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516
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 199 (10 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...
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 98 (19 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.
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 74 (10 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...
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 non ..."
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Cited by 70 (26 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...
Asynchronous Variational Integrators
 ARCH. RATIONAL MECH. ANAL.
, 2003
"... We describe a new class of asynchronous variational integrators (AVI) for nonlinear elastodynamics. The AVIs are distinguished by the following attributes: (i) The algorithms permit the selection of independent time steps in each element, and the local time steps need not bear an integral relation t ..."
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Cited by 66 (11 self)
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We describe a new class of asynchronous variational integrators (AVI) for nonlinear elastodynamics. The AVIs are distinguished by the following attributes: (i) The algorithms permit the selection of independent time steps in each element, and the local time steps need not bear an integral relation to each other; (ii) the algorithms derive from a spacetime form of a discrete version of Hamilton’s variational principle. As a consequence of this variational structure, the algorithms conserve local momenta and a local discrete multisymplectic structure exactly. To guide the development of the discretizations, a spacetime multisymplectic formulation of elastodynamics is presented. The variational principle used incorporates both configuration and spacetime reference variations. This allows a unified treatment of all the conservation properties of the system. A discrete version of reference configuration is also considered, providing a natural definition of a discrete energy. The possibilities for discrete energy conservation are evaluated. Numerical tests reveal that, even when local energy balance is not enforced exactly, the global and local energy behavior of the AVIs is quite remarkable, a property which can probably be traced to the symplectic nature of the algorithm.
Moving Least Square Reproducing Kernel Method (III): Wavelet Packet Its Applications
, 1997
"... This work is a natural extension of the work done in Part II of this series. A new partition of unity  the synchronized reproducing kernel (SRK) interpolantis proposed within the framework of moving least square reproducing kernel representation. It is a further development and generalization ..."
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Cited by 64 (13 self)
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This work is a natural extension of the work done in Part II of this series. A new partition of unity  the synchronized reproducing kernel (SRK) interpolantis proposed within the framework of moving least square reproducing kernel representation. It is a further development and generalization of the reproducing kernel particle method (RKPM), which demonstrates some superior computational capability in multiple scale numerical simulations. To form such an interpolant, a class of new wavelet functions are introduced in an unconventional way, and they form an independent sequence that is referred to as the wavelet packet. By choosing different combinations in the wavelet series expansion, the desirable synchronized convergence effect in interpolation can be achieved. Based upon the builtin consistency conditions, the differential consistency conditions for the wavelet functions are derived. It serves as an indispensable instrument in establishing the interpolation error estimate, w...
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 62 (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 59 (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.
What Is a Good Linear Finite Element?  Interpolation, Conditioning, Anisotropy, and Quality Measures
 In Proc. of the 11th International Meshing Roundtable
, 2002
"... When a mesh of simplicial elements (triangles or tetrahedra) is used to form a piecewise linear approximation of a function, the accuracy of the approximation depends on the sizes and shapes of the elements. In finite element methods, the conditioning of the stiffness matrices also depends on the si ..."
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Cited by 59 (0 self)
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When a mesh of simplicial elements (triangles or tetrahedra) is used to form a piecewise linear approximation of a function, the accuracy of the approximation depends on the sizes and shapes of the elements. In finite element methods, the conditioning of the stiffness matrices also depends on the sizes and shapes of the elements. This article explains the mathematical connections between mesh geometry, interpolation errors, discretization errors, and stiffness matrix conditioning. These relationships are expressed by error bounds and element quality measures that determine the fitness of a triangle or tetrahedron for interpolation or for achieving low condition numbers. Unfortunately, the quality measures for these purposes do not fully agree with each other; for instance, small angles are bad for matrix conditioning but not for interpolation or discretization. The upper and lower bounds on interpolation error and element stiffness matrix conditioning given here are tighter than those usually seen in the literature, so the quality measures are likely to be unusually precise indicators of element fitness. Bounds are included for anisotropic cases, wherein long, thin elements perform better than equilateral ones. Surprisingly, there are circumstances wherein interpolation, conditioning, and discretization error are each best served by elements of different aspect ratios or orientations.
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 56 (13 self)
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this paper to collect wellknown results on 3D mesh refinement