23862247
The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids
An abstract framework ofauxiliary space method is proposed and, as an application, an optimal multigrid technique is developed for general unstructured grids. The auxiliary space method is a (nonnested) two level preconditioning technique based on a simple relaxation scheme (smoother) and an auxiliary space (that may be roughly understood as a nonnested coarser space). An optimal multigrid preconditioner is then obtained for a discretized partial differential operator defined on an unstructured grid by using an auxiliary space defined on a more structured grid in which a furthernested multigrid method can be naturally applied. This new technique makes it possible to apply multigrid methods to general unstructured grids without too much more programming effort than traditional solution methods. Some simple examples are also given to illustrate the abstract theory and for instance the Morley finite element space is used as an auxiliary space to construct a preconditioner for Argyris element for biharmonic equations. Some numerical results are also given to demonstrate the efficiency of using structured grid for auxiliary space to precondition unstructured grids.ZusammenfassungDas abstrakte Konzept derHilfsraummethode wird vorgeschlagen, und als Anwendung eine optimale Mehrgittertechnik für allgemeine unstrukturierte Gitter entwickelt. Die Hilfsraummethode ist eine (nichtgeschachtelte) Zweigitter-Präkonditionierungstechnik, bestehend aus einem einfachen Glättungsverfahren und einem Hilfsgitter, das etwa einem nichtgeschachtelten Grobgitter entspricht. Zu einer partiellen Differentialgleichung, die auf einem unstrukturierten Gitter diskretisiert ist, wird eine optimale Mehrgittermethode konstruiert, die einen Hilfsraum mit einem besser strukturierten Gitter verwendet. In diesem ist ein Mehrgitterverfahren mit geschachtelten Räumen anwendbar. Die neue Technik ermöglicht es, Mehrgittermethoden für unstrukturierte Gitter anzuwenden, die nur wenig mehr Programmieraufwand erfordern als traditionelle Methoden. Einige einfache Beispiele sollen die abstrakte Theorie illustrieren. Als Beispiel wird der Morley-FE-Raum als Hilfsraum zur Präkonditionierung der Argyris-Elemente für die biharmonische Gleichung benutzt. Einige numerische Resultate demonstrieren die Effizienz des Einsatzes des strukturierten Hilfsraumes.
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.
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.
15045053
A monotone finite element scheme for convection-diffusion equations
A simple technique is given in this paper for the construction and analysis of a class of finite element discretizations for convection-diffusion problems in any spatial dimension by properly averaging the PDE coefficients on element edges. The resulting finite element stiffness matrix is an M-matrix under some mild assumption for the underlying (generally unstructured) finite element grids. As a consequence the proposed edge-averaged finite element scheme is particularly interesting for the discretization of convection dominated problems. This scheme admits a simple variational formulation, it is easy to analyze, and it is also suitable for problems with a relatively smooth flux variable. Some simple numerical examples are given to demonstrate its effectiveness for convection dominated problems.
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 ;
27008432
Traveling Salesman Problem of Segments
In this paper, we present a polynomial time approximation scheme (PTAS) for a variant of the traveling salesman problem (called segment TSP) in which a traveling salesman tour is sought to traverse a set of n Ɛ-separated segments in two dimensional space. Our results are based on a number of geometric observations and an interesting generalization of Arora's technique [5] for Euclidean TSP (of a set of points). The randomized version of our algorithm takes O(n2(log n)O(1/Ɛ4)) time to compute a (1 + Ɛ)-approximation with probability ≥ 1/2, and can be derandomized with an additional factor of O(n2). Our technique is likely applicable to TSP problems of certain Jordan arcs and related problems.
6867693
Searching Documents Based on Relevance and Type
This paper extends previous work on document retrieval and document type classification, addressing the problem of 'typed search'. Specifically, given a query and a designated document type, the search system retrieves and ranks documents not only based on the relevance to the query, but also based on the likelihood of being the designated document type. The paper formalizes the problem in a general framework consisting of 'relevance model' and 'type model'. The relevance model indicates whether or not a document is relevant to a query. The type model indicates whether or not a document belongs to the designated document type. We consider three methods for combing the models: linear combination of scores, thresholding on the type score, and a hybrid of the previous two methods. We take course page search and instruction document search as examples and have conducted a series of experiments. Experimental results show our proposed approaches can significantly outperform the baseline methods.
16907782
Large-scale IP traceback in high-speed Internet: practical techniques and theoretical foundation
Tracing attack packets to their sources, known as IP traceback, is an important step to counter distributed denial-of-service (DDoS) attacks. In this paper, we propose a novel packet logging based (i.e., hash-based) traceback scheme that requires an order of magnitude smaller processing and storage cost than the hash-based scheme proposed by Snoeren, et al. (2001), thereby being able to scalable to much higher link speed (e.g., OC-768). The baseline idea of our approach is to sample and log a small percentage (e.g., 3.3%) of packets. The challenge of this low sampling rate is that much more sophisticated techniques need to be used for traceback. Our solution is to construct the attack tree using the correlation between the attack packets sampled by neighboring routers. The scheme using naive independent random sampling does not perform well due to the low correlation between the packets sampled by neighboring routers. We invent a sampling scheme that improves this correlation and the overall efficiency significantly. Another major contribution of this work is that we introduce a novel information-theoretic framework for our traceback scheme to answer important questions on system parameter tuning and the fundamental trade-off between the resource used for traceback and the traceback accuracy. Simulation results based on real-world network topologies (e.g. Skitter) match very well with results from the information-theoretic analysis. The simulation results also demonstrate that our traceback scheme can achieve high accuracy, and scale very well to a large number of attackers (e.g., 5000+).
16173548
Robust traffic matrix estimation with imperfect information: making use of multiple data sources
Estimation of traffic matrices, which provide critical input for network capacity planning and traffic engineering, has recently been recognized as an important research problem. Most of the previous approaches infer traffic matrix from either SNMP link loads or sampled NetFlow records. In this work, we design novel inference techniques that, by statistically correlating SNMP link loads and sampled NetFlow records, allow for much more accurate estimation of traffic matrices than obtainable from either information source alone, even when sampled NetFlow records are available at only a subset of ingress. Our techniques are practically important and useful since both SNMP and NetFlow are now widely supported by vendors and deployed in most of the operational IP networks. More importantly, this research leads us to a new insight that SNMP link loads and sampled NetFlow records can serve as "error correction codes" to each other. This insight helps us to solve a challenging open problem in traffic matrix estimation, "How to deal with dirty data (SNMP and NetFlow measurement errors due to hardware/software/transmission problems)?" We design techniques that, by comparing notes between the above two information sources, identify and remove dirty data, and therefore allow for accurate estimation of the traffic matrices with the cleaned dat.We conducted experiments on real measurement data obtained from a large tier-1 ISP backbone network. We show that, when full deployment of NetFlow is not available, our algorithm can improve estimation accuracy significantly even with a small fraction of NetFlow data. More importantly, we show that dirty data can contaminate a traffic matrix, and identifying and removing them can reduce errors in traffic matrix estimation by up to an order of magnitude. Routing changes is another a key factor that affects estimation accuracy. We show that using them as the a priori, the traffic matrices can be estimated much more accurately than those omitting the routing change. To the best of our knowledge, this work is the first to offer a comprehensive solution which fully takes advantage of using multiple readily available data sources. Our results provide valuable insights on the effectiveness of combining flow measurement and link load measurement.