Showing papers in "Mathematics of Computation in 2006"

Simulating Hamiltonian dynamics.

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Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces

Abstract: We reformulate the original component-by-component algorithm for rank-1 lattices in a matrix-vector notation so as to highlight its structural properties. For function spaces similar to a weighted Korobov space, we derive a technique which has construction cost O(snlog(n)), in contrast with the original algorithm which has construction cost O(sn 2 ). Herein s is the number of dimensions and n the number of points (taken prime). In contrast to other approaches to speed up construction, our fast algorithm computes exactly the same quantity as the original algorithm. The presented algorithm can also be used to construct randomly shifted lattice rules in weighted Sobolev spaces.

Spectral residual method without gradient information for solving large-scale nonlinear systems of equations

Abstract: A fully derivative-free spectral residual method for solving largescale nonlinear systems of equations is presented. It uses in a systematic way the residual vector as a search direction, a spectral steplength that produces a nonmonotone process and a globalization strategy that allows for this nonmonotone behavior. The global convergence analysis of the combined scheme is presented. An extensive set of numerical experiments that indicate that the new combination is competitive and frequently better than well-known Newton-Krylov methods for largescale problems is also presented.

High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems

Abstract: This paper is concerned with the development of high order methods for the numerical approximation of one-dimensional nonconservative hyperbolic systems In particular, we are interested in high order extensions of the generalized Roe methods introduced by I Toumi in 1992, based on WENO reconstruction of states We also investigate the well-balanced properties of the resulting schemes Finally, we will focus on applications to shallow-water systems

Convolution quadrature time discretization of fractional diffusion-wave equations

Abstract: We propose and study a numerical method for time discretization of linear and semilinear integro-partial differential equations that are intermediate between diffusion and wave equations, or are subdiffusive. The method uses convolution quadrature based on the second-order backward differentiation formula. Second-order error bounds of the time discretization and regularity estimates for the solution are shown in a unified way under weak assumptions on the data in a Banach space framework. Numerical experiments illustrate the theoretical results.

The efficient evaluation of the hypergeometric function of a matrix argument

Abstract: We present new algorithms that efficiently approximate the hypergeometric function of a matrix argument through its expansion as a series of Jack functions. Our algorithms exploit the combinatorial properties of the Jack function, and have complexity that is only linear in the size of the matrix.

A variant of the level set method and applications to image segmentation

Abstract: In this paper we propose a variant of the level set formulation for identifying curves separating regions into different phases. In classical level set approaches, the sign of n level set functions are utilized to identify up to 2" phases. The novelty in our approach is to introduce a piecewise constant level set function and use each constant value to represent a unique phase. If 2" phases should be identified, the level set function must approach 2" predetermined constants. We just need one level set function to represent 2" unique phases, and this gains in storage capacity. Further, the reinitializing procedure requested in classical level set methods is superfluous using our approach. The minimization functional for our approach is locally convex and differentiable and thus avoids some of the problems with the nondifferentiability of the Delta and Heaviside functions. Numerical examples are given, and we also compare our method with related approaches.

Fast algorithms for computing the Boltzmann collision operator

Abstract: The development of accurate and fast numerical schemes for the five fold Boltzmann collision integral represents a challenging problem in scientific computing. For a particular class of interactions, including the so-called hard spheres model in dimension three, we are able to derive spectral methods that can be evaluated through fast algorithms. These algorithms are based on a suitable representation and approximation of the collision operator. Explicit expressions for the errors in the schemes are given and spectral accuracy is proved. Parallelization properties and adaptivity of the algorithms are also discussed.

The 192 solutions of the Heun equation

Abstract: A machine-generated list of 192 local solutions of the Heun equation is given. They are analogous to Kummer's 24 solutions of the Gauss hypergeometric equation, since the two equations are canonical Fuchsian differential equations on the Riemann sphere with four and three singular points, respectively. Tabulation is facilitated by the identification of the automorphism group of the equation with n singular points as the Coxeter group D n . Each of the 192 expressions is labeled by an element of D 4 . Of the 192, 24 are equivalent expressions for the local Heun function Hl, and it is shown that the resulting order-24 group of transformations of Hl is isomorphic to the symmetric group S 4 . The isomorphism encodes each transformation as a permutation of an abstract four-element set, not identical to the set of singular points.

Iterated function systems, Ruelle operators, and invariant projective measures

Abstract: We introduce a Fourier-based harmonic analysis for a class of discrete dynamical systems which arise from Iterated Function Systems. Our starting point is the following pair of special features of these systems. (1) We assume that a measurable space X comes with a finite-to-one endomorphism r: X → X which is onto but not one-to-one. (2) In the case of affine Iterated Function Systems (IFSs) in R d , this harmonic analysis arises naturally as a spectral duality defined from a given pair of finite subsets B, L in R d of the same cardinality which generate complex Hadamard matrices. Our harmonic analysis for these iterated function systems (IFS) (X,μ) is based on a Markov process on certain paths. The probabilities are determined by a weight function W on X. From W we define a transition operator R w acting on functions on X, and a corresponding class H of continuous R w -harmonic functions. The properties of the functions in H are analyzed, and they determine the spectral theory of L 2 (μ). For affine IFSs we establish orthogonal bases in L 2 (μ). These bases are generated by paths with infinite repetition of finite words. We use this in the last section to analyze tiles in R d .

Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems

Abstract: We derive a posteriori error estimates for fully discrete approximations to solutions of linear parabolic equations. The space discretization uses finite element spaces that are allowed to change in time. Our main tool is an appropriate adaptation of the elliptic reconstruction technique, introduced by Makridakis and Nochetto. We derive novel a posteriori estimates for the norms of L∞(0, T; L2(Ω)) and the higher order spaces, L∞(0, T;H1(Ω)) and H1(0, T; L2(Ω)), with optimal orders of convergence.

Energy norm a posteriori error estimates for mixed finite element methods

Abstract: This paper deals with the a posteriori error analysis of mixed finite element methods for second order elliptic equations. It is shown that a reliable and efficient error estimator can be constructed using a postprocessed solution of the method. The analysis is performed in two different ways: under a saturation assumption and using a Helmholtz decomposition for vector fields.

Error reduction and convergence for an adaptive mixed finite element method

Abstract: An adaptive mixed finite element method (AMFEM) is designed to guarantee an error reduction, also known as saturation property: after each refinement step, the error for the fine mesh is strictly smaller than the error for the coarse mesh up to oscillation terms. This error reduction property is established here for the Raviart-Thomas finite element method with a reduction factor p < 1 uniformly for the L 2 norm of the flux errors. Our result allows for linear convergence of a proper adaptive mixed finite element algorithm with respect to the number of refinement levels. The adaptive algorithm surprisingly does not require any particular mesh design, unlike the conforming finite element method. The new arguments are a discrete local efficiency and a quasi-orthogonality estimate. The proof does not rely on duality or on regularity.

Regularization of some linear ill-posed problems with discretized random noisy data

Abstract: For linear statistical ill-posed problems in Hilbert spaces we introduce an adaptive procedure to recover the unknown so- lution from indirect discrete and noisy data.. This procedure is shown to be order optimal for a large class of problems. Smooth- ness of the solution is measured in terms of general source condi- tions. The concept of operator monotone functions turns out to be an important tool for the analysis.

Analysis of finite element approximations of a phase field model for two-phase fluids

Abstract: This paper studies a phase field model for the mixture of two immiscible and incompressible fluids. The model is described by a nonlinear parabolic system consisting of the nonstationary Stokes equations coupled with the Allen-Cahn equation through an extra phase induced stress term in the Stokes equations and a fluid induced transport term in the Allen-Cahn equation. Both semi-discrete and fully discrete finite element methods are developed for approximating the parabolic system. It is shown that the proposed numerical methods satisfy a discrete energy law which mimics the basic energy law for the phase field model. Error estimates are derived for the semi-discrete method, and the convergence to the phase field model and to its sharp interface limiting model are established for the fully discrete finite element method by making use of the discrete energy law. Numerical experiments are also presented to validate the theory and to show the effectiveness of the combined phase field and finite element approach.

An optimal adaptive wavelet method without coarsening of the iterands

Abstract: In this paper, an adaptive wavelet method for solving linear operator equations is constructed that is a modification of the method from [Math. Comp, 70 (2001), pp. 27-75] by Cohen, Dahmen and DeVore, in the sense that there is no recurrent coarsening of the iterands. Despite this, it will be shown that the method has optimal computational complexity. Numerical results for a simple model problem indicate that the new method is more efficient than an existing alternative adaptive wavelet method.

Analysis of a finite pml approximation for the three dimensional time-harmonic maxwell and acoustic scattering problems

Abstract: We consider the approximation of the frequency domain three-dimensional Maxwell scattering problem using a truncated domain perfectly matched layer (PML). We also treat the time-harmonic PML approximation to the acoustic scattering problem. Following work of Lassas and Somersalo in 1998, a transitional layer based on spherical geometry is defined, which results in a constant coefficient problem outside the transition. A truncated (computational) domain is then defined, which covers the transition region. The truncated domain need only have a minimally smooth outer boundary (e.g., Lipschitz continuous). We consider the truncated PML problem which results when a perfectly conducting boundary condition is imposed on the outer boundary of the truncated domain. The existence and uniqueness of solutions to the truncated PML problem will be shown provided that the truncated domain is sufficiently large, e.g., contains a sphere of radius R t . We also show exponential (in the parameter R t ) convergence of the truncated PML solution to the solution of the original scattering problem inside the transition layer. Our results are important in that they are the first to show that the truncated PML problem can be posed on a domain with nonsmooth outer boundary. This allows the use of approximation based on polygonal meshes. In addition, even though the transition coefficients depend on spherical geometry, they can be made arbitrarily smooth and hence the resulting problems are amenable to numerical quadrature. Approximation schemes based on our analysis are the focus of future research.

On mixed and componentwise condition numbers for Moore–Penrose inverse and linear least squares problems

Abstract: Classical condition numbers are normwise: they measure the size of both input perturbations and output errors using some norms. To take into account the relative of each data component, and, in particular, a possible data sparseness, componentwise condition numbers have been increasingly considered. These are mostly of two kinds: mixed and componentwise. In this paper, we give explicit expressions, computable from the data, for the mixed and componentwise condition numbers for the computation of the Moore-Penrose inverse as well as for the computation of solutions and residues of linear least squares problems. In both cases the data matrices have full column (row) rank.

Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics.

Abstract: The efficient numerical treatment of incompressible fluid dynamics is a formidable challenge in computational mathematics. To develop an efficient treatment, one needs to understand the basic equations of fluid dynamics; i.e., the Stokes and Navier–Stokes equations, together with the boundary conditions that give wellposedness. Next, for variational formulations, one needs a knowledge of finite element discretizations that are stable or finite element stabilization techniques, in order to generate accurate numerical solutions. Finally, one needs a grasp of fast iterative solution schemes, both linear and nonlinear, so that an overall efficient method can be constructed. To gain acquaintance with such a cutting-edge development, prior to the publication of this book, the main reference sources were technical journal articles on each of these treatment components. Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics, by Elman, Silvester, and Wathen, now gives a thorough presentation of the state-ofthe-art techniques in one volume. The authors are acclaimed in this research field. Indeed, this book presents many of their own research results, but in a readable form accessible to a broad audience of both professionals and students of engineering, mathematics, and interdisciplinary computational science. The book is self-contained. Only basic knowledge of discretization methods for partial differential equations, fundamental functional analysis, and computational linear algebra are needed. Presentation of the basic partial differential equations underlying fluid dynamics, of finite element discretization, and of iterative linear system schemes start from the basics. The book then systematically progresses to more advanced theoretical and computational techniques for incompressible fluid dynamics. These advanced topics include continuous and discrete inf-sup stability and finite elements that satisfy them, stabilized finite element techniques and the patch-test to verify stability, Krylov iterative methods and the theory behind them, construction and justification of efficient preconditioners for the Stokes and Navier–Stokes equations, and a posteriori adaptive procedures. Even though these topics are often accessible only to experts in the field, the authors give a comprehendible yet thorough presentation, with concrete test-suite examples and appropriate analytical and computational exercises to give the reader a grasp of these concepts.

Computing the Ehrhart quasi-polynomial of a rational simplex

Abstract: We present a polynomial time algorithm to compute any fixed number of the highest coefficients of the Ehrhart quasi-polynomial of a rational simplex. Previously such algorithms were known for integer simplices and for rational polytopes of a fixed dimension. The algorithm is based on the formula relating the kth coefficient of the Ehrhart quasi-polynomial of a rational polytope to volumes of sections of the polytope by affine lattice subspaces parallel to k-dimensional faces of the polytope. We discuss possible extensions and open questions.

Locking-free adaptive discontinuous Galerkin FEM for linear elasticity problems

Abstract: An adaptive discontinuous Galerkin finite element method for linear elasticity problems is presented. We develop an a posteriori error estimate and prove its robustness with respect to nearly incompressible materials (absence of volume locking). Furthermore, we present some numerical experiments which illustrate the performance of the scheme on adaptively refined meshes.

A locally divergence-free nonconforming finite element method for the time-harmonic maxwell equations

Abstract: A new numerical method for computing the divergence-free part of the solution of the time-harmonic Maxwell equations is studied in this paper. It is based on a discretization that uses the locally divergence-free Crouzeix-Raviart nonconforming P 1 vector fields and includes a consistency term involving the jumps of the vector fields across element boundaries. Optimal convergence rates (up to an arbitrary positive e) in both the energy norm and the L 2 norm are established on graded meshes. The theoretical results are confirmed by numerical experiments.

Numerical differentiation from a viewpoint of regularization theory

Abstract: In this paper, we discuss the classical ill-posed problem of numerical differentiation, assuming that the smoothness of the function to be differentiated is unknown. Using recent results on adaptive regularization of general ill-posed problems, we propose new rules for the choice of the stepsize in the finite-difference methods, and for the regularization parameter choice in numerical differentiation regularized by the iterated Tikhonov method. These methods are shown to be effective for the differentiation of noisy functions, and the order-optimal convergence results for them are proved.

Quadrature methods for multivariate highly oscillatory integrals using derivatives

Abstract: While there exist effective methods for univariate highly oscillatory quadrature, this is not the case in a multivariate setting. In this paper we embark on a project, extending univariate theory to more variables. Inter alia, we demonstrate that, in the absence of critical points and subject to a nonresonance condition, an integral over a simplex can be expanded asymptotically using only function values and derivatives at the vertices, a direct counterpart of the univariate case. This provides a convenient avenue towards the generalization of asymptotic and Filon-type methods, as formerly introduced by the authors in a single dimension, to simplices and, more generally, to polytopes. The nonresonance condition is bound to be violated once the boundary of the domain of integration is smooth: in effect, its violation is equivalent to the presence of stationary points in a single dimension. We further explore this issue and propose a technique that often can be used in this situation. Yet, much remains to be done to understand more comprehensively the influence of resonance on the asymptotics of highly oscillatory integrals.

On polynomial selection for the general number field sieve

Abstract: The general number field sieve (GNFS) is the asymptotically fastest algorithm for factoring large integers. Its runtime depends on a good choice of a polynomial pair. In this article we present an improvement of the polynomial selection method of Montgomery and Murphy which has been used in recent GNFS records.