Showing papers by "Courant Institute of Mathematical Sciences published in 1997"
TL;DR: In this article, a numerical method for solving incompressible viscous flow problems is introduced, which uses the velocities and the pressure as variables and is equally applicable to problems in two and three space dimensions.
2,797 citations
TL;DR: A new version of the Fast Multipole Method for the evaluation of potential fields in three dimensions is introduced based on a new diagonal form for translation operators and yields high accuracy at a reasonable cost.
Abstract: We introduce a new version of the Fast Multipole Method for the evaluation of potential fields in three dimensions. It is based on a new diagonal form for translation operators and yields high accuracy at a reasonable cost.
896 citations
TL;DR: In this article, the effective dielectric function for a medium of anisotropic inclusions embedded in an isotropic host is calculated using the Maxwell Garnett approximation for a model of quasi-one-dimensional organic polymers.
Abstract: The effective dielectric function ${\ensuremath{\epsilon}}_{e}$ for a medium of anisotropic inclusions embedded in an isotropic host is calculated using the Maxwell Garnett approximation. For uniaxial inclusions, ${\ensuremath{\epsilon}}_{e}$ depends on how well the inclusions are aligned. We apply this approximation to study ${\ensuremath{\epsilon}}_{e}$ for a model of quasi-one-dimensional organic polymers. The polymer is assumed to be made up of small single crystals embedded in an isotropic host of randomly oriented polymer chains. The host dielectric function is calculated using the effective-medium approximation (EMA). The resulting frequency-dependent ${\ensuremath{\epsilon}}_{e}(\ensuremath{\omega})$ closely resembles experiment. Specifically, $\mathrm{Re}{\ensuremath{\epsilon}}_{e}(\ensuremath{\omega})$ is negative over a wide frequency range, while $\mathrm{Im}{\ensuremath{\epsilon}}_{e}(\ensuremath{\omega})$ exhibits a broad ``surface plasmon'' band at low frequencies, which results from localized electronic excitations within the crystallites. If the host is above the conductivity percolation threshold, $\mathrm{Im}{\ensuremath{\epsilon}}_{e}(\ensuremath{\omega})$ has a low-frequency Drude peak in addition to the surface plasmon band, and $\mathrm{Re}{\ensuremath{\epsilon}}_{e}(\ensuremath{\omega})$ is negative over an even wider frequency range. We also calculate the cubic nonlinear susceptibility ${\ensuremath{\chi}}_{e}(\ensuremath{\omega})$ of the polymer, using a nonlinear EMA. At certain frequencies, ${\ensuremath{\chi}}_{e}(\ensuremath{\omega})$ is found to be strongly enhanced above that of the corresponding single crystals. Our results suggest that the electromagnetic properties of conducting polymers can be understood by viewing the material as randomly inhomogeneous on a small scale such that the quasistatic limit is applicable.
331 citations
TL;DR: It is shown that primal and dual nondegeneracy and strict complementarity all hold generically and Numerical experiments suggest probability distributions for the ranks ofX andZ which are consistent with the nondEGeneracy conditions.
Abstract: Primal and dual nondegeneracy conditions are defined for semidefinite programming. Given the existence of primal and dual solutions, it is shown that primal nondegeneracy implies a unique dual solution and that dual nondegeneracy implies a unique primal solution. The converses hold if strict complementarity is assumed. Primal and dual nondegeneracy assumptions do not imply strict complementarity, as they do in LP. The primal and dual nondegeneracy assumptions imply a range of possible ranks for primal and dual solutions $X$ and $Z$. This is in contrast with LP where nondegeneracy assumptions exactly determine the number of variables which are zero. It is shown that primal and dual nondegeneracy and strict complementarity all hold generically. Numerical experiments suggest probability distributions for the ranks of $X$ and $Z$ which are consistent with the nondegeneracy conditions.
277 citations
262 citations
TL;DR: In this paper, a two-dimensional eddy-resolving model is used to study the transition from the stratocumulus topped boundary layer to the trade cumulus boundary layer.
Abstract: A two-dimensional eddy-resolving model is used to study the transition from the stratocumulus topped boundary layer to the trade cumulus boundary layer. The 10-day simulations use an idealized Lagrangian trajectory representative of summertime climatological conditions in the subtropical northeastern Pacific. The sea surface temperature is increased steadily at 1.5 K day−1, reflecting the southwestward advection of the subtropical marine boundary layer by the trade winds, while the free tropospheric temperature remains unchanged. Results from simulations with both a fixed diurnally averaged shortwave radiative forcing and a diurnally varying shortwave forcing are presented. A two-stage model for the boundary layer evolution consistent with these simulations is proposed. In the first stage, decoupling is induced by increased latent heat fluxes in the deepening boundary layer. After decoupling, cloud cover remains high, but the cloudiness regime changes from a single stratocumulus layer to sporadic...
248 citations
TL;DR: In this article, a family of one-dimensional nonlinear dispersive wave equations is introduced as a model for assessing the validity of weak turbulence theory for random waves in an unambiguous and transparent fashion.
Abstract: A family of one-dimensional nonlinear dispersive wave equations is introduced as a model for assessing the validity of weak turbulence theory for random waves in an unambiguous and transparent fashion. These models have an explicitly solvable weak turbulence theory which is developed here, with Kolmogorov-type wave number spectra exhibiting interesting dependence on parameters in the equations. These predictions of weak turbulence theory are compared with numerical solutions with damping and driving that exhibit a statistical inertial scaling range over as much as two decades in wave number.
235 citations
TL;DR: In this paper, it was shown that for certain class of compactly supported C˜ initial data, smooth solutions of the unsteady Prandtl's equation blow up in nite time.
Abstract: We prove that for certain class of compactly supported C˜ initial data, smooth solutions of the unsteady Prandtl's equation blow up in nite time
220 citations
TL;DR: The traditional “BigNumber” package that forms the work-horse for exact computation must be reinvented to take advantage of many features found in geometric algorithms to make robustness a non-issue by computing exactly.
Abstract: Exact computation is assumed in most algorithms in computational geometry. In practice, implementors perform computation in some fixed-precision model, usually the machine floating-point arithmetic. Such implementations have many well-known problems, here informally called “robustness issues”. To reconcile theory and practice, authors have suggested that theoretical algorithms ought to be redesigned to become robust under fixed-precision arithmetic. We suggest that in many cases, implementors should make robustness a non-issue by computing exactly. The advantages of exact computation are too many to ignore. Many of the presumed difficulties of exact computation are partly surmountable and partly inherent with the robustness goal. This paper formulates the theoretical framework for exact computation based on algebraic numbers. We then examine the practical support needed to make the exact approach a viable alternative. It turns out that the exact computation paradigm encompasses a rich set of computational tactics. Our fundamental premise is that the traditional “BigNumber” package that forms the work-horse for exact computation must be reinvented to take advantage of many features found in geometric algorithms. Beyond this, we postulate several other packages to be built on top of the BigNumber package.
219 citations
TL;DR: A detailed description of fractional kinetics is given in connection to islands' topology in the phase space of a system, using the method of renormalization group to obtain characteristic exponents of the fractional space and time derivatives, and an analytic expression for the transport exponents.
Abstract: A detailed description of fractional kinetics is given in connection to islands’ topology in the phase space of a system. The method of renormalization group is applied to the fractional kinetic equation in order to obtain characteristic exponents of the fractional space and time derivatives, and an analytic expression for the transport exponents. Numerous simulations for the web-map and standard map demonstrate different results of the theory. Special attention is applied to study the singular zone, a domain near the island boundary with a self-similar hierarchy of subislands. The birth and collapse of islands of different types are considered.
177 citations
TL;DR: In this paper, the authors studied the continuum scaling limit of convex Hamiltonians, which are gradient perturbations of a massless free field, and proved that their long distance behavior is identical to a new (homogenized) continuum massless field.
Abstract: We study the continuum scaling limit of some statistical mechanical models defined by convex Hamiltonians which are gradient perturbations of a massless free field. By proving a central limit theorem for these models, we show that their long distance behavior is identical to a new (homogenized) continuum massless free field. We shall also obtain some new bounds on the 2-point correlation functions of these models.
TL;DR: It is shown that the maximum number of edges of a quasi-planar graph withn vertices is O(n), where n is the number of vertices in the graph.
Abstract: A graph is calledquasi-planar if it can be drawn in the plane so that no three of its edges are pairwise crossing. It is shown that the maximum number of edges of a quasi-planar graph withn vertices isO(n).
TL;DR: In this article, it was shown that any Palais-smale sequence of the energy functional on surfaces with uniformly L 2-bounded tension fields converges pointwise, by taking a subsequence if necessary, to a map from connected (possibly singular) surfaces, which consist of the original surfaces and finitely many bubble trees.
Abstract: In this article we prove that any Palais-Smale sequence of the energy functional on surfaces with uniformly L2-bounded tension fields converges pointwise, by taking a subsequence if necessary, to a map from connected (possibly singular) surfaces, which consist of the original surfaces and finitely many bubble trees. We therefore get the corresponding results about how the solutions of heat flow for harmonic maps from surfaces form singularities at infinite time. © 1997 John Wiley & Sons, Inc.
TL;DR: In this article, a new and direct approach for analyzing the scaling properties of the various distribution functions for the random forced Burgers equation is proposed, and the authors consider the problem of the growth of random surfaces.
Abstract: Statistical properties of solutions of the random forced Burgers equation have been a subject of intensive studies recently (see Refs. [1–6]). Of particular interest are the asymptotic properties of probability distribution functions associated with velocity gradients and velocity increments. Aside from the fact that such issues are of direct interest to a large number of problems such as the growth of random surfaces [1], it is also hoped that the field-theoretic techniques developed for the Burgers equation will eventually be useful for understanding more complex phenomena such as turbulence. In this paper, we propose a new and direct approach for analyzing the scaling properties of the various distribution functions for the random forced Burgers equation. We will consider the problem
TL;DR: In this article, the Cα regularity of weak solutions of a class of quasi-linear equations in nilpotent stratified Lie groups of step two was proved and a Liouville type theorem for 1-quasi-conformal mappings between domains of the Heisenberg group.
Abstract: We prove the Cα regularity of the gradient of weak solutions of a class of quasi-linear equations in nilpotent stratified Lie groups of step two. As applications, we prove higher regularity theorems and a Liouville type theorem for 1-quasi-conformal mappings between domains of the Heisenberg group. © 1997 John Wiley & Sons, Inc.
TL;DR: For every fixed k ≥ 3, there exists a constantc ≥ 3 matching covering all vertices with the following property: if k = 3, then there exists at most O(k −1/(k−1)
Abstract: For every fixedk≥3 there exists a constantc
k
with the following property. LetH be ak-uniform,D-regular hypergraph onN vertices, in which no two edges contain more than one common vertex. Ifk>3 thenH contains a matching covering all vertices but at mostc
k
ND
−1/(k−1). Ifk=3, thenH contains a matching covering all vertices but at mostc
3
ND
−1/2ln3/2
D. This improves previous estimates and implies, for example, that any Steiner Triple System onN vertices contains a matching covering all vertices but at mostO(N
1/2ln3/2
N), improving results by various authors.
TL;DR: An algorithm for computing the 2-center of a set S of n points in the plane (that is, a pair of congruent disks of smallest radius whose union covers S), improving the previous O(n^2\log n) -time algorithm.
Abstract: We present an $O(n\log^{9}n)$ -time algorithm for computing the 2-center of a set S of n points in the plane (that is, a pair of congruent disks of smallest radius whose union covers S), improving the previous $O(n^2\log n)$ -time algorithm of [10].
TL;DR: This paper describes the parallel implementation of the immersed boundary method on a shared-memory machine such as the Cray C-90 computer and an application to the computer simulation of blood flow in the heart is presented.
Abstract: This paper describes the parallel implementation of the immersed boundary method on a shared-memory machine such as the Cray C-90 computer. In this implementation, outer loops are parallelized and inner loops are vectorized. The sustained computation rates achieved are 0.258 Gflops with a single processor, 1.89 Gflops with 8 processors, and 2.50 Gflops with 16 processors. An application to the computer simulation of blood flow in the heart is presented.
06 Jan 1997
TL;DR: In this paper, the authors proposed a robust Cartesian mesh generation algorithm based on a robust geometry engine that uses adaptive precision arithmetic and automatically handles geometric degenerades with an algorithmic tie-breaking routine.
Abstract: This work documents a new method for rapid and robust Cartesian mesh generation for component-based geometry. The new algorithm adopts a novel strategy that first intersects the components to extract the wetted surface before proceeding with volume mesh generation in a second phase. The intersection scheme is based on a robust geometry engine that uses adaptive precision arithmetic and automatically and consistently handles geometric degenerades with an algorithmic tie-breaking routine. The intersection procedure has worst-case computational complexity of O(N log N) and is demonstrated on test cases with up to 121 overlapping and intersecting components, including a variety of geometric degeneracies. The volume mesh generation takes the intersected surface triangulation as input and generates the mesh through cell division of an initially uniform coarse grid. In refining hexagonal cells to resolve the geometry, the new approach preserves the ability to directionally divide cells that are well aligned with local geometry. The mesh generation scheme has linear asymptotic complexity with memory requirements that total approximately 14-17 words/cell. The mesh generation speed is approximately 10 6 cells/minute on a typical engineering workstation
TL;DR: In this article, a new family of first-passage percolation (FPP) models in the context of Poisson-Voronoi tesselations of ℝ d was introduced.
Abstract: We introduce a new family of first-passage percolation (FPP) models in the context of Poisson-Voronoi tesselations of ℝ d . Compared to standard FPP on ℤ d , these models have some technical complications but also have the advantage of statistical isotropy. We prove two almost sure results: a shape theorem (where isotropy implies an exact Euclidean ball for the asymptotic shape) and nonexistence of certain doubly infinite geodesics (where isotropy yields a stronger result than in standard FPP).
TL;DR: In this article, the authors propose a different approach, in which an appropriate thermodynamic description of such a system is instead based on a metastate, which is an ensemble of (possibly mixed) thermodynamic states.
Abstract: In realistic disordered systems, such as the Edwards-Anderson (EA) spin glass, no order parameter, such as the Parisi overlap distribution, can be both translation-invariant and non-self-averaging. The standard mean-field picture of the EA spin glass phase can therefore not be valid in any dimension and at any temperature. Further analysis shows that, in general, when systems have many competing (pure) thermodynamic states, a single state which is a mixture of many of them (as in the standard mean-field picture) contains insufficient information to reveal the full thermodynamic structure. We propose a different approach, in which an appropriate thermodynamic description of such a system is instead based on a metastate, which is an ensemble of (possibly mixed) thermodynamic states. This approach, modeled on chaotic dynamical systems, is needed when chaotic size dependence (of finite volume correlations) is present. Here replicas arise in a natural way, when a metastate is specified by its (meta)correlations. The metastate approach explains, connects, and unifies such concepts as replica symmetry breaking, chaotic size dependence and replica nonindependence. Furthermore, it replaces the older idea of non-self-averaging as dependence on the bulk couplings with the concept of dependence on the state within the metastate at fixed coupling realization. We use these ideas to classify possible metastates for the EA model, and discuss two scenarios introduced by us earlier---a nonstandard mean-field picture and a picture intermediate between that and the usual scaling-droplet picture.
TL;DR: For any 2-coloring of the segments determined by n points in general position in the plane, at least one of the color classes contains a non-self-intersecting spanning tree.
Abstract: For any 2-coloring of the ${n \choose 2}$ segments determined by n points in general position in the plane, at least one of the color classes contains a non-self-intersecting spanning tree. Under the same assumptions, we also prove that there exist $\lfloor (n+1)/3 \rfloor$ pairwise disjoint segments of the same color, and this bound is tight. The above theorems were conjectured by Bialostocki and Dierker. Furthermore, improving an earlier result of Larman et al., we construct a family of m segments in the plane, which has no more than $m^{\log 4/\log 27}$ members that are either pairwise disjoint or pairwise crossing. Finally, we discuss some related problems and generalizations.
TL;DR: A two-level overlapping Schwarz method is considered for a Nédélec finite element approximation of 3D Maxwell's equations, generalizing well–known results for conforming finite elements for second order elliptic scalar equations.
Abstract: Two-level overlapping Schwarz methods are considered for finite element problems of 3D Maxwell''s equations. Nedelec elements built on tetrahedra and hexahedra are considered. Once the relative overlap is fixed, the condition number of the additive Schwarz method is bounded, independently of the diameter of the triangulation and the number of subregions. A similar result is obtained for a multiplicative method. These bounds are obtained for quasi-uniform triangulations. In addition, for the Dirichlet problem, the convexity of the domain has to be assumed. Our work generalizes well-known results for conforming finite elements for second order elliptic scalar equations.
TL;DR: In this article, a new connection between the Fokker-planck equation with gradient drift term and an associated free energy functional was established, which can be interpreted as a gradient flux or a steepest descent with respect to a certain metric.
05 Jan 1997
TL;DR: In this article, the authors established a near-cubic upper bound on the complexity of the space of line transversals of a collection of n balls in 3D space.
Abstract: We establish a near-cubic upper bound on the complexity of the space of line transversals of a collection of n balls in three dimensions, and show that the bound is almost tight, in the worst case. We apply this bound to obtain a near-cubic algorithm for computing a smallest infinite cylinder enclosing a given set of points or balls in 3-space. We also present an approximation algorithm for computing a smallest enclosing cylinder.
TL;DR: In this article, the authors consider 4-dimensional, real, analytic Hamiltonian systems with a saddle center equilibrium and a homoclinic orbit to it and find conditions for the existence of transversal homocliic orbits to periodic orbits of long period in every energy level sufficiently close to the energy level of the saddle centre equilibrium.
Abstract: We consider 4-dimensional, real, analytic Hamiltonian systems with a saddle center equilibrium (related to a pair of real and a pair of imaginary eigenvalues) and a homoclinic orbit to it We find conditions for the existence of transversal homoclinic orbits to periodic orbits of long period in every energy level sufficiently close to the energy level of the saddle center equilibrium We also consider one-parameter families of reversible, 4-dimensional Hamiltonian systems We prove that the set of parameter values where the system has homoclinic orbits to a saddle center equilibrium has no isolated points We also present similar results for systems with heteroclinic orbits to saddle center equilibria © 1997 John Wiley & Sons, Inc
TL;DR: It is proved that the Arnoldi basis vectors begin to lose their linear independenceonly after the GMRES residual norm has been reduced to almost its final level of accuracy, which is proportional to κ(A)ε, where λ is the condition number of A and ε is the machine precision.
Abstract: In [6] the Generalized Minimal Residual Method (GMRES) which constructs the Arnoldi basis and then solves the transformed least squares problem was studied. It was proved that GMRES with the Householder orthogonalization-based implementation of the Arnoldi process (HHA), see [9], is backward stable. In practical computations, however, the Householder orthogonalization is too expensive, and it is usually replaced by the modified Gram-Schmidt process (MGSA). Unlike the HHA case, in the MGSA implementation the orthogonality of the Arnoldi basis vectors is not preserved near the level of machine precision. Despite this, the MGSA-GMRES performs surprisingly well, and its convergence behaviour and the ultimately attainable accuracy do not differ significantly from those of the HHA-GMRES. As it was observed, but not explained, in [6], it is thelinear independence of the Arnoldi basis, not the orthogonality near machine precision, that is important. Until the linear independence of the basis vectors is nearly lost, the norms of the residuals in the MGSA implementation of GMRES match those of the HHA implementation despite the more significant loss of orthogonality.
TL;DR: In this article, the authors studied the homogenization of Hamiltonian systems with a strong potential forcing the solutions to oscillate on a very small time scale, where the size ϵ tends to zero but the velocity components remain oscillating with an amplitude variation of the order O(1).
01 Aug 1997
TL;DR: The foveation transform of an image is introduced and a new method for foveating images based on wavelets is introduced, based on the multiresolution framework of Mallat.
Abstract: Motivated by applications of foveated images in visualization, we introduce the foveation transform of an image. We study the basic properties of these transforms using the multiresolution framework of Mallat. We also consider practical methods of realizing such transforms. In particular, we introduce a new method for foveating images based on wavelets. Preliminary experimental results are shown.