Approximation orders of shift-invariant subspaces of W2s(Rd)W^s_2({\Bbb R}^d)

We extend the existing theory of approximation orders provided by shift-invariant subspaces of $L_2$ to the setting of Sobolev spaces, provide treatment of $L_2$ cases that have not been covered before, and apply our results to determine approximation order of solutions to a refinement equation with a higher-dimensional solution space.Comment: 49 page

Subdivision Directional Fields

We present a novel linear subdivision scheme for face-based tangent directional fields on triangle meshes. Our subdivision scheme is based on a novel coordinate-free representation of directional fields as halfedge-based scalar quantities, bridging the finite-element representation with discrete exterior calculus. By commuting with differential operators, our subdivision is structure-preserving: it reproduces curl-free fields precisely, and reproduces divergence-free fields in the weak sense. Moreover, our subdivision scheme directly extends to directional fields with several vectors per face by working on the branched covering space. Finally, we demonstrate how our scheme can be applied to directional-field design, advection, and robust earth mover's distance computation, for efficient and robust computation

Stability and Independence for Multivariate Refinable Distributions

AbstractDue to their so-called time-frequency localization properties, wavelets have become a powerful tool in signal analysis and image processing. Typical constructions of wavelets depend on the stability of the shifts of an underlying refinable function. In this paper, we derive necessary and sufficient conditions for the stability of the shifts of certain compactly supported refinable functions. These conditions are in terms of the zeros of the refinement mask. Our results are actually applicable to more general distributions which are not of function type, if we generalize the notion of stability appropriately. We also provide a similar characterization of the (global) linear independence of the shifts. We present several examples illustrating our results, as well as one example in which known results on box splines are derived using the theorems of this paper

Multivariate orthonormal interpolating scaling vectors

AbstractIn this paper we introduce an algorithm for the construction of interpolating scaling vectors on Rd with compact support and orthonormal integer translates. Our method is substantiated by constructing several examples of bivariate scaling vectors for quincunx and box–spline dilation matrices. As the main ingredients of our recipe we derive some implementable conditions for accuracy and orthonormality of an interpolating scaling vector in terms of its mask

Construction of interpolating and orthonormal multigenerators and multiwavelets on the interval

In den letzten Jahren haben sich Wavelets zu einem hochwertigen Hilfsmittel in der angewandten Mathematik entwickelt. Eine Waveletbasis ist im Allgemeinen ein System von Funktionen, das durch die Skalierung, Translation und Dilatation einer endlichen Menge von Funktionen, den sogenannten Mutterwavelets, entsteht. Wavelets wurden sehr erfolgreich in der digitalen Signal- und Bildanalyse, z. B. zur Datenkompression verwendet. Ein weiteres wichtiges Anwendungsfeld ist die Analyse und die numerische Behandlung von Operatorgleichungen. Insbesondere ist es gelungen, adaptive numerische Algorithmen basierend auf Wavelets für eine riesige Klasse von Operatorgleichungen, einschließlich Operatoren mit negativer Ordnung, zu entwickeln. Der Erfolg der Wavelet- Algorithmen ergibt sich als Konsequenz der folgenden Fakten: - Gewichtete Folgennormen von Wavelet-Expansionskoeffizienten sind in einem bestimmten Bereich (abhängig von der Regularität der Wavelets) äquivalent zu Glättungsnormen wie Besov- oder Sobolev-Normen. - Für eine breite Klasse von Operatoren ist ihre Darstellung in Wavelet-Koordinaten nahezu diagonal. - Die verschwindenden Momente von Wavelets entfernen den glatten Teil einer Funktion und führen zu sehr effizienten Komprimierungsstrategien. Diese Fakten können z. B. verwendet werden, um adaptive numerische Strategien mit optimaler Konvergenzgeschwindigkeit zu konstruieren, in dem Sinne, dass diese Algorithmen die Konvergenzordnung der besten N-Term-Approximationsschemata realisieren. Die maßgeblichen Ergebnisse lassen sich für lineare, symmetrische, elliptische Operatorgleichungen erzielen. Es existiert auch eine Verallgemeinerung für nichtlineare elliptische Gleichungen. Hier verbirgt sich jedoch eine ernste Schwierigkeit: Jeder numerische Algorithmus für diese Gleichungen erfordert die Auswertung eines nichtlinearen Funktionals, welches auf eine Wavelet-Reihe angewendet wird. Obwohl einige sehr ausgefeilte Algorithmen existieren, erweisen sie sich als ziemlich langsam in der Praxis. In neueren Studien wurde gezeigt, dass dieses Problem durch sogenannte Interpolanten verbessert werden kann. Dabei stellt sich heraus, dass die meisten bekannten Basen der Interpolanten keine stabilen Basen in L2[a,b] bilden. In der vorliegenden Arbeit leisten wir einen wesentlichen Beitrag zu diesem Problem und konstruieren neue Familien von Interpolanten auf beschränkten Gebieten, die nicht nur interpolierend, sondern auch stabil in L2[a,b] sind. Da dies mit nur einem Generator schwer (oder vielleicht sogar unmöglich) zu erreichen ist, werden wir mit Multigeneratoren und Multiwavelets arbeiten.In recent years, wavelets have become a very powerful tools in applied mathematics. In general, a wavelet basis is a system of functions that is generated by scaling, translating and dilating a finite set of functions, the so-called mother wavelets. Wavelets have been very successfully applied in image/signal analysis, e.g., for denoising and compression purposes. Another important field of applications is the analysis and the numerical treatment of operator equations. In particular, it has been possible to design adaptive numerical algorithms based on wavelets for a huge class of operator equations including operators of negative order. The success of wavelet algorithms is an ultimative consequence of the following facts: - Weighted sequence norms of wavelet expansion coefficients are equivalent in a certain range (depending on the regularity of the wavelets) to smoothness norms such as Besov or Sobolev norms. - For a wide class of operators their representation in wavelet coordinates is nearly diagonal. -The vanishing moments of wavelets remove the smooth part of a function. These facts can, e.g., be used to construct adaptive numerical strategies that are guaranteed to converge with optimal order, in the sense that these algorithms realize the convergence order of best N-term approximation schemes. The most far-reaching results have been obtained for linear, symmetric elliptic operator equations. Generalization to nonlinear elliptic equations also exist. However, then one is faced with a serious bottleneck: every numerical algorithm for these equations requires the evaluation of a nonlinear functional applied to a wavelet series. Although some very sophisticated algorithms exist, they turn out to perform quite slowly in practice. In recent studies, it has been shown that this problem can be ameliorated by means of so called interpolants. However, then the problem occurs that most of the known bases of interpolants do not form stable bases in L2[a,b]. In this PhD project, we intend to provide a significant contribution to this problem. We want to construct new families of interpolants on domains that are not only interpolating, but also stable in L2[a,b]or even orthogonal. Since this is hard to achieve (or maybe even impossible) with just one generator, we worked with multigenerators and multiwavelets

A Hermite interpolatory subdivision scheme for C2C^2-quintics on the Powell-Sabin 12-split

In order to construct a $C^1$-quadratic spline over an arbitrary triangulation, one can split each triangle into 12 subtriangles, resulting in a finer triangulation known as the Powell-Sabin 12-split. It has been shown previously that the corresponding spline surface can be plotted quickly by means of a Hermite subdivision scheme. In this paper we introduce a nodal macro-element on the 12-split for the space of quintic splines that are locally $C^3$ and globally $C^2$. For quickly evaluating any such spline, a Hermite subdivision scheme is derived, implemented, and tested in the computer algebra system Sage. Using the available first derivatives for Phong shading, visually appealing plots can be generated after just a couple of refinements.Comment: 17 pages, 7 figure