7214751
A conforming finite element method for overlapping and nonmatching grids
In this paper we propose a finite element method for nonmatching overlapping grids based on the partition of unity. Both overlapping and nonoverlapping cases are considered. We prove that the new method admits an optimal convergence rate. The error bounds are in terms of local mesh sizes and they depend on neither the overlapping size of the subdomains nor the ratio of the mesh sizes from different subdomains. Our results are valid for multiple subdomains and any spatial dimensions.
2513560
An Algebraic Multigrid Method for Higher-order Finite Element Discretizations
In this paper, we will design and analyze a class of new algebraic multigrid methods for algebraic systems arising from the discretization of second order elliptic boundary value problems by high-order finite element methods. For a given sparse stiffness matrix from a quadratic or cubic Lagrangian finite element discretization, an algebraic approach is carefully designed to recover the stiffness matrix associated with the linear finite element disretization on the same underlying (but nevertheless unknown to the user) finite element grid. With any given classical algebraic multigrid solver for linear finite element stiffness matrix, a corresponding algebraic multigrid method can then be designed for the quadratic or higher order finite element stiffness matrix by combining with a standard smoother for the original system. This method is designed under the assumption that the sparse matrix to be solved is associated with a specific higher order, quadratic for example, finite element discretization on a finite element grid but the geometric data for the underlying grid is unknown. The resulting new algebraic multigrid method is shown, by numerical experiments, to be much more efficient than the classical algebraic multigrid method which is directly applied to the high-order finite element matrix. Some theoretical analysis is also provided for the convergence of the new method.
9688726
Motion Editing in 3D Video Database
As the next generation of media, 3D video is attracting increased attention. 3D video is a sequence of three dimensional mesh models, captured and generated for a real dynamic object. In this paper, we present a simple framework of motion editing in 3D video database to re-use 3D video data. Our system is composed of two modules. In the first module, a motion database is automatically set up from original 3D video sequences off-line by analyzing the feature vectors of each frame. It is observed that our original 3D video sequences have a two-level temporal structure. A fine-to-coarse method is proposed to extract such a structure. 3D video is segmented into the fine-level structure by a three reference frame strategy and then is clustered into the coarse-level structure. In the second module, users can synthesize the motions to edit a new 3D video sequence online. A cost function is optimized to transit between two motions with the users' requirements. All the algorithms in the system are based on the analysis in feature vector space and the edited 3D video sequence is played using OpenGL.
11854232
Local and parallel finite element algorithms based on two-grid discretizations
A number of new local and parallel discretization and adaptive nite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems. These algorithms are motivated by the observation that, for a solution to some elliptic problems, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a ne grid by some local and parallel procedure. The theoretical tools for analyzing these methods are some local a priori and a posteriori estimates that are also obtained in this paper for nite element solutions on general shape-regular grids. Some numerical experiments are also presented to support the theory. In this paper, we will propose some new parallel techniques for nite element computation. These techniques are based on our understanding of the local and global properties of a nite element solution to some elliptic problems. Simply speaking, the global behavior of a solution is mostly governed by low frequency components while the local behavior is mostly governed by high frequency compo- nents. The main idea of our new algorithms is to use a coarse grid to approximate the low frequencies and then to use a ne grid to correct the resulted residue (which contains mostly high frequencies) by some local/parallel procedures. Let us now give a somewhat more detailed but informal (and hopefully infor- mative) description of the main ideas and results in this paper. We consider the following very simple model problem posed on a convex polygonal domain R 2 : ( u + bru = f; in ;
36507
Ranking definitions with supervised learning methods
This paper is concerned with the problem of definition search. Specifically, given a term, we are to retrieve definitional excerpts of the term and rank the extracted excerpts according to their likelihood of being good definitions. This is in contrast to the traditional approaches of either generating a single combined definition or simply outputting all retrieved definitions. Definition ranking is essential for the task. Methods for performing definition ranking are proposed in this paper, which formalize the problem as either classification or ordinal regression. A specification for judging the goodness of a definition is given. We employ SVM as the classification model and Ranking SVM as the ordinal regression model respectively, such that they rank definition candidates according to their likelihood of being good definitions. Features for constructing the SVM and Ranking SVM models are defined. An enterprise search system based on this method has been developed and has been put into practical use. Experimental results indicate that the use of SVM and Ranking SVM can significantly outperform the baseline methods of using heuristic rules or employing the conventional information retrieval method of Okapi. This is true both when the answers are paragraphs and when they are sentences. Experimental results also show that SVM or Ranking SVM models trained in one domain can be adapted to another domain, indicating that generic models for definition ranking can be constructed.
30458595
Finding an Optimal Path without Growing the Tree
In this paper, we study a class of optimal path problems with the following phenomenon: The space complexity of the algorithms for reporting the lengths of single-source optimal paths for these problems is asymptotically smaller than the space complexity of the "standard" treegrowing algorithms for finding actual optimal paths. We present a general and efficient algorithmic paradigm for finding an actual optimal path for such problems without having to grow a single-source optimal path tree. Our paradigm is based on the "marriage-before-conquer" strategy, the prune-and-search technique, and a data structure called clipped trees. The paradigm enables us to compute an actual path for a number of optimal path problems and dynamic programming problems in computational geometry, graph theory, and combinatorial optimization. Our algorithmic solutions improve the space bounds (in certain cases, the time bounds as well) of the previously best known algorithms, and settle some open problems. Our techniques are likely to be applicable to other problems.
8946964
Extending Network Lifetime for Precision-Constrained Data Aggregation in Wireless Sensor Networks
This paper exploits the tradeoff between data quality and energy consumption to extend the lifetime of wireless sensor networks. We consider the applications that require some aggregate form of sensed data with precision guarantees. Our key idea is to differentiate the precisions of data collected from different sensor nodes to balance their energy consumption. This is achieved by partitioning the precision constraint of data aggregation and allocating error bounds to individual sensor nodes in a coordinated fashion. Three factors affecting the lifetime of sensor nodes are identified: (1) the changing pattern of sensor readings; (2) the residual energy of sensor nodes; and (3) the communication cost between the sensor nodes and the base station. We analyze the optimal precision allocation in terms of network lifetime and propose an adaptive precision allocation scheme that dynamically adjusts the error bounds of sensor nodes. Experimental results using real data traces show that the proposed scheme significantly improves network lifetime compared to existing methods.