Regularization of the backward stochastic heat conduction problem

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nguyen Huy Tuan ◽  
Daniel Lesnic ◽  
Tran Ngoc Thach ◽  
Tran Bao Ngoc
Keyword(s):  
Heat Conduction ◽  
Regularization Method ◽  
Continuous Dependence ◽  
Input Data ◽  
Convergence Rates ◽  
Heat Conduction Problem ◽  
A Priori ◽  
Backward Problem ◽  
Ill Posed ◽  
Stochastic Setting

Abstract In this paper, we study the backward problem for the stochastic parabolic heat equation driven by a Wiener process. We show that the problem is ill-posed by violating the continuous dependence on the input data. In order to restore stability, we apply a filter regularization method which is completely new in the stochastic setting. Convergence rates are established under different a priori assumptions on the sought solution.

scholarly journals A Mollification Regularization Method for a Fractional-Diffusion Inverse Heat Conduction Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Zhi-Liang Deng ◽  
Xiao-Mei Yang ◽  
Xiao-Li Feng
Keyword(s):  
Regularization Method ◽  
Heat Conduction Problem ◽  
A Priori ◽  
Dirichlet Kernel ◽  
Inverse Heat Conduction Problem ◽  
Inverse Heat Conduction ◽  
Linear Diffusion ◽  
Ill Posed ◽  
Transient Data ◽  
Parameter Choice Rules

The ill-posed problem of attempting to recover the temperature functions from one measured transient data temperature at some interior point of a one-dimensional semi-infinite conductor when the governing linear diffusion equation is of fractional type is discussed. A simple regularization method based on Dirichlet kernel mollification techniques is introduced. We also proposea priorianda posterioriparameter choice rules and get the corresponding error estimate between the exact solution and its regularized approximation. Moreover, a numerical example is provided to verify our theoretical results.

scholarly journals Identical Approximation Operator and Regularization Method for the Cauchy problem of 2D Heat Conduction Equation

2020 ◽  
Author(s):  
Shangqin He ◽  
Xiufang Feng
Keyword(s):  
Cauchy Problem ◽  
Heat Conduction ◽  
Regularization Method ◽  
Heat Conduction Equation ◽  
Convergence Rates ◽  
Regularization Parameter ◽  
Numerical Results ◽  
Conduction Equation ◽  
The Cauchy Problem ◽  
Ill Posed

In this paper, an identical approximate regularization method is extended to the Cauchy problem of two-dimensional heat conduction equation, this kind of problem is severely ill-posed. The convergence rates are obtained under a priori regularization parameter choice rule. Numerical results are presented for two examples with smooth and continuous but not smooth boundaries, and compared the identical approximate regularization solutions which are displayed in paper. The numerical results show that our method is effective, accurate and stable to solve the ill-posed Cauchy problems.

scholarly journals Optimal Tikhonov approximation for a sideways parabolic equation

2005 ◽  
Vol 2005 (8) ◽  
pp. 1221-1237 ◽  
Author(s):  
Chu-Li Fu ◽  
Hong-Fang Li ◽  
Xiang-Tuan Xiong ◽  
Peng Fu
Keyword(s):  
Heat Conduction ◽  
Parabolic Equation ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Heat Conduction Problem ◽  
Inverse Heat Conduction Problem ◽  
Tikhonov Regularization Method ◽  
Inverse Heat Conduction ◽  
Convection Term ◽  
Ill Posed

We consider an inverse heat conduction problem with convection term which appears in some applied subjects. This problem is ill posed in the sense that the solution (if it exists) does not depend continuously on the data. A generalized Tikhonov regularization method for this problem is given, which realizes the best possible accuracy.

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.

METHOD OF FUNDAMENTAL SOLUTION FOR AN INVERSE HEAT CONDUCTION PROBLEM WITH VARIABLE COEFFICIENTS

2013 ◽  
Vol 10 (02) ◽  
pp. 1341009 ◽  
Author(s):  
MING LI ◽  
XIANG-TUAN XIONG ◽  
YAN LI
Keyword(s):  
Heat Conduction ◽  
Heat Conduction Problem ◽  
Fundamental Solutions ◽  
Variable Coefficients ◽  
Inverse Heat Conduction Problem ◽  
Method Of Fundamental Solutions ◽  
Inverse Heat Conduction ◽  
Modified Method ◽  
On Line ◽  
Ill Posed

In this paper, we consider an inverse heat conduction problem with variable coefficient a(t). In many practical situations such as an on-line testing, we cannot know the initial condition for example because we have to estimate the problem for the heat process which was already started. Based on the method of fundamental solutions, we give a numerical scheme for solving the reconstruction problem. Since the governing equation contains variable coefficients, modified method of fundamental solutions was used to solve this kind of ill-posed problems. Some numerical examples are given for verifying the efficiency and accuracy of the presented method.

scholarly journals The Impact of the Discrepancy Principle on the Tikhonov-Regularized Solutions with Oversmoothing Penalties

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 331
Author(s):  
Bernd Hofmann ◽  
Christopher Hofmann
Keyword(s):  
Case Studies ◽  
Convergence Rates ◽  
Regularization Parameter ◽  
A Priori ◽  
Discrepancy Principle ◽  
Parameter Choice ◽  
Operator Equations ◽  
Analytical Results ◽  
Ill Posed ◽  
The Impact

This paper deals with the Tikhonov regularization for nonlinear ill-posed operator equations in Hilbert scales with oversmoothing penalties. One focus is on the application of the discrepancy principle for choosing the regularization parameter and its consequences. Numerical case studies are performed in order to complement analytical results concerning the oversmoothing situation. For example, case studies are presented for exact solutions of Hölder type smoothness with a low Hölder exponent. Moreover, the regularization parameter choice using the discrepancy principle, for which rate results are proven in the oversmoothing case in in reference (Hofmann, B.; Mathé, P. Inverse Probl. 2018, 34, 015007) is compared to Hölder type a priori choices. On the other hand, well-known analytical results on the existence and convergence of regularized solutions are summarized and partially augmented. In particular, a sketch for a novel proof to derive Hölder convergence rates in the case of oversmoothing penalties is given, extending ideas from in reference (Hofmann, B.; Plato, R. ETNA. 2020, 93).

Analysis of Order of the Sequential Regularization Solutions of Inverse Heat Conduction Problems

1989 ◽  
Vol 111 (2) ◽  
pp. 218-224 ◽  
Author(s):  
E. P. Scott ◽  
J. V. Beck
Keyword(s):  
Heat Conduction ◽  
Regularization Method ◽  
Heat Conduction Problem ◽  
Regularization Parameter ◽  
Inverse Heat Conduction Problem ◽  
Regularization Methods ◽  
Inverse Heat Conduction ◽  
Random Errors ◽  
Internal Temperature ◽  
Inverse Heat Conduction Problems

Various methods have been proposed to solve the inverse heat conduction problem of determining a boundary condition at the surface of a body from discrete internal temperature measurements. These include function specification and regularization methods. This paper investigates the various components of the regularization method using the sequential regularization method proposed by Beck and Murio (1986). Specifically, the effects of the regularization order and the influence of the regularization parameter are analyzed. It is shown that as the order of regularization increases, the bias errors decrease and the variance increases. Comparatively, the zeroth regularization has higher bias errors and the second-order regularization is more sensitive to random errors. As the regularization parameter decreases, the sensitivity of the estimator to random errors is shown to increase; on the other hand, the bias errors are shown to decrease.

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.

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