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1.
Numerical integration using sparse grids 总被引:4,自引:0,他引:4
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. 相似文献
2.
Approximation by translates of refinable functions 总被引:23,自引:0,他引:23
Summary.
The functions
are
refinable if they are
combinations of the rescaled and translated functions
.
This is very common in scientific computing on a regular mesh.
The space of approximating functions with meshwidth
is a
subspace of with meshwidth
.
These refinable spaces have refinable basis functions.
The accuracy of the computations
depends on , the
order of approximation, which is determined by the degree of
polynomials
that lie in .
Most refinable functions (such as scaling functions in the theory
of wavelets) have no simple formulas.
The functions
are known only through the coefficients
in the refinement equation – scalars in the traditional case,
matrices for multiwavelets.
The scalar "sum rules" that determine
are well known.
We find the conditions on the matrices
that
yield approximation of order
from .
These are equivalent to the Strang–Fix conditions on the Fourier
transforms
, but for refinable
functions they can be explicitly verified from
the .
Received
August 31, 1994 / Revised version received May 2, 1995 相似文献
3.
U. Fidalgo Prieto J. R. Illán González G. López Lagomasino 《Numerische Mathematik》2007,106(1):99-128
We discuss the convergence and numerical evaluation of simultaneous quadrature formulas which are exact for rational functions.
The problem consists in integrating a single function with respect to different measures using a common set of quadrature
nodes. Given a multi-index n, the nodes of the integration rule are the zeros of the multi-orthogonal Hermite–Padé polynomial with respect to (S, α, n), where S is a collection of measures, and α is a polynomial which modifies the measures in S. The theory is based on the connection between Gauss-type simultaneous quadrature formulas of rational type and multipoint
Hermite–Padé approximation. The numerical treatment relies on the technique of modifying the integrand by means of a change
of variable when it has real poles close to the integration interval. The output of some tests show the power of this approach
in comparison with other ones in use. 相似文献
4.
Takashi Nakamura 《The Ramanujan Journal》2010,21(3):285-302
In this paper, we consider multiplication formulas and their inversion formulas for Hurwitz–Lerch zeta functions. Inversion
formulas give simple proofs of known results, and also show generalizations of those results. Next, we give a generalization
of digamma and gamma functions in terms of Hurwitz–Lerch zeta functions, and consider its properties. In all the sections,
various results are always proved by multiplication and inversion formulas. 相似文献
5.
We consider Smolyak's construction for the numerical integration over the d‐dimensional unit cube. The underlying class of integrands is a tensor product space consisting of functions that are analytic
in the Cartesian product of ellipses. The Kronrod–Patterson quadrature formulae are proposed as the corresponding basic sequence
and this choice is compared with Clenshaw–Curtis quadrature formulae. First, error bounds are derived for the one‐dimensional
case, which lead by a recursion formula to error bounds for higher dimensional integration. The applicability of these bounds
is shown by examples from frequently used test packages. Finally, numerical experiments are reported.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
6.
Kai Diethelm 《Numerische Mathematik》1996,73(1):53-63
Summary.
We show that, if
(),
the error term of
every modified positive interpolatory quadrature rule for
Cauchy principal value integrals of the type
,
, fulfills
uniformly for all
, and hence it is
of optimal
order of magnitude in the classes
().
Here, is a weight function with the property
.
We give explicit upper bounds for the Peano-type error
constants of such rules.
This improves and completes earlier results by
Criscuolo and Mastroianni
(Calcolo 22 (1985), 391–441 and Numer. Math.
54 (1989), 445–461)
and Ioakimidis (Math. Comp. 44 (1985), 191–198).
For the special case of the Gaussian rule, we
show that the restriction
can be dropped.
The results are based on a new representation of the
Peano kernels of these formulae via the Peano kernels of the underlying
classical quadrature formulae. This representation may also be
useful in connection with some different problems.
Received November 21, 1994 相似文献
7.
Chia-chi Tung 《Complex Analysis and Operator Theory》2011,5(3):701-730
Formulas for the Euler vector fields, the Neumann derivatives, and the Euler as well as Dirichlet product are derived. Extensions
to a Riemann domain of the Gauss operator, the Gauss’ lemma and the related jump formulas are given, and the Gauss–Helmholtz
representation with ramifications proved. Examples of elementary solutions to certain modified Laplace operators, applications
to pseudospherical harmonics, and characterizations of pseudoradial, pseudospherical, nearly holomorphic, and holomorphic
functions, are obtained, and constancy criterion for locally Lipschitz, semiharmonic, respectively, weakly holomorphic functions
are given. 相似文献
8.
The authors develop an algorithm for the numerical evaluation of the finite Hilbert transform, with respect to non-standard
weight functions, by a product quadrature rule. In particular, this algorithm allows us to deal with the weight functions
with algebraic and/or logarithmic singularities in the interval [−1, 1], by using the Chebyshev points as quadrature nodes.
The practical application of the rule is shown to be straightforward and to yield satisfactory numerical results. Convergence
theorems are also given, when the nodes are the zeros of certain classical Jacobi polynomials and the weight is defined as
a generalized Ditzian-Totik weight.
This work was supported by the Ministero dell'Università e della Ricerca Scientifica e Tecnologica (first author) and by the
Italian Research Council (second author). 相似文献
9.
C. Krattenthaler 《Israel Journal of Mathematics》1995,90(1-3):67-79
New bideterminantal formulas for the irreducible symplectic and orthogonal characters are given that generalize the classical
bideterminantal formulas. These formulas are analogous to Regev’s (Israel J. Math. 80 (1992), 155–160) bideterminantal formulas
for Schur functions, the irreducible general linear characters. Also, new bideterminantal formulas for Proctor’s intermediate
symplectic characters are derived. 相似文献
10.
In this paper we consider second order scalar elliptic boundary value problems posed over three–dimensional domains and their
discretization by means of mixed Raviart–Thomas finite elements [18]. This leads to saddle point problems featuring a discrete
flux vector field as additional unknown. Following Ewing and Wang [26], the proposed solution procedure is based on splitting
the flux into divergence free components and a remainder. It leads to a variational problem involving solenoidal Raviart–Thomas
vector fields. A fast iterative solution method for this problem is presented. It exploits the representation of divergence
free vector fields as s of the –conforming finite element functions introduced by Nédélec [43]. We show that a nodal multilevel splitting of these finite
element spaces gives rise to an optimal preconditioner for the solenoidal variational problem: Duality techniques in quotient
spaces and modern algebraic multigrid theory [50, 10, 31] are the main tools for the proof.
Received November 4, 1996 / Revised version received February 2, 1998 相似文献
11.
This paper considers the Kipriyanov–Radon transform constructed as a special Radon transform adopted for dealing with singular
Bessel differential operators of the corresponding indices acting on a part of the variables. The authors obtain inversion
formulas generalizing the classical formulas for the Radon transform of axially-symmetric functions and relating to the integro-differentiation
of fractional order in a one-dimensional parameter.
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 54, Suzdal
Conference–2006, Part 2, 2008. 相似文献
12.
13.
Working within Bishop’s constructive framework, we examine the connection between a weak version of the Heine–Borel property,
a property antithetical to that in Specker’s theorem in recursive analysis, and the uniform continuity theorem for integer-valued
functions. The paper is a contribution to the ongoing programme of constructive reverse mathematics. 相似文献
14.
We study cubature formulas for d -dimensional integrals with arbitrary weight function of tensor product form. We present a construction that yields a high
polynomial exactness: for fixed degree, the number of knots depends on the dimension in an order-optimal way. The cubature
formulas are universal: the order of convergence is almost optimal for two different scales of function spaces. The construction
is simple: a small number of arithmetical operations is sufficient to compute the knots and the weights of the formulas.
August 25, 1997. Date revised: December 3, 1998. Date accepted: March 3, 1999. 相似文献
15.
This paper deals with the boundary behavior of functions in the de Branges–Rovnyak spaces. First, we give a criterion for
the existence of radial limits for the derivatives of functions in the de Branges–Rovnyak spaces. This criterion generalizes
a result of Ahern–Clark. Then we prove that the continuity of all functions in a de Branges–Rovnyak space on an open arc I of the boundary is enough to ensure the analyticity of these functions on I. We use this property in a question related to Bernstein’s inequality.
Received: May 10, 2007. Revised: August 8, 2007. Accepted: August 8, 2007. 相似文献
16.
Abstract. We consider polynomials which are orthogonal with respect to weight functions, which are defined in terms of the modified
Bessel function I
ν
and which are related to the noncentral χ
2
-distribution. It turns out that it is the most convenient to use two weight functions with indices ν and ν+1 and to study orthogonality with respect to these two weights simultaneously. We show that the corresponding multiple orthogonal
polynomials of type I and type II exist and give several properties of these polynomials (differential properties, Rodrigues
formula, explicit formulas, recurrence relation, differential equation, and generating functions). 相似文献
17.
Miodrag M. Spalevi? Miroslav S. Prani? 《Journal of Computational and Applied Mathematics》2010,234(4):1049-5756
We study the kernel of the remainder term of Gauss quadrature rules for analytic functions with respect to one class of Bernstein-Szegö weight functions. The location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective error bounds of the corresponding Gauss quadratures. 相似文献
18.
The De Donder–Weyl (DW) Hamilton–Jacobi equation is investigated in this paper, and the connection between the DW Hamilton–Jacobi
equation and multi-symplectic Hamiltonian system is established. Based on the DW Hamilton–Jacobi theory, generating functions
for multi-symplectic Runge–Kutta (RK) methods and partitioned Runge–Kutta (PRK) methods are presented.
The work is supported by the Foundation of ICMSEC, LSEC, AMSS and CAS, the NNSFC (No.10501050, 19971089 and 10371128) and
the Special Funds for Major State Basic Research Projects of China (2005CB321701). 相似文献
19.
J.R. Illán González 《Journal of Computational and Applied Mathematics》2009,233(3):745-748
A flexible treatment of Gaussian quadrature formulas based on rational functions is given to evaluate the integral , when f is meromorphic in a neighborhood V of the interval I and W(x) is an ill-scaled weight function. Some numerical tests illustrate the power of this approach in comparison with Gautschi’s method. 相似文献
20.
We derive residual based a posteriori error estimates of the flux in L
2-norm for a general class of mixed methods for elliptic problems. The estimate is applicable to standard mixed methods such
as the Raviart–Thomas–Nedelec and Brezzi–Douglas–Marini elements, as well as stabilized methods such as the Galerkin-Least
squares method. The element residual in the estimate employs an elementwise computable postprocessed approximation of the
displacement which gives optimal order. 相似文献