Showing papers in "Mathematics of Computation in 1992"

Reconstruction algorithms in irregular sampling

Abstract: A constructive solution of the irregular sampling problem for band- limited functions is given. We show how a band-limited function can be com- pletely reconstructed from any random sampling set whose density is higher than the Nyquist rate, and give precise estimates for the speed of convergence of this iteration method. Variations of this algorithm allow for irregular sampling with derivatives, reconstruction of band-limited functions from local averages, and irregular sampling of multivariate band-limited functions. In the irregular sampling problem one is asked whether and how a band- limited function / can be completely reconstructed from its irregularly sam- pled values f(xi). This has many applications in signal and image processing, seismology, meteorology, medical imaging, etc. Finding constructive solutions of this problem has received considerable attention among mathematicians and engineers. The mathematical literature provides several uniqueness results (1, 2, 17, 18, 19). It is now part of the folklore that for stable sampling the sampling rate must be at least the Nyquist rate (18). These results, as deep as they are, have had little impact for the applied sciences, because they were not constructive. If the sampling set is just a perturbation of the regular oversampling, then a reconstruction method has been obtained in a seminal paper by Duffin and Schaeffer (6) (see also (29)): if for some L > 0, a > 0, and o > 0 the sampling points xk , k e Z , satisfy (a) \xk - ok\ a, k ^ I, then the norm equivalence A iR \f(x)\2dx ) with w < n/o. This norm equivalence implies that it is possible to reconstruct / through an iterative procedure, the so-called frame method. Most of the later work on constructive methods consists of variations of this method (3, 21, 22, 26). The above conditions on the sampling set exclude random irregular sampling sets, e.g., sets with regions of higher sampling density. A partial, but undesirable remedy, to handle highly irregular sampling sets, would be to force the above conditions by throwing away information on part of the points and accept a very slow convergence of the iteration.

Linear finite element methods for planar linear elasticity

Abstract: A linear nonconforming (conforming) displacement finite element method for the pure displacement (pure traction) problem in two-dimensional linear elasticity for a homogeneous isotropic elastic material is considered. In the case of a convex polygonal configuration domain, error estimates in the energy (L[sup 2]) norm are obtained. The convergence rate does not deteriorate for nearly incompressible material. Furthermore, the convergence analysis does not rely on the theory of saddle point problems. 22 refs.

Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation

Abstract: A finite element method for the Cahn-Hilliard equation (a semilinear parabolic equation of fourth order) is analyzed, both in a spatially semidiscrete case and in a completely discrete case based on the backward Euler method. Error bounds of optimal order over a finite time interval are obtained for solutions with smooth and nonsmooth initial data. A detailed study of the regularity of the exact solution is included. The analysis is based on local Lipschitz conditions for the nonlinearity with respect to Sobolev norms, and the existence of a Ljapunov functional for the exact and the discretized equations is essential. A result concerning the convergence of the attractor of the corresponding approximate nonlinear semigroup (upper semicontinuity with respect to the discretization parameters) is obtained as a simple application of the nonsmooth data error estimate.

Reconstruction techniques for classical inverse Sturm-Liouville problems

Abstract: This paper gives constructive algorithms for the classical inverse Sturm-Liouville problem. It is shown that many of the formulations of this problem are equivalent to solving an overdetermined boundary value problem for a certain hyperbolic operator. Two methods of solving this latter problem are then provided, and numerical examples are presented.

Rank-Revealing QR Factorizations and the Singular Value Decomposition

Abstract: T. Chan has noted that, even when the singular value decomposition of a matrix A is known, it is still not obvious how to find a rank-revealing QR factorization (RRQR) of A if A has numerical rank deficiency. This paper offers a constructive proof of the existence of the RRQR factorization of any matrix A of size m x n with numerical rank r . The bounds derived in this paper that guarantee the existence of RRQR are all of order f i ,in comparison with Chan's 0(2"-') . It has been known for some time that if A is only numerically rank-one deficient, then the column permutation l7 of A that guarantees a small rnn in the QR factorization of A n can be obtained by inspecting the size of the elements of the right singular vector of A corresponding to the smallest singular value of A . To some extent, our paper generalizes this well-known result. We consider the interplay between two important matrix decompositions: the singular value decomposition and the QR factorization of a matrix A . In particular, we are interested in the case when A is singular or nearly singular. It is well known that for any A E R m X n (a real matrix with rn rows and n columns, where without loss of generality we assume rn > n) there are orthogonal matrices U and V such that where C is a diagonal matrix with nonnegative diagonal elements: We assume that a, 2 a2 2 . . 2 on 2 0 . The decomposition (0.1) is the singular value decomposition (SVD) of A , and the ai are the singular values of A . The columns of V are the right singular vectors of A , and the columns of U are the left singular vectors of A . Mathematically, in terms of the singular values, Received December 1, 1990; revised February 8, 199 1. 199 1 Mathematics Subject Classification. Primary 65F30, 15A23, 15A42, 15A15.

Computational aspects of polynomial interpolation in several variables

Abstract: The pair (6, P) of a point set 8 C R¿ and a polynomial space P on Rd is correct if the restriction map P —» E8 : p t-> P|Q is invertible, i.e., if there is, for any / defined on 0 , a unique p G P which matches /on 9. We discuss here a particular assignment 6 >-+ Fie , introduced by us previously, for which (8, lie) is always correct, and provide an algorithm for the construction of a basis for ne , which is related to Gauss elimination applied to the Vandermonde matrix (öa)öee a£Zd for 8. We also discuss some attractive properties of the above assignment and algorithmic details, and present some bivariate examples. We say that the pair (6, P) of a (finite) point set OcK1' and a (polynomial) space P of functions on Rd is correct if the restriction map

Analysis of locally stabilized mixed finite-element methods for the stokes problem

Abstract: In this paper, a locally stabilized finite element formulation of the Stokes problem is analyzed. A macroelement condition which is sufficient for the stability of (locally stabilized) mixed methods based on a piecewise constant pressure approximation is introduced. By satisfying this condition, the stability of the QI Po quadrilateral, and the PI Po triangular element, can be established.

Blossoming begets B -spline bases built better by B -patches

Abstract: The concept of symmetric recursive algorithm leads to new, sdimensional spline spaces. We present a general scheme for constructing a collection of multivariate B-splines with k-I continuous derivatives whose linear span contains all polynomials of degree at most k. This scheme is different from the one developed earlier by Dahmen and Micchelli and, independently, by H6llig, which was based on combinatorial principles and the geometric interpretation of the B-spline. The new spline space introduced here seems to offer possibilities for economizing the computation for evaluating linear combinations of B-splines.

Geometry Processing for Design and Manufacturing

Abstract: This text includes papers covering topics in geometry processing applications, such as surface-surface intersections and offset surfaces. Present methods fundamental to geometric modelling are highlighted.

On mixed finite element methods for the Reissner-Mindlin plate model

Abstract: In this paper we analyze the convergence of mixed finite element approximations to the solution of the Reissner-Mindlin plate problem. We show that several known elements fall into our analysis, thus providing a unified approach. We also introduce a low-order triangular element which is optimalorder convergent uniformly in the plate thickness.

Primitive polynomials over finite fields

Abstract: In this note we extend the range of previously published tables of primitive polynomials over finite fields. For each p" < 1050 with p < 97 we provide a primitive polynomial of degree n over Fp . Moreover, each polynomial has the minimal number of nonzero coefficients among all primitives of degree n over Fp .

Searching for primitive roots in finite fields

Abstract: Let GF(pn) be the finite field with pn elements, where p is prime. We consider the problem of how to deterministically generate in polynomial time a subset of GF(pn) that contains a primitive root, i.e., an element that generates the multiplicative group of nonzero elements in GF(pn) . We present three results. First, we present a solution to this problem for the case where p is small, i.e., p = n-d) . Second, we present a solution to this problem under the assumption of the Extended Riemann Hypothesis (ERH) for the case where p is large and n = 2. Third, we give a quantitative improvement of a theorem of Wang on the least primitive root for GF(p), assuming the ERH.

Finite element approximation of a parabolic integro-differential equation with a weakly singular kernel

Abstract: We give error estimates for the numerical solution by means of the Galerkin finite element method of an integro-differential equation of parabolic type with a memory term containing a weakly singular kernel. Optimal-order estimates are shown for spatially semidiscrete and completely discrete methods. Special attention is paid to the regularity of the exact solution.

The analysis of smoothers for multigrid algorithms

Abstract: The purpose of this paper is to provide a general technique for defining and analyzing smoothing operators for use in multigrid algorithms. The smoothing operators considered are based on subspace decomposition and include point, line, and block versions of Jacobi and Gauss-Seidel iteration as well as generalizations. We shall show that these smoothers will be effective in multigrid algorithms provided that the subspace decomposition satisfies two simple conditions. In many applications, these conditions are trivial to verify.

Scientific Computing with Automatic Result Verification

Abstract: Automatic Result Verification.- I. Numerical Methods with Result Verification.- A Method for Producing Verified Results for Two-point Boundary Value Problems.- A Kind of Difference Method for Enclosing Solutions of Ordinary Linear Boundary Value Problems.- A Self-validating Method for Solving Linear Programming Problems with Interval Input Data.- Enclosing the Solutions of Linear Equations by Interval Iterative Processes.- Errorbounds for Quadratic Systems of Nonlinear Equations Using the Precise Scalar Product.- Inclusion of Eigenvalues of General Eigenvalue Problems of Matrices.- Verified Inclusion for Eigenvalues of Certain Difference and Differential Equations.- II. Applications in the Technical Sciences.- VIB - Verified Inclusions of Critical Bending Vibrations.- Stability Test for Periodic Differential Equations on Digital Computers with Applications.- The Periodic Solutions of the Oregonator and Verification of Results.- On Arithmetical Problems of Geometric Algorithms in the Plane.- III. Improving the Tools.- Precise Evaluation of Polynomials in Several Variables.- Evaluation of Arithmetic Expressions with Guaranteed High Accuracy.- Standard Functions for Real and Complex Point and Interval Arguments with Dynamic Accuracy.- Inverse Standard Functions for Real and Complex Point and Interval Arguments with Dynamic Accuracy.- Inclusion Algorithms with Functions as Data.- FORTRAN-SC. A Study of a FORTRAN Extension for Engineering/Scientific Computation with Access to ACRITH.

Clothoid spline transition spirals

Abstract: Highway and railway designers use clothoid splines (planar G2 curves consisting of straight line segments, circular arcs, and clothoid segments) as center lines in route location. This paper considers the problem of finding a clothoid spline transition spiral which joins two given points and matches given curvatures and unit tangents at the two points. Conditions are given for the existence and uniqueness of the clothoid spline transition spirals, and algorithms for finding them are outlined.

Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension

Abstract: An energy-preserving, linearly implicit finite difference scheme is presented for approximating solutions to the periodic Cauchy problem for the one-dimensional Zakharov system of two nonlinear partial differential equations. First-order convergence estimates are obtained in a standard "energy" norm in terms of the initial errors and the usual discretization errors.

Curve and surface design

Abstract: Part I. Curve Design: Properties of Minimal Energy Splines G. Brunnett Minimal Energy splines with Various End Constraints E. Jou and W. Han Interval Weighted Tau- splines D. Lasser and H. Hagen Curve and surface Interpolation using Quintic Weight Tau-splines D. Neuser Weighted splines Based on Piecewise Polynomial Weighted Functions K. Salkauskas Algorithms for Geometric spline Curves M. Eck On the Problem of Determining the distance Between parametric curves F. Fritsch and G. Nielson Part II. Non-Tensor Product Surfaces: A survey of scattered Data Fitting Using Triangular Interpolants T. De Rose Free-form surfaces from Partial Differential Equations M. I. G. Bloor and M. J. Wilson Modeling with Box spline surfaces M. Daehlen.

Knot insertion and deletion algorithms for B-spline curves and surfaces

Abstract: 1. An Introduction to Blossoming Phillip J. Barry 2. Algorithms for Progressive Curves: Extending B-Spline and Blossoming Techniques to the Monomial, Power, and Newton Dual Bases Phillip J. Barry and Ronald N. Goldman 3. Factored Knot Insertion Phillip J. Barry and Ronald N. Goldman 4. Knot Insertion Algorithms Phillip J. Barry and Ronald N. Goldman 5. Conversion Between B-Spline Bases Using the Generalized Oslo Algorithm Tom Lyche, Knut Morken and Kyrre Strom 6. How Much Can the Size of the B-Spline Coefficients be Reduced by Inserting One Knot? Tom Lyche and Knut Morken 7. An Envelope Approach to a Sketching Editor for Hierarchical Free-form Curve Design and Modification Michael J. Banks, Elaine Cohen and Timothy I. Mueller Index.

On the distribution of a scaled condition number

Abstract: In this note, we give the exact distribution of a scaled condition number used by Demmel to model the probability that matrix inversion is difficult. Specifically, consider a random matrix A and the scaled condition number KCD(A) = IIAlIF * IIA1 1. Demmel provided bounds for the condition number distribution when A has real or complex normally distributed elements. Here, we give the exact formula.

Families of modular eigenforms

Abstract: This article is an expansion of the notes to a one-hour lecture for an MSRI workshop on computational number theory. The editors of Mathematics of Computation kindly asked us to submit these notes for publication, and we are enormously pleased to do so. Our original audience did not consist of experts in the field of modular forms, and we have tried to keep this article accessible to nonexperts. We have made an experimental investigation of certain arithmetic conjec- tures using MACSYMA. This investigation requires a search for certain mod- ular eigenforms of high weight. These computations pose problems which we feel may be interesting on their own. We are novices here, and we seek advice from people more experienced in making computations of an analogous sort. We are, in fact, deeply grateful to J. F. Mestre, who came to our aid and who vastly extended our computations using PARI.1 Mestre has graciously allowed us to present his computations in this article. The kind of families we have in mind, as in the title of this lecture, has as its prototypical example the standard family of classical Eisenstein series of level 1 and weight k for k = 4, 6, 8, ... , whose Fourier expansions are given by oo

Computational Physics--Fortran Version.

Abstract: * Basic Mathematical Operations * Ordinary Differential Equations * Boundary Value and Eigenvalue Problems * Special Functions and Gaussian Quadrature * Matrix Operations * Elliptic Partial Differential Equations * Parabolic Partial Differential Equations * Monte Carlo Methods

Approximation theory and functional analysis.

Abstract: Part 1 Research and survey articles: on Bernstein-Durrineyer polynomials with Jacob weights, H.Berens and Y.Xu an overview of wavelets, C.Chui a note on weak inequalities in Orlicz and Lorentz spaces, G.A.Edgar and L.Sucheston bivariate Birkhoff interpolation - a survey, R.A.Lorentz linear approximations of functions with several restricted derivatives, Y.Makovoz some characterizations theorems for measures associated with orthogonal polynomials on the unit circle, K.Pan and E.B.Saff box splines, cardinal series, and wavelets, S.D.Reimenschneider and Z.Shen some aspects of the subspace structure of infinite dimensional banach space, H.Rosenthal fairness and monotone curvature, J.Roulier et al projections on 2-dimensional spaces, N.Tomczak-Jacgermann real versus complex best rational approximation, R.S.Varga and A.Ruttan.

Runge-Kutta methods for partial differential equations and fractional orders of convergence

Abstract: We apply Runge-Kutta methods to linear partial differential equations of the form u¡(x, t) =5?(x, d)u(x, t)+f(x, t). Under appropriate assumptions on the eigenvalues of the operator 5C and the (generalized) Fourier coefficients of /, we give a sharp lower bound for the order of convergence of these methods. We further show that this order is, in general, fractional and that it depends on the //-norm used to estimate the global error. The analysis also applies to systems arising from spatial discretization of partial differential equations by finite differences or finite element techniques. Numerical examples illustrate the results.

Power series expansions of Riemann’s function

Abstract: We show how high-precision values of the coefficients of power series expansions of functions related to Riemann's í function may be calculated. We also show how the Stieltjes constants can be evaluated using this scheme and how the Riemann hypothesis can be expressed in terms of the behavior of two of the sequences of coefficients. High-precision values for the coefficients of these power series are found using Mathematica TM .

Computations of Siegel modular forms of genus two

Abstract: We explain the basic notions and theorems for doing computations in the theory of Siegel modular forms of degree two, on the full modular group and of even weight. This synopsis concludes with a handy and computationally realistic algorithm for tabulating the Fourier coefficients of such forms and the Euler factors of their Spinor zeta functions. In the second part of this paper we present and discuss some of the results of actual computations which we performed following this algorithm. We point out two (theoretically) striking phenomena that are implied by the results of these computations.

The approximation of the exact boundary conditions at an artificial boundary for linear elastic equations and its application

Abstract: The exterior boundary value problems of linear elastic equations are considered. A sequence of approximations to the exact boundary conditions at an artificial boundary is given. Then the original problem is reduced to a boundary value problem on a bounded domain. Furthermore, a finite element approximation of this problem and optimal error estimates are obtained. Finally, a numerical example shows the effectiveness of this method.

On the distribution of quadratic residues and nonresidues modulo a prime number

Abstract: Let P be a prime number and al, at be distinct integers modulo P. Let x be chosen at random with uniform distribution in Zp , and let yi = x + ai . We prove that the joint distribution of the quadratic characters of the yi 's deviates from the distribution of independent fair coins by no more than t(3 + xfi-)/P. That is, the probability of (Yi, ...Y, t) matching any particular quadratic character sequence of length t is in the range (I )t i t(3 + v/ii)/P. We establish the implications of this bound on the number of occurrences of arbitrary patterns of quadratic residues and nonresidues modulo P. We then explore the randomness complexity of finding these patterns in polynomial time. We give (exponentially low) upper bounds for the probability of failure achievable in polynomial time using, as a source of randomness, no more than one random number modulo P.