Point Placement for Observation of the Heat Equation on the Sphere

Abstract

In [14] Wallace and I studied discrete observability for invariant evolution equations on compact homogeneous spaces, e.g. for the heat equation on the sphere. The observations there were given by simultaneous measurements, corresponding to function evaluations. The initial data was observed as a limit of truncations deduced from a finite number of measurements. That procedure naturally involves two types of errors. Observations in the form of evaluation functionals are restricted to a finite part of the Fourier Peter Weyl expansion; that restriction implicitly involves a convolution. This discretization error occurs in the head of the approximation. The other error is the truncation error; that is the error in the tail of the approximation. In a later paper [15], we showed that the head (discretization) error depends linearly on the tail (truncation) error, and we investigated the consequences of various spectral growth conditions on the rate of vanishing of the tail error. Here I return to the rate of dependence of the head error on the tail error. This is a matter of placing the observation points, and the results to date are just the outcome of numerical experiments.KeywordsHeat EquationObservation PointIrreducible Unitary RepresentationProduct ArrayRiemannian Symmetric SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.