Smooth approximation of data on the sphere with splines

A computable function, defined over the sphere, is constructed, which is of classC1 at least and which approximates a given set of data. The construction is based upon tensor product spline basisfunctions, while at the poles of the spherical system of coordinates modified basisfunctions, suggested by the spherical harmonics expansion, are introduced to recover the continuity order at these points. Convergence experiments, refining the grid, are performed and results are compared with similar results available in literature.\ud \ud The approximation accuracy is compared with that of the expansion in terms of spherical harmonics. The use of piecewise approximation, with locally supported basisfunctions, versus approximation with spherical harmonics is discussed

Quadratic spline wavelets with arbitrary simple knots on the sphere

AbstractIn this paper, we extend the method for fitting functions on the sphere, described in Lyche and Schumaker (SIAM J. Sci. Comput. 22 (2) (2000) 724) to the case of nonuniform knots. We present a multiresolution method leading to C1-functions on the sphere, which is based on tensor products of quadratic polynomial splines and trigonometric splines of order three with arbitrary simple knot sequences. We determine the decomposition and reconstruction matrices corresponding to the polynomial and trigonometric spline spaces. We describe the tensor product decomposition and reconstruction algorithms in matrix forms which are convenient for the compression of surfaces. We give the different steps of computer implementation and finally we present a test example by using two knot sequences: a uniform one and a sequence of Chebyshev points

An algorithm for fitting data over a circle using tensor product splines

AbstractAn algorithm is described for surface fitting over a circle by using tensor product splines which satisfy certain boundary conditions. This algorithm is an extension of an existing one for fitting data over a rectangle. The knots of the splines are chosen automatically but a single parameter must be specified to control the tradeoff between closeness of fit and smoothness of fit. The algorithm can easily be generalized for fitting data over any domain that can be described in polar coordinates. Constraints at the boundaries of this approximation domain can be imposed

A Characteristic Mapping Method for Transport on the Sphere

A semi-Lagrangian method for the solution of the transport equation on a sphere is presented. The method evolves the inverse flow-map using the Characteristic Mapping (CM) [1] and Gradient-Augmented Level Set (GALS) [2] frameworks. Transport of the advected quantity is then computed by composition with the numerically approximated inverse flow-map. This framework allows for the advection of complicated sets and multiple quantities with arbitrarily fine-features using a coarse computational grid. We discuss the CM method for linear transport on the sphere and its computational implementation. Standard test cases for solid body rotation, deformational and divergent flows, and numerical mixing are presented. The unique features of the method are demonstrated by the transport of a multi-scale function and a fractal set in a complex flow environment

Fast increased fidelity approximate Gibbs samplers for Bayesian Gaussian process regression

The use of Gaussian processes (GPs) is supported by efficient sampling algorithms, a rich methodological literature, and strong theoretical grounding. However, due to their prohibitive computation and storage demands, the use of exact GPs in Bayesian models is limited to problems containing at most several thousand observations. Sampling requires matrix operations that scale at $\mathcal{O}(n^3),$ where $n$ is the number of unique inputs. Storage of individual matrices scales at $\mathcal{O}(n^2),$ and can quickly overwhelm the resources of most modern computers. To overcome these bottlenecks, we develop a sampling algorithm using $\mathcal{H}$ matrix approximation of the matrices comprising the GP posterior covariance. These matrices can approximate the true conditional covariance matrix within machine precision and allow for sampling algorithms that scale at \mathcal{O}(n \ \mbox{log}^2 n) time and storage demands scaling at \mathcal{O}(n \ \mbox{log} \ n). We also describe how these algorithms can be used as building blocks to model higher dimensional surfaces at \mathcal{O}(d \ n \ \mbox{log}^2 n), where $d$ is the dimension of the surface under consideration, using tensor products of one-dimensional GPs. Though various scalable processes have been proposed for approximating Bayesian GP inference when $n$ is large, to our knowledge, none of these methods show that the approximation's Kullback-Leibler divergence to the true posterior can be made arbitrarily small and may be no worse than the approximation provided by finite computer arithmetic. We describe $\mathcal{H}-$matrices, give an efficient Gibbs sampler using these matrices for one-dimensional GPs, offer a proposed extension to higher dimensional surfaces, and investigate the performance of this fast increased fidelity approximate GP, FIFA-GP, using both simulated and real data sets

Splines and local approximation of the earth's gravity field

Bibliography: pages 214-220.The Hilbert space spline theory of Delvos and Schempp, and the reproducing kernel theory of L. Schwartz, provide the conceptual foundation and the construction procedure for rotation-invariant splines on Euclidean spaces, splines on the circle, and splines on the sphere and harmonic outside the sphere. Spherical splines and surface splines such as multi-conic functions, Hardy's multiquadric functions, pseudo-cubic splines, and thin-plate splines, are shown to be largely as effective as least squares collocation in representing geoid heights or gravity anomalies. A pseudo-cubic spline geoid for southern Africa is given, interpolating Doppler-derived geoid heights and astro-geodetic deflections of the vertical. Quadrature rules are derived for the thin-plate spline approximation (over a circular disk, and to a planar approximation) of Stokes's formula, the formulae of Vening Meinesz, and the L₁ vertical gradient operator in the analytical continuation series solution of Molodensky's problem