On extrapolation of Jagerman and Stetter rules

In this paper quadrature rules introduced by Jagerman [1] and Stetter [2] are considered and asymptotic expansions for the error given. This allows to make use of the Romberg extrapolation process. Such rules can be viewed as generalizations of the well-known mid-point rule. Thus, numerical examples comparing these rules are finally presented.     

Good Lattice Rules in Weighted Korobov Spaces with General Weights

We study the problem of multivariate integration and the construction of good lattice rules in weighted Korobov spaces with general weights. These spaces are not necessarily tensor products of spaces of univariate functions. Sufficient conditions for tractability and strong tractability of multivariate integration in such weighted function spaces are found. These conditions are also necessary if the weights are such that the reproducing kernel of the weighted Korobov space is pointwise non-negative. The existence of a lattice rule which achieves the nearly optimal convergence order is proven. A component-by-component (CBC) algorithm that constructs good lattice rules is presented. The resulting lattice rules achieve tractability or strong tractability error bounds and achieve nearly optimal convergence order for suitably decaying weights. We also study special weights such as finite-order and order-dependent weights. For these special weights, the cost of the CBC algorithm is polynomial. Numerical computations show that the lattice rules constructed by the CBC algorithm give much smaller worst-case errors than the mean worst-case errors over all quasi-Monte Carlo rules or over all lattice rules, and generally smaller worst-case errors than the best Korobov lattice rules in dimensions up to hundreds. Numerical results are provided to illustrate the efficiency of CBC lattice rules and Korobov lattice rules (with suitably chosen weights), in particular for high-dimensional finance problems.     

High-order quadratures for integral operators with singular kernels

A numerical integration method that has rapid convergence for integrands with known singularities is presented. Based on endpoint corrections to the trapezoidal rule, the quadratures are suited for the discretization of a variety of integral equations encountered in mathematical physics. The quadratures are based on a technique introduced by Rokhlin (1990). The present modification controls the growth of the quadrature weights and permits higher-order rules in practice. Several numerical examples are included.     

Asymptotic expansions for trapezoidal type product integration rules

This paper is concerned with the numerical integration of functions by piecewise polynomial product integration rules followed by application of extrapolation procedures. The studied rules can be considered as generalizations of the conventional trapezoidal rule. Euler-MacLaurin type asymptotic expansions are obtained with only even powers. Furthermore, numerical examples are given in order to show the effectiveness of these methods and a comparison with rules of similar characteristics is also made.     

Construction algorithms for polynomial lattice rules for multivariate integration

We introduce a new construction algorithm for digital nets for integration in certain weighted tensor product Hilbert spaces. The first weighted Hilbert space we consider is based on Walsh functions. Dick and Pillichshammer calculated the worst-case error for integration using digital nets for this space. Here we extend this result to a special construction method for digital nets based on polynomials over finite fields. This result allows us to find polynomials which yield a small worst-case error by computer search. We prove an upper bound on the worst-case error for digital nets obtained by such a search algorithm which shows that the convergence rate is best possible and that strong tractability holds under some condition on the weights.

We extend the results for the weighted Hilbert space based on Walsh functions to weighted Sobolev spaces. In this case we use randomly digitally shifted digital nets. The construction principle is the same as before, only the worst-case error is slightly different. Again digital nets obtained from our search algorithm yield a worst-case error achieving the optimal rate of convergence and as before strong tractability holds under some condition on the weights. These results show that such a construction of digital nets yields the until now best known results of this kind and that our construction methods are comparable to the construction methods known for lattice rules.

We conclude the article with numerical results comparing the expected worst-case error for randomly digitally shifted digital nets with those for randomly shifted lattice rules.

     

Geometric demonstration of the fundamental theorems of the calculus

After the monumental discovery of the fundamental theorems of the calculus nearly 350 years ago, it became possible to answer extremely complex questions regarding the natural world. Here, a straightforward yet profound demonstration, employing geometrically symmetric functions, describes the validity of the general power rules for integration and differentiation. Differentiation and integration are readily seen to be reverse operations that compute slopes and under-areas of curves, without requiring tedious infinitesimal limits or infinite summation algebraic procedures. The areas under any two symmetric curves within a square combine to equal its square measure. Corresponding evaluated integrals of any symmetric pair were also found to add to that same area. The general power rules and the fundamental theorems are confirmed for an infinite number of functions containing exponents from the entire real number line, rational or irrational. Any particular equation represents the slope of its own under-area formula, as first discovered by Isaac Newton, where the rate that area accumulates at a point under a curve, traced at constant horizontal velocity, is the value of the curve at that point. Applications of the calculus in mathematics, physics and chemistry elucidated the orbital structure of the atom, vast scientific formula and secrets of the nature of light and gravity.     

Symbolic-numeric Gaussian cubature rules

It is well known that Gaussian cubature rules are related to multivariate orthogonal polynomials. The cubature rules found in the literature use common zeroes of some linearly independent set of products of basically univariate polynomials. We show how a new family of multivariate orthogonal polynomials, so-called spherical orthogonal polynomials, leads to symbolic-numeric Gaussian cubature rules in a very natural way. They can be used for the integration of multivariate functions that in addition may depend on a vector of parameters and they are exact for multivariate parameterized polynomials. Purely numeric Gaussian cubature rules for the exact integration of multivariate polynomials can also be obtained.We illustrate their use for the symbolic-numeric solution of the partial differential equations satisfied by the Appell function F₂, which arises frequently in various physical and chemical applications. The advantage of a symbolic-numeric formula over a purely numeric one is that one obtains a continuous extension, in terms of the parameters, of the numeric solution. The number of symbolic-numeric nodes in our Gaussian cubature rules is minimal, namely m for the exact integration of a polynomial of homogeneous degree 2m−1.In Section 1 we describe how the symbolic-numeric rules are constructed, in any dimension and for any order. In Sections 2, 3 and 4 we explicit them on different domains and for different weight functions. An illustration of the new formulas is given in Section 5 and we show in Section 6 how numeric cubature rules can be derived for the exact integration of multivariate polynomials. From Section 7 it is clear that there is a connection between our symbolic-numeric cubature rules and numeric cubature formulae with a minimal (or small) number of nodes.     

On the relative merits of three-point integration rules for six-node triangles

There exist two three-point integration rules for triangular elements. Both rules are precise up to the second order and used for evaluating the six-node triangles. While one of rules has its sampling stations inside the triangle, that of the other coincide with the edge nodes. Though the former is commonly employed, it will be seen in this short paper that latter is indeed more favourable in view of element accuracy.     

Product integration rules for volterra integral equations of the first kind

The numerical solution of Volterra integral equations of the first kind can be achieved via product integration. This paper establishes the asymptotic error expansions of certain product integration rules. The rectangular rules are found to produce expansions containing all powers ofh, and the midpoint product method is found to produce even powers ofh. Extrapolation to the limit is then applied.     

Existence and construction of shifted lattice rules with an arbitrary number of points and bounded weighted star discrepancy for general decreasing weights

We study the problem of constructing shifted rank-1 lattice rules for the approximation of high-dimensional integrals with a low weighted star discrepancy, for classes of functions having bounded weighted variation, where the weighted variation is defined as the weighted sum of Hardy–Krause variations over all lower-dimensional projections of the integrand. Under general conditions on the weights, we prove the existence of rank-1 lattice rules such that for any δ>0, the general weighted star discrepancy is O(n^−1+δ) for any number of points n>1 (not necessarily prime), any shift of the lattice, general (decreasing) weights, and uniformly in the dimension. We also show that these rules can be constructed by a component-by-component strategy. This implies in particular that a single infinite-dimensional generating vector can be used for integrals in any number of dimensions, and even for infinite-dimensional integrands when they have bounded weighted variation. These same lattices are also good with respect to the worst-case error in weighted Korobov spaces with the same types of general weights. Similar results were already available for various special cases, such as general weights and prime n, or arbitrary n and product weights, but not for the most general combination of n composite, general weights, arbitrary shift, and star discrepancy, considered here. Our results imply tractability or strong tractability of integration for classes of integrands with finite weighted variation when the weights satisfy the conditions we give. These classes are a strict superset of those covered by earlier sufficient tractability conditions.     

Blending product-type quadrature rules

The idea of blending which was originally used for bivariate approximation is utilized for the numerical integration of the product of two functions. The combination of three product-type quadrature rules results in a rule with a lower error than each of the original rules. Rules of different exactness degrees as well as compounded rules of different step sizes can be taken for such a combination. Two explicit rules are constructed for demonstration; numerical examples confirm the asymptotic rates of convergence of these rules.     

Local asymptotic normality and asymptotical minimax efficiency of the MLE under random censorship

Here we study the problems of local asymptotic normality of the parametric family of distributions and asymptotic minimax efficient estimators when the observations are subject to right censoring. Local asymptotic normality will be established under some mild regularity conditions. A lower bound for local asymptotic minimax risk is given with respect to a bowl-shaped loss function, and furthermore a necessary and sufficient condition is given in order to achieve this lower bound. Finally, we show that this lower bound can be attained by the maximum likelihood estimator in the censored case and hence it is local asymptotic minimax efficient.     

Admissible solutions of k-extended finite state set and sequence compound decision problems

We treat situations in which independent structurally identical decision problems are to be faced either simultaneously or serially. Recent work on such compound decision problems has centered on finding procedures that satisfy the strengthened asymptotic optimality property of Gilliland and Hannan (Ann. Math. Statist.40 (1969), 1536–1541) and on providing decision rules that at once satisfy an admissibility property and the classical asymptotic optimality property. We suggest a simplified method of accomplishing the first of these goals in general situations and then provide decision rules for finite state components simultaneously admissible and satisfying the strengthened optimality property.     

分部积分与Taylor公式

利用定积分的分部积分法由简到繁的推导得到了微分学中的Taylor公式从而给出了Taylor公式的另一种证法,并利用这种方法还可得到复变函数或泛函分析中的Taylor公式及某些函数的渐进级数和更广泛的函数展开.     

Gaussian Quadrature Rules with Simple Node-Weight Relations

In this paper, we consider the symmetric Gaussian and L-Gaussian quadrature rules associated with twin periodic recurrence relations with possible variations in the initial coefficient. We show that the weights of the associated Gaussian quadrature rules can be given as rational functions in terms of the corresponding nodes where the numerators and denominators are polynomials of degree at most 4. We also show that the weights of the associated L-Gaussian quadrature rules can be given as rational functions in terms of the corresponding nodes where the numerators and denominators are polynomials of degree at most 5. Special cases of these quadrature rules are given. Finally, an easy to implement procedure for the evaluation of the nodes is described.     

On Gregory- and modified Gregory-type corrections to Newton—Cotes quadrature

Gregory-type formulae associated with the class of composite Newton—Cotes quadrature rules of the closed type are established. Furthermore, it is shown how these formulae can be extended by introducing mixed interpolation functions which contain a polynomial and a trigonometric part. The case of the modified Gregory rules associated with the composite Simpson quadrature rule is worked out in detail. Also the error term is analysed and the obtained rules are numerically tested.     

Bayesian Stopping Rules for Greedy Randomized Procedures

A greedy randomized adaptive search procedure (GRASP) is proposed for the approximate solution of general mixed binary programming problems (MBP). Examples are provided of practical applications that can be formulated as MBP requiring the solution of a large number of problem instances. This justifies, from both a practical and a theoretical perspective, the development of stopping rules aimed at controlling the number of iterations in a GRASP. To this end, a bayesian framework is laid down, two different prior distributions are proposed and stopping conditions are explicitly derived in analytical form. Numerical evidence shows that the stopping rules lead to an optimal trade-off between accuracy and computational effort, saving from unneeded iterations and still achieving good approximations.     

On obtaining higher order convergence for smooth periodic functions

We present two algorithms for multivariate numerical integration of smooth periodic functions. The cubature rules on which these algorithms are based use fractional parts of multiples of irrationals in combination with certain weights. Previous work led to algorithms with quadratic and cubic error convergence. We generalize these algorithms so that one can use them to obtain general higher order error convergence. The algorithms are open in the sense that extra steps can easily be taken in order to improve the result. They are also linear in the number of steps and their memory cost is low.     

A comparison of numerical integration rules for the meshless local Petrov–Galerkin method

The meshless local Petrov–Galerkin (MLPG) method is a mesh-free procedure for solving partial differential equations. However, the benefit in avoiding the mesh construction and refinement is counterbalanced by the use of complicated non polynomial shape functions with subsequent difficulties, and a potentially large cost, when implementing numerical integration schemes. In this paper we describe and compare some numerical quadrature rules with the aim at preserving the MLPG solution accuracy and at the same time reducing its computational cost.