by James V. Beck
reproduced from IPNet Digest Vol 2, No 01, Jan. 1995

There is apparently some disagreement regarding what the inverse heat conduction problem (IHCP) is. One definition is the determination of the surface heat flux (or temperature) from measured transient temperatures inside a heat conducting body. In this definition the initial temperature distribution is considered known.

Another definition estimates the surface heat flux from transient measured interior temperatures and simultaneously the initial temperature distribution.

I would appreciate receiving messages regarding which definition that you use and if it makes a difference. If you have a different definition, I would appreciate learning about it.


The Burggraf and other exact solutions for the linear IHCP show that the solution does not depend upon the initial condition. See Beck, Blackwell and St. Clair for the Burggraf analysis or Murio (The Mollification Method ...., Wiley-Interscience, 1993 ) for the reference and related discussion.

This behavior is also shown by many linear methods including function specification, Tikhonov regularization (whether or not the adjoint equations are used with conjugate gradients,provided the unique minimum for the given objective function is found), mollification and space marching. This is only for linear problems. Another stipulation is that the calculations start a significant time before the heating/cooling starts. For all of the above cases the solution must be linear and can be investigated by having a single error at a given time. This then leads to a filter algorithm of the form of q(M) = sum i from M -m1 to M +m2 of f(M-i)Y(i). The f(j) values are the filter coefficients for a single unit error at any time (except near the beginning or the end). The Y(i) values are the measured temperatures. (Beck, Blackwell and St. Clair, p. 197). This equation says that the estimates are independent of the initial conditions and that one can find the values at any time without finding previous or subsequent values. The difference between the methods is the contained in the precise form of f(i). For a finite body which is heated at x= 0 and insulated at x = L, the above is true. It may be more general.

I welcome any comments. My E-mail address is beck@egr.msu.edu.