scholarly journals A Stable Approach for Numerical Differentiation by Local Regularization Method with its Regularization Parameter Selection Strategies

2020 ◽  
pp. 27-35
Author(s):  
Huilin Xu ◽  
Xiaoyan Xiang ◽  
Yanling He
Keyword(s):  
Regularization Method ◽  
Regularization Parameter ◽  
A Priori ◽  
Numerical Differentiation ◽  
Selection Strategy ◽  
Numerical Comparison ◽  
Local Regularization ◽  
Selection Strategies ◽  
The Given ◽  
Regularization Parameter Selection

The local regularization method for solving the first-order numerical differentiation problem is considered in this paper. The a-priori and a-posteriori selection strategy of the regularization parameter is introduced, and the convergence rate of local regularization solution under some assumption of the exact derivative is also given. Numerical comparison experiments show that the local regularization method can reflect sharp variations and oscillations of the exact derivative while suppress the noise of the given data effectively.

scholarly journals Fourier Truncation Regularization Method for a Time-Fractional Backward Diffusion Problem with a Nonlinear Source

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 865 ◽  
Author(s):  
Fan Yang ◽  
Ping Fan ◽  
Xiao-Xiao Li ◽  
Xin-Yi Ma
Keyword(s):  
Regularization Method ◽  
Regularization Parameter ◽  
A Priori ◽  
Fractional Diffusion ◽  
Diffusion Problem ◽  
Nonlinear Source ◽  
Convergence Error ◽  
Strip Domain ◽  
Ill Posed ◽  
Regularization Parameter Selection

In present paper, we deal with a backward diffusion problem for a time-fractional diffusion problem with a nonlinear source in a strip domain. We all know this nonlinear problem is severely ill-posed, i.e., the solution does not depend continuously on the measurable data. Therefore, we use the Fourier truncation regularization method to solve this problem. Under an a priori hypothesis and an a priori regularization parameter selection rule, we obtain the convergence error estimates between the regular solution and the exact solution at 0 ≤ x < 1 .

scholarly journals A Tikhonov-Type Regularization Method for Identifying the Unknown Source in the Modified Helmholtz Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Jinghuai Gao ◽  
Dehua Wang ◽  
Jigen Peng
Keyword(s):  
Helmholtz Equation ◽  
Regularization Method ◽  
Regularization Parameter ◽  
A Priori ◽  
Inverse Source Problem ◽  
A Posteriori ◽  
Modified Helmholtz Equation ◽  
Numerical Tests ◽  
Theoretical Frame ◽  
Set Up

An inverse source problem in the modified Helmholtz equation is considered. We give a Tikhonov-type regularization method and set up a theoretical frame to analyze the convergence of such method. A priori and a posteriori choice rules to find the regularization parameter are given. Numerical tests are presented to illustrate the effectiveness and stability of our proposed method.

scholarly journals SOLUTION OF THE PROBLEM OF COMPUTATIONAL STABILITY AND CONSISTENCY OF SAMPLE ESTIMATES OF THE CORRELATION MATRIX OF OBSERVATIONS BY THE METHOD OF DYNAMIC REGULARIZATION

2019 ◽  
Vol 26 ◽  
pp. 47-60
Author(s):  
V. SKACHKOV ◽  
Keyword(s):  
Inverse Problem ◽  
Regularization Method ◽  
Correlation Matrix ◽  
Regularization Parameter ◽  
A Priori ◽  
Computational Cost ◽  
Spectral Composition ◽  
Adaptive Antenna ◽  
Computational Stability ◽  
A Priori Uncertainty

The problem of forming sample estimates of the correlation matrix of observations that satisfy the criterion "computational stability – consistency" is considered. The variants in which the direct and inverse asymptotic forms of the correlation matrix of observations are approximated by various types of estimates formed from a sample of a fixed volume are investigated. The consistency of computationally stable estimates of the correlation matrix for their static regularization was analyzed. The contradiction inherent in the problem of regularization of the estimates with a fixed parameter is revealed. The dynamic regularization method as an alternative approach is proposed, which is based on the uniqueness theorem for solving the inverse problem with perturbed initial data. An optimal mean-square approximation algorithm has been developed for dynamic regularization of sample estimates of the correlation matrix of observations, using the law of monotonic decrease in the regularizing parameter with increasing sample size. An optimal dynamic regularization function was obtained for sample estimates of the correlation matrix under conditions of a priori uncertainty with respect to their spectral composition. The preference of this approach to the regularization of sample estimates of the correlation matrix under conditions of a priori uncertainty is proved, which allows to exclude the domain of computational instability from solving the inverse problem and obtain its solution in real time without involving prediction data and additional computational cost for finding the optimal value of the regularization parameter. The application of the dynamic regularization method is shown for solving the problem of detecting a signal at the output of an adaptive antenna array in a nondeterministic clutter and jamming environment. The results of a computational experiment that confirm the main conclusions are presented.

scholarly journals Inverse Problem Solution and Regularization Parameter Selection for Current Distribution Reconstruction in Switching Arcs by Inverting Magnetic Fields

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Jinlong Dong ◽  
Guogang Zhang ◽  
Zhiqiang Zhang ◽  
Yingsan Geng ◽  
Jianhua Wang
Keyword(s):  
Inverse Problem ◽  
Density Distribution ◽  
Current Density ◽  
Cross Validation ◽  
Regularization Parameter ◽  
Current Density Distribution ◽  
Optimality Criteria ◽  
Parameter Selection ◽  
Selection Strategies ◽  
Regularization Parameter Selection

Current density distribution in electric arcs inside low voltage circuit breakers is a crucial parameter for us to understand the complex physical behavior during the arcing process. In this paper, we investigate the inverse problem of reconstructing the current density distribution in arcs by inverting the magnetic fields. A simplified 2D arc chamber is considered. The aim of this paper is the computational side of the regularization method, regularization parameter selection strategies, and the estimation of systematic error. To address the ill-posedness of the inverse problem, Tikhonov regularization is analyzed, with the regularization parameter chosen by Morozov’s discrepancy principle, the L-curve, the generalized cross-validation, and the quasi-optimality criteria. The provided range of regularization parameter selection strategies is much wider than in the previous works. Effects of several features on the performance of these criteria have been investigated, including the signal-to-noise ratio, dimension of measurement space, and the measurement distance. The numerical simulations show that the generalized cross-validation and quasi-optimality criteria provide a more satisfactory performance on the robustness and accuracy. Moreover, an optimal measurement distance can be expected when using a planner sensor array to perform magnetic measurements.

A quasi-boundary regularization method for identifying the initial value of time-fractional diffusion equation on spherically symmetric domain

2019 ◽  
Vol 27 (5) ◽  
pp. 609-621 ◽  
Author(s):  
Fan Yang ◽  
Ni Wang ◽  
Xiao-Xiao Li ◽  
Can-Yun Huang
Keyword(s):  
Diffusion Equation ◽  
Regularization Method ◽  
Regularization Parameter ◽  
A Priori ◽  
Symmetric Domain ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Spherically Symmetric ◽  
Initial Value ◽  
Time Fractional Diffusion Equation

Abstract In this paper, an inverse problem to identify the initial value for high dimension time fractional diffusion equation on spherically symmetric domain is considered. This problem is ill-posed in the sense of Hadamard, so the quasi-boundary regularization method is proposed to solve the problem. The convergence estimates between the regularization solution and the exact solution are presented under the a priori and a posteriori regularization parameter choice rules. Numerical examples are provided to show the effectiveness and stability of the proposed method.

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.

A New Computational Inverse Method and Application to the Identification of Dynamic Loads

2013 ◽  
Vol 631-632 ◽  
pp. 1298-1302
Author(s):  
Lin Jun Wang ◽  
You Xiang Xie ◽  
Hai Hua Wu
Keyword(s):  
Tikhonov Regularization ◽  
Regularization Method ◽  
Present Method ◽  
Inverse Method ◽  
Regularization Parameter ◽  
A Priori ◽  
Identification Problem ◽  
Dynamic Loads ◽  
Tikhonov Regularization Method ◽  
Simply Supported

In this paper, we propose a new computational inverse method for solving the identification of multi-source dynamic loads acting on a simply supported plate. Using a priori choosing appropriate regularization parameter, the present method can obtain higher optimum asymptotic order of the regularized solution than ordinary Tikhonov regularization method. In the numerical simulations, the identification problem of multi-source dynamic loads on a surface of simply supported plate is successfully solved by the present method. Meanwhile, most of its performances are better than ordinary Tikhonov regularization 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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