Stabilizing of an ill-posed inverse problem by using smoothing splines and hyperbolic heat equation

2008 ◽  
Vol 16 (2) ◽  
pp. 233-247 ◽  
Author(s):  
Khalid Masood ◽  
MT Mustafa
Keyword(s):  
Inverse Problem ◽  
Heat Equation ◽  
Smoothing Splines ◽  
Ill Posed ◽  
Hyperbolic Heat Equation

scholarly journals Determination of conductivity in a heat equation

2000 ◽  
Vol 24 (9) ◽  
pp. 589-594 ◽  
Author(s):  
Ping Wang ◽  
Kewang Zheng
Keyword(s):  
Numerical Simulation ◽  
Inverse Problem ◽  
Approximate Solution ◽  
Heat Equation ◽  
Ill Posed ◽  
Smooth Data

We consider the problem of determining the conductivity in a heat equation from overspecified non-smooth data. It is an ill-posed inverse problem. We apply a regularization approach to define and construct a stable approximate solution. We also conduct numerical simulation to demonstrate the accuracy of our approximation.

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 Simultaneous Identification of Thermal Conductivity and Heat Source in the Heat Equation

2021 ◽  
pp. 1968-1978
Author(s):  
M. J. Huntul ◽  
M.S. Hussein
Keyword(s):  
Thermal Conductivity ◽  
Inverse Problem ◽  
Heat Source ◽  
Heat Equation ◽  
Mathematical Formulation ◽  
Approximate Solutions ◽  
Ill Posed ◽  
The One ◽  
Nicolson Scheme ◽  
Noisy Measurements

This paper presents a numerical solution to the inverse problem consisting of recovering time-dependent thermal conductivity and  heat source coefficients  in the one-dimensional  parabolic heat equation.   This  mathematical  formulation  ensures that the inverse problem  has a unique  solution.   However, the problem  is still  ill-posed since small errors  in the input data lead to a drastic  amount  of errors in the output coefficients.  The  finite  difference method  with  the Crank-Nicolson  scheme is adopted  as a direct  solver of the problem in a fixed domain.   The inverse problem is solved subjected to both exact and noisy measurements  by using the MATLAB  optimization  toolbox  routine  lsqnonlin , which is also applied to minimize the nonlinear  Tikhonov  regularization functional.  The thermal conductivity and heat source coefficients are reconstructed using heat flux measurements. The root mean squares error is used to assess the accuracy of the approximate solutions of the problem. A couple of  numerical  examples are presented to verify the accuracy and stability of the solutions.

AN INVERSE PROBLEM FOR THE HYPERBOLIC HEAT EQUATION

1992 ◽  
Vol 02 (01) ◽  
pp. 113-120
Author(s):  
E.G. SAVATEEV ◽  
L.M. DE SOCIO
Keyword(s):  
Inverse Problem ◽  
Heat Equation ◽  
Existence And Uniqueness ◽  
Local Existence ◽  
One Dimensional ◽  
Constructive Solution ◽  
Local Existence And Uniqueness ◽  
Diffusive Term ◽  
Hyperbolic Heat Equation

In this paper we prove a theorem of local existence and uniqueness for the solution of the hyperbolic heat equation in the case where the coefficient of the diffusive term is unknown. The problem is one-dimensional in space and the ratio of the two characteristics times, upon which the physics depends, is small. The demonstration relies on a constructive solution.

scholarly journals Bayesian Reconstruction through Adaptive Image Notion

Proceedings ◽  
2019 ◽  
Vol 33 (1) ◽  
pp. 21
Author(s):  
Fabrizia Guglielmetti ◽  
Eric Villard ◽  
Ed Fomalont
Keyword(s):  
Inverse Problem ◽  
Image Plane ◽  
Source Distribution ◽  
Bayesian Probability ◽  
Model Parameters ◽  
Uncertain Information ◽  
Source Detection ◽  
Component Mixture ◽  
Ill Posed ◽  
Image Planes

A stable and unique solution to the ill-posed inverse problem in radio synthesis image analysis is sought employing Bayesian probability theory combined with a probabilistic two-component mixture model. The solution of the ill-posed inverse problem is given by inferring the values of model parameters defined to describe completely the physical system arised by the data. The analysed data are calibrated visibilities, Fourier transformed from the ( u , v ) to image planes. Adaptive splines are explored to model the cumbersome background model corrupted by the largely varying dirty beam in the image plane. The de-convolution process of the dirty image from the dirty beam is tackled in probability space. Probability maps in source detection at several resolution values quantify the acquired knowledge on the celestial source distribution from a given state of information. The information available are data constrains, prior knowledge and uncertain information. The novel algorithm has the aim to provide an alternative imaging task for the use of the Atacama Large Millimeter/Submillimeter Array (ALMA) in support of the widely used Common Astronomy Software Applications (CASA) enhancing the capabilities in source detection.

scholarly journals Numerical method of identification of an unknown source term in a heat equation

2002 ◽  
Vol 8 (2) ◽  
pp. 161-168 ◽  
Author(s):  
Afet Golayoğlu Fatullayev
Keyword(s):  
Inverse Problem ◽  
Heat Equation ◽  
Numerical Method ◽  
Minimization Problem ◽  
Unknown Function ◽  
Source Term ◽  
Numerical Procedure ◽  
Numerical Examples ◽  
Unknown Source

A numerical procedure for an inverse problem of identification of an unknown source in a heat equation is presented. Approach of proposed method is to approximate unknown function by polygons linear pieces which are determined consecutively from the solution of minimization problem based on the overspecified data. Numerical examples are presented.

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