Showing papers in "Mathematics of Computation in 1995"

A polyhedral method for solving sparse polynomial systems

Abstract: A continuation method is presented for computing all isolated roots of a semimixed sparse system of polynomial equations. We introduce mixed subdivisions of Newton polytopes, and we apply them to give a new proof and algorithm for Bernstein's theorem on the expected number of roots. This results in a numerical homotopy with the optimal number of paths to be followed. In this homotopy there is one starting system for each cell of the mixed subdivision, and the roots of these starting systems are obtained by an easy combinatorial construction.

Interior maximum-norm estimates for finite element methods, part II

Abstract: We consider bilinear forms A(.,.) connected with second-order elliptic problems and assume that for u h in a finite element space S h , we have A(u - u h , x) = F(X) for X in S h with local compact support. We give local estimates for u - U h in L∞ and W∞ 1 of the type local best approximation plus weak outside influences plus the local size of F .

Polynomial roots from companion matrix eigenvalues

Abstract: In classical linear algebra, the eigenvalues of a matrix are sometimes defined as the roots of the characteristic polynomial. An algorithm to compute the roots of a polynomial by computing the eigenvalues of the corresponding companion matrix turns the tables on the usual definition. We derive a first- order error analysis of this algorithm that sheds light on both the underlying geometry of the problem as well as the numerical error of the algorithm. Our error analysis expands on work by Van Dooren and Dewilde in that it states that the algorithm is backward normwise stable in a sense that must be defined carefully. Regarding the stronger concept of a small componentwise backward error, our analysis predicts a small such error on a test suite of eight random polynomials suggested by Toh and Trefethen. However, we construct examples for which a small componentwise relative backward error is neither predicted nor obtained in practice. We extend our results to polynomial matrices, where the result is essentially the same, but the geometry becomes more complicated.

Fast Gaussian elimination with partial pivoting for matrices with displacement structure

Abstract: Fast O(n 2 ) implementation of Gaussian elimination with partial pivoting is designed for matrices possessing Cauchy-like displacement structure. We show how Toeplitz-like, Toeplitz-plus-Hankel-like and Vandermonde-like matrices can be transformed into Cauchy-like matrices by using Discrete Fourier, Cosine or Sine Transform matrices. In particular this allows us to propose a new fast O(n 2 ) Toeplitz solver GKO, which incorporates partial pivoting. A large set of numerical examples showed that GKO demonstrated stable numerical behavior and can be recommended for solving linear systems, especially with nonsymmetric, indefinite and ill-conditioned positive definite Toeplitz matrices. It is also useful for block Toeplitz and mosaic Toeplitz ( Toeplitz-block ) matrices. The algorithms proposed in this paper suggest an attractive alternative to look-ahead approaches, where one has to jump over ill-conditioned leading submatrices, which in the worst case requires O(n 3 ) operations.

Computational Geometry in C.

Abstract: From the Publisher: This is the newly revised and expanded edition of a popular introduction to the design and implementation of geometry algorithms arising in areas such as computer graphics, robotics, and engineering design. The basic techniques used in computational geometry are all covered: polygon triangualtions, convex hulls, Voronoi diagrams, arrangements, geometric searching, and motion planning. The self-contained treatment presumes only an elementary knowledge of mathematics, but it reaches topics on the frontier of current research. Thus professional programmers will find it a useful tutorial.

Summation by parts, projections, and stability. II

Abstract: We have derived stability results for high-order finite difference approximations of mixed hyperbolic-parabolic initial-boundary value problems (IBVP). The results are obtained using summation by parts and a new way of representing general linear boundary conditions as an orthogonal projection. By slightly rearranging the analytic equations, we can prove strict stability for hyperbolic-parabolic IBVP. Furthermore, we generalize our technique so as to yield strict stability on curvilinear non-smooth domains in two space dimensions. Finally, we show how to incorporate inhomogeneous boundary data while retaining strict stability. Using the same procedure one can prove strict stability in higher dimensions as well.

Hermite interpolation by Pythagorean hodograph quintics

Abstract: The Pythagorean hodograph (PH) curves are polynomial parametric curves {x(t),y(t)} whose hodograph (derivative) components satisfy the Pythagorean condition x' 2 (t) + y' 2 (t) = σ 2 (t) for some polynomial σ(t). Thus, unlike polynomial curves in general, PH curves have arc lengths and offset curves that admit exact rational representations. The lowest-order PH curves that are sufficiently flexible for general interpolation/approximation problems are the quintics. While the PH quintics are capable of matching arbitrary first-order Hermite data, the solution procedure is not straightforward and furthermore does not yield a unique result-there are always four distinct interpolants (of which only one, in general, has acceptable shape characteristics). We show that formulating PH quintics as complex-valued functions of a real parameter leads to a compact Hermite interpolation algorithm and facilitates an identification of the good interpolant (in terms of minimizing the absolute rotation number). This algorithm establishes the PH quintics as a viable medium for the design or approximation of free-form curves, and allows a one-for-one substitution of PH quintics in lieu of the widely-used ordinary cubics.

On multivariate Lagrange interpolation

Abstract: Lagrange interpolation by polynomials in several variables is studied through a finite difference approach. We establish an interpolation formula analogous to that of Newton and a remainder formula, both of them in terms of finite differences. We prove that the finite difference admits an integral representation involving simplex spline functions. In particular, this provides a remainder formula for Lagrange interpolation of degree n of a function f, which is a sum of integrals of certain (n + 1)st directional derivatives of f multiplied by simplex spline functions. We also provide two algorithms for the computation of Lagrange interpolants which use only addition, scalar multiplication, and point evaluation of polynomials.

Numerical methods and applications

Abstract: Iterative Methods Based on Linearization for Nonlinear Elliptic Grid Systems, E.G. Dyakonov How to Solve Stiff Systems of Differential Equations by Explicit Methods, V.I. Lebedev On Numerical Methods of Solving Navier-Stokes Equations in "Velocity-Pressure" Variables, G.M. Kobelkov Stiff Systems of Ordinary Differential Equations, R.P. Fedorenko Convergence Rate Estimates of Finite Element Methods for Second-Order Hyperbolic Equations, A.A. Zlotnik Fictitious Domain Methods and Computation of Homogenized Properties of Composites, N.S. Bakhvalov and A.V. Knyazev

On the implementation of mixed methods as nonconforming methods for second-order elliptic problems

Abstract: In this paper we show that mixed finite element methods for a fairly general second-order elliptic problem with variable coefficients can be given a nonmixed formulation. (Lower-order terms are treated, so our results apply also to parabolic equations.) We define an approximation method by incorporating some projection operators within a standard Galerkin method, which we call a projection finite element method. It is shown that for a given mixed method, if the projection method's finite element space M h satisfies three conditions, then the two approximation methods are equivalent. These three conditions can be simplified for a single element in the case of mixed spaces possessing the usual vector projection operator. We then construct appropriate nonconforming spaces M h for the known triangular and rectangular elements. The lowest-order Raviart-Thomas mixed solution on rectangular finite elements in R 2 and R 3 , on simplices, or on prisms, is then implemented as a nonconforming method modified in a simple and computationally trivial manner. This new nonconforming solution is actually equivalent to a postprocessed version of the mixed solution. A rearrangement of the computation of the mixed method solution through this equivalence allows us to design simple and optimal-order multigrid methods for the solution of the linear system.

Balancing domain decomposition for mixed finite elements

Abstract: The rate of convergence of the Balancing Domain Decomposition method applied to the mixed finite element discretization of second order elliptic equations is analyzed. The Balancing Domain Decomposition method, introduced by Mandel in [24], is a substructuring method that involves at each iteration the solution of a local problem with Dirichlet data, a local problem with Neumann data, and a "coarse grid" problem to propagate information globally and to insure the consistency of the Neumann problems. It is shown that the condition number grows at worst like the logarithm squared of the ratio of the subdomain size to the element size, in both two and three dimensions and for elements of arbitrary order. The bounds are uniform with respect to coefficient jumps of arbitrary size between subdomains. The key component of our analysis is the demonstration of an equivalence between the norm induced by the bilinear form on the interface and the H^{1/2}-norm of an interplant of the boundary data. Computational results from a message passing parallel implementation on an INTEL-Delta machine demonstrate the scalability properties of the method and show almost optimal linear observed speed-up for up to 64 processors.

Accurate reconstructions of functions of finite regularity from truncated Fourier series expansions

Abstract: Kowledge of a truncated Fourier series expansion for a 2π-periodic function of finite regularity, which is assumed to be piecewise smooth in each period, is used to accurately reconstruct the corresponding function. An algebraic equation of degree M is constructed for the M singularity locations in each period for the function in question. The M coefficients in this algebraic equation are obtained by solving an algebraic system of M equations determined by the coefficients in the known truncated expansion. If discontinuities in the derivatives of the function are considered, in addition to discontinuities in the function itself, that algebraic system will be nonlinear with respect to the M unknown coefficients. The degree of the algebraic system will depend on the desired order of accuracy for the reconstruction, i.e., a higher degree will normally lead to a more accurate determination of the singularity locations. By solving an additional linear algebraic system for the jumps of the function and its derivatives up to the arbitrarily specified order at the calculated singularity locations, we are able to reconstruct the 2π-periodic function of finite regularity as the sum of a piecewise polynomial function and a function which is continuously differentiab1e up to the specified order

Analysis of Coppersmith's block Wiedemann algorithm for the parallel solution of sparse linear systems

Abstract: By using projections by a block of vectors in place of a single vector it is possible to parallelize the outer loop of iterative methods for solving sparse linear systems. We analyze such a scheme proposed by Coppersmith for Wiedemann's coordinate recurrence algorithm, which is based in part on the Krylov subspace approach. We prove that by use of certain randomizations on the input system the parallel speed up is roughly by the number of vectors in the blocks when using as many processors. Our analysis is valid for fields of entries that have sufficiently large cardinality. Our analysis also deals with an arising subproblem of solving a singular block Toeplitz system by use of the theory of Toeplitz-like matrices

Recent advances in wavelet analysis

Abstract: This book covers recent advances in wavelet analysis and applications in areas including wavelets on bounded intervals, wavelet decomposition of special interest to statisticians, wavelets approach to differential and integral equations, analysis of subdivision operators, and wavelets related to problems in engineering and physics.

On the Gibbs phenomenon IV: recovering exponential accuracy in a subinterval from a Gegenbauer partial sum of a piecewise analytic function

Abstract: We continue our investigation of overcoming Gibbs phenomenon, i.e., to obtain exponential accuracy at all points (including at the discontinuities themselves), from the knowledge of a spectral partial sum of a discontinuous but piecewise analytic function. We show that if we are given the first N Gegenbauer expansion coefficients, based on the Gegenbauer polynomials C(x) with the weight function (1-x^2)- for any constant 0, of an L_1 function f(x), we can construct an exponentially convergent approximation to the point values of f(x) in any sub-interval in which the function is analytic. The proof covers the cases of Chebyshev or Legendre partial sums, which are most common in applications.

Runge-Kutta approximation of quasi-linear parabolic equations

Abstract: We study the convergence properties of implicit Runge-Kutta meth- ods applied to time discretization of parabolic equations with time- or solution- dependent operator. Error bounds are derived in the energy norm. The con- vergence analysis uses two different approaches. The first, technically simpler approach relies on energy estimates and requires algebraic stability of the Runge- Kutta method. The second one is based on estimates for linear time-invariant equations and uses Fourier and perturbation techniques. It applies to A(9)- stable Runge-Kutta methods and yields the precise temporal order of conver- gence. This order is noninteger in general and depends on the type of boundary conditions.

Error estimators for nonconforming finite element approximations of the Stokes problem

Abstract: In this paper we define and analyze a posteriori error estimators for nonconforming approximations of the Stokes equations. We prove that these estimators are equivalent to an appropriate norm of the error. For the case of piecewise linear elements we define two estimators. Both of them are easy to compute, but the second is simpler because it can be computed using only the right-hand side and the approximate velocity. We show how the first estimator can be generalized to higher-order elements. Finally, we present several numerical examples in which one of our estimators is used for adaptive refinement.

Finite difference method for generalized Zakharov equations

Abstract: . A conservative difference scheme is presented for the initial-boundary value problem for generalized Zakharov equations. The scheme canbe implicit or semiexplicit depending on the choice of a parameter. On thebasis of a priori estimates and an inequality about norms, convergence of thedifference solution is proved in order 0(h2 + t2) , which is better than previous results. IntroductionThe Zakharov equations [20](1.1) iEt + Exx-NE = 0,(1.2) ^Ntt-{N+\E\2)xx = 0describe the propagation of Langmuir waves in plasmas. Here the complexunknown function E is the slowly varying envelope of the highly oscillatoryelectric field, and the unknown real function N denotes the fluctuation of the ion density about its equilibrium value.The global existence of a weak solution for the Zakharov equations in one dimension is proved in [19], and existence and uniqueness of a smooth solutionfor the equations are obtained provided smooth initial data are prescribed.Numerical methods for the Zakharov equations are studied only in [5, 9, 10,

Primitive divisors of Lucas and Lehmer sequences

Abstract: Stewart reduced the problem of determining all Lucas and Lehmer sequences whose nth element does not have a primitive divisor to solving certain Thue equations. Using the method of Tzanakis and de Weger for solving Thue equations, we determine such sequences for n ≤ 30. Further computations lead us to conjecture that, for n > 30, the nth element of such sequences always has a primitive divisor

A second-order accurate linearized difference scheme for the two-dimensional Cahn-Hilliard equation

Abstract: The Cahn-Hilliard equation is a nonlinear evolutionary equation that is of fourth order in space. In this paper a linearized finite difference scheme is derived by the method of reduction of order. It is proved that the scheme is uniquely solvable and convergent with the convergence rate of order two in a discrete L2-norm. The coefficient matrix of the difference system is symmetric and positive definite, so many well-known iterative methods (e.g. Gauss-Seidel, SOR) can be used to solve the system.

Pointwise a posteriori error estimates for elliptic problems on highly graded meshes

Abstract: Pointwise a posteriori error estimates are derived for linear second-order elliptic problems over general polygonal domains in 2D. The analysis carries over regardless of convexity, accounting even for slit domains, and applies to highly graded unstructured meshes as well. A key ingredient is a new asymptotic a priori estimate for regularized Green's functions. The estimators lead always to upper bounds for the error in the maximum norm, along with lower bounds under very mild regularity and nondegeneracy assumptions. The effect of both point and line singularities is examined. Three popular local estimators for the energy norm are shown to be equivalent, when suitably interpreted, to those introduced here

Computing the Hilbert transform on the real line

Abstract: We introduce a new method for computing the Hilbert transform on the real line. It is a collocation method, based on an expansion in rational eigenfunctions of the Hilbert transform operator, and implemented through the Fast Fourier Transform. An error analysis is given, and convergence rates for some simple classes of functions are established. Numerical tests indicate that the method compares favorably with existing methods

A posteriori error estimates for boundary element methods

Abstract: This paper deals with a general framework for a posteriori error estimates in boundary element methods which is specified for three examples, namely Symm's integral equation, an integral equation with a hypersingular operator, and a boundary integral equation for a transmission problem. Based on these estimates, an analog of Eriksson and Johnson's adaptive finite element method is proposed for the h -version of the Galerkin boundary element method for integral equations of the first kind. The efficiency of the approach is shown by numerical experiments which yield almost optimal convergence rates even in the presence of singularities. The construction of an adaptive mesh refinement procedure is of very high practical importance in the numerical analysis of partial differential equations, and we refer to the pioneering work of Babuska and Miller (3) and Eriksson and Johnson (10, 11). Whereas the main features ofadaptivity for finite element methods now seem to be visible, and the door is open to implementation (15), comparably little is known for boundary element methods for integral equations (see e.g. (1, 13, 18, 19,24)). In this paper a new adaptive «-version of the Galerkin discretization for the boundary element method is presented based on a posteriori error estimates. A general framework for these a posteriori error estimates is derived in §2, and three examples are discussed in §§3-5 involving the Dirichlet problem, the Neumann problem (for a closed and an open surface), and a transmission problem for the Laplacian, leading to integral equations with strongly elliptic pseudodifferential operators. Even for smooth data the lack of regularity of the solution near corners (of a polygonal domain Q) leads to poor solutions of the numerical schemes unless appropriate singular functions are incorporated in the trial space or a suitable mesh refinement is used. In practical problems such information is missing, e.g., when we have singular (or nearly singular) data and the main problem is how to balance a graded mesh refinement towards singularities and a global

Domain decomposition with nonmatching grids: augmented Lagrangian approach

Abstract: We propose and study a domain decomposition method which treats the constraint of displacement continuity at the interfaces by augmented La- grangian techniques and solves the resulting problem by a parallel version of the Peaceman-Rachford algorithm. We prove that this algorithm is equivalent to the fictitious overlapping method introduced by P.L. Lions. We also prove its linear convergence independently of the discretization step h , even if the finite element grids do not match at the interfaces. A new preconditioner using fictitious overlapping and well adapted to three-dimensional elasticity problems is also introduced and is validated on several numerical examples.

A finite element model for the time-dependent Joule heating problem

Abstract: We study a spatially semidiscrete and a completely discrete finite element model for a nonlinear system consisting of an elliptic and a parabolic partial differential equation describing the electric heating of a conducting body. We prove error bounds of optimal order under minimal regularity assumptions when the number of spatial variables d < 3. We establish the existence of solutions with the required regularity over arbitrarily long intervals of time when d < 2 .

The sharpness of Kuznetsov's O D x L 1 -error estimate for monotone difference schemes

Abstract: We derive a lower error bound for monotone difference schemes to the solution of the linear advection equation with BV initial data. A rigorous analysis shows that for any monotone difference scheme the lower L 1 -error bound is O(√Δx), where Δx is the spatial stepsize

Locking effects in the finite element approximation of plate models

Abstract: We analyze the robustness of various standard finite element schemes for a hierarchy of plate models and obtain asymptotic convergence estimates that are uniform in terms of the thickness d. We identify h version schemes that show locking, i.e., for which the asymptotic convergence rate deteriorates as d→0, and also show that the p version is free of locking. In order to isolate locking erects from boundary layer erects (which also arise as d →0), our analysis is carried out for the periodic case, which is free of boundary layers. We analyze in'detail the lowest model of the hierarchy, the well-known Reissner-Mindlin model, and show that the locking and robustness of finite element schemes for higher models of the hierarchy are essentially identical to the Riessner-Mindlin case

On orders of optimal normal basis generators

Abstract: In this paper we give some experimental results on the multiplicative orders of optimal normal basis generators in F 2 n over F 2 for n < 1200 whenever the complete factorization of 2 n - 1 is known. Our results show that a subclass of optimal normal basis generators always have high multiplicative orders, at least O((2 n - 1)/n), and are very often primitive. For a given optimal normal basis generator a in F 2 n and an arbitrary integer e, we show that α e can be computed in O(n. v(e)) bit operations, where v(e) is the number of 1's in the binary representation of e.

Cubature formulas and modern analysis : an introduction

Abstract: BASIC CONCEPTS AND FORMULATIONS - Concepts in Linear Algebra - Point Latices - Some Functional Spaces - Generalized Derivatives and Pdm@ve Functions - Densi@ of Fin4e Fundons - P@ncipal Embedding Theorems for We@hted Spaces - Fundons of a Discrete kgument - Generalized Functions - Fou@er Transformabon of Generalized Functions - Periodic Functions and Generalized Functions - Generalized Spherical Harmonics - POLYHARMONIC EQUATION - Green's Formula - Fundamental Solution of a Polyharmonic Equation - Differential Prope@ies of Solutons of a Polyharmonic Equation - Behavior of Polyharmonic Functions in the Neighborhood of Infinity - Almansi's Theorem and Kelvin Transformation - P@ncipal Solutions of a Polyharmonic Equation - Expansion of Polyharmonic Functions in Principal Solutions - Polyharmonic Functons from W,"@ in the Neighborhood of Intini@ - Boundary Value Problems for a Polyharmonic Equation in a Bounded Domain - Kelvin of bodies adrature s revised of these s topics Problem in An Infinite Domain - Exterior Variational Problem for a Polyharmonic Equation - Extension of a Funct ion from the Region i2 on R' with the Least Norm - Values of Functions from Wpm) at the Lattice Points Explicit Method of Regularization of Divergent Integrals - TWO SIMPLE PROBLEMS OF THE THEORY OF COMPUTATIONS - Interpolation Construction oi Cubature Formulas - Functional Formulation of the Problems, Extremal Function of a Cubature Formula - Error Functional in W'2 n, (RI Square of the Norm of the Error Functional - Deviation of Error of a Cubature Formula from the Optimal - ORDER OF CONVERGENCE OF CUBATURE FORMULAS - Lower Estimate of the Norm of the Error Functional Approximate Upper Estimate of the Norm of the Error Functional CUBATURE FORMULAS CONSIDERING A REGULAR BOUNDARY LAYER Formulas for Periodic Functions - Norm of the Error Functional for Periodic Functions - Composition of Formulas with Small Suppoas - Error for Finite Functions - Construction of Formulas with a Regular Boundary Layer - Norm of the Error Functional of Cubature Formulas with a Regular Boundary Layer in the Space L(')(R) - Norm of the Error of Formulas with Regular Boundary 2 Layer in L(al(Q) - OPTIMAL FORMULAS - Formulation of the Problem on 2 Optimal Coefficients - Fourier Transformation of a Discrete Potential - Prope@ies of the Operator D(m@[pl' - Discrete Analog of a Polyharmonic h" Operator - Optimal Coefficients of One-Dimensional Formulas - CONVERGENCE OF CUBATURE FORMULAS IN VARIOUS CLASSES AND DIFFERENT FUNCTIONS - Functional Class (D(PIA) - Functional Class T(plo) - Cubature Formulas for Infinitely Differentiable Functions - Convergence oi Cubature Formulas for an Arbitrary Function (P(x) c @ll - CUBATURE FORMULAS FOR RATIONAL POLYHEDRA - Convex Polyhedra Euler's Formula - Rational Polyhedra - Structure of Formulas for Rational Polyhedra - Cubature Formulas for a Polyhedron and Its Solid Angles - Formulas with a Formal Boundary Layer