Fast and accurate tensor approximation of a multivariate convolution with linear scaling in dimension

In the present paper we present the tensor-product approximation of a multidimensional convolution transform discretized via a collocation-projection scheme on uniform or composite refined grids. Examples of convolving kernels are provided by the classical Newton, Slater (exponential) and Yukawa potentials, 1/‖x‖, and with x∈R^d. For piecewise constant elements on the uniform grid of size n^d, we prove quadratic convergence O(h²) in the mesh parameter h=1/n, and then justify the Richardson extrapolation method on a sequence of grids that improves the order of approximation up to O(h³). A fast algorithm of complexity O(dR₁R₂nlogn) is described for tensor-product convolution on uniform/composite grids of size n^d, where R₁,R₂ are tensor ranks of convolving functions. We also present the tensor-product convolution scheme in the two-level Tucker canonical format and discuss the consequent rank reduction strategy. Finally, we give numerical illustrations confirming: (a) the approximation theory for convolution schemes of order O(h²) and O(h³); (b) linear-logarithmic scaling of 1D discrete convolution on composite grids; (c) linear-logarithmic scaling in n of our tensor-product convolution method on an n×n×n grid in the range n≤16384.     

Estimation of convergence orders in repeated richardson extrapolation

LetA(h) be an approximation depending on a single parameterh to a fixed quantityA, and assume thatA–A(h)=c ₁ h k ₁ +c ₂ h ^k ₂ +.... Given a sequence ofh-valuesh ₁>h ₂>...>h _n and corresponding computed approximationsA(h _i ), the orders for repeated Richardson extrapolation are estimated, and the repeated extrapolation is performed. Hence in this case the algorithm described here can do the same work as Brezinski'sE-algorithm and at the same time provide a check whether repeated extrapolation is justified.     

Numerical solution of delay differential equations by uniform corrections to an implicit Runge-Kutta method

Summary In this paper we develop a class of numerical methods to approximate the solutions of delay differential equations. They are essentially based on a modified version, in a predictor-corrector mode, of the one-step collocation method atn Gaussian points. These methods, applied to ODE's, provide a continuous approximate solution which is accurate of order 2n at the nodes and of ordern+1 uniformly in the whole interval. In order to extend the methods to delay differential equations, the uniform accuracy is raised to the order 2n by some a posteriori corrections. Numerical tests and comparisons with other methods are made on real-life problems.This work was supported by CNR within the Progetto Finalizzato Informatica-Sottopr. P1-SOFMAT     

A capacitance matrix method for Dirichlet problem on polygon region

Summary An efficient algorithm for the solution of linear equations arising in a finite element method for the Dirichlet problem is given. The cost of the algorithm is proportional toN ²log₂ N (N=1/h) where the cost of solving the capacitance matrix equations isNlog₂ N on regular grids andN ^3/2log₂ N on irregular ones.     

PreconditioningP 1 nonconforming finite elements: Condition numbers and singular value distributions

Summary This work deals with the condition numbers and the distribution of theB _h singular values of the preconditioned operators {B _h ^–1 A _h }_{0, whereA h andB h are discretizations of second order elliptic operatorsA andB usingP 1 nonconforming finite elements of Crouzeix and Raviart.B is also assumed to be self-adjoint and positive definite. For conforming finite elements, Parter and Wong have shown that the singular values cluster in a positive finite interval. These reults are being extended to the aforementioned nonconforming finite elements. It will be shown that, for quasiuniform grids, theB h singular values are bounded above and below by positive constants which are independent of the grid sizeh. Moreover, they also cluster in a smaller but usually estimable interval. Issues of implementation are also discussed.This research was supported by the National Science Foundation under grant number DMS-8913091}     

Asymptotic expansions and acceleration of convergence for higher order iteration processes

Summary We analyze the convergence behavior of sequences of real numbers {x _n }, which are defined through an iterative process of the formx _n :=T(x _n –1), whereT is a suitable real function. It will be proved that under certain mild assumptions onT, these numbersx _n possess an asymptotic (error) expansion, where the type of this expansion depends on the derivative ofT in the limit point ; this generalizes a result of G. Meinardus [6].It is well-known that the convergence of sequences, which possess an asymptotic expansion, can be accelerated significantly by application of a suitable extrapolation process. We introduce two types of such processes and study their main properties in some detail. In addition, we analyze practical aspects of the extrapolation and present the results of some numerical tests. As we shall see, even the convergence of Newton's method can be accelerated using the very simple linear extrapolation process.Dedicated to Professor Dr. Günter Meinardus on the occasion of his 65^th birthday     

A generalized conjugate gradient algorithm for minimization

Summary A generalized conjugate gradient algorithm which is invariant to a nonlinear scaling of a strictly convex quadratic function is described, which terminates after at mostn steps when applied to scaled quadratic functionsf: R _n R₁ of the formf(x)=h(F(x)) withF(x) strictly convex quadratic andhC ¹ (R₁) an arbitrary strictly monotone functionh. The algorithm does not suppose the knowledge ofh orF but only off(x) and its gradientg(x).     

Error saturation in Gaussian radial basis functions on a finite interval

Radial basis function (RBF) interpolation is a “meshless” strategy with great promise for adaptive approximation. One restriction is “error saturation” which occurs for many types of RBFs including Gaussian RBFs of the form ?(x;α,h)=exp(−α²(x/h)²): in the limit h→0 for fixed α, the error does not converge to zero, but rather to E_S(α). Previous studies have theoretically determined the saturation error for Gaussian RBF on an infinite, uniform interval and for the same with a single point omitted. (The gap enormously increases E_S(α).) We show experimentally that the saturation error on the unit interval, x∈[−1,1], is about 0.06exp(−0.47/α²)‖f‖_∞ — huge compared to the O(2π/α²)exp(−π²/[4α²]) saturation error for a grid with one point omitted. We show that the reason the saturation is so large on a finite interval is that it is equivalent to an infinite grid which is uniform except for a gap of many points. The saturation error can be avoided by choosing α?1, the “flat limit”, but the condition number of the interpolation matrix explodes as O(exp(π²/[4α²])). The best strategy is to choose the largest α which yields an acceptably small saturation error: If the user chooses an error tolerance δ, then .     

Extraneous fixed points,basin boundaries and chaotic dynamics for Schröder and König rational iteration functions

Summary The Schröder and König iteration schemes to find the zeros of a (polynomial) functiong(z) represent generalizations of Newton's method. In both schemes, iteration functionsf _m (z) are constructed so that sequencesz _n+1 =f _m (z _n ) converge locally to a rootz ^* ofg(z) asO(|z _n –z ^*|^m). It is well known that attractive cycles, other than the zerosz ^*, may exist for Newton's method (m=2). Asm increases, the iteration functions add extraneous fixed points and cycles. Whether attractive or repulsive, they affect the Julia set basin boundaries. The König functionsK _m (z) appear to minimize such perturbations. In the case of two roots, e.g.g(z)=z ²–1, Cayley's classical result for the basins of attraction of Newton's method is extended for allK _m (z). The existence of chaotic {z _n } sequences is also demonstrated for these iteration methods.     

Directional approximation of the jacobians in ROW-methods

In this paper a new technique for avoiding exact Jacobians in ROW methods is proposed. The Jacobiansf' _n are substituted by matricesA _n satisfying a directional consistency conditionA _n f _n =f' _n f _n +O(h). In contrast to generalW-methods this enables us to reduce the number of order conditions and we construct a 2-stage method of order 3 and families of imbedded 4-stage methods of order 4(3). The directional approximation of the Jacobians has been realized via rank-1 updating as known from quasi-Newton methods.     

The numerical stability of evaluation schemes for polynomials based on the lagrange interpolation form

For evaluation schemes based on the Lagrangian form of a polynomial with degreen, a rigorous error analysis is performed, taking into account that data, computation and even the nodes of interpolation might be perturbed by round-off. The error norm of the scheme is betweenn ² andn ²+(3n+7) _n , where _n denotes the Lebesgue constant belonging to the nodes. Hence, the error norm is of least possible orderO(n ²) if, for instance, the nodes are chosen to be the Chebyshev points or the Fekete points.     

Maximal order for quadratures usingn evaluations

The problem is to find approximationsI (f; h) to the integralI(f; h)= ₀ ^h f. Such an approximation has local orderp ifI(f; h)–I (f; h)=O(h ^p ) ash0. Let(n) denote the maximal local order possible for a method usingn evaluations of a function or its derivatives. We show that (n)=2n+1 if the information used is Hermitian. This is conjectured to be true in general. The conjecture is established for all methods using three or fewer evaluations.This research was supported in part by the National Science Foundation under Grant MCS75-222-55 and the Office of Naval Research under Contract N00014-76-C-0370, NR 044-422. Numerical results reported in this paper were obtained through the computing facilities of the University of Maryland.     

Lösung von Differentialgleichungen mit Splinefunktionen. Eine Störungstheorie

Summary In this paper non-linear splines (depending onn+1 parameters) are used to patch up the solution of an initial value problem in intervals of stepsizeh. The elements of the solution are fixed byq smoothness conditions andd conditions derived from the differential equation in an appropriate setup. The feasibility of the method can be connected to that of the polynomial spline method by a perturbation type argument. Thus the question of convergence forh0 is closely connected to the linear (polynomial) case.A new elementary prove is given for divergence of the polynomial splines ifq is larger thand+1, as was done by Mülthei [4] with other techniques.A byproduct is an extention of the famous result for polynomial interpolation by Runge on equidistant grids that interpolation of a given function by splines of too high smoothness can cause divergence forh0.

Diese Arbeit ist mit Unterstützung des von der Deutschen Forschungsgemeinschaft getragenen Sonderforschungsbereiches 72 entstanden     

A quadrature-based approach to improving the collocation method

Summary The collocation method is a popular method for the approximate solution of boundary integral equations, but typically does not achieve the high order of convergence reached by the Galerkin method in appropriate negative norms. In this paper a quadrature-based method for improving upon the collocation method is proposed, and developed in detail for a particular case. That case involves operators with even symbol (such as the logarithmic potential) operating on 1-periodic functions; a smooth-spline trial space of odd degree, with constant mesh spacingh=1/n; and a quadrature rule with 2n points (where ann-point quadrature rule would be equivalent to the collocation method). In this setting it is shown that a special quadrature rule (which depends on the degree of the splines and the order of the operator) can yield a maximum order of convergence two powers ofh higher than the collocation method.     

The Euler Maclaurin expansion for the Cauchy Principal Value integral

Summary The classical Euler Maclaurin Summation Formula expresses the difference between a definite integral over [0, 1] and its approximation using the trapezoidal rule with step lengthh=1/m as an asymptotic expansion in powers ofh together with a remainder term. Many variants of this exist some of which form the basis of extrapolation methods such as Romberg Integration. in this paper a variant in which the integral is a Cauchy Principal Value integral is derived. The corresponding variant of the Fourier Coefficient Asymptotic Expansion is also derived. The possible role of the former in numerical quadrature is discussed.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under contract W-31-109-Eng-38     

Expectation of random polytopes

A random polytopeP _n in a convex bodyC is the convex hull ofn identically and independently distributed points inC. Its expectation is a convex body in the interior ofC. We study the deviation of the expectation ofP _n fromC asn→∞: while forC of classC ^k+1,k≥1, precise asymptotic expansions for the deviation exist, the behaviour of the deviation is extremely irregular for most convex bodiesC of classC ¹. Dedicated to my teacher and friend Professor Edmund Hlawka on the occasion of his 80^th birthday     

Hypergraphs with no special cycles

A special cycle in a hypergraph is a cyclex ₁ E ₁ x ₂ E ₂ x ₃ ...x _n E _n x ₁ ofn distinct verticesx _i andn distinct edgesE _j (n≧3) whereE _i ∩{x ₁,x ₂, ...,x _n }={x _i ,x _i+1} (x _n+1=x ₁). In the equivalent (0, 1)-matrix formulation, a special cycle corresponds to a square submatrix which is the incidence matrix of a cycle of size at least 3. Hypergraphs with no special cycles have been called totally balanced by Lovász. Simple hypergraphs with no special cycles onm vertices can be shown to have at most ( ₂ ^m )+m+1 edges where the empty edge is allowed. Such hypergraphs with the maximum number of edges have a fascinating structure and are called solutions. The main result of this paper is an algorithm that shows that a simple hypergraph on at mostm vertices with no special cycles can be completed (by adding edges) to a solution. Support provided by NSERC.     

Extremal hypergraph problems and convex hulls

Theprofile of a hypergraph onn vertices is (f ₀, f₁, ...,f _n) wheref _i denotes the number ofi-element edges. The extreme points of the set of profiles is determined for certain hypergraph classes. The results contain many old theorems of extremal set theory as particular cases (Sperner. Erdős—Ko—Rado, Daykin—Frankl—Green—Hilton).     

Boundary element preconditioners for a hypersingular integral equation on an interval

We propose an almost optimal preconditioner for the iterative solution of the Galerkin equations arising from a hypersingular integral equation on an interval. This preconditioning technique, which is based on the single layer potential, was already studied for closed curves [11,14]. For a boundary element trial space, we show that the condition number is of order (1 + | log h _min|)², where h _min is the length of the smallest element. The proof requires only a mild assumption on the mesh, easily satisfied by adaptive refinement algorithms.     

The endomorphism spectrum of an ordered set

Theendomorphism spectrum of an ordered setP, spec(P)={|f(P)|:f End(P)} andspectrum number, sp(P)=max(spec(P)\{|P|}) are introduced. It is shown that |P|>(1/2)n(n – 1) ^{n – 1} implies spec(P) = {1, 2, ...,n} and that if a projective plane of ordern exists, then there is an ordered setP of size 2n ²+2n+2 with spec(P)={1, 2, ..., 2n+2, 2n+4}. Lettingh(n)=max{|P|: sp(P)n}, it follows thatc ₁ n ²h(n)c ₂ n ⁿ⁺¹ for somec ₁ andc ₂. The lower bound disproves the conjecture thath(n)2n. It is shown that if |P| – 1 spec(P) thenP has a retract of size |P| – 1 but that for all there is a bipartite ordered set with spec(P) = {|P| – 2, |P| – 4, ...} which has no proper retract of size|P| – . The case of reflexive graphs is also treated.Partially supported by a grant from the NSERC.Partially supported by a grant from the NSERC.