Recovery of an initial temperature from discrete sampling
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The problem of recovering the initial temperature of a body from discrete temperature measurements made at later times is studied. While this problem has a general formulation, the results of this paper are only given in the simplest setting of a finite (one-dimensional), constant coefficient, linear rod. It is shown that with a judicious placement of a thermometer on this rod, the initial temperature profile of the rod can be completely determined by later time measurements. The paper then studies the number of measurements that are needed to recover the initial profile to a prescribed accuracy and provides an optimal reconstruction algorithm under the assumption that the initial profile is in a Sobolev class.