scholarly journals Extract the information from big data with randomly distributed noise

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jin Cheng ◽  
Jiantang Zhang ◽  
Min Zhong
Keyword(s):  
Big Data ◽  
Confidence Interval ◽  
Unique Solvability ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Data Driven ◽  
Numerical Examples ◽  
Ill Posed ◽  
General Data ◽  
Bound Estimation

Abstract In this manuscript, a purely data-driven statistical regularization method is proposed for extracting the information from big data with randomly distributed noise. Since the variance of the noise may be large, the method can be regarded as a general data preprocessing method in ill-posed problems, which is able to overcome the difficulty that the traditional regularization method is unable to solve, and has superior advantage in computing efficiency. The unique solvability of the method is proved, and a number of conditions are given to characterize the solution. The regularization parameter strategy is discussed, and the rigorous upper bound estimation of the confidence interval of the error in the L 2 L^{2} norm is established. Some numerical examples are provided to illustrate the appropriateness and effectiveness of the method.

scholarly journals EXTRAPOLATION OF TIKHONOV REGULARIZATION METHOD

2010 ◽  
Vol 15 (1) ◽  
pp. 55-68 ◽  
Author(s):  
Uno Hämarik ◽  
Reimo Palm ◽  
Toomas Raus
Keyword(s):  
Approximate Solution ◽  
Tikhonov Regularization ◽  
Noise Level ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Noisy Data ◽  
Tikhonov Regularization Method ◽  
Guaranteed Accuracy ◽  
Numerical Examples ◽  
Ill Posed

We consider regularization of linear ill‐posed problem Au = f with noisy data fδ, ¦fδ - f¦≤ δ . The approximate solution is computed as the extrapolated Tikhonov approximation, which is a linear combination of n ≥ 2 Tikhonov approximations with different parameters. If the solution u* belongs to R((A*A) n ), then the maximal guaranteed accuracy of Tikhonov approximation is O(δ 2/3) versus accuracy O(δ 2n/(2n+1)) of corresponding extrapolated approximation. We propose several rules for choice of the regularization parameter, some of these are also good in case of moderate over‐ and underestimation of the noise level. Numerical examples are given.

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.

The Regularization Method Based on Complex Collinearity Diagnostics and Metrics

2013 ◽  
Vol 416-417 ◽  
pp. 1393-1398
Author(s):  
Chao Zhong Ma ◽  
Yong Wei Gu ◽  
Ji Fu ◽  
Yuan Lu Du ◽  
Qing Ming Gui
Keyword(s):  
Tikhonov Regularization ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Measurement Data ◽  
Tikhonov Regularization Method ◽  
Parameter Determination ◽  
Selection Methods ◽  
Ill Posed ◽  
Collinearity Diagnostics ◽  
Regularization Matrix

In a large number of measurement data processing, the ill-posed problem is widespread. For such problems, this paper introduces the solution of ill-posed problem of the unity of expression and Tikhonov regularization method, and then to re-collinearity diagnostics and metrics based on proposed based on complex collinearity diagnostics and the metric regularization method is given regularization matrix selection methods and regularization parameter determination formulas. Finally, it uses a simulation example to verify the effectiveness of the method.

scholarly journals A Regularization Method to Solve a Cauchy Problem for the Two-Dimensional Modified Helmholtz Equation

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 360 ◽  
Author(s):  
Shangqin He ◽  
Xiufang Feng
Keyword(s):  
Helmholtz Equation ◽  
Regularization Method ◽  
Numerical Approximation ◽  
Regularization Parameter ◽  
Two Dimensional ◽  
Approximation Solution ◽  
Modified Helmholtz Equation ◽  
Strip Domain ◽  
Ill Posed ◽  
De La Vallée Poussin

In this paper, the ill-posed problem of the two-dimensional modified Helmholtz equation is investigated in a strip domain. For obtaining a stable numerical approximation solution, a mollification regularization method with the de la Vallée Poussin kernel is proposed. An error estimate between the exact solution and approximation solution is given under suitable choices of the regularization parameter. Two numerical experiments show that our procedure is effective and stable with respect to perturbations in the data.

Elastoplasticity 2D Problems: Numerical Applications of the Tikhonov Regularization Method

2015 ◽  
Vol 751 ◽  
pp. 109-117
Author(s):  
Thales Augusto Barbosa Pinto Silva ◽  
Hilbeth Parente Azikri de Deus ◽  
Claudio Roberto Ávila da Silva
Keyword(s):  
Tikhonov Regularization ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Numerical Approach ◽  
Theoretical Development ◽  
Tikhonov Regularization Method ◽  
Equilibrium Curve ◽  
Numerical Examples ◽  
Elastoplastic Materials ◽  
Ill Conditioning

The numerical simulation is widely used, in now days, to verify the viability and to optimize structural mechanic designs. The numerical approach of elastoplastic materials can found some problems related to ill-conditioning of matrices (from FEM systems), associated to the critical points from the snap through or snap back shape of the equilibrium curve. Aiming to overcome this misfortune it is proposed a strategy via Tikhonov regularization method in association with L-curve technique to determine the regularization parameter. This strategy can be used in many numerical applications for structural analysis. The theoretical development about these Some numerical examples are presented to attest the efficiency of this proposed approach.

A wavelet method for numerical fractional derivative with noisy data

2016 ◽  
Vol 14 (05) ◽  
pp. 1650038 ◽  
Author(s):  
Xiangtuan Xiong ◽  
Qiang Cheng ◽  
Yanfeng Kong ◽  
Jin Wen
Keyword(s):  
Fractional Derivative ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Large Change ◽  
Stability Estimates ◽  
Parameter Choice ◽  
Wavelet Regularization ◽  
Ill Posed ◽  
Meyer Wavelet ◽  
Parameter Choice Rules

Numerical fractional differentiation is a classical ill-posed problem in the sense that a small perturbation in the data can cause a large change in the fractional derivative. In this paper, we consider a wavelet regularization method for solving a reconstruction problem for numerical fractional derivative with noise. A Meyer wavelet projection regularization method is given, and the Hölder-type stability estimates under both apriori and aposteriori regularization parameter choice rules are obtained. Some numerical examples show that the method works well.

An Improved Blind Image Restoration Based on Bi-Spectral Reconstruction

2014 ◽  
Vol 644-650 ◽  
pp. 4229-4232
Author(s):  
Li Li Liu ◽  
Jian Song Tian
Keyword(s):  
Image Restoration ◽  
High Frequency ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Low Frequency ◽  
Blind Image Restoration ◽  
Spectral Reconstruction ◽  
Blind Restoration ◽  
Ill Posed ◽  
Blind Image

Image blind restoration is very important in our life. The image restoration is a ill-posed question so the regularization is much better method. For the regularization method, the most important is to select the regularization parameter [1]. If the parameter is bigger, to be smooth the edge or detail, but smaller, not to be smooth the noise [2], In this paper, we present a new method. Firstly, decomposing the image using wavelet transform, the high frequency information is corresponding to the edge and noise, the low frequency is the flat .We denoise using the bi-spectral reconstruction in high frequency, for the low frequency, we recover by the regularization method .This method has advantage in holding the edge and is simple to choose the parameter of regularization .Experimental results show the good performance, this method is very effective for the image polluted by the symmetry noise.

On The Analog Of The Monotone Error Rule For Parameter Choice In The (Iterated) Lavrentiev Regularization

2008 ◽  
Vol 8 (3) ◽  
pp. 237-252 ◽  
Author(s):  
U HAMARIK ◽  
R. PALM ◽  
T. RAUS
Keyword(s):  
Hilbert Spaces ◽  
Noise Level ◽  
Regularization Parameter ◽  
Optimal Error Estimates ◽  
Discrepancy Principle ◽  
Parameter Choice ◽  
Optimal Error ◽  
Numerical Examples ◽  
Lavrentiev Regularization ◽  
Ill Posed

AbstractWe consider linear ill-posed problems in Hilbert spaces with a noisy right hand side and a given noise level. To solve non-self-adjoint problems by the (it-erated) Tikhonov method, one effective rule for choosing the regularization parameter is the monotone error rule (Tautenhahn and Hamarik, Inverse Problems, 1999, 15, 1487– 1505). In this paper we consider the solution of self-adjoint problems by the (iterated) Lavrentiev method and propose for parameter choice an analog of the monotone error rule. We prove under certain mild assumptions the quasi-optimality of the proposed rule guaranteeing convergence and order optimal error estimates. Numerical examples show for the proposed rule and its modifications much better performance than for the modified discrepancy principle.

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 IDENTIFICATION OF THE SOURCE FOR FULL PARABOLIC EQUATIONS

2021 ◽  
Vol 26 (3) ◽  
pp. 339-357
Author(s):  
Guillermo Federico Umbricht
Keyword(s):  
Parabolic Equations ◽  
Regularization Parameter ◽  
Estimation Error ◽  
Noisy Data ◽  
Stability And Convergence ◽  
Parameter Choice ◽  
Independent Source ◽  
Numerical Examples ◽  
Ill Posed ◽  
The Stability

In this work, we consider the problem of identifying the time independent source for full parabolic equations in Rn from noisy data. This is an ill-posed problem in the sense of Hadamard. To compensate the factor that causes the instability, a family of parametric regularization operators is introduced, where the rule to select the value of the regularization parameter is included. This rule, known as regularization parameter choice rule, depends on the data noise level and the degree of smoothness that it is assumed for the source. The proof for the stability and convergence of the regularization criteria is presented and a Hölder type bound is obtained for the estimation error. Numerical examples are included to illustrate the effectiveness of this regularization approach.

Sign in / Sign up

Export Citation Format

Share Document