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264
Spectral Partitioning Works: Planar graphs and finite element meshes
 In IEEE Symposium on Foundations of Computer Science
, 1996
"... Spectral partitioning methods use the Fiedler vectorthe eigenvector of the secondsmallest eigenvalue of the Laplacian matrixto find a small separator of a graph. These methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extr ..."
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Cited by 144 (8 self)
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Spectral partitioning methods use the Fiedler vectorthe eigenvector of the secondsmallest eigenvalue of the Laplacian matrixto find a small separator of a graph. These methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extremely well. In this paper, we show that spectral partitioning methods work well on boundeddegree planar graphs and finite element meshes the classes of graphs to which they are usually applied. While naive spectral bisection does not necessarily work, we prove that spectral partitioning techniques can be used to produce separators whose ratio of vertices removed to edges cut is O( p n) for boundeddegree planar graphs and twodimensional meshes and O i n 1=d j for wellshaped ddimensional meshes. The heart of our analysis is an upper bound on the secondsmallest eigenvalues of the Laplacian matrices of these graphs. 1. Introduction Spectral partitioning has become one of the mos...
Fast Nonsymmetric Iterations and Preconditioning for NavierStokes Equations
 SIAM J. Sci. Comput
, 1994
"... Discretization and linearization of the steadystate NavierStokes equations gives rise to a nonsymmetric indefinite linear system of equations. In this paper, we introduce preconditioning techniques for such systems with the property that the eigenvalues of the preconditioned matrices are bounded i ..."
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Cited by 68 (9 self)
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Discretization and linearization of the steadystate NavierStokes equations gives rise to a nonsymmetric indefinite linear system of equations. In this paper, we introduce preconditioning techniques for such systems with the property that the eigenvalues of the preconditioned matrices are bounded independently of the mesh size used in the discretization. We confirm and supplement these analytic results with a series of numerical experiments indicating that Krylov subspace iterative methods for nonsymmetric systems display rates of convergence that are independent of the mesh parameter. In addition, we show that preconditioning costs can be kept small by using iterative methods for some intermediate steps performed by the preconditioner. * This work was supported by the U. S. Army Research Office under grant DAAL0392G0016 and the U. S. National Science Foundation under grant ASC8958544 at the University of Maryland, and the Science and Engineering Research Council of Great Britain V...
Penalty Methods For American Options With Stochastic Volatility
, 1998
"... The American early exercise constraint can be viewed as transforming the two dimensional stochastic volatility option pricing PDE into a differential algebraic equation (DAE). Several methods are described for forcing the algebraic constraint by using a penalty source term in the discrete equations. ..."
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Cited by 62 (18 self)
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The American early exercise constraint can be viewed as transforming the two dimensional stochastic volatility option pricing PDE into a differential algebraic equation (DAE). Several methods are described for forcing the algebraic constraint by using a penalty source term in the discrete equations. The resulting nonlinear algebraic equations are solved using an approximate Newton iteration. The solution of the Jacobian is obtained using an incomplete LU (ILU) preconditioned PCG method. Some example computations are presented for option pricing problems based on a stochastic volatility model, including an exotic American chooser option written on a put and call with discrete double knockout barriers and discrete dividends.
Combinatorial preconditioners for sparse, symmetric, diagonally dominant linear systems
, 1996
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Dynamic Partitioning of NonUniform Structured Workloads with Spacefilling Curves
 IEEE Transactions on Parallel and Distributed Systems
, 1995
"... We discuss Inverse Spacefilling Partitioning (ISP), a partitioning strategy for nonuniform scientific computations running on distributed memory MIMD parallel computers. We consider the case of a dynamic workload distributed on a uniform mesh, and compare ISP against Orthogonal Recursive Bisectio ..."
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Cited by 56 (2 self)
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We discuss Inverse Spacefilling Partitioning (ISP), a partitioning strategy for nonuniform scientific computations running on distributed memory MIMD parallel computers. We consider the case of a dynamic workload distributed on a uniform mesh, and compare ISP against Orthogonal Recursive Bisection (ORB) and a Median of Medians variant of ORB, ORBMM. We present two results. First, ISP and ORBMM are superior to ORB in rendering balanced workloadsbecause they are more finegrained and incur communication overheads that are comparable to ORB. Second, ISP is more attractive than ORBMM from a software engineering standpoint because it avoids elaborate bookkeeping. Whereas ISP partitionings can be described succinctly as logically contiguous segments of the line, ORBMM's partitionings are inherently unstructured. We describe the general ddimensional ISP algorithm and report empirical results with two and threedimensional, nonhierarchical particle methods. Scott B. Bad...
Preconditioning For The SteadyState NavierStokes Equations With Low Viscosity
 SIAM J. SCI. COMPUT
, 1996
"... We introduce a preconditioner for the linearized NavierStokes equations that is effective when either the discretization mesh size or the viscosity approaches zero. For constant coefficient problems with periodic boundary conditions, we show that the preconditioning yields a system with a single ei ..."
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Cited by 50 (10 self)
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We introduce a preconditioner for the linearized NavierStokes equations that is effective when either the discretization mesh size or the viscosity approaches zero. For constant coefficient problems with periodic boundary conditions, we show that the preconditioning yields a system with a single eigenvalue equal to one, so that performance is independent of both viscosity and mesh size. For other boundary conditions, we demonstrate empirically that convergence depends only mildly on these parameters and we give a partial analysis of this phenomenon. We also show that some expensive subsidiary computations required by the new method can be replaced by inexpensive approximate versions of these tasks based on iteration, with virtually no degradation of performance.
Besov Regularity for Elliptic Boundary Value Problems
 Appl. Math. Lett
, 1995
"... This paper studies the regularity of solutions to boundary value problems for Laplace's equation on Lipschitz domains\Omega in R d and its relationship with adaptive and other nonlinear methods for approximating these solutions. The smoothness spaces which determine the efficiency of such nonlinea ..."
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Cited by 48 (17 self)
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This paper studies the regularity of solutions to boundary value problems for Laplace's equation on Lipschitz domains\Omega in R d and its relationship with adaptive and other nonlinear methods for approximating these solutions. The smoothness spaces which determine the efficiency of such nonlinear approximation in L p(\Omega\Gamma are the Besov spaces B ff ø (L ø(\Omega\Gamma16 ø := (ff=d + 1=p) \Gamma1 . Thus, the regularity of the solution in this scale of Besov spaces is investigated with the aim of determining the largest ff for which the solution is in B ff ø (L ø(\Omega\Gamma21 The regularity theorems given in this paper build upon the recent results of Jerison and Kenig [JK]. The proof of the regularity theorem uses characterizations of Besov spaces by wavelet expansions. Key Words: Besov spaces, elliptic boundary value problems, potential theory, adaptive methods, nonlinear approximation, wavelets AMS Subject classification: primary 35B65, secondary 31B10, 41A46, 46E...
Quality Mesh Generation in Higher Dimensions
, 1996
"... We consider the problem of triangulating a ddimensional region. Our mesh generation algorithm, called QMG, is a quadtreebased algorithm that can triangulate any polyhedral region including nonconvex regions with holes. Furthermore, our algorithm guarantees a bounded aspect ratio triangulation ..."
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Cited by 48 (7 self)
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We consider the problem of triangulating a ddimensional region. Our mesh generation algorithm, called QMG, is a quadtreebased algorithm that can triangulate any polyhedral region including nonconvex regions with holes. Furthermore, our algorithm guarantees a bounded aspect ratio triangulation provided that the input domain itself has no sharp angles. Finally, our algorithm is guaranteed never to overrefine the domain in the sense that the number of simplices produced by QMG is bounded above by a factor times the number produced by any competing algorithm, where the factor depends on the aspect ratio bound satisfied by the competing algorithm. The QMG algorithm has been implemented in C++ and is used as a mesh generator for the finite element method.
WellSpaced Points for Numerical Methods
, 1997
"... mesh generation, mesh coarsening, multigrid Abstract A numerical method for the solution of a partial differential equation (PDE) requires the following steps: (1) discretizing the domain (mesh generation); (2) using an approximation method and the mesh to transform the problem into a linear system; ..."
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Cited by 44 (2 self)
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mesh generation, mesh coarsening, multigrid Abstract A numerical method for the solution of a partial differential equation (PDE) requires the following steps: (1) discretizing the domain (mesh generation); (2) using an approximation method and the mesh to transform the problem into a linear system; (3) solving the linear system. The approximation error and convergence of the numerical method depend on the geometric quality of the mesh, which in turn depends on the size and shape of its elements. For example, the shape quality of a triangular mesh is measured by its element's aspect ratio. In this work, we shift the focus to the geometric properties of the nodes, rather than the elements, of well shaped meshes. We introduce the concept of wellspaced points and their spacing functions, and show that these enable the development of simple and efficient algorithms for the different stages of the numerical solution of PDEs. We first apply wellspaced point sets and their accompanying technology to mesh coarsening, a crucial step in the multigrid solution of a PDE. A good aspectratio coarsening sequence of an unstructured mesh M0 is a sequence of good aspectratio meshes M1; : : : ; Mk such that Mi is an approximation of Mi\Gamma 1 containing fewer nodes and elements. We present a new approach to coarsening that guarantees the sequence is also of optimal size and width up to a constant factor the first coarsening method that provides these guarantees. We also present experimental results, based on an implementation of our approach, that substantiate the theoretical claims.
Adaptive Multilevel Methods in Three Space Dimensions
 Int. J. Numer. Methods Eng
, 1993
"... this paper to collect wellknown results on 3D mesh refinement ..."
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Cited by 42 (6 self)
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this paper to collect wellknown results on 3D mesh refinement