Oleg Davydov

Smooth approximation and rendering of large scattered data sets

Presents an efficient method to automatically compute a smooth approximation of large functional scattered data sets given over arbitrarily shaped planar domains. Our approach is based on the construction of a C/sup 1/-continuous bivariate cubic spline and our method offers optimal approximation order. Both local variation and nonuniform distribution of the data are taken into account by using local polynomial least squares approximations of varying degree. Since we only need to solve small linear systems and no triangulation of the scattered data points is required, the overall complexity of the algorithm is linear in the total number of points. Numerical examples dealing with several real-world scattered data sets with up to millions of points demonstrate the efficiency of our method. The resulting spline surface is of high visual quality and can be efficiently evaluated for rendering and modeling. In our implementation we achieve real-time frame rates for typical fly-through sequences and interactive frame rates for recomputing and rendering a locally modified spline surface.

Scattered data fitting by direct extension of local polynomials to bivariate splines

We present a new scattered data fitting method, where local approximating polynomials are directly extended to smooth (C1 or C2) splines on a uniform triangulation Δ (the four-directional mesh). The method is based on designing appropriate minimal determining sets consisting of whole triangles of domain points for a uniformly distributed subset of Δ. This construction allows to use discrete polynomial least squares approximations to the local portions of the data directly as parts of the approximating spline. The remaining Bernstein–Bézier coefficients are efficiently computed by extension, i.e., using the smoothness conditions. To obtain high quality local polynomial approximations even for difficult point constellations (e.g., with voids, clusters, tracks), we adaptively choose the polynomial degrees by controlling the smallest singular value of the local collocation matrices. The computational complexity of the method grows linearly with the number of data points, which facilitates its application to large data sets. Numerical tests involving standard benchmarks as well as real world scattered data sets illustrate the approximation power of the method, its efficiency and ability to produce surfaces of high visual quality, to deal with noisy data, and to be used for surface compression.

Bernstein-Bézier Finite Elements of Arbitrary Order and Optimal Assembly Procedures

Algorithms are presented that enable the element matrices for the standard finite element space, consisting of continuous piecewise polynomials of degree

Local hybrid approximation for scattered data fitting with bivariate splines

n

Adaptive meshless centres and RBF stencils for Poisson equation

on simplicial elements in

Approximation order of bivariate spline interpolation for arbitrary smoothness

\mathbb{R}^d

Numerical solution of fully nonlinear elliptic equations by Böhmer’s method

, to be computed in optimal complexity

Nonlinear approximation from differentiable piecewise polynomials

\mathcal{O}(n^{2d})

Interpolation by bivariate linear splines

. The algorithms (i) take into account numerical quadrature; (ii) are applicable to nonlinear problems; and (iii) do not rely on precomputed arrays containing values of one-dimensional basis functions at quadrature points (although these can be used if desired). The elements are based on Bernstein polynomials and are the first to achieve optimal complexity for the standard finite element spaces on simplicial elements.

Error bounds for kernel-based numerical differentiation

We suggest a local hybrid approximation scheme based on polynomials and radial basis functions, and use it to improve the scattered data fitting algorithm of (Davydov, O., Zeilfelder, F., 2004. Scattered data fitting by direct extension of local polynomials to bivariate splines. Adv. Comp. Math. 21, 223-271). Similar to that algorithm, the new method has linear computational complexity and is therefore suitable for large real world data. Numerical examples suggest that it can produce high quality artifact-free approximations that are more accurate than those given by the original method where pure polynomial local approximations are used.