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The P-splines of Eilers and Marx (1996) combine a B-spline basis with a discrete quadratic penalty on the basis coefficients, to produce a reduced rank spline like smoother. P-splines have three properties that make them very popular as…
We present a robust and efficient multigrid method for single-patch isogeometric discretizations using tensor product B-splines of maximum smoothness. Our method is based on a stable splitting of the spline space into a large subspace of…
B-spline models are a powerful way to represent scientific data sets with a functional approximation. However, these models can suffer from spurious oscillations when the data to be approximated are not uniformly distributed. Model…
A new efficient orthogonalization of the B-spline basis is proposed and contrasted with some previous orthogonalized methods. The resulting orthogonal basis of splines is best visualized as a net of functions rather than a sequence of them.…
B-spline models are a powerful way to represent scientific data sets with a functional approximation. However, these models can suffer from spurious oscillations when the data to be approximated are not uniformly distributed. Model…
This work introduces and analyzes B-spline approximation spaces defined on general geometric domains obtained through a mapping from a parameter domain. These spaces are constructed as sparse-grid tensor products of univariate spaces in the…
We outline the construction of compatible B-splines on 3D surfaces that satisfy the continuity requirements for electromagnetic scattering analysis with the boundary element method (method of moments). Our approach makes use of Non-Uniform…
In this paper, a method is proposed to solve the problem of monotone smoothing splines using general linear systems. This problem, also called monotone control theoretic splines, has been solved only when the curve generator is modeled by…
Tensor B-spline methods are a high-performance alternative to solve partial differential equations (PDEs). This paper gives an overview on the principles of Tensor B-spline methodology, shows their use and analyzes their performance in…
Spline functions have long been used in numerically solving differential equations. Recently it revives as isogeometric analysis, which uses NURBS for both parametrization and element functions. In this paper, we introduce some multivariate…
In the absence of symmetry assumptions most numerical relativity simulations adopt Cartesian coordinates. While Cartesian coordinates have some desirable properties, spherical polar coordinates appear better suited for certain applications,…
We formulate as an inverse problem the construction of sparse parametric continuous curve models that fit a sequence of contour points. Our prior is incorporated as a regularization term that encourages rotation invariance and sparsity. We…
Solutions of partial differential equations can often be written as surface integrals having a kernel related to a singular fundamental solution. Special methods are needed to evaluate the integral accurately at points on or near the…
In simulation technology, computationally expensive objective functions are often replaced by cheap surrogates, which can be obtained by interpolation. Full grid interpolation methods suffer from the so-called curse of dimensionality,…
Normal multi-scale transform [4] is a nonlinear multi-scale transform for representing geometric objects that has been recently investigated [1, 7, 10]. The restrictive role of the exact order of polynomial reproduction $P_e$ of the…
This paper addresses asymptotic properties of general penalized spline estimators with an arbitrary B-spline degree and an arbitrary order difference penalty. The estimator is approximated by a solution of a linear differential equation…
In this paper, we describe a general class of $C^1$ smooth rational splines that enables, in particular, exact descriptions of ellipses and ellipsoids - some of the most important primitives for CAD and CAE. The univariate rational splines…
Recently the author and U. Reif introduced the concept of diversification of uniform tensor product B-splines. Based on this concept, we give a new constructive modification of non-uniform B-splines. The resulting spline spaces are…
In this paper, we develop approximation error estimates as well as corresponding inverse inequalities for B-splines of maximum smoothness, where both the function to be approximated and the approximation error are measured in standard…
We show that isogeometric Galerkin discretizations of eigenvalue problems related to the Laplace operator subject to any standard type of homogeneous boundary conditions have no outliers in certain optimal spline subspaces. Roughly…