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 On an initial inverse problem for a diffusion equation with a conformable derivative

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Tran Thanh Binh ◽  
Nguyen Hoang Luc ◽  
Donal O’Regan ◽  
Nguyen H. Can
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Regularization Method ◽  
Conformable Derivative ◽  
Parameter Choice ◽  
Backward Problem ◽  
Ill Posed ◽  
Two Parameter ◽  
Parameter Choice Rules ◽  
Regularized Solution

AbstractIn this paper, we consider the initial inverse problem for a diffusion equation with a conformable derivative in a general bounded domain. We show that the backward problem is ill-posed, and we propose a regularizing scheme using a fractional Landweber regularization method. We also present error estimates between the regularized solution and the exact solution using two parameter choice rules.

Dual Regularized Total Least Squares And Multi-Parameter Regularization

2008 ◽  
Vol 8 (3) ◽  
pp. 253-262 ◽  
Author(s):  
S. LU ◽  
S.V. PEREVERZEV ◽  
U. TAUTENHAHN
Keyword(s):  
Least Squares ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Differential Operators ◽  
Total Least Squares ◽  
Right Hand ◽  
Regularization Parameters ◽  
Ill Posed ◽  
Regularized Total Least Squares ◽  
Parameter Regularization

AbstractIn this paper we continue our study of solving ill-posed problems with a noisy right-hand side and a noisy operator. Regularized approximations are obtained by Tikhonov regularization with differential operators and by dual regularized total least squares (dual RTLS) which can be characterized as a special multi-parameter regularization method where one of the two regularization parameters is negative. We report on order optimality results for both regularized approximations, discuss compu-tational aspects, provide special algorithms and show by experiments that dual RTLS is competitive to Tikhonov regularization with differential operators.

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.

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.

scholarly journals Regularization and error estimates for an inverse heat problem under the conformable derivative

2018 ◽  
Vol 16 (1) ◽  
pp. 999-1011 ◽  
Author(s):  
Ho Vu ◽  
Donal O’Regan ◽  
Ngo Van Hoa
Keyword(s):  
Estimation Error ◽  
The Other ◽  
Time Interval ◽  
Conformable Derivative ◽  
Time Problem ◽  
Regularization Parameters ◽  
Boundary Value Method ◽  
Ill Posed ◽  
Nonhomogeneous Heat ◽  
Inverse Heat Problem

AbstractIn this paper we study an inverse time problem for the nonhomogeneous heat equation under the conformable derivative which is a severely ill-posed problem. Using the quasi-boundary value method with two regularization parameters (one related to the error in a measurement process and the other is related to the regularity of the solution) we regularize this problem and obtain a Hölder-type estimation error for the whole time interval. Numerical results are presented to illustrate the accuracy and efficiency of the method.

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.

A modified integral equation method of the semilinear backward heat problem

2011 ◽  
Vol 217 (12) ◽  
pp. 5177-5185
Author(s):  
Nguyen Huy Tuan ◽  
Dang Duc Trong ◽  
Pham Hoang Quan
Keyword(s):  
Integral Equation ◽  
Integral Equation Method ◽  
Equation Method ◽  
Heat Problem ◽  
Backward Heat Problem

On an asymmetric backward heat problem with the space and time-dependent heat source on a disk

2019 ◽  
Vol 27 (1) ◽  
pp. 103-115
Author(s):  
Triet Minh Le ◽  
Quan Hoang Pham ◽  
Phong Hong Luu
Keyword(s):  
Exact Solution ◽  
Heat Source ◽  
Stable Solution ◽  
Time Dependent ◽  
Polar Coordinates ◽  
Heat Problem ◽  
Space And Time ◽  
Boundary Value Method ◽  
Ill Posed ◽  
Backward Heat Problem

Abstract In this article, we investigate the backward heat problem (BHP) which is a classical ill-posed problem. Although there are many papers relating to the BHP in many domains, considering this problem in polar coordinates is still scarce. Therefore, we wish to deal with this problem associated with a space and time-dependent heat source in polar coordinates. By modifying the quasi-boundary value method, we propose the stable solution for the problem. Furthermore, under some initial assumptions, we get the Hölder type of error estimates between the exact solution and the approximated solution. Eventually, a numerical experiment is provided to prove the effectiveness and feasibility of our method.

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