Identification of source locations in two-dimensional heat equations

In this paper, we show the uniqueness of the identification of unknown source locations in two-dimensional heat equations from scattered measurements. Based on the assumption that the unknown source function is a sum of some known functions, we prove that one measurement point is sufficient to identify the number of sources and three measurement points are sufficient to determine all unknown source locations. For verification, we propose a numerical reconstruction scheme for recovering the number of unknown sources and all source locations.

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

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

[3]  H. Park,et al.  A sequential method of solving inverse natural convection problems , 2002 .

[4]  An Inverse Problem for the Heat Equation II , 2002 .

[5]  E. Lieb,et al.  Analysis, Second edition , 2001 .

[6]  A El Badia,et al.  Some remarks on the problem of source identification from boundary measurements , 1998 .

[7]  Michael A. Golberg,et al.  The method of fundamental solutions for Poisson's equation , 1995 .

[8]  J. J. Moré,et al.  Levenberg--Marquardt algorithm: implementation and theory , 1977 .

[9]  A. G. Ramm.  An inverse problem for the heat equation , 1993 .

[10]  John R. Cannon,et al.  Uniqueness and stability of 3D heat sources , 1991 .

[11]  Ching-Shyang Chen,et al.  The Method Of Fundamental Solutions For Solving Poisson Problems , 2002 .

[12]  D. Littlewood,et al.  A university algebra , 1952 .

[13]  Masahiro Yamamoto,et al.  REVERSE CONVOLUTION INEQUALITIES AND APPLICATIONS TO INVERSE HEAT SOURCE PROBLEMS , 2002 .

[14]  J. Leblond,et al.  Recovery of pointwise sources or small inclusions in 2D domains and rational approximation , 2005 .

[15]  D. Marquardt.  An Algorithm for Least-Squares Estimation of Nonlinear Parameters , 1963 .

[16]  Xiaoqun Wang.  Improving the rejection sampling method in quasi-Monte Carlo methods , 2000 .

[17]  Y. Hon,et al.  Inverse source identification for Poisson equation , 2005 .

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