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This article presents a method for estimating functions on topologically and/or geometrically complex surfaces from possibly noisy observations. Our approach is an extension of spline smoothing, using a finite element method. The article has a substantial tutorial component: we start by reviewing smoothness measures for functions defined on surfaces, simplicial surfaces and differentiable structures on such surfaces, subdivison functions, and subdivision surfaces. After describing our method, we show results of an experiment comparing finite element approximations to exact smoothing splines on the sphere, and we give examples suggesting that generalized cross-validation is an effective way of determining the optimal degree of smoothing for function estimation on surfaces.
Duchamp et al. (Sun,) studied this question.