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Numerical Solution of a Cauchy Problem for a Parabolic Equation in Two or more Space Dimensions by the Arnoldi Method
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
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 [en]
Cauchy problem, inverse problem, ill-posed, iterative method, Arnoldi method, Schur decomposition, parabolic PDE
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-54299OAI: oai:DiVA.org:liu-54299DiVA: diva2:302604
Available from: 2010-03-08 Created: 2010-03-08 Last updated: 2011-03-09Bibliographically approved
In thesis
1. Numerical Solution of Ill-posed Cauchy Problems for Parabolic Equations
Open this publication in new window or tab >>Numerical Solution of Ill-posed Cauchy Problems for Parabolic Equations
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:nbn:se:liu:diva-54300 (URN)978-91-7393-443-5 (ISBN)
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

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Ranjbar, Zohreh

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