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Showing papers in "Mathematics of Computation in 1998"


Journal ArticleDOI
TL;DR: A class of high order TVD (total variation diminishing) Runge-Kutta time discretization initialized in Shu& Osher (1988), suitable for solving hyperbolic conservation laws with stable spatial discretizations is explored, verifying the claim that TVD runge-kutta methods are important for such applications.
Abstract: In this paper we further explore a class of high order TVD (total variation diminishing) Runge-Kutta time discretization initialized in Shu& Osher (1988), suitable for solving hyperbolic conservation laws with stable spatial discretizations. We illustrate with numerical examples that non-TVD but linearly stable Runge-Kutta time discretization can generate oscillations even for TVD (total variation diminishing) spatial discretization, verifying the claim that TVD Runge-Kutta methods are important for such applications. We then explore the issue of optimal TVD Runge-Kutta methods for second, third and fourth order, and for low storage Runge-Kutta methods.

2,146 citations


Journal ArticleDOI
David Levin1
TL;DR: The interpolation approximation in R d is shown to be a C∞ function, and an approximation order result is proven for quasi-uniform sets of data points.
Abstract: A general method for near-best approximations to functionals on R d , using scattered-data information is discussed. The method is actually the moving least-squares method, presented by the Backus-Gilbert approach. It is shown that the method works very well for interpolation, smoothing and derivatives' approximations. For the interpolation problem this approach gives Mclain's method. The method is near-best in the sense that the local error is bounded in terms of the error of a local best polynomial approximation. The interpolation approximation in R d is shown to be a C∞ function, and an approximation order result is proven for quasi-uniform sets of data points.

778 citations


Journal ArticleDOI
TL;DR: An error bound for multidimensional quadrature is derived that includes the Koksma-Hlawka inequality as a special case and includes as special cases the L p -star discrepancy and P α that arises in the study of lattice rules.
Abstract: An error bound for multidimensional quadrature is derived that includes the Koksma-Hlawka inequality as a special case. This error bound takes the form of a product of two terms. One term, which depends only on the integrand, is defined as a generalized variation. The other term, which depends only on the quadrature rule, is defined as a generalized discrepancy. The generalized discrepancy is a figure of merit for quadrature rules and includes as special cases the L p -star discrepancy and P α that arises in the study of lattice rules.

693 citations


Journal ArticleDOI
TL;DR: It is shown that for box constrained variational inequalities if the involved function is P- uniform, the iteration sequence generated by the smoothing Newton method will converge to the unique solution of the problem globally and superlinearly (quadratically).
Abstract: The smoothing Newton method for solving a system of nonsmooth equations F(x) = 0, which may arise from the nonlinear complementarity problem, the variational inequality problem or other problems, can be regarded as a variant of the smoothing method. At the kth step, the nonsmooth function F is approximated by a smooth function f(.,∈ κ ), and the derivative of f(.,∈ κ ) at x k is used as the Newton iterative matrix. The merits of smoothing methods and smoothing Newton methods are global convergence and convenience in handling. In this paper, we show that the smoothing Newton method is also superlinearly convergent if F is semismooth at the solution and f satisfies a Jacobian consistency property. We show that most common smooth functions, such as the Gabriel-More function, have this property. As an application, we show that for box constrained variational inequalities if the involved function is P- uniform, the iteration sequence generated by the smoothing Newton method will converge to the unique solution of the problem globally and superlinearly (quadratically).

288 citations


Journal ArticleDOI
TL;DR: It is shown that to solve the discrete log problem in a subgroup of order p of an elliptic curve over the finite field of characteristic p one needs O(ln p) operations in this field.
Abstract: We show that to solve the discrete log problem in a subgroup of order p of an elliptic curve over the finite field of characteristic p one needs O(ln p) operations in this field.

239 citations


Journal ArticleDOI
TL;DR: The approximation properties of multivariate refinable functions are investigated and a characterization for the approximation order provided by a refinable function in terms of the order of the sum rules satisfied by the refinement mask is given.
Abstract: Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the approximation properties of multivariate refinable functions. We give a characterization for the approximation order provided by a refinable function in terms of the order of the sum rules satisfied by the refinement mask. We connect the approximation properties of a refinable function with the spectral properties of the corresponding subdivision and transition operators. Finally, we demonstrate that a refinable function in W k-1 1 (R s ) provides approximation order k.

154 citations


Journal ArticleDOI
TL;DR: The convergence of the discontinuous Galerkin method for the nonlinear (cubic) Schrodinger equation is analyzed and the existence of the resulting approximations and optimal order error estimates in L∞(L 2 ).
Abstract: The convergence of the discontinuous Galerkin method for the nonlinear (cubic) Schrodinger equation is analyzed in this paper. We show the existence of the resulting approximations and prove optimal order error estimates in L∞(L 2 ). These estimates are valid under weak restrictions on the space-time mesh.

134 citations


Journal ArticleDOI
TL;DR: New probabilistic algorithms are presented for factoring univariate polynomials over finite fields, using fast matrix multiplication techniques and the new baby step/giant step techniques.
Abstract: New probabilistic algorithms are presented for factoring univariate polynomials over finite fields. The algorithms factor a polynomial of degree n over a finite field of constant cardinality in time O(n 1.815 ). Previous algorithms required time Θ(n 2+o(1) ). The new algorithms rely on fast matrix multiplication techniques. More generally, to factor a polynomial of degree n over the finite field F q with q elements, the algorithms use O(n 1.815 log q) arithmetic operations in F q . The new baby step/giant step techniques used in our algorithms also yield new fast practical algorithms at super-quadratic asymptotic running time, and subquadratic-time methods for manipulating normal bases of finite fields.

130 citations


Journal ArticleDOI
A. H. Schatz1
TL;DR: These sharpen known quasi-optimal L∞ and W 1∼ ∞ estimates for the error on irregular quasi-uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution u, noting that in general the higher order finite element spaces exhibit more local behavior than lower order spaces.
Abstract: This part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in R N . In a sense to be discussed below these sharpen known quasi-optimal L∞ and W 1∼ ∞ estimates for the error on irregular quasi-uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution u. We note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces. As a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities. These expansion inequalities are valid for large classes of finite elements defined on irregular grids in R N and have applications to superconvergence and extrapolation and a posteriori estimates. Part II of this series will contain local estimates applicable to non-smooth problems.

116 citations


Journal ArticleDOI
TL;DR: The analysis presented allows variable time steps which, as will be shown, call then efficiently be selected to match singularities in the solution induced by singularity in the kernel of the memory term or by nonsmooth initial data.
Abstract: The numerical solution of a parabolic equation with memory is considered. The equation is first discretized in time by means of the discontinuous Galerkin method with piecewise constant or piecewise linear approximating functions. The analysis presented allows variable time steps which, as will be shown, call then efficiently be selected to match singularities in the solution induced by singularities in the kernel of the memory term or by nonsmooth initial data. The combination with finite element discretizat in space is also studied.

112 citations


Journal ArticleDOI
TL;DR: A new fast algorithm for the computation of the matrix-vector product Pa in O(N log 2 N) arithmetical operations is presented, which divides into a fast transform which replaces Pa with C N+1 ā and a subsequent fast cosine transform.
Abstract: Consider the Vandermonde-like matrix P:= (P k (cos jπ/N)) j,k=0 N , where the polynomials P k satisfy a three-term recurrence relation. If P k are the Chebyshev polynomials T k , then P coincides with C N+1 := (cos jkπ/N) j,k=0 N . This paper presents a new fast algorithm for the computation of the matrix-vector product Pa in O(N log 2 N) arithmetical operations. The algorithm divides into a fast transform which replaces Pa with C N+1 ā and a subsequent fast cosine transform. The first and central part of the algorithm is realized by a straightforward cascade summation based on properties of associated polynomials and by fast polynomial multiplications. Numerical tests demonstrate that our fast polynomial transform realizes Pa with almost the same precision as the Clenshaw algorithm, but is much faster for N ≥ 128.

Journal ArticleDOI
TL;DR: The theory of fiberization is applied to yield compactly supported tight affine frames (wavelets) in L 2 (R d) from box splines that exhibit a wealth of symmetries, and have a relatively small support.
Abstract: The theory of fiberization is applied to yield compactly supported tight affine frames (wavelets) in L 2 (R d ) from box splines. The wavelets obtained are smooth piecewise-polynomials on a simple mesh; furthermore, they exhibit a wealth of symmetries, and have a relatively small support. The number of mother wavelets, however, increases with the increase of the required smoothness. Two bivariate constructions, of potential practical value, are highlighted. In both, the wavelets are derived from four-direction mesh box splines that are refinable with respect to the dilation matrix ( 1 1 1 -1 ).

Journal ArticleDOI
TL;DR: The main result characterizes the convergence of a subdivision scheme associated with the mask a in terms of the joint spectral radius of two finite matrices derived from the mask, which leads to a class of continuous orthogonal double wavelets with symmetry.
Abstract: We consider solutions of a system of refinement equations written in the form formula math where the vector of functions Φ = (Φ 1 ,...,Φ r ) T is in (L p (R)) r and a is a finitely supported sequence of r × r matrices called the refinement mask. Associated with the mask a is a linear operator Q a defined on (L p (R)) r by Q a f:= Σ α ∈ z a(α)f(2.-α). This paper is concerned with the convergence of the subdivision scheme associated with a, i.e., the convergence of the sequence (Q a n f) n=1,2... in the L p -norm. Our main result characterizes the convergence of a subdivision scheme associated with the mask a in terms of the joint spectral radius of two finite matrices derived from the mask. Along the way, properties of the joint spectral radius and its relation to the subdivision scheme are discussed. In particular, the L 2 -convergence of the subdivision scheme is characterized in terms of the spectral radius of the transition operator restricted to a certain invariant subspace. We analyze convergence of the subdivision scheme explicitly for several interesting classes of vector refinement equations. Finally, the theory of vector subdivision schemes is used to characterize orthonormality of multiple refinable functions. This leads us to construct a class of continuous orthogonal double wavelets with symmetry.

Journal ArticleDOI
TL;DR: By splitting a given singular function into a relatively smooth part and a specially structured singular part, it is shown how the traditional Fourier method can be modified to give numerical methods of high order for calculating derivatives and integrals.
Abstract: By splitting a given singular function into a relatively smooth part and a specially structured singular part, it is shown how the traditional Fourier method can be modified to give numerical methods of high order for calculating derivatives and integrals. Singular functions with various types of singularities of importance in applications are considered. Relations between the discrete and the continuous Fourier series for the singular functions are established. Of particular interest are piecewise smooth functions, for which various important applications are indicated, and for which numerous numerical results are presented.

Journal ArticleDOI
TL;DR: The results are applied to the Kuramoto-Sivashinsky and the Cahn-Hilliard equations in one dimension, as well as to a class of reaction diffusion equations in R v, v = 2,3.
Abstract: We approximate the solution of initial boundary value problems for nonlinear parabolic equations. In space we discretize by finite element methods. The discretization in time is based on linear multistep schemes. One part of the equation is discretized implicitly and the other explicitly. The resulting schemes are stable, consistent and very efficient, since their implementation requires at each time step the solution of a linear system with the same matrix for all time levels. We derive optimal order error estimates. The abstract results are applied to the Kuramoto-Sivashinsky and the Cahn-Hilliard equations in one dimension, as well as to a class of reaction diffusion equations in R v , v = 2,3.

Journal ArticleDOI
TL;DR: A residual a posteriori error estimator for space-time finite element discretizations of quasilinear parabolic pdes gives global upper and local lower bounds on the error of the numerical solution.
Abstract: Using the abstract framework of [9] we analyze a residual a posteriori error estimator for space-time finite element discretizations of quasilinear parabolic pdes. The estimator gives global upper and local lower bounds on the error of the numerical solution. The finite element discretizations in particular cover the so-called 0-scheme, which includes the implicit and explicit Euler methods and the Crank-Nicholson scheme.

Journal ArticleDOI
TL;DR: It is proven that for some explicit parameter values the Lorenz equations exhibit chaotic dynamics.
Abstract: Details of a new technique for obtaining rigorous results concerning the global dynamics of nonlinear systems is described. The technique combines abstract existence results based on the Conley index theory with rigorous computer assisted computations. As an application of these methods it is proven that for some explicit parameter values the Lorenz equations exhibit chaotic dynamics.

Journal ArticleDOI
TL;DR: This work proposes to introduce first a simple smoothing change of variable, and then to apply classical numerical methods such as product-integration and collocation based on global polynomial approximations to solve one-dimensional linear weakly singular integral equations on bounded intervals.
Abstract: To solve one-dimensional linear weakly singular integral equations on bounded intervals, with input functions which may be smooth or not, we propose to introduce first a simple smoothing change of variable, and then to apply classical numerical methods such as product-integration and collocation based on global polynomial approximations. The advantage of this approach is that the order of the methods can be arbitrarily high and that the associated linear systems one has to solve are very well-conditioned.

Journal ArticleDOI
TL;DR: It is found that a simple modification can improve the performance of the Newton iteration dramatically and the convergence is quadratic if the Frechet derivative is invertible at the solution.
Abstract: When Newton's method is applied to find the maximal symmetric solution of an algebraic Riccati equation, convergence can be guaranteed under moderate conditions. In particular, the initial guess need not be close to the solution. The convergence is quadratic if the Frechet derivative is invertible at the solution. In this paper we examine the behaviour of the Newton iteration when the derivative is not invertible at the solution. We find that a simple modification can improve the performance of the Newton iteration dramatically.

Journal ArticleDOI
TL;DR: It is shown how the regularization of the backward parabolic problem leads very naturally to a reformulation of the problem as a second-kind Fredholm integral equation, which can be very easily approximated using methods previously developed by Ames and Epperson.
Abstract: We consider numerical methods for a quasi-boundary value regularization of the backward parabolic problem given by {u t +Au=0, 0

Journal ArticleDOI
TL;DR: This paper contains error estimates for covolume discretizations of Maxwell's equations in three space dimensions and some of the unstructured mesh results are specialized to rectangular meshes, both uniform and nonuniform.
Abstract: This paper contains error estimates for covolume discretizations of Maxwell's equations in three space dimensions. Several estimates are proved. First, an estimate for a semi-discrete scheme is given. Second, the estimate is extended to cover the classical interlaced time marching technique. Third, some of our unstructured mesh results are specialized to rectangular meshes, both uniform and nonuniform. By means of some additional analysis it is shown that the spatial convergence rate is one order higher than for the unstructured case.

Journal ArticleDOI
TL;DR: In this paper, the Trotter-Kato theorem is used for approximation of linear C 0 -semigroups which provide very useful framework when convergence of numerical approximations to solutions of PDEs are studied.
Abstract: We present formulations of the Trotter-Kato theorem for approximation of linear C 0 -semigroups which provide very useful framework when convergence of numerical approximations to solutions of PDEs are studied. Applicability of our results is demonstrated using a first order hyperbolic equation, a wave equation and Stokes' equation as illustrative examples.

Journal ArticleDOI
TL;DR: All polynomials with degree at most 24 and Mahler measure less than 1.3 are determined, and none has measure smaller than that of Lehmer's degree 10 polynomial.
Abstract: We describe several searches for polynomials with integer coefficients and small Mahler measure. We describe the algorithm used to test Mahler measures. We determine all polynomials with degree at most 24 and Mahler measure less than 1.3, test all reciprocal and antireciprocal polynomials with height 1 and degree at most 40, and check certain sparse polynomials with height 1 and degree as large as 181. We find a new limit point of Mahler measures near 1.309, four new Salem numbers less than 1.3, and many new polynomials with small Mahler measure. None has measure smaller than that of Lehmer's degree 10 polynomial.

Journal ArticleDOI
TL;DR: An algorithm to compute approximate kth roots and it is proved, using Loxton's theorem on multiple linear forms in logarithms, that this perfect-power decomposition algorithm runs in time (log n) 1+o(1) .
Abstract: This paper (1) gives complete details of an algorithm to compute approximate kth roots; (2) uses this in an algorithm that, given an integer n > 1, either writes n as a perfect power or proves that n is not a perfect power; (3) proves, using Loxton's theorem on multiple linear forms in logarithms, that this perfect-power decomposition algorithm runs in time (log n) 1+o(1) .

Journal ArticleDOI
TL;DR: This paper replaces the matrix multiplications in Clausen's algorithm with sums indexed by combinatorial objects that generalize Young tableaux, and writes the result in a form similar to Horner's rule.
Abstract: This paper introduces new techniques for the efficient computation of Fourier transforms on symmetric groups and their homogeneous spaces. We replace the matrix multiplications in Clausen's algorithm with sums indexed by combinatorial objects that generalize Young tableaux, and write the result in a form similar to Horner's rule. The algorithm we obtain computes the Fourier transform of a function on S n in no more than 3/4n(n-1)|S n | multiplications and the same number of additions. Analysis of our algorithm leads to several combinatorial problems that generalize path counting. We prove corresponding results for inverse transforms and transforms on homogeneous spaces.

Journal ArticleDOI
TL;DR: This paper deals with a new class of parallel asynchronous iterative algorithms for the solution of nonlinear systems of equations with the possibility of flexible commmunication between processors.
Abstract: This paper deals with a new class of parallel asynchronous iterative algorithms for the solution of nonlinear systems of equations. The main feature of the new class of methods preseted here is the possibility of flexible commmunication between processors. In particular partial updates can be exchanged. Approximation of the asssociated fixed point mapping is also considered. A detailed convergengence study is presented. A connection with the Schwarz alternating method is made for nonlinear boundary value problems. Computational results on a shared memory multiprocessotr IBM 3090 are presented.

Journal ArticleDOI
TL;DR: A posteriori error estimates are derived for the approximation of linear elliptic problems on domains with piecewise smooth boundary on a Finite Element mesh, whose boundary vertices are located on the boundary of the continuous problem.
Abstract: We derive a posteriori error estimates for the approximation of linear elliptic problems on domains with piecewise smooth boundary. The numerical solution is assumed to be defined on a Finite Element mesh, whose boundary vertices are located on the boundary of the continuous problem. No assumption is made on a geometrically fitting shape. A posteriori error estimates are given in the energy norm and the L 2 -norm, and efficiency of the adaptive algorithm is proved in the case of a saturated boundary approximation. Furthermore, a strategy is presented to compute the effect of the non-discretized part of the domain on the error starting from a coarse mesh. This especially implies that parts of the domain, where the measured error is small, stay non-discretized. The presented algorithm includes a stable path following to supply a sufficient polygonal approximation of the boundary, the reliable computation of the a posteriori estimates and a mesh adaptation based on Delaunay techniques. Numerical examples illustrate that errors outside the initial discretization will be detected.

Journal ArticleDOI
TL;DR: This work presents a general elementary approach, based on an idea of Cohn and the theory of the Pell equation, that solves many Diophantine equations, which arise when looking for integer points on an elliptic curve with a rational 2-torsion point.
Abstract: Consider the system of Diophantine equations x 2 - ay 2 = b, P(x, y) = z 2 , where P is a given integer polynomial. Historically, such systems have been analyzed by using Baker's method to produce an upper bound on the integer solutions. We present a general elementary approach, based on an idea of Cohn and the theory of the Pell equation, that solves many such systems. We apply the approach to the cases P(x, y) = cy 2 + d and P(x, y) = cx + d, which arise when looking for integer points on an elliptic curve with a rational 2-torsion point.

Journal ArticleDOI
TL;DR: A new algorithm is presented for computing the structure of a finite abelian group, which has to store only a fixed, small number of group elements, independent of the group order, and it is proved that the expected run time is O(√n) and the O-constants are determined.
Abstract: We present a new algorithm for computing the structure of a finite abelian group, which has to store only a fixed, small number of group elements, independent of the group order. We estimate the computational complexity by counting the group operations such as multiplications and equality checks. Under some plausible assumptions, we prove that the expected run time is O(√n) (with n denoting the group order), and we explicitly determine the O-constants. We implemented our algorithm for ideal class groups of imaginary quadratic orders and present experimental results.