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Numerical Solution of Ill-posed Cauchy Problems for Parabolic Equations
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
2010 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Ill-posed mathematical problem occur in many interesting scientific and engineering applications. The solution of such a problem, if it exists, may not depend continuously on the observed data. For computing a stable approximate solution it is necessary to apply a regularization method. The purpose of this thesis is to investigate regularization approaches and develop numerical methods for solving certain ill-posed problems for parabolic partial differential equations. In thermal engineering applications one wants to determine the surface temperature of a body when the surface itself is inaccessible to measurements. This problem can be modelled by a sideways heat equation. The mathematical and numerical properties of the sideways heat equation with constant convection and diffusion coefficients is first studied. The problem is reformulated as a Volterra integral equation of the first kind with smooth kernel. The influence of the coefficients on the degree of ill-posedness are also studied. The rate of decay of the singular values of the Volterra integral operator determines the degree of ill-posedness. It is shown that the sign of the coefficient in the convection term influences the rate of decay of the singular values.

Further a sideways heat equation in cylindrical geometry is studied. The equation is a mathematical model of the temperature changes inside a thermocouple, which is used to approximate the gas temperature in a combustion chamber. The heat transfer coefficient at the surface of thermocouple is also unknown. This coefficient is approximated via a calibration experiment. Then the gas temperature in the combustion chamber is computed using the convection boundary condition. In both steps the surface temperature and heat flux are approximated using Tikhonov regularization and the method of lines.

Many existing methods for solving sideways parabolic equations are inadequate for solving multi-dimensional problems with variable coefficients. A new iterative regularization technique for solving a two-dimensional sideways parabolic equation with variable coefficients is proposed. A preconditioned Generalized Minimum Residuals Method (GMRS) is used to regularize the problem. The preconditioner is based on a semi-analytic solution formula for the corresponding problem with constant coefficients. Regularization is used in the preconditioner as well as truncating the GMRES algorithm. The computed examples indicate that the proposed PGMRES method is well suited for this problem.

In this thesis also a numerical method is presented for the solution of a Cauchy problem for a parabolic equation in multi-dimensional space, where the domain is cylindrical in one spatial direction. The formal solution is written as a hyperbolic cosine function in terms of a parabolic unbounded operator. The ill-posedness is dealt with by truncating the large eigenvalues of the operator. The approximate solution is computed by projecting onto a smaller subspace generated by the Arnoldi algorithm applied on the inverse of the operator. A well-posed parabolic problem is solved in each iteration step. Further the hyperbolic cosine is evaluated explicitly only for a small triangular matrix. Numerical examples are given to illustrate the performance of the method.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press , 2010. , 15 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1300
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-54300ISBN: 978-91-7393-443-5 (print)OAI: oai:DiVA.org:liu-54300DiVA: diva2:302609
Public defence
2010-03-29, C3, C-huset, Campus Valla, Linköpings universitet, Linköping, 13:15 (English)
Opponent
Supervisors
Available from: 2010-03-08 Created: 2010-03-08 Last updated: 2013-08-30Bibliographically approved
List of papers
1. Numerical analysis of an ill-posed Cauchy problem for a convection - Diffusion equation
Open this publication in new window or tab >>Numerical analysis of an ill-posed Cauchy problem for a convection - Diffusion equation
2007 (English)In: Inverse Problems in Science and Engineering, ISSN 1741-5977, Vol. 15, no 3, 191-211 p.Article in journal (Refereed) Published
Abstract [en]

The mathematical and numerical properties of an ill-posed Cauchy problem for a convection - diffusion equation are investigated in this study. The problem is reformulated as a Volterra integral equation of the first kind with a smooth kernel. The rate of decay of the singular values of the integral operator determines the degree of ill-posedness. The purpose of this article is to study how the convection term influences the degree of ill-posedness by computing numerically the singular values. It is also shown that the sign of the coefficient in the convection term determines the rate of decay of the singular values. Some numerical examples are also given to illustrate the theory.

Keyword
Cauchy problem, Convection - diffusion equation, Ill-posed, Inverse problem, Singular value decomposition, Volterra integral operator
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-50032 (URN)10.1080/17415970600557299 (DOI)
Available from: 2009-10-11 Created: 2009-10-11 Last updated: 2013-08-30
2. A Sideways Heat Equation Applied to the Measurement of the Gas Temperature in a Combustion Chamber
Open this publication in new window or tab >>A Sideways Heat Equation Applied to the Measurement of the Gas Temperature in a Combustion Chamber
2010 (English)Report (Other academic)
Abstract [en]

We consider a Cauchy problem for a parabolic equation as a mathematical model of the temperature development inside a suction pyrometer. Such devices are often used to calibrate the temperature sensor in a combustion chamber. Mathematically the problem is severely ill-posed and needs to be regularized. The model is simplified to make it one-dimensional in space. The temperature measurements are done in two steps. First, the heat transfer coefficient is approximated via a calibration experiment. Then the gas temperature in the combustion chamber is computed using a convection boundary condition. In both steps one computes the surface temperature and heat flux based on interior measurements in the thermocouple. Numerical examples are presented to test the proposed approach.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2010. 27 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2010:2
Keyword
Conduction, convective boundary condition, heat transfer coefficient, ill-posed, Volterra integral operator
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-54294 (URN)
Available from: 2010-03-08 Created: 2010-03-08 Last updated: 2013-08-30Bibliographically approved
3. A Preconditioned GMRES Method for Solving a Sideways Parabolic Equation in Two Space Dimensions
Open this publication in new window or tab >>A Preconditioned GMRES Method for Solving a Sideways Parabolic Equation in Two Space Dimensions
2010 (English)Report (Other academic)
Abstract [en]

The sideways parabolic equation (SPE) is a model of the problem of determining the temperature on the surface of a body from the interior measurements. Mathematically it can be formulated as a non-characteristic Cauchy problem for a parabolic partial differential equation. This problem is severely ill-posed: the solution does not depend continuously on the data. We consider both one and two-dimensional SPE with both constant and variable coefficients. We apply the preconditioned Generalized Minimum Residuals Method (GMRES) on these problems. Preconditioners are chosen in ways that allow efficient implementation using the Fast Fourier Transform (FFT). Regularization is used in the preconditioner as well as truncating the GMRES algorithm. Numerical experiments demonstrate that the proposed method works well.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2010. 36 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2010:3
Keyword
Cauchy problem, inverse problem, ill-posed, iterative methods, GMRES preconditioning, FFT, parabolic PDE
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-54298 (URN)
Available from: 2010-03-08 Created: 2010-03-08 Last updated: 2013-08-30Bibliographically approved
4. Numerical Solution of a Cauchy Problem for a Parabolic Equation in Two or more Space Dimensions by the Arnoldi Method
Open this publication in new window or tab >>Numerical Solution of a Cauchy Problem for a Parabolic Equation in Two or more Space Dimensions by the Arnoldi Method
2010 (English)Report (Other academic)
Abstract [en]

We consider the numerical solution of a Cauchy problem for a parabolic equation in multi-dimensional space with cylindrical domain in one spatial space direction. It is desired to find the lower boundary values from the Cauchy data on the upper boundary. This problem is severely ill-posed. The formal solution is written as a hyperbolic cosine function in terms of a multidimensional parabolic (unbounded) operator. We compute an approximate solution by projecting onto a smaller subspace generated via the Arnoldi algorithm applied on the discretized inverse of the operator. Further we regularize the projected problem. The hyperbolic cosine is evaluated explicitly on a low-dimensional subspace. In each iteration step of the Arnoldi method a well-posed parabolic problem is solved. Numerical examples are given to illustrate the performance of the method.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2010. 23 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2010:4
Keyword
Cauchy problem, inverse problem, ill-posed, iterative method, Arnoldi method, Schur decomposition, parabolic PDE
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-54299 (URN)
Available from: 2010-03-08 Created: 2010-03-08 Last updated: 2011-03-09Bibliographically approved

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Ranjbar, Zohreh

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