Identification of an unknown time-dependent heat source term from overspecified Dirichlet boundary data by conjugate gradient method

The inverse problem of identifying the unknown time-dependent heat source H(t) of the variable coefficient heat equation u"t=(k(x)u"x)"x+F(x)H(t), with separable sources of the form F(x)H(t), from supplementary temperature measurement h(t)@?u(0,t) at the left end of the rod, is investigated. The Fourier method is employed to illustrate the comparison of spacewise (F(x)) and time-dependent (H(t) heat source identification problems. An explicit formula for the Frechet gradient of the cost functional J(H)=@?u(0,@?;H)-h@?"L"""2"("0","T"""f")^2 is derived via the unique solution of the appropriate adjoint problem. The Conjugate Gradient Algorithm, based on the gradient formula for the cost functional, is then proposed for numerical solution of the inverse source problem. The algorithm is examined through numerical examples related to reconstruction of continuous and discontinuous heat sources H(t), when heat is transferred through non-homogeneous as well as composite structures. Numerical analysis of the algorithm applied to the inverse source problem in typical classes of source functions is presented. Computational results, obtained for random noisy output data, show how the iteration number of the Conjugate Gradient Algorithm can be estimated. Based on these results it is shown that this iteration number plays a role of a regularization parameter. Numerical results illustrate bounds of applicability of the proposed algorithm, and also its efficiency and accuracy.

[1]  V. Isakov.  Appendix -- Function Spaces , 2017 .

[2]  Daniel Lesnic,et al.  The boundary-element method for the determination of a heat source dependent on one variable , 2006 .

[3]  A. A. Samarskii,et al.  The Theory of Difference Schemes , 2001 .

[4]  Alemdar Hasanov,et al.  Identification of spacewise and time dependent source terms in 1D heat conduction equation from temperature measurement at a final time , 2012 .

[5]  Masahiro Yamamoto,et al.  Conditional stability in determining a heat source , 2004 .

[6]  Chein-Shan Liu.  A two-stage LGSM to identify time-dependent heat source through an internal measurement of temperature , 2009 .

[7]  A. Kirsch.  An Introduction to the Mathematical Theory of Inverse Problems , 1996, Applied Mathematical Sciences.

[8]  D. A. Murio,et al.  Source term identification in 1-D IHCP , 2004 .

[9]  Leevan Ling,et al.  Identification of source locations in two-dimensional heat equations , 2006 .

[10]  T. Ha-Duong,et al.  On an inverse source problem for the heat equation. Application to a pollution detection problem , 2002 .

[11]  Otmar Scherzer,et al.  Uniqueness and stable determination of forcing terms in linear partial differential equations with overspecified boundary data , 1994 .

[12]  J. Cannon.  Determination of an Unknown Heat Source from Overspecified Boundary Data , 1968 .

[13]  J. I. Frankel,et al.  Residual-minimization least-squares method for inverse heat conduction , 1996 .

[14]  Alemdar Hasanov,et al.  Simultaneous determination of source terms in a linear parabolic problem from the final overdetermination: Weak solution approach , 2007 .

[15]  Mehdi Dehghan,et al.  Inverse problem of time-dependent heat sources numerical reconstruction , 2011, Math. Comput. Simul..

[16]  Liu Yang,et al.  Optimization method for the inverse problem of reconstructing the source term in a parabolic equation , 2009, Math. Comput. Simul..

[17]  D. Reeve,et al.  Determination of a source term in the linear diffusion equation , 1994 .

[18]  Alemdar Hasanov,et al.  Inverse heat conduction problems with boundary and final time measured output data , 2011 .

[19]  Chu-Li Fu,et al.  Identification of an unknown source depending on both time and space variables by a variational method , 2012 .

[20]  Per Christian Hansen,et al.  Regularization methods for large-scale problems , 1993 .

[21]  A. I. Prilepko,et al.  Inverse problem for a parabolic equation with integral overdetermination , 2003 .

[22]  Dinh Nho Hào,et al.  Methods for inverse heat conduction problems , 1998 .

[23]  I Bushuyev,et al.  Global uniqueness for inverse parabolic problems with final observation , 1995 .

[24]  J. Cannon,et al.  Structural identification of an unknown source term in a heat equation , 1998 .

[25]  C. Fu,et al.  The method of fundamental solutions for the inverse heat source problem. , 2008 .

[26]  O. A. Ladyzhenskai︠a︡,et al.  Linear and Quasi-linear Equations of Parabolic Type , 1995 .

[27]  D. Lesnic,et al.  Determination of a spacewise dependent heat source , 2007 .

[28]  O. Ladyzhenskaya.  The Boundary Value Problems of Mathematical Physics , 1985 .

[29]  Åke Björck,et al.  Numerical methods for least square problems , 1996 .

[30]  P. Hansen.  Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion , 1987 .

[31]  Alemdar Hasanov,et al.  An inverse source problem with single Dirichlet type measured output data for a linear parabolic equation , 2011, Appl. Math. Lett..

[32]  E. G. SAVATEEV,et al.  On problems of determining the source function in a parabolic equation , 1995 .