scholarly journals Inverse Estimates for Nonhomogeneous Backward Heat Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Tao Min ◽  
Weimin Fu ◽  
Qiang Huang
Keyword(s):  
Genetic Algorithm ◽  
Integral Equation ◽  
Regularization Method ◽  
Initial Temperature ◽  
Regularization Parameter ◽  
Final Temperature ◽  
Ill Posed ◽  
Tikhonov’S Regularization ◽  
Nonhomogeneous Heat ◽  
Backward Heat Problem

We investigate the inverse problem in the nonhomogeneous heat equation involving the recovery of the initial temperature from measurements of the final temperature. This problem is known as the backward heat problem and is severely ill-posed. We show that this problem can be converted into the first Fredholm integral equation, and an algorithm of inversion is given using Tikhonov's regularization method. The genetic algorithm for obtaining the regularization parameter is presented. We also present numerical computations that verify the accuracy of our approximation.

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.

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.

An optimal regularization method for convolution equations on the sourcewise represented set

2015 ◽  
Vol 23 (5) ◽  
Author(s):  
Ye Zhang ◽  
Dmitry V. Lukyanenko ◽  
Anatoly G. Yagola
Keyword(s):  
Integral Equation ◽  
Regularization Method ◽  
A Priori ◽  
Noisy Data ◽  
Optimal Recovery ◽  
Multidimensional Case ◽  
A Priori Information ◽  
Ill Posed ◽  
Priori Information ◽  
Convolution Equations

AbstractIn this article, we consider an inverse problem for the integral equation of the convolution type in a multidimensional case. This problem is severely ill-posed. To deal with this problem, using a priori information (sourcewise representation) based on optimal recovery theory we propose a new method. The regularization and optimization properties of this method are proved. An optimal minimal a priori error of the problem is found. Moreover, a so-called optimal regularized approximate solution and its corresponding error estimation are considered. Efficiency and applicability of this method are demonstrated in a numerical example of the image deblurring problem with noisy data.

Investigation of the Initial Inverse Problem in the Heat Equation

2004 ◽  
Vol 126 (2) ◽  
pp. 294-296 ◽  
Author(s):  
Khalid Masood ◽  
F. D. Zaman
Keyword(s):  
Inverse Problem ◽  
Heat Conduction ◽  
Heat Equation ◽  
Initial Temperature ◽  
Final Temperature ◽  
Conduction Model ◽  
Heat Conduction Model ◽  
Ill Posed

We investigate the inverse problem in the heat equation involving the recovery of the initial temperature from measurements of the final temperature. This problem is extremely ill-posed and it is believed that only information in the first few modes can be recovered by classical methods. We will consider this problem with a regularizing parameter which approximates and regularizes the heat conduction model.

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.

Preliminary Results of Integrating Tikhonov’s Regularization in Forward-Backward Time-Stepping Technique for Object Detection

2016 ◽  
Vol 833 ◽  
pp. 170-175 ◽  
Author(s):  
Andrew Sia Chew Chie ◽  
Kismet Anak Hong Ping ◽  
Yong Guang ◽  
Ng Shi Wei ◽  
Nordiana Rajaee
Keyword(s):  
Image Reconstruction ◽  
Inverse Scattering ◽  
Time Domain ◽  
Regularization Method ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Time Stepping ◽  
Ill Posed ◽  
Tikhonov’S Regularization

The inverse scattering in time domain known as Forward-Backward Time-Stepping (FBTS) technique is applied to determine the sizes, shape and location of the embedded objects. Tikhonov’s regularization method has been proposed in order to improve or solve the ill-posed of FBTS inverse scattering problem. The reconstructed results showed that FBTS technique can detect the presence of embedded objects. The reconstructed results of FBTS technique utilizing with the Tikhonov’s regularization method shown better results than the results only applied FBTS technique. Tikhonov’s regularization combined with FBTS technique to improve the quality of image reconstruction.

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.

scholarly journals A two-dimensional backward heat problem with statistical discrete data

2018 ◽  
Vol 26 (1) ◽  
pp. 13-31 ◽  
Author(s):  
Nguyen Dang Minh ◽  
Khanh To Duc ◽  
Nguyen Huy Tuan ◽  
Dang Duc Trong
Keyword(s):  
Regression Models ◽  
Initial Temperature ◽  
Least Squares Method ◽  
Random Noise ◽  
Heat Problem ◽  
Formula Group ◽  
Noise Data ◽  
Ill Posed ◽  
Grid Points ◽  
Backward Heat Problem

AbstractWe focus on the nonhomogeneous backward heat problem of finding the initial temperature {\theta=\theta(x,y)=u(x,y,0)} such that\left\{\begin{aligned} \displaystyle u_{t}-a(t)(u_{xx}+u_{yy})&\displaystyle=f% (x,y,t),&\hskip 10.0pt(x,y,t)&\displaystyle\in\Omega\times(0,T),\\ \displaystyle u(x,y,t)&\displaystyle=0,&\hskip 10.0pt(x,y)&\displaystyle\in% \partial\Omega\times(0,T),\\ \displaystyle u(x,y,T)&\displaystyle=h(x,y),&\hskip 10.0pt(x,y)&\displaystyle% \in\overline{\Omega},\end{aligned}\right.\vspace*{-0.5mm}where {\Omega=(0,\pi)\times(0,\pi)}. In the problem, the source {f=f(x,y,t)} and the final data {h=h(x,y)} are determined through random noise data {g_{ij}(t)} and {d_{ij}} satisfying the regression models\displaystyle g_{ij}(t)=f(X_{i},Y_{j},t)+\vartheta\xi_{ij}(t),\displaystyle d_{ij}=h(X_{i},Y_{j})+\sigma_{ij}\varepsilon_{ij},where {(X_{i},Y_{j})} are grid points of Ω. The problem is severely ill-posed. To regularize the instable solution of the problem, we use the trigonometric least squares method in nonparametric regression associated with the projection method. In addition, convergence rate is also investigated numerically.

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.

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