Numerical integration formulas of degree two

Numerical integration formulas in n-dimensional nonsymmetric Euclidean space of degree two, consisting of n+1 equally weighted points, are discussed, for a class of integrals often encountered in statistics. This is an extension of Stroud's theory [A.H. Stroud, Remarks on the disposition of points in numerical integration formulas, Math. Comput. 11 (60) (1957) 257–261; A.H. Stroud, Numerical integration formulas of degree two, Math. Comput. 14 (69) (1960) 21–26]. Explicit formulas are given for integrals with nonsymmetric weights. These appear to be new results and include the Stroud's degree two formula as a special case.     

Numerical integration over polygons using an eight-node quadrilateral spline finite element

In this paper, a cubature formula over polygons is proposed and analysed. It is based on an eight-node quadrilateral spline finite element [C.-J. Li, R.-H. Wang, A new 8-node quadrilateral spline finite element, J. Comp. Appl. Math. 195 (2006) 54–65] and is exact for quadratic polynomials on arbitrary convex quadrangulations and for cubic polynomials on rectangular partitions. The convergence of sequences of the above cubatures is proved for continuous integrand functions and error bounds are derived. Some numerical examples are given, by comparisons with other known cubatures.     

Numerical indefinite integration by double exponential sinc method

We present a numerical method for approximating an indefinite integral by the double exponential sinc method. The approximation error of the proposed method with integrand function evaluations is

for a reasonably wide class of integrands, including those with endpoint singularities. The proposed method compares favorably with the existing formulas based on the ordinary sinc method. Computational results show the accordance of the actual convergence rates with the theoretical estimate.

     

Discretizing continuous wavelet transforms using integrated wavelets

We present integrated wavelets as a method for discretizing the continuous wavelet transform. Using the language of group theory, the results are presented for wavelet transforms over semidirect product groups. We obtain tight wavelet frames for these wavelet transforms. Further integrated wavelets yield tight families of convolution operators independent of the choice of discretization of scale and orientation parameters. Thus these families can be adapted to specific problems. The method is more flexible than the well-known dyadic wavelet transform. We state an exact algorithm for implementing this transform. As an application the enhancement of digital mammograms is presented.     

Numerical treatment of a twisted tail using extrapolation methods

Highly oscillatory integral, called a twisted tail, is proposed as a challenge in The SIAM 100-digit challenge. A Study in High-Accuracy Numerical Computing, where Drik Laurie developed numerical algorithms based on the use of Aitken’s Δ²-method, complex integration and transformation to a Fourier integral. Another algorithm is developed by Walter Gautschi based on Longman’s method; Newton’s method for solving a nonlinear equation; Gaussian quadrature; and the epsilon algorithm of Wynn for accelerating the convergence of infinite series. In the present work, nonlinear transformations for improving the convergence of oscillatory integrals are applied to the integration of this wildly oscillating function. Specifically, the transformation and its companion the W algorithm, and the G transformation are all used in the analysis of the integral. A Fortran program is developed employing each of the methods, and accuracies of up to 15 correct digits are reached in double precision.     

Numerical integration of constrained Hamiltonian systems using Dirac brackets

We study the numerical properties of the equations of motion of constrained systems derived with Dirac brackets. This formulation is compared with one based on the extended Hamiltonian. As concrete examples, a pendulum in Cartesian coordinates and a chain molecule are treated.

     

Some algorithms for numerical quadrature using the derivatives of the integrand in the integration interval

Some quadrature formulae using the derivatives of the integrand are discussed. As special cases are obtained generalizations of both the ordinary and the modified Romberg algorithms. In all cases the error terms are expressed in terms of Bernoulli polynomials and functions.     

On the holes of a class of bidimensional nonseparable wavelets

Let I be the 2×2 identity matrix, and M a 2×2 dilation matrix with M²=2I. Since one can explicitly construct M-basic wavelets from an MRA related to M, and many applications employ wavelet bases in R², M-wavelets and wavelet frames have been extensively discussed. This paper focuses on dilation matrices M satisfying M²=2I. For any matrix M integrally similar to , an optimal estimate on the boundary of the holes of M-wavelets is obtained. This result tells us the holes cannot be too large. Contrast to this result, when the modulus of the Fourier transform of an M-wavelet is, up to a constant, a characteristic function on some set, a property of this set is obtained, which shows the holes of this kind of wavelets cannot be too small.     

A fourth degree integration formula for the n-dimensional simplex

A fourth degree integration formula is given for the n-dimensional simplex for all n2, which is invariant under the group G of all affine transformations of T_n onto itself. The formula contains (n²+5n+6)/2 nodes.     

Numerical approximation of weakly singular integrals on the half line

This paper deals with the numerical approximation of a weakly singular integral transform by means of Laguerre nodes. Error estimates in a weighted uniform norm and some numerical tests are given.     

Numerical integration using sparse grids

We present new and review existing algorithms for the numerical integration of multivariate functions defined over d-dimensional cubes using several variants of the sparse grid method first introduced by Smolyak [49]. In this approach, multivariate quadrature formulas are constructed using combinations of tensor products of suitable one-dimensional formulas. The computing cost is almost independent of the dimension of the problem if the function under consideration has bounded mixed derivatives. We suggest the usage of extended Gauss (Patterson) quadrature formulas as the one‐dimensional basis of the construction and show their superiority in comparison to previously used sparse grid approaches based on the trapezoidal, Clenshaw–Curtis and Gauss rules in several numerical experiments and applications. For the computation of path integrals further improvements can be obtained by combining generalized Smolyak quadrature with the Brownian bridge construction. This revised version was published online in August 2006 with corrections to the Cover Date.     

Estimation of dimension functions of band-limited wavelets

The dimension function D_ψ of a band-limited wavelet ψ is bounded by n if its Fourier transform is supported in [−(2ⁿ⁺²/3)π,(2ⁿ⁺²/3)π]. For each and for each , 0<<δ=δ(n), we construct a wavelet ψ with supp

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such that D_ψ>n on a set of positive measure, which proves that [−(2ⁿ⁺²/3)π,(2ⁿ⁺²/3)π] is the largest symmetric interval for estimating the dimension function by n. This construction also provides a family of (uncountably many) wavelet sets each consisting of infinite number of intervals.     

A sequentially optimal algorithm for numerical integration

For the class of functions of one variable, satisfying the Lipschitz condition with a fixed constant, an optimal passive algorithm for numerical integration (an optimal quadrature formula) has been found by Nikol'skii. In this paper, a sequentially optimal algorithm is constructed; i.e., the algorithm on each step makes use in an optimal way of all relevant information which was accumulated on previous steps. Using the algorithm, it is necessary to solve an integer program at each step. An effective algorithm for solving these problems is given.The author is indebted to Professor S. E. Dreyfus, Department of Industrial Engineering and Operations Research, University of California, Berkeley, California, for his helpful attention to this paper.     

Direct limits, multiresolution analyses, and wavelets

A multiresolution analysis for a Hilbert space realizes the Hilbert space as the direct limit of an increasing sequence of closed subspaces. In a previous paper, we showed how, conversely, direct limits could be used to construct Hilbert spaces which have multiresolution analyses with desired properties. In this paper, we use direct limits, and in particular the universal property which characterizes them, to construct wavelet bases in a variety of concrete Hilbert spaces of functions. Our results apply to the classical situation involving dilation matrices on L²(Rn), the wavelets on fractals studied by Dutkay and Jorgensen, and Hilbert spaces of functions on solenoids.     

Numerical differentiation for high orders by an integration method

This paper mainly studies the numerical differentiation by integration method proposed first by Lanczos. New schemes of the Lanczos derivatives are put forward for reconstructing numerical derivatives for high orders from noise data. The convergence rate of these proposed methods is as the noise level δ→0. Numerical examples show that the proposed methods are stable and efficient.     

Orthonormal wavelets and shift invariant generalized multiresolution analyses

All wavelets can be associated to a multiresolution-like structure, i.e. an increasing sequence of subspaces of . We consider the interaction of a wavelet and the shift operator in terms of which of the subspaces in this multiresolution-like structure are invariant under the shift operator. This action defines the notion of the shift invariance property of order . In this paper we show that wavelets of all levels of shift invariance exist, first for the classic case of dilation by 2, and then for arbitrary integral dilation factors.

     

Numerical integration by using nonlinear techniques

In this paper several nonlinear techniques, mainly based on the use of Padé approximation and rational interpolation, are given for computing the value of a definite integral. Some convergence properties of these methods are proved. A comparison is made between these non-linear techniques and classical linear techniques on the base of some numerical examples.     

Parseval wavelets on hierarchical graphs

Wavelets on graphs have been studied for the past few years, and in particular, several approaches have been proposed to design wavelet transforms on hierarchical graphs. Although such methods are computationally efficient and easy to implement, their frames are highly restricted. In this paper, we propose a general framework for the design of wavelet transforms on hierarchical graphs. Our design is guaranteed to be a Parseval tight frame, which preserves the $l_2$ norm of any input signals. To demonstrate the potential usefulness of our approach, we perform several experiments, in which we learn a wavelet frame based on our framework, and show, in inpainting experiments, that it performs better than a Haar-like hierarchical wavelet transform and a learned treelet. We also show with category theory that the algebraic properties of the proposed transform have a strong relationship with those of the hierarchical graph that represents the structure of the given data.