Cubic Spline Wavelet Bases of Sobolev Spaces and Multilevel Interpolation

AbstractIn this paper, a semi-orthogonal cubic spline wavelet basis of homogeneous Sobolev spaceH20(I) is constructed, which turns out to be a basis of the continuous spaceC0(I). At the same time, the orthogonal projections on the wavelet subspaces inH20(I) are extended to the interpolating operators on the corresponding wavelet subspaces inC0(I). A fast discrete wavelet transform (FWT) for functions inC0(I) is also given, which is different from the pyramid algorithm and easy to perform using a parallel algorithm. Finally, it is shown that the singularities of a function can be traced from its wavelet coefficients, which provide an adaptive approximation scheme allowing us to reduce the operation time in computation

Efficient Subgraph Matching on Billion Node Graphs

The ability to handle large scale graph data is crucial to an increasing number of applications. Much work has been dedicated to supporting basic graph operations such as subgraph matching, reachability, regular expression matching, etc. In many cases, graph indices are employed to speed up query processing. Typically, most indices require either super-linear indexing time or super-linear indexing space. Unfortunately, for very large graphs, super-linear approaches are almost always infeasible. In this paper, we study the problem of subgraph matching on billion-node graphs. We present a novel algorithm that supports efficient subgraph matching for graphs deployed on a distributed memory store. Instead of relying on super-linear indices, we use efficient graph exploration and massive parallel computing for query processing. Our experimental results demonstrate the feasibility of performing subgraph matching on web-scale graph data.Comment: VLDB201

Smooth Subdivision Surfaces: Mesh Blending and Local Interpolation

Subdivision surfaces are widely used in computer graphics and animation. Catmull-Clark subdivision (CCS) is one of the most popular subdivision schemes. It is capable of modeling and representing complex shape of arbitrary topology. Polar surface, working on a triangle-quad mixed mesh structure, is proposed to solve the inherent ripple problem of Catmull-Clark subdivision surface (CCSS). CCSS is known to be C1 continuous at extraordinary points. In this work, we present a G2 scheme at CCS extraordinary points. The work is done by revising CCS subdivision step with Extraordinary-Points-Avoidance model together with mesh blending technique which selects guiding control points from a set of regular sub-meshes (named dominative control meshes) iteratively at each subdivision level. A similar mesh blending technique is applied to Polar extraordinary faces of Polar surface as well. Both CCS and Polar subdivision schemes are approximating. Traditionally, one can obtain a CCS limit surface to interpolate given data mesh by iteratively solving a global linear system. In this work, we present a universal interpolating scheme for all quad subdivision surfaces, called Bezier Crust. Bezier Crust is a specially selected bi-quintic Bezier surface patch. With Bezier Crust, one can obtain a high quality interpolating surface on CCSS by parametrically adding CCSS and Bezier Crust. We also show that with a triangle/quad conversion process one can apply Bezier Crust on Polar surfaces as well. We further show that Bezier Crust can be used to generate hollowed 3D objects for applications in rapid prototyping. An alternative interpolating approach specifically designed for CCSS is developed. This new scheme, called One-Step Bi-cubic Interpolation, uses bicubic patches only. With lower degree polynomial, this scheme is appropriate for interpolating large-scale data sets. In sum, this work presents our research on improving surface smoothness at extraordinary points of both CCS and Polar surfaces and present two local interpolating approaches on approximating subdivision schemes. All examples included in this work show that the results of our research works on subdivision surfaces are of high quality and appropriate for high precision engineering and graphics usage

Higher Order Difference Schemes for Heat Equation

In this paper, we construct the explicit difference schemes for the heat equation with arbitrary high orders. We also show the validity of the new schemes by numerical simulations

Only rational homology spheres admit Ω(f)\Omega(f) to be union of DE attractors

If there exists a diffeomorphism $f$ on a closed, orientable $n$-manifold $M$ such that the non-wandering set $\Omega(f)$ consists of finitely many orientable $(\pm)$ attractors derived from expanding maps, then $M$ must be a rational homology sphere; moreover all those attractors are of topological dimension $n-2$. Expanding maps are expanding on (co)homologies.Comment: 23 pages, 2 figure

An Iterative Scheme for Leverage-based Approximate Aggregation

The current data explosion poses great challenges to the approximate aggregation with an efficiency and accuracy. To address this problem, we propose a novel approach to calculate the aggregation answers with a high accuracy using only a small portion of the data. We introduce leverages to reflect individual differences in the samples from a statistical perspective. Two kinds of estimators, the leverage-based estimator, and the sketch estimator (a "rough picture" of the aggregation answer), are in constraint relations and iteratively improved according to the actual conditions until their difference is below a threshold. Due to the iteration mechanism and the leverages, our approach achieves a high accuracy. Moreover, some features, such as not requiring recording the sampled data and easy to extend to various execution modes (e.g., the online mode), make our approach well suited to deal with big data. Experiments show that our approach has an extraordinary performance, and when compared with the uniform sampling, our approach can achieve high-quality answers with only 1/3 of the same sample size.Comment: 17 pages, 9 figure