Accuracy and Conditioning in the Inversion of the Heat Equation

Abstract

In the paper [2] an exponentially accurate scheme was developed to recover the initial temperature distribution in an infinite or semi-infinite rod. As is well known in this inverse heat conduction problem as well as others [1,2,4,6], these problems are ill-posed in the sense of Hadamard. Hence, even in the case when the problem has a unique solution, a numerical scheme for the inverse problem is a notoriously more difficult problem than is the numerical solution of the forward problem. As is often the case in the numerical treatment of such problems, the ill-posedness is manifest in the conditioning of the matrix problem defining the approximate inverse problem. The last statement is in contrast to the procedure outLined in [2] wherein the conditioning of the discrete inverse problem was shown to be bounded by exp(l/4) (see (3.30) below). This excellent conditioning was obtained at the expense of changing the sampling time with increasing number of spatial sensors. Specifically as the number of spatial sensors is increased, the time at which the temperature is sampled is decreased. This in turn implied that the method, for high order accuracy, required sampled temperatures after small durations.