Construction of Piecewise Linear Wavelets.

It is well known that in many areas of computational mathematics, wavelet based algorithms are becoming popular for modeling and analyzing data and for providing efficient means for hierarchical data decomposition of function spaces into mutually orthogonal wavelet spaces. Wavelet construction in more than one-dimensional setting is a very challenging and important research topic. In this thesis, we first introduce the method of construction wavelets by using semi-wavelets. Second, we construct piecewise linear wavelets with smaller support over type-2 triangulations. Then, parameterized wavelets are constructed using the orthogonality conditions

Adaptive boundary element methods with convergence rates

This paper presents adaptive boundary element methods for positive, negative, as well as zero order operator equations, together with proofs that they converge at certain rates. The convergence rates are quasi-optimal in a certain sense under mild assumptions that are analogous to what is typically assumed in the theory of adaptive finite element methods. In particular, no saturation-type assumption is used. The main ingredients of the proof that constitute new findings are some results on a posteriori error estimates for boundary element methods, and an inverse-type inequality involving boundary integral operators on locally refined finite element spaces.Comment: 48 pages. A journal version. The previous version (v3) is a bit lengthie

Hierarchical Riesz bases for Hs(Omega), 1 < s < 5/2

On arbitrary polygonal domains $Omega subset RR^2$, we construct $C^1$ hierarchical Riesz bases for Sobolev spaces $H^s(Omega)$. In contrast to an earlier construction by Dahmen, Oswald, and Shi (1994), our bases will be of Lagrange instead of Hermite type, by which we extend the range of stability from $s in (2,frac{5}{2})$ to $s in (1,frac{5}{2})$. Since the latter range includes $s=2$, with respect to the present basis, the stiffness matrices of fourth-order elliptic problems are uniformly well-conditioned

Algorithms and error bounds for multivariate piecewise constant approximation

We review the surprisingly rich theory of approximation of functions of many vari- ables by piecewise constants. This covers for example the Sobolev-Poincar´e inequalities, parts of the theory of nonlinear approximation, Haar wavelets and tree approximation, as well as recent results about approximation orders achievable on anisotropic partitions

A 10-point interpolatory recursive subdivision algorithm for the generation of parametric surfaces

In this paper, an interpolatory subdivision algorithm for surfaces over arbitrary triangulations is introduced and its properties over uniform triangulations studied. The Butterfly Scheme, which is introduced by Dyn, Gregory and Levin is a special case of this algorithm. In our analysis, the matrix approach is employed and the idea of "Cross Difference of Directional Divided Difference" analysis is presented. This method is a generalization of the technique used by Dyn, Gregory and Levin etc. to analyse univariate subdivision algorithms. It is proved that the algorithm produces smooth surfaces provided the shape parameters are kept within an appropriate range

Graph Wedgelets: Adaptive Data Compression on Graphs based on Binary Wedge Partitioning Trees and Geometric Wavelets

We introduce graph wedgelets - a tool for data compression on graphs based on the representation of signals by piecewise constant functions on adaptively generated binary graph partitionings. The adaptivity of the partitionings, a key ingredient to obtain sparse representations of a graph signal, is realized in terms of recursive wedge splits adapted to the signal. For this, we transfer adaptive partitioning and compression techniques known for 2D images to general graph structures and develop discrete variants of continuous wedgelets and binary space partitionings. We prove that continuous results on best m-term approximation with geometric wavelets can be transferred to the discrete graph setting and show that our wedgelet representation of graph signals can be encoded and implemented in a simple way. Finally, we illustrate that this graph-based method can be applied for the compression of images as well.Comment: 12 pages, 10 figure

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

Adaptive multiresolution analysis based on anisotropic triangulations

A simple greedy refinement procedure for the generation of data-adapted triangulations is proposed and studied. Given a function of two variables, the algorithm produces a hierarchy of triangulations and piecewise polynomial approximations on these triangulations. The refinement procedure consists in bisecting a triangle T in a direction which is chosen so as to minimize the local approximation error in some prescribed norm between the approximated function and its piecewise polynomial approximation after T is bisected. The hierarchical structure allows us to derive various approximation tools such as multiresolution analysis, wavelet bases, adaptive triangulations based either on greedy or optimal CART trees, as well as a simple encoding of the corresponding triangulations. We give a general proof of convergence in the Lp norm of all these approximations. Numerical tests performed in the case of piecewise linear approximation of functions with analytic expressions or of numerical images illustrate the fact that the refinement procedure generates triangles with an optimal aspect ratio (which is dictated by the local Hessian of of the approximated function in case of C2 functions).Comment: 19 pages, 7 figure

A quadratic finite element wavelet Riesz basis

In this paper, continuous piecewise quadratic finite element wavelets are constructed on general polygons in $\mathbb{R}^2$. The wavelets are stable in $H^s$ for $|s|<\frac{3}{2}$ and have two vanishing moments. Each wavelet is a linear combination of 11 or 13 nodal basis functions. Numerically computed condition numbers for $s \in \{-1,0,1\}$ are provided for the unit square.Comment: 13 page