A stopping rule for the conjugate gradient regularization method for ill‐posed problems

2002 ◽  
Vol 112 (5) ◽  
pp. 2381-2381
Author(s):  
Thomas DeLillo ◽  
Tomasz Hrycak
Keyword(s):  
Conjugate Gradient ◽  
Regularization Method ◽  
Stopping Rule ◽  
Ill Posed

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 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.

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 Two-Parameter Regularization Method for a Nonlinear Backward Heat Problem with a Conformable Derivative

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Vu Ho ◽  
Donal O’Regan ◽  
Hoa Ngo Van
Keyword(s):  
Integral Equation ◽  
Regularization Method ◽  
Time Variable ◽  
Conformable Derivative ◽  
Heat Problem ◽  
Regularization Parameters ◽  
Ill Posed ◽  
Parameter Regularization ◽  
Two Parameter ◽  
Backward Heat Problem

In this paper, we consider the nonlinear inverse-time heat problem with a conformable derivative concerning the time variable. This problem is severely ill posed. A new method on the modified integral equation based on two regularization parameters is proposed to regularize this problem. Numerical results are presented to illustrate the efficiency of the proposed method.

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