Showing papers in "Mathematics of Computation in 2001"
TL;DR: The main result of the paper is the construction of an adaptive scheme which produces an approximation to u with error O(N -s ) in the energy norm, whenever such a rate is possible by N-term approximation.
Abstract: This paper is concerned with the construction and analysis of wavelet-based adaptive algorithms for the numerical solution of elliptic equations. These algorithms approximate the solution u of the equation by a linear combination of N wavelets, Therefore, a benchmark for their performance is provided by the rate of best approximation to u by an arbitrary linear combination of N wavelets (so called N-term approximation), which would be obtained by keeping the N largest wavelet coefficients of the real solution (which of course is unknown). The main result of the paper is the construction of an adaptive scheme which produces an approximation to u with error O(N -s ) in the energy norm, whenever such a rate is possible by N-term approximation. The range of s > 0 for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator. Moreover, it is shown that the number of arithmetic operations needed to compute the approximate solution stays proportional to N. The adaptive algorithm applies to a wide class of elliptic problems and wavelet bases. The analysis in this paper puts forward new techniques for treating elliptic problems as well as the linear systems of equations that arise from the wavelet discretization.
488 citations
TL;DR: A two-grid discretization scheme is proposed for solving eigenvalue problems, including both partial differential equations and integral equations, and maintains an asymptotically optimal accuracy.
Abstract: A two-grid discretization scheme is proposed for solving eigenvalue problems, including both partial differential equations and integral equations. With this new scheme, the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a much coarser grid, and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy.
251 citations
TL;DR: Here, quadrature formulas are obtained that are exact for spherical harmonics of a fixed order, have nonnegative weights, and are based on function values at scattered sites for the unit sphere embedded in R q.
Abstract: Geodetic and meteorological data, collected via satellites for example, are genuinely scattered and not confined to any special set of points. Even so, known quadrature formulas used in numerically computing integrals involving such data have had restrictions either on the sites (points) used or, more significantly, on the number of sites required. Here, for the unit sphere embedded in R q , we obtain quadrature formulas that are exact for spherical harmonics of a fixed order, have nonnegative weights, and are based on function values at scattered sites. To be exact, these formulas require only a number of sites comparable to the dimension of the space. As a part of the proof, we derive L 1 -Marcinkiewicz-Zygmund inequalities for such sites.
213 citations
TL;DR: The extent to which this order of convergence can be improved is investigated, and it is found that better approximations are possible for the case of additive noise, but for multiplicative noise it is shown that no improvements are possible.
Abstract: We consider the numerical solution of the stochastic partial differential equation ∂u/∂t = ∂ 2 u/∂x 2 + σ(u)W(x,t), where W is space-time white noise, using finite differences. For this equation Gyongy has obtained an estimate of the rate of convergence for a simple scheme, based on integrals of W over a rectangular grid. We investigate the extent to which this order of convergence can be improved, and find that better approximations are possible for the case of additive noise (σ(u) = 1) if we wish to estimate space averages of the solution rather than pointwise estimates, or if we are permitted to generate other functionals of the noise. But for multiplicative noise (σ(u) = u) we show that no such improvements are possible.
181 citations
TL;DR: If the active constraints satisfy an independence condition and the Lagrangian satisfies a coercivity condition, then locally there exists a solution to the Euler discretization, and the error is bounded by a constant times the mesh size.
Abstract: We analyze the Euler approximation to a state constrained control problem. We show that if the active constraints satisfy an independence condition and the Lagrangian satisfies a coercivity condition, then locally there exists a solution to the Euler discretization, and the error is bounded by a constant times the mesh size. The proof couples recent stability results for state constrained control problems with results established here on discrete-time regularity. The analysis utilizes mappings of the discrete variables into continuous spaces where classical finite element estimates can be invoked.
152 citations
TL;DR: This paper studies a new, larger class of smooth radial functions of compact support which contains other compactly supported ones that were proposed earlier in the literature.
Abstract: Radial basis functions are well-known and successful tools for the interpolation of data in many dimensions. Several radial basis functions of compact support that give rise to nonsingular interpolation problems have been proposed, and in this paper we study a new, larger class of smooth radial functions of compact support which contains other compactly supported ones that were proposed earlier in the literature.
149 citations
TL;DR: A new theoretical Evans function condition is used as the basis of a numerical test of viscous shock wave stability, and the need to incorporate features from the analytic Evans function theory for purposes of numerical stability is found.
Abstract: A new theoretical Evans function condition is used as the basis of a numerical test of viscous shock wave stability. Accuracy of the method is demonstrated through comparison against exact solutions, a convergence study, and evaluation of approximate error equations. Robustness is demonstrated by applying the method to waves for which no current analytic results apply (highly nonlinear waves from the cubic model and strong shocks from gas dynamics). An interesting aspect of the analysis is the need to incorporate features from the analytic Evans function theory for purposes of numerical stability. For example, we find it necessary, for numerical accuracy, to solve ODEs on the space of wedge products.
144 citations
TL;DR: In this article, two different approaches for numeri- cal differentiation are considered based on a regularized Volterra equation and disretized version of the regularized VOLTERRA equation.
Abstract: Based on a regularized Volterra equation, two different approaches for numeri- cal differentiation are considered. The first approach consists of solving a regularized Volterra equation while the second approach is based on solving a disretized version of the regularized Volterra equation. Numerical experiments show that these methods are efficient and compete fa- vorably with the variational regularization method for stable calculating the derivatives of noisy functions.
144 citations
TL;DR: In this article, a new integer relation algorithm designed for parallel computer systems is presented, but as a bonus it also gives superior results on single processor systems, together with performance results on a parallel computer system.
Abstract: Let {x1,x2,···,xn} be a vector of real numbers. An integer relation algorithm is a computational scheme to find the n integers ak, if they exist, such that a1x1 + a2x2 + ··· + anxn = 0. In the past few years, integer relation algorithms have been utilized to discover new results in mathematics and physics. Existing programs for this purpose require very large amounts of computer time, due in part to the requirement for multiprecision arithmetic, yet are poorly suited for parallel processing. This paper presents a new integer relation algorithm designed for parallel computer systems, but as a bonus it also gives superior results on single processor systems. Single- and multi-level implementations of this algorithm are described, together with performance results on a parallel computer system. Several applications of these programs are discussed, including some new results in number theory, quantum field theory and chaos theory.
144 citations
TL;DR: It is shown that a class of walks that lead to the same performance as expected in the random case are introduced, thus making Pollard's rho method for prime group orders about 20% faster than before.
Abstract: We consider Pollard's rho method for discrete logarithm computation. Usually, in the analysis of its running time the assumption is made that a random walk in the underlying group is simulated. We show that this assumption does not hold for the walk originally suggested by Pollard: its performance is worse than in the random case. We study alternative walks that can be efficiently applied to compute discrete logarithms. We introduce a class of walks that lead to the same performance as expected in the random case. We show that this holds for arbitrarily large prime group orders, thus making Pollard's rho method for prime group orders about 20% faster than before.
133 citations
TL;DR: It is proved that a discrete maximum principle holds for continuous piecewise linear finite element approximations for the Poisson equation with the Dirichlet boundary condition also under a condition of the existence of some obtuse internal angles between faces of terahedra of triangulations of a given space domain.
Abstract: We prove that a discrete maximum principle holds for continuous piecewise linear finite element approximations for the Poisson equation with the Dirichlet boundary condition also under a condition of the existence of some obtuse internal angles between faces of terahedra of triangulations of a given space domain. This result represents a weakened form of the acute type condition for the three-dimensional case.
TL;DR: It is shown that under a natural hypothesis - called the uniform separation condition - the Ritz pairs (N,X) converge to the eigenpair (L,X), and certain refined Ritz vectors whose convergence is guaranteed, even when the uniformseparation condition is not satisfied.
Abstract: This paper concerns the Rayleigh- Ritz method for computing an approximation to an eigenspace X of a general matrix A from a subspace W that contains an approximation to X. The method produces a pair (N,X) that purports to approximate a pair (L,X), where X is a basis for X and AX = XL. In this paper we consider the convergence of (N,X) as the sine e of the angle between X and W approaches zero. It is shown that under a natural hypothesis - called the uniform separation condition - the Ritz pairs (N,X) converge to the eigenpair (L,X). When one is concerned with eigenvalues and eigenvectors, one can compute certain refined Ritz vectors whose convergence is guaranteed, even when the uniform separation condition is not satisfied. An attractive feature of the analysis is that it does not assume that A has distinct eigenvalues or is diagonalizable.
TL;DR: It is proposed to use a nonconforming H 2 -element which is H 1 -conforming as an alternative to the Morley element and it is shown that the new finite element method converges in the energy norm uniformly in the perturbation parameter.
Abstract: Finite element methods for some elliptic fourth order singular perturbation problems are discussed. We show that if such problems are discretized by the nonconforming Morley method, in a regime close to second order elliptic equations, then the error deteriorates. In fact, a counterexample is given to show that the Morley method diverges for the reduced second order equation. As an alternative to the Morley element we propose to use a nonconforming H 2 -element which is H 1 -conforming. We show that the new finite element method converges in the energy norm uniformly in the perturbation parameter.
TL;DR: In this article, a cyclic reduction based quadratically convergent algorithm is proposed to solve the extreme solutions of the matrix equations X + A*X -1 A = Q and X - A* X -1A = Q.
Abstract: We propose a new quadratically convergent algorithm, having a low computational cost per step and good numerical stability properties, which allows the simultaneous approximation of the extreme solutions of the matrix equations X + A*X -1 A = Q and X - A*X -1 A = Q. The algorithm is based on the cyclic reduction method.
TL;DR: This is the first proof of convergence of the eigenvalue problem for general edge elements, and it extends and unifies the theory for both problems, based on the discrete compactness property of edge element due to Kikuchi.
Abstract: We analyze the use of edge finite element methods to approximate Maxwell's equations in a bounded cavity. Using the theory of collectively compact operators, we prove h-convergence for the source and eigenvalue problems. This is the first proof of convergence of the eigenvalue problem for general edge elements, and it extends and unifies the theory for both problems. The convergence results are based on the discrete compactness property of edge element due to Kikuchi. We extend the original work of Kikuchi by proving that edge elements of all orders possess this property.
TL;DR: It is shown that a regular solution is guaranteed to exist for sufficiently small stepsize Δ, provided that certain technical assumptions are satisfied, and any given curve, which is assumed to be G 2 and to consist of analytical segments can approximately be converted into polynomial PH form.
Abstract: Polynomial Pythagorean hodograph (PH) curves form a remarkable subclass of polynomial parametric curves; they are distinguished by having a polynomial arc length function and rational offsets (parallel curves). Many related references can be found in the article by Farouki and Neff on C 1 Hermite interpolation with PH quintics. We extend the C 1 Hermite interpolation scheme by taking additional curvature information at the segment boundaries into account. As a result we obtain a new construction of curvature continuous polynomial PH spline curves. We discuss Hermite interpolation of G 2 [C 1 ] boundary data (points, first derivatives, and curvatures) with PH curves of degree 7. It is shown that up to eight possible solutions can be found by computing the roots of two quartic polynomials, With the help of the canonical Taylor expansion of planar curves, we analyze the existence and shape of the solutions. More precisely, for Hermite data which are taken from an analytical curve, we study the behaviour of the solutions for decreasing stepsize Δ. It is shown that a regular solution is guaranteed to exist for sufficiently small stepsize Δ, provided that certain technical assumptions are satisfied. Moreover, this solution matches the shape of the original curve; the approximation order is 6. As a consequence, any given curve, which is assumed to be G 2 (curvature continuous) and to consist of analytical segments can approximately be converted into polynomial PH form. The latter assumption is automatically satisfied by the standard curve representations of Computer Aided Geometric Design, such as Bezier or B-spline curves. The conversion procedure acts locally, without any need for solving a global system of equations, It produces G 2 polynomial PH spline curves of degree 7.
TL;DR: A class of a posteriori estimators is studied for the error in the maximum-norm of the gradient on single elements when the finite element method is used to approximate solutions of second order elliptic problems.
Abstract: A class of a posteriori estimators is studied for the error in the maximum-norm of the gradient on single elements when the finite element method is used to approximate solutions of second order elliptic problems. The meshes are unstructured and, in particular, it is not assumed that there are any known superconvergent points. The estimators are based on averaging operators which are approximate gradients, recovered gradients, which are then compared to the actual gradient of the approximation on each element. Conditions are given under which they are asympotically exact or equivalent estimators on each single element of the underlying meshes. Asymptotic exactness is accomplished by letting the approximate gradient operator average over domains that are large, in a controlled fashion to be detailed below, compared to the size of the elements.
TL;DR: Computable a posteriori error bounds and related adaptive meshrefining algorithms are provided for the numerical treatment of monotone stationary flow problems with a quite general class of conforming and non-conforming finite element methods.
Abstract: Computable a posteriori error bounds and related adaptive meshrefining algorithms are provided for the numerical treatment of monotone stationary flow problems with a quite general class of conforming and non-conforming finite element methods. A refined residual-based error estimate generalises the works of Verfurth; Dari, Duran and Padra; Bao and Barrett. As a consequence, reliable and efficient averaging estimates can be established on unstructured grids. The symmetric formulation of the incompressible flow problem models certain nonNewtonian flow problems and the Stokes problem with mixed boundary conditions. A Helmholtz decomposition avoids any regularity or saturation assumption in the mathematical error analysis. Numerical experiments for the partly nonconforming method analysed by Kouhia and Stenberg indicate efficiency of related adaptive mesh-refining algorithms.
TL;DR: This article proves the convergence of a splitting scheme of high order for a reaction-diffusion system of the form u t - MΔu + F(u) = 0 where M is an m × m matrix whose spectrum is included in {Rz > 0}.
Abstract: In this article, we prove the convergence of a splitting scheme of high order for a reaction-diffusion system of the form u t - MΔu + F(u) = 0 where M is an m × m matrix whose spectrum is included in {Rz > 0}. This scheme is obtained by applying a Richardson extrapolation to a Strang formula.
TL;DR: The principal tool is an estimate related to the Carmichael function λ(m), the size of the largest cyclic subgroup of the multiplicative group of residues modulo m, which shows that for any Δ ≥ (log log N) 3 , the authors have λ (m) ≥ N exp(-Δ) for all integers m with l ≤ m ≤ N, apart from at most N exp (-0.69(Δ log Δ) 1/3 ) exceptions.
Abstract: Consider the pseudorandom number generator u n ≡ u e n-1 (mod m), 0 ≤ u n ≤ m - 1, n = 1,2,..., where we are given the modulus m, the initial value u 0 = and the exponent e. One case of particular interest is when the modulus m is of the form pl, where p, I are different primes of the same magnitude. It is known from work of the first and third authors that for moduli m = pl, if the period of the sequence (u n ) exceeds m 3/4+e , then the sequence is uniformly distributed. We show rigorously that for almost all choices of p, l it is the case that for almost all choices of , e, the period of the power generator exceeds (pl) 1-e . And so, in this case, the power generator is uniformly distributed. We also give some other cryptographic applications, namely, to ruling-out the cycling attack on the RSA cryptosystem and to so-called time-release crypto. The principal tool is an estimate related to the Carmichael function λ(m), the size of the largest cyclic subgroup of the multiplicative group of residues modulo m. In particular, we show that for any Δ ≥ (log log N) 3 , we have λ(m) ≥ N exp(-Δ) for all integers m with l ≤ m ≤ N, apart from at most N exp (-0.69(Δ log Δ) 1/3 ) exceptions.
TL;DR: This paper provides empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves and relates six quantities associated to a Jacobian over the rational numbers to the size of the Shafarevich-Tate group.
Abstract: This paper provides empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves. The second of these conjectures relates six quantities associated to a Jacobian over the rational numbers. One of these six quantities is the size of the Shafarevich-Tate group. Unable to compute that, we computed the five other quantities and solved for the last one. In all 32 cases, the result is very close to an integer that is a power of 2. In addition, this power of 2 agrees with the size of the 2-torsion of the Shafarevich-Tate group, which we could compute.
TL;DR: This paper presents the first nontrivial bounds on the discrepancy of individual sequences of inversive congruential pseudorandom numbers in parts of the period.
Abstract: The inversive congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. In this paper we present the first nontrivial bounds on the discrepancy of individual sequences of inversive congruential pseudorandom numbers in parts of the period. The proof is based on a new bound for certain incomplete exponential sums.
TL;DR: The convergence rate of an asynchronous space decomposition method for constrained convex minimization in a reflexive Banach space is analyzed, which generalizes the additive Schwarz domain decomposition methods to allow for asynchronous updates.
Abstract: We analyze the convergence rate of an asynchronous space decomposition method for constrained convex minimization in a reflexive Banach space. This method includes as special cases parallel domain decomposition methods and multigrid methods for solving elliptic partial differential equations. In particular, the method generalizes the additive Schwarz domain decomposition methods to allow for asynchronous updates. It also generalizes the BPX multigrid method to allow for use as solvers instead of as preconditioners, possibly with asynchronous updates, and is applicable to nonlinear problems. Applications to an overlapping domain decomposition for obstacle problems are also studied. The method of this work is also closely related to relaxation methods for nonlinear network flow. Accordingly, we specialize our convergence rate results to the above methods. The asynchronous method is implementable in a multiprocessor system, allowing for communication and computation delays among the processors.
TL;DR: Stable wavelet bases for the stream function spaces H(curl) and H(div) are constructed and analysed and discrete (orthogonal) Hodge decompositions are obtained.
Abstract: Some years ago, compactly supported divergence-free wavelets have been constructed which also give rise to a stable (biorthogonal) wavelet splitting of H(div). These bases have successfully been used both in the analysis and numerical treatment of the Stokes- and Navier-Stokes equations. In this paper, we construct stable wavelet bases for the stream function spaces H(curl). Moreover, curl-free vector wavelets are constructed and analysed. The relationship between H(div) and H(curl) are expressed in terms of these wavelets. We obtain discrete (orthogonal) Hodge decompositions. Our construction works independently of the space dimension, but in terms of general assumptions on the underlying wavelet systems in L^2(\Omega) that are used as building blocks. We give concrete examples of such bases for tensor product and certain more general domains $\Omega\subset\er^n$. As an application, we obtain wavelet multilevel preconditioners in H(div) and H(curl). EMAIL:: urban@dragon.ian.pv.cnr.it KEYWORDS:: H(div), H(curl), stream function spaces, wavelets
TL;DR: An algorithm for the computation of generating polynomials for all extensions K/k of a given degree and discriminant for p-adic field k is presented.
Abstract: Let k be a p-adic field. It is well-known that k has only finitely many extensions of a given finite degree. Krasner has given formulae for the number of extensions of a given degree and discriminant. Following his work, we present an algorithm for the computation of generating polynomials for all extensions K/k of a given degree and discriminant.
TL;DR: Using a carefully optimized segmented sieve and an efficient checking algorithm, the Goldbach conjecture has been verified and is now known to be true up to 4.10 14 .
Abstract: Using a carefully optimized segmented sieve and an efficient checking algorithm, the Goldbach conjecture has been verified and is now known to be true up to 4.10 14 . The program was distributed to various workstations. It kept track of maximal values of the smaller prime p in the minimal partition of the even numbers, where a minimal partition is a representation 2n = p + q with 2n - p' being composite for all p' < p. The maximal prime p needed in the considered interval was found to be 5569 and is needed for the partition 389965026819938 = 5569 + 389965026814369.
TL;DR: It is shown that the n-dimensional tensor-product of Gauss-Lobatto quadrature points are also Fekete points, which suggests a way to generalize spectral methods based on Gaussian points to non-tensor-product domains, since Fekede points are known to exist and have been computed in the triangle and tetrahedron.
Abstract: Tensor products of Gauss-Lobatto quadrature points are frequently used as collocation points in spectral element methods. Unfortunately, it is not known if Gauss-Lobatto points exist in non-tensor-product domains like the simplex. In this work, we show that the n-dimensional tensor-product of Gauss-Lobatto quadrature points are also Fekete points. This suggests a way to generalize spectral methods based on Gauss-Lobatto points to non-tensor-product domains, since Fekete points are known to exist and have been computed in the triangle and tetrahedron. In one dimension this result was proved by Fejer in 1932, but the extension to higher dimensions in non-trivial.
TL;DR: This work constructs symmetric cubature formulae of degrees in the 13-39 range for the surface measure on the unit sphere by exploiting a recently published correspondence between cubatureformulae on the sphere and on the triangle.
Abstract: We construct symmetric cubature formulae of degrees in the 13-39 range for the surface measure on the unit sphere. We exploit a recently published correspondence between cubature formulae on the sphere and on the triangle. Specifically, a fully symmetric cubature formula for the surface measure on the unit sphere corresponds to a symmetric cubature formula for the triangle with weight function (u 1 u 2 u 3 ) -1/2 , where u 1 , u 2 , and u 3 are homogeneous coordinates.
TL;DR: A boundary integral method for time-dependent, three-dimensional, doubly periodic water waves is designed and it is proved that it converges with O(h 3 ) accuracy, without restriction on amplitude.
Abstract: We design a boundary integral method for time-dependent, three-dimensional, doubly periodic water waves and prove that it converges with O(h 3 ) accuracy, without restriction on amplitude. The moving surface is represented by grid points which are transported according to a computed velocity. An integral equation arising from potential theory is solved for the normal velocity. A new method is developed for the integration of singular integrals, in which the Green's function is regularized and an efficient local correction to the trapezoidal rule is computed. The sums replacing the singular integrals are treated as discrete versions of pseudodifferential operators and are shown to have mapping properties like the exact operators. The scheme is designed so that the error is governed by evolution equations which mimic the structure of the original problem, and in this way stability can be assured. The wavelike character of the exact equations of motion depends on the positivity of the operator which assigns to a function on the surface the normal derivative of its harmonic extension; similarly, the stability of the scheme depends on maintaining this property for the discrete operator. With n grid points, the scheme can be implemented with essentially O(n) operations per time step.
TL;DR: An a posteriori error bound is derived for the Lagrange-Galerkin discretisation of an unsteady (linear) convection-diffusion problem, assuming only that the underlying space-time mesh is nondegenerate.
Abstract: In this paper we derive an a posteriori error bound for the Lagrange-Galerkin discretisation of an unsteady (linear) convection-diffusion problem, assuming only that the underlying space-time mesh is nondegenerate. The proof of this error bound is based on strong stability estimates of an associated dual problem, together with the Galerkin orthogonality of the finite element method. Based on this a posteriori bound, we design and implement the corresponding adaptive algorithm to ensure global control of the error with respect to a user-defined tolerance.