Abstract
We consider the problem of approximating a function f from an Euclidean domain to a manifold M by scattered samples (f(xi _i))_{iin mathcal {I}}, where the data sites (xi _i)_{iin mathcal {I}} are assumed to be locally close but can otherwise be far apart points scattered throughout the domain. We introduce a natural approximant based on combining the moving least square method and the Karcher mean. We prove that the proposed approximant inherits the accuracy order and the smoothness from its linear counterpart. The analysis also tells us that the use of Karcher’s mean (dependent on a Riemannian metric and the associated exponential map) is inessential and one can replace it by a more general notion of ‘center of mass’ based on a general retraction on the manifold. Consequently, we can substitute the Karcher mean by a more computationally efficient mean. We illustrate our work with numerical results which confirm our theoretical findings.
Highlights
Let f : ⊂ Rs → M with M a Riemannian manifold be an unknown function and we only know its values at a set of distinct pointsi∈I ⊂ ̄
We are concerned with finding an approximant to f. Such an approximation problem for manifold-valued data arises in numerical geometric integration [23], diffusion tensor interpolation [7] and more recently in fast online methods for dimensionality reduced-order models [5]
In the aforementioned references it was implicitly assumed that the data points ( f)i∈I ∈ M are close enough so that they can all be mapped to a single tangent space Tp M by the inverse exponential map log
Summary
Let f : ⊂ Rs → M with M a Riemannian manifold be an unknown function and we only know its values at a set of distinct points (ξi )i∈I ⊂ ̄. To approximate the value f (x) ∈ M, use any standard linear method (polynomial, spline, radial basis function etc.) to interpolate the values (log( p, ( f (ξi )))i∈I ⊂ Tp M at the abscissa (ξi )i∈I , evaluate the interpolant Q at the desired value x, and apply the exponential map to get the approximation f (x) ∼ exp( p, Q(x)) This ‘push-interpolate-pull’ technique only works when all the available data points ( f (ξi ))i∈I fall within the injectivity radius of the point p, and the method only provides an approximation for f (x) if x is near p. In this case the problem is local, and the topology of the manifold plays no role. We examine an application to the interpolation of reduced order models and compare our method to the method introduced in [6] where it turns our that our method delivers superior approximation power
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