большой размерности. Features of numerical solution of the three-dimensional ill-posed problem devoted to the inversion of full tensor magnetic gradient data are considered. This problem is simulated by a system of two three-dimensional Fredholm integral equations of the first kind. The Tikhonov regularization is applied to solve this ill-posed problem. The conjugate gradient method is used as a minimization method. The choice of the regularization parameter is realized according to the generalized residual principle with consideration of round-off errors capable of affecting the final result of calculations significantly.

scholarly journals Simplification method for nonlinear equations of monotone type in Banach space

2021 ◽  
Vol 23 (2) ◽  
pp. 185-192
Author(s):  
Irina P. Ryazantseva
Keyword(s):  
Banach Space ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Original Equation ◽  
Ill Posed ◽  
Simple Operator ◽  
The Right ◽  
Dual Mapping ◽  
The Given ◽  
Monotone Type

Abstract. In a Banach space, we study an operator equation with a monotone operator T. The operator is an operator from a Banach space to its conjugate, and T=AC, where A and C are operators of some classes. The considered problem belongs to the class of ill-posed problems. For this reason, an operator regularization method is proposed to solve it. This method is constructed using not the operator T of the original equation, but a more simple operator A, which is B-monotone, B=C−1. The existence of the operator B is assumed. In addition, when constructing the operator regularization method, we use a dual mapping with some gauge function. In this case, the operators of the equation and the right-hand side of the given equation are assumed to be perturbed. The requirements on the geometry of the Banach space and on the agreement conditions for the perturbation levels of the data and of the regularization parameter are established, which provide a strong convergence of the constructed approximations to some solution of the original equation. An example of a problem in Lebesgue space is given for which the proposed method is applicable.

Peristaltic Carreau-Yasuda nanofluid flow and mixed heat transfer analysis in an asymmetric vertical and tapered wavy wall channel

2020 ◽  
Vol 1 (1) ◽  
pp. 128-140 ◽  
Author(s):  
Mohammad Hatami ◽  
◽  
D Jing ◽  
Keyword(s):  
Heat Transfer ◽  
Least Square Method ◽  
Weissenberg Number ◽  
Least Square ◽  
Heat Transfer Analysis ◽  
Phase Method ◽  
Nanofluid Flow ◽  
Two Phase ◽  
Nanoparticles Concentration ◽  
The Right

In this study, two-phase asymmetric peristaltic Carreau-Yasuda nanofluid flow in a vertical and tapered wavy channel is demonstrated and the mixed heat transfer analysis is considered for it. For the modeling, two-phase method is considered to be able to study the nanoparticles concentration as a separate phase. Also it is assumed that peristaltic waves travel along X-axis at a constant speed, c. Furthermore, constant temperatures and constant nanoparticle concentrations are considered for both, left and right walls. This study aims at an analytical solution of the problem by means of least square method (LSM) using the Maple 15.0 mathematical software. Numerical outcomes will be compared. Finally, the effects of most important parameters (Weissenberg number, Prandtl number, Brownian motion parameter, thermophoresis parameter, local temperature and nanoparticle Grashof numbers) on the velocities, temperature and nanoparticles concentration functions are presented. As an important outcome, on the left side of the channel, increasing the Grashof numbers leads to a reduction in velocity profiles, while on the right side, it is the other way around.

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.

scholarly journals Regularization of backward time-fractional parabolic equations by Sobolev-type equations

2020 ◽  
Vol 28 (5) ◽  
pp. 659-676
Author(s):  
Dinh Nho Hào ◽  
Nguyen Van Duc ◽  
Nguyen Van Thang ◽  
Nguyen Trung Thành
Keyword(s):  
Error Estimates ◽  
Parabolic Equations ◽  
Regularization Parameter ◽  
A Priori ◽  
Sobolev Type ◽  
A Posteriori ◽  
Parameter Choice ◽  
Numerical Tests ◽  
Ill Posed ◽  
Parameter Choice Rules

AbstractThe problem of determining the initial condition from noisy final observations in time-fractional parabolic equations is considered. This problem is well known to be ill-posed, and it is regularized by backward Sobolev-type equations. Error estimates of Hölder type are obtained with a priori and a posteriori regularization parameter choice rules. The proposed regularization method results in a stable noniterative numerical scheme. The theoretical error estimates are confirmed by numerical tests for one- and two-dimensional equations.

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.

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