Showing papers in "Mathematics of Computation in 1990"
TL;DR: In this article, a modified Lagrange type interpolation operator is proposed to approximate functions in Sobolev spaces by continuous piecewise polynomials, and the combination of averaging and interpolation is shown to be a projection, and optimal error estimates are proved for the projection error.
Abstract: In this paper, we propose a modified Lagrange type interpolation operator to approximate functions in Sobolev spaces by continuous piecewise polynomials. In order to define interpolators for "rough" functions and to preserve piecewise polynomial boundary conditions, the approximated functions are averaged appropriately either on dor (d 1)-simplices to generate nodal values for the interpolation operator. This combination of averaging and interpolation is shown to be a projection, and optimal error estimates are proved for the projection error.
1,648 citations
TL;DR: The two-dimensional version of the Runge- Kutta Local Projection Discontinuous Galerkin (RKDG) methods are studied, which can easily handle the boundary conditions, verify maximum principles, and are formally uniformly high-order accurate.
Abstract: In this paper we study the two-dimensional version of the Runge- Kutta Local Projection Discontinuous Galerkin (RKDG) methods, already de- fined and analyzed in the one-dimensional case. These schemes are defined on general triangulations. They can easily handle the boundary conditions, verify maximum principles, and are formally uniformly high-order accurate. Prelimi- nary numerical results showing the performance of the schemes on a variety of initial-boundary value problems are shown.
1,583 citations
TL;DR: Simulated Annealing in practice as discussed by the authors : Simulated annealing is an algorithm for parallel simulated Annealing algorithms with Boltzmann machines, which can be found in the literature.
Abstract: SIMULATED ANNEALING. Combinatorial Optimization. Simulated Annealing. Asymptotic Convergence. Finite-Time Approximation. Simulated Annealing in Practice. Parallel Simulated Annealing Algorithms. BOLTZMANN MACHINES. Neural Computing. Boltzmann Machines. Combinatorial Optimization and Boltzmann Machines. Classification and Boltzmann Machines. Learning and Boltzmann Machines. Appendix. Bibliography. Indices.
1,238 citations
TL;DR: This paper discusses an approach for developing completely parallel multilevel preconditioners and describes the simplest application of the technique to a model elliptic problem.
Abstract: In this paper, we shall report on some techniques for the development of preconditioners for the discrete systems which arise in the approximation of solutions to elliptic boundary value problems. Here we shall only state the resulting theorems. It has been demonstrated that preconditioned iteration techniques often lead to the most computationally effective algorithms for the solution of the large algebraic systems corresponding to boundary value problems in two and three dimensional Euclidean space. The use of preconditioned iteration will become even more important on computers with parallel architecture. This paper discusses an approach for developing completely parallel multilevel preconditioners. In order to illustrate the resulting algorithms, we shall describe the simplest application of the technique to a model elliptic problem.
626 citations
TL;DR: In this paper, the Fourier transform was used to analyze the variational framework for multivariate interpolation and obtained error estimates of arbitrarily high order for a class of interpolation methods that includes multiquadrics.
Abstract: We continue an earlier study of certain spaces that provide a variational framework for multivariate interpolation. Using the Fourier transform to analyze these spaces, we obtain error estimates of arbitrarily high order for a class of interpolation methods that includes multiquadrics
556 citations
292 citations
TL;DR: In this article, it was shown that if the Extended Riemann Hypothesis holds, a composite number m has a witness for its compositeness (in the sense of Miller or Solovay-Strassen) that is at most 2 log 2m.
Abstract: Many number-theoretic algorithms rely on a result of Ankeny, which states that if the Extended Riemann Hypothesis (ERH) is true, any nontrivial multiplicative subgroup of the integers modulo m omits a number that is O(log2 m) . This has been generalized by Lagarias, Montgomery, and Odlyzko to give a similar bound for the least prime ideal that does not split completely in an abelian extension of number fields. This paper gives a different proof of this theorem, in which explicit constants are supplied. The bounds imply that if the ERH holds, a composite number m has a witness for its compositeness (in the sense of Miller or Solovay-Strassen) that is at most 2 log2m .
235 citations
219 citations
TL;DR: In this paper, the authors gave a generalization to Abelian varieties over finite fields of the algorithm of Schoof for elliptic curves, and showed how to count the rational points on the reductions modulo primes p of a fixed curve over Q in time polynomial in logp.
Abstract: "If 'twere done when 'tis done, then 'twere well/ It were done quickly."-Macbeth. Abstract. We give a generalization to Abelian varieties over finite fields of the algorithm of Schoof for elliptic curves. Schoof showed that for an elliptic curve E over F , given by a Weierstrass equation, one can compute the number of Q F -rational points of E in time 0((log
210 citations
TL;DR: In this paper, a shock-capturing term was proposed to add artificial viscosity depending on the local absolute value of the residual of the finite element solution and the meh size.
Abstract: We extend our previous analysis of streamline diffusion finite element methods for hyperbolic systems of conservation laws to include a shock-capturing term adding artificial viscosity depending on the local absolute value of the residual of the finite element solution and the meh size. With this term present, we prove a maximum norm bound for finite element solutionsof Burgers' equation an thus complete an earlier convergence proof for this equation. We further prove, using entropy variables, that a strong limit of finite element solutions is a weak solution of the system of conservation laws and satisfies the entropy inequality asociated with the entropy variables. Results of some numerical experiments for the time-dependent compressible Euler equations in two dimensions are also reported.
204 citations
TL;DR: It is proved that the stronger result that the problem of finding irreducible polynomials of specified degree over a finite field is deterministic polynomial-time reducible to theproblem of factoring polynmials over the prime field is true.
Abstract: We present a new algorithm for finding an irreducible polynomial of specified degree over a finite field. Our algorithm is deterministic, and it runs in polynomial time for fields of small characteristic. We in fact prove the stronger result that the problem of finding irreducible polynomials of specified degree over a finite field is deterministic polynomial-time reducible to the problem of factoring polynomials over the prime field
TL;DR: This work reduces several computations with Hilbert and Vandermonde type matrices to matrix computations of the Hankel-Toeplitz type (and vice versa), which enables us to compute the inverses and the determinants of n x n matrices of VandermondE and Hilbert types for the cost of 0(n log2 n) arithmetic operations.
Abstract: We reduce several computations with Hilbert and Vandermonde type matrices to matrix computations of the Hankel-Toeplitz type (and vice versa). This unifies various known algorithms for computations with dense structured matrices and enables us to extend any progress in computations with matrices of one class to the computations with other classes of matrices. In particular, this enables us to compute the inverses and the determinants of n x n matrices of Vandermonde and Hilbert types for the cost of 0(n log2 n) arithmetic operations. (Previously, such results were only known for the more narrow class of Vandermonde and generalized Hilbert matrices.)
TL;DR: In this article, an explicit estimate for the difference between the Weil height and the canonical height of points on elliptic curves is given in terms of the j-invariant and discriminant of the elliptic curve.
Abstract: Estimates for the difference of the Weil height and the canonical height of points on elliptic curves are used for many purposes, both theoretical and computational. In this note we give an explicit estimate for this difference in terms of the j-invariant and discriminant of the elliptic curve. The method of proof, suggested by Serge Lang, is to use the decomposition of the canonical height into a sum of local heights. We illustrate one use for our estimate by computing generators for the Mordell-Weil group in three examples. Let E be an elliptic curve defined over a number field K, say given by a Weierstrass equation (1) y2 =x +Ax+B with A and B in the ring of integers of K. The canonical height on E is a quadratic form h: E(K)-+ R. (For the definition and basic properties of h, see [10, Chapter VIII, ?9 or 6, Chapter VI].) The canonical height is determined by this property together with the fact that the difference
TL;DR: A survey of real analysis can be found in this paper, where the authors present a survey of results from complex analysis in higher dimensions, including linear iterative methods and matrix and vector analysis.
Abstract: Preface to the second edition Preface to the first edition 1. Hyperbolic partial differential equations 2. Analysis of finite difference Schemes 3. Order of accuracy of finite difference schemes 4. Stability for multistep schemes 5. Dissipation and dispersion 6. Parabolic partial differential equations 7. Systems of partial differential equations in higher dimensions 8. Second-order equations 9. Analysis of well-posed and stable problems 10. Convergence estimates for initial value problems 11. Well-posed and stable initial-boundary value problems 12. Elliptic partial differential equations and difference schemes 13. Linear iterative methods 14. The method of steepest descent and the conjugate gradient method Appendix A. Matrix and vectoranalysis Appendix B. A survey of real analysis Appendix C. A Survey of results from complex analysis References Index.
TL;DR: A survey of boundary integral equation methods for the Numerical solution of Laplace's Equation in three dimensions can be found in this article, along with a discussion of the convergence theorem for singular integral equation.
Abstract: 1. A Survey of Boundary Integral Equation Methods for the Numerical Solution of Laplace's Equation in Three Dimensions.- 2. Superconvergence.- 3. Perturbed Projection Methods for Various Classes of Operator and Integral Equations.- 4. Numerical Solution of Parallel Processors of Two-Point Boundary-Value Problems of Astrodynamics.- 5. Introduction to the Numerical Solution of Cauchy Singular Integral Equations.- 6. Convergence Theorems for Singular Integral Equations.- 7. Planing Surfaces.- 8. Abel Integral Equations.
TL;DR: In this paper, a Poisson equation on a rectangular domain is solved by coupling two methods: the domain is divided in two squares, a finite element approximation is used on the first square and a spectral discretization on the second one.
Abstract: A Poisson equation on a rectangular domain is solved by coupling two methods: the domain is divided in two squares, a finite element approximation is used on the first square and a spectral discretization is used on the second one. Two kinds of matching conditions on the interface are presented and compared. In both cases, error estimates are proved.
TL;DR: In this paper, the optimal order of convergence for two-dimensional finite element methods for the Stokes equations was proved for the triangular P3 P2 element and the Qk Qk-1 I k > 2, family of quadrilateral elements.
Abstract: We prove the optimal order of convergence for some two-dimensional finite element methods for the Stokes equations. First we consider methods of the Taylor-Hood type: the triangular P3 P2 element and the Qk Qk-1 I k > 2, family of quadrilateral elements. Then we introduce two new low-order methods with piecewise constant approximations for the pressure. The analysis is performed using our macroelement technique, which is reviewed in a slightly altered form.
TL;DR: In this article, the convection-diffusion equa- tion method is used for solving linear systems of the type arising from two-cyclic discretizations of non-self-adjoint two-dimensional ellip-tic partial differential equations.
Abstract: We study iterative methods for solving linear systems of the type arising from two-cyclic discretizations of non-self-adjoint two-dimensional ellip- tic partial differential equations. A prototype is the convection-diffusion equa- tion. The methods consist of applying one step of cyclic reduction, resulting in a "reduced system" of half the order of the original discrete problem, com- bined with a reordering and a block iterative technique for solving the reduced system. For constant-coefficient problems, we present analytic bounds on the spectral radii of the iteration matrices in terms of cell Reynolds numbers that show the methods to be rapidly convergent. In addition, we describe numerical experiments that supplement the analysis and that indicate that the methods compare favorably with methods for solving the "unreduced" system.
TL;DR: In this paper, the authors presented the results of a search to find optimal maximal period multipliers for multiplicative congruential random number generators with moduli 2 32 and 2 48.
Abstract: This paper presents the results of a search to find optimal maximal period multipliers for multiplicative congruential random number generators with moduli 2 32 and 2 48 . Here a multiplier is said to be optimal if the distance between adjacent parallel hyperplanes on which k-tuples lie does not exceed the minimal achievable distance by more than 25 percent for k=2, ..., 6
TL;DR: In this article, explicit formulas for sums of the form ǫ-ZakHk or Ã-ZaHkHn_k are derived, where the ak are simple functions of k. These identities are generalized by means of generating functions.
Abstract: Let Hn denote the nth harmonic number. Explicit formulas for sums of the form ^ZakHk or ^ZakHkHn_k are derived, where the ak are simple functions of k . These identities are generalized in a natural way by means of generating functions.
TL;DR: In this paper, various pointwise estimates for solutions of linear ellip- tic difference inequalities with random coefficients are presented. But none of these estimates are pointwise estimators of the maximum principle of Aleksandrov and Harnack in equalities.
Abstract: We prove various pointwise estimates for solutions of linear ellip- tic difference inequalities with random coefficients. These estimates include discrete versions of the maximum principle of Aleksandrov and Harnack in- equalities and Holder estimates of Krylov and Safonov for elliptic differential operators with bounded coefficients.
TL;DR: In this paper, a sensitivity analysis of numerical problems with independent, non-correlated data for general systems of linear and nonlinear equations is presented, with the additional property that, due to an automatic error control mechanism, every computed result is guaranteed to be correct.
Abstract: Methods are presented for performing a rigorous sensitivity analysis of numerical problems with independent, noncorrelated data for general systems of linear and nonlinear equations. The methods may serve for the following two purposes. First, to bound the dependency of the solution on changes in the input data. In contrast to condition numbers a componentwise sensitivity analysis of the solution vector is performed. Second, to estimate the true solution set for problems the input data of which are afflicted with tolerances. The methods presented are very effective with the additional property that, due to an automatic error control mechanism, every computed result is guaranteed to be correct. Examples are given for linear systems demonstrating that the computed bounds are in general very sharp. Interesting comparisons to traditional condition numbers are given.
TL;DR: In this article, certain projection operators on the space of all polynomials of degree less than or equal to N with respect to a class of one-dimensional weighted Sobolev spaces are proven.
Abstract: Results are proven concerning certain projection operators on the space of all polynomials of degree less than or equal to N with respect to a class of one-dimensional weighted Sobolev spaces. The results are useful in the theory of the approximation of partial differential equations with spectral methods.
TL;DR: In this paper, a polynomial-time algorithm for finding a complete set of irreducible representations over the field of complex numbers of a finite group given by its multiplication table is presented.
Abstract: The bit complexity of computing irreducible representations of finite groups is considered. Exact computations in algebraic number fields are performed symbolically. A polynomial-time algorithm for finding a complete set of inequivalent irreducible representations over the field of complex numbers of a finite group given by its multiplication table is presented. It follows that some representative of each equivalence class of irreducible representations admits a polynomial-size description. The problem of decomposing a given representation V of the finite group G over an algebraic number field F into absolutely irreducible constituents is considered. It is shown that this can be done in deterministic polynomial time if V is given by the list of matrices (V(g); g in G) and in randomized (Las Vegas) polynomial time under the more concise input (V(g); g in S), where S is a set of generators of G. >
TL;DR: In this article, various estimates for the error in adaptive approximation were obtained and a relationship between adaptive approximation and free-knot spline approximation was established, and the relationship between the two was established.
Abstract: We obtain various estimates for the error in adaptive approximation and also establish a relationship between adaptive approximation and free-knot spline approximation.
TL;DR: This chapter discusses linear systems of Equations, Trigonometirc Approximation and the Fast Fourier Transorm, and the solution of Nonlinear Equations.
Abstract: 1. Introduction. 2. Computer Arithmetic and Computational Errors. 3. Linear systems of Equations. 4. Interpolation. 5. Numerical Quadrature. 6. Linear Least-Square Data Fitting. 7. Solution of Nonlinear Equations. 8. Ordinary Diernetial Equations. 9. Optimization and Nonlinear Least Squares. 10. Simulation and Random Numbers. 11. Trigonometirc Approximation and the Fast Fourier Transorm. Bibliography.
TL;DR: An optimal-order W-cycle multigrid method for solving the stationary Stokes equations is developed, using P1 nonconforming divergence-free finite elements.
Abstract: An optimal-order W-cycle multigrid method for solving the stationary Stokes equations is developed, using P1 nonconforming divergence-free finite elements.
TL;DR: In this paper, the authors present a method for homoclinic and heterocliic continuation and bifurcation analysis of equilibria up to codimension three.
Abstract: Bifurcation to rotating waves from non-trivial steady-states.- Use of approximate inertial manifolds in bifurcation calculations.- Understanding steady-state bifurcation diagrams for a model reaction-diffusion system.- Bifurcations, chaos and self-organization in reaction-diffusion systems.- Eigenvalue problems with the symmetry of a group and bifurcations.- Steady-state/steady-state mode interaction in nonlinear equations with Z2-symmetry.- Symbolic computation and bifurcation methods.- Bifurcation analysis: a combined numerical and analytical approach.- to the numerical solution of symmetry-breaking bifurcation problems.- A computational method and path following for periodic solutions with symmetry.- Global bifurcations and their numerical computation.- Computation of invariant manifold bifurcations.- A method for homoclinic and heteroclinic continuation in two and three dimensions.- The global attractor under discretisation.- The numerical detection of Hopf bifurcation points.- A Newton-like method for simple bifurcation problems with application to large sparse systems.- Aspects of continuation software.- Interactive system for studies in nonlinear dynamics.- LINLBF: A program for continuation and bifurcation analysis of equilibria up to codimension three.- On the topology of three-dimensional separations, A guide for classification.- The construction of cubature formulae using continuation and bifurcation software.- Determining an Organizing Center for Passive Optical Systems.- Optimization by continuation.- Stability of Marangoni convection in a microgravity environment.- Computing with reaction-diffusion systems: applications in image processing.- Bifurcation of Codimension 2 for a discrete map.- Bifurcation of periodic solutions in PDE's: Numerical techniques and applications (abstract).- Bifurcation and chaos in Chua's circuit family (abstract).- Continuation and collocation for parameter dependent boundary value problems (abstract).- Porous medium combustion (abstract).- Block elimination and the computation of simple turning points (abstract).- Bifurcation into gaps in the essential spectrum (abstract).- Application of numerical continuation in aerospace problems (abstract).- Application of a reduced basis method in structural analysis (abstract).- Some applications of bifurcation theory in engineering (abstract).- Higher order predictors in numerical path following schemes (abstract).
TL;DR: In this article, the Christoffel number for ultraspherical weight functions w,A, WA(X) = (1 x 2 1/2 ) was investigated using only elementary functions, and new inequalities, monotonicity properties and asymptotic approximations were obtained.
Abstract: In this paper the Christoffel numbers a(2)G for ultraspherical weight functions w,A, WA(X) = (1 x2 1/2 , are investigated. Using only elementary functions, we state new inequalities, monotonicity properties and asymptotic approximations, which improve several known results. In particular, denoting by 0(i) the trigonometric representation of the Gaussian nodes, we obtain for A E [0, 11 the inequalities n v,fn 2(n + A)2 sin2 (A) (AG 7r 2A1 3.) av,n 1/2, a(A)G 7r 2A (A) __________ ~~~~~ sin 0 ~ ~ in()' v,n n + A /f n 2(n + A)2sin 2 ) v,n A ( I A) [3(A + I )(A 2) + 4 sin 2 0(A)n (8(n?+ A)4 sin 46~ +O(n )