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1.
We consider the numerical integration of multivariate functions defined over the unit hypercube. Here, we especially address the high–dimensional case, where in general the curse of dimension is encountered. Due to the concentration of measure phenomenon, such functions can often be well approximated by sums of lower–dimensional terms. The problem, however, is to find a good expansion given little knowledge of the integrand itself. The dimension–adaptive quadrature method which is developed and presented in this paper aims to find such an expansion automatically. It is based on the sparse grid method which has been shown to give good results for low- and moderate–dimensional problems. The dimension–adaptive quadrature method tries to find important dimensions and adaptively refines in this respect guided by suitable error estimators. This leads to an approach which is based on generalized sparse grid index sets. We propose efficient data structures for the storage and traversal of the index sets and discuss an efficient implementation of the algorithm. The performance of the method is illustrated by several numerical examples from computational physics and finance where dimension reduction is obtained from the Brownian bridge discretization of the underlying stochastic process. 相似文献
2.
In this paper, we will design and analyze a class of new algebraic multigrid methods for algebraic systems arising from the
discretization of second order elliptic boundary value problems by high-order finite element methods. For a given sparse stiffness
matrix from a quadratic or cubic Lagrangian finite element discretization, an algebraic approach is carefully designed to
recover the stiffness matrix associated with the linear finite element disretization on the same underlying (but nevertheless
unknown to the user) finite element grid. With any given classical algebraic multigrid solver for linear finite element stiffness
matrix, a corresponding algebraic multigrid method can then be designed for the quadratic or higher order finite element stiffness
matrix by combining with a standard smoother for the original system. This method is designed under the assumption that the
sparse matrix to be solved is associated with a specific higher order, quadratic for example, finite element discretization
on a finite element grid but the geometric data for the underlying grid is unknown. The resulting new algebraic multigrid
method is shown, by numerical experiments, to be much more efficient than the classical algebraic multigrid method which is
directly applied to the high-order finite element matrix. Some theoretical analysis is also provided for the convergence of
the new method. 相似文献
3.
Sparse grid methods represent a powerful and efficient technique for the representation and approximation of functions and
particularly the solutions of partial differential equations in moderately high space dimensions. To extend the approach to
truly high-dimensional problems as they arise in quantum chemistry, an additional property has to be brought into play, the
symmetry or antisymmetry of the functions sought there.
In the present article, an adaptive sparse grid refinement scheme is developed that takes full advantage of such symmetry
properties and for which the amount of work and storage remains strictly proportional to the number of degrees of freedom.
To overcome the problems with the approximation of the inherently complex antisymmetric functions, augmented sparse grid spaces
are proposed. 相似文献
4.
With respect to a wavelet basis, singular integral operators can be well approximated by sparse matrices, and in Found. Comput. Math. 2: 203–245 (2002) and SIAM J. Math. Anal. 35: 1110–1132 (2004), this property was used to prove certain optimal complexity results in the context of adaptive wavelet
methods. These results, however, were based upon the assumption that, on average, each entry of the approximating sparse matrices
can be computed at unit cost. In this paper, we confirm this assumption by carefully distributing computational costs over
the matrix entries in combination with choosing efficient quadrature schemes. 相似文献
5.
The central results of this paper are bounds for the second Peano kernels of the Gaussian quadrature formulae. Hence, for
these quadrature formulae, we derive asymptotically optimal error constants for classes of functions with a bounded second
derivative or with a first derivative of bounded variation, or for a class of convex functions. To obtain these bounds, we
first prove inequalities related to MacMahon's expansion and some further results on the Bessel function J
o
, as well as some “trapezoid theorems for the weights of Gaussian formulae” (cf. Davis and Rabinowitz [6]) with explicit bounds
for the error term. 相似文献
6.
The quadrature formulae of Newton-Cotes type for the computation of hypersingular integrals with second order singularity on interval are discussed. We improve the estimates given by Linz [22] such that the Newton-Cotes method is valid with less restriction on the location of the singular point. We also present a new Newton-Cotes formula which is applicable when the singular point coincides with a mesh point, while the classical Newton-Cotes method fails in this case. Error analysis for the new formula is given. Numerical experiments are presented to validate the analysis. 相似文献
7.
The extended differential quadrature (EDQ) has been proposed. A certain order derivative or partial derivative of the variable function with respect to the coordinate variables at an arbitrary discrete point is expressed as a weighted linear sum of the values of function and/or its possible derivatives at all grid nodes. The grid pattern can be fixed while the selection of discrete points for defining discrete fundamental relations is flexible. This method can be used to the differential quadrature element and generalized differential quadrature element analyses. 相似文献
8.
In addition to the well-known and widely-used adaptive strategies for region subdivision and the choice of local quadrature
rules, there still exists a third possibility for doing numerical integration adaptively, when interval computation is considered.
Based on the Peano's kernels theorem and interval arithmetic, we are able to estimate the truncation error of a quadrature
rule by means of different derivatives of the integrand, where the available orders of the derivatives depend on the degree
of smoothness of the integrand and the exactness degree of the underlying quadrature rule. We classify the methods as the
adaptive orders strategies, if they make use of the derivatives of different orders to improve each single local error estimation.
In this paper, alternatives for adaptive error estimation are discussed. Moreover, a practical way for realization of the
optimal error estimation is suggested. Numerical results for integrands of different classes as well as numerical comparisons
of different methods are given. 相似文献
9.
We integrate optimal quadratic and cubic spline collocation methods for second-order two-point boundary value problems with
adaptive grid techniques, and grid size and error estimators. Some adaptive grid techniques are based on the construction
of a mapping function that maps uniform to non-uniform points, placed appropriately to minimize a certain norm of the error.
One adaptive grid technique for cubic spline collocation is mapping-free and resembles the technique used in COLSYS (COLNEW)
[2], [4]. Numerical results on a variety of problems, including problems with boundary or interior layers, and singular perturbation
problems indicate that, for most problems, the cubic spline collocation method requires less computational effort for the
same error tolerance, and has equally reliable error estimators, when compared to Hermite piecewise cubic collocation. Comparison
results with quadratic spline collocation are also presented. 相似文献
10.
We consider gaussian quadrature rules for the numerical approximation of the derivatives of Cauchy principal value integrals. Convergence results are given and some estimates of the remainder are established in relation to the regularity of the «density» function. 相似文献
11.
It is well-known that for the ultraspherical weight function (1-x 2) λ-1/2 there exist no Chebyshev quadrature formulae in the strict sense having n nodes, where n is sufficiently large and λ>0, whereas
on the other hand for λ=0 every Gaussian quadrature formulae is a Chebyshev formula in the strict sense. In this paper we
study the open question of Chebyshev quadrature for λ <0. It is shown that there exists no Chebyshev quadrature formula in
the strict sense having more than two nodes for λ≤λ 0=-.30056... (for definition of λ 0 see (1.8) below). Moreover, results are obtained for Chebyshev-type formulae and Chebyshev formulae of closed type. For the
remaining values of λ (λ 0<λ<0) numerical results are given. 相似文献
12.
We present a method for discretizing and solving general elliptic partial differential equations on sparse grids employing higher order finite elements. On the one hand, our approach is charactarized by its simplicity. The calculation of the occurring functionals is composed of basic pointwise or unidirectional algorithms. On the other hand, numerical experiments prove our method to be robust and accurate. Discontinuous coefficients can be treated as well as curvilinearly bounded domains. When applied to adaptively refined sparse grids, our discretization results to be highly efficient, yielding balanced errors on the computational domain. 相似文献
13.
O ( h
n
−1
n
d
−1) instead of O( h
n
−d
) grid points and unknowns are involved. Here d denotes the dimension of the feature space and h
n
= 2
−n
gives the mesh size. To be precise, we suggest to use the sparse grid combination technique [42] where the classification
problem is discretized and solved on a certain sequence of conventional grids with uniform mesh sizes in each coordinate direction.
The sparse grid solution is then obtained from the solutions on these different grids by linear combination. In contrast to
other sparse grid techniques, the combination method is simpler to use and can be parallelized in a natural and straightforward
way. We describe the sparse grid combination technique for the classification problem in terms of the regularization network
approach. We then give implementational details and discuss the complexity of the algorithm. It turns out that the method
scales only linearly with the number of instances, i.e. the amount of data to be classified. Finally we report on the quality
of the classifier built by our new method. Here we consider standard test problems from the UCI repository and problems with
huge synthetical data sets in up to 9 dimensions. It turns out that our new method achieves correctness rates which are competitive
to that of the best existing methods.
Received April 25, 2001 相似文献
14.
In this paper, we describe a matrix-free iterative algorithm based on the GMRES method for solving electromagnetic scattering
problems expressed in an integral formulation. Integral methods are an interesting alternative to differential equation solvers
for this problem class since they do not require absorbing boundary conditions and they mesh only the surface of the radiating
object giving rise to dense and smaller linear systems of equations. However, in realistic applications the discretized systems
can be very large and for some integral formulations, like the popular Electric Field Integral Equation, they become ill-conditioned
when the frequency increases. This means that iterative Krylov solvers have to be combined with fast methods for the matrix-vector
products and robust preconditioning to be affordable in terms of CPU time. In this work we describe a matrix-free two-grid
preconditioner for the GMRES solver combined with the Fast Multipole Method. The preconditioner is an algebraic two-grid cycle
built on top of a sparse approximate inverse that is used as smoother, while the grid transfer operators are defined using
spectral information of the preconditioned matrix. Experiments on a set of linear systems arising from real radar cross section
calculation in industry illustrate the potential of the proposed approach for solving large-scale problems in electromagnetism. 相似文献
15.
In this paper we show how, under minimal conditions, a combination extrapolation can be introduced for an adaptive sparse
grid. We apply this technique for the solution of a two-dimensional model singular perturbation problem, defined on the domain
exterior of a circle.
Received October 18, 1999 相似文献
16.
To improve the filtering effect of the sparse grid quadrature filter (SGQF) under non-Gaussian conditions, the Gaussian sum technique is introduced, and the Gaussian sum sparse grid quadrature filter (GSSGQF) is developed. We present a systematic formulation of the SGQF and extend it to the discrete-time nonlinear system with the non-Gaussian noise. The proposed algorithm approximates the non-Gaussian probability densities by a finite number of weighted sums of Gaussian densities, and takes the SGQF as the Gaussian sub-filter to conduct the time and measurement update for each Gaussian component. An application in the discrete-time nonlinear system with the non-Gaussian noise has been shown to demonstrate the accuracy of the GSSGQF. It outperforms the unscented Kalman filter (UKF), the cubature Kalman filter (CKF) and the SGQF. Theoretical analysis and simulation results prove that the GSSGQF provides significant performance improvement in the calculation accuracy for nonlinear non-Gaussian filtering problems. 相似文献
17.
In this paper we introduce the Boundary Element Tearing and Interconnecting (BETI) methods as boundary element counterparts of the well-established Finite Element Tearing and Interconnecting (FETI) methods. In some practical important applications such as far field computations, handling of singularities and moving parts etc., BETI methods have certainly some advantages over their finite element counterparts. This claim is especially true for the sparse versions of the BETI preconditioners resp. methods. Moreover, there is an unified framework for coupling, handling, and analyzing both methods. In particular, the FETI methods can benefit from preconditioning components constructed by boundary element techniques. The first numerical results confirm the efficiency and the robustness predicted by our analysis. 相似文献
18.
In this paper, to obtain an efficient parallel algorithm to solve sparse block-tridiagonal linear systems, stair matrices are used to construct some parallel polynomial approximate inverse preconditioners. These preconditioners are suitable when the desired goal is to maximize parallelism. Moreover, some theoretical results concerning these preconditioners are presented and how to construct preconditioners effectively for any nonsingular block tridiagonal H-matrices is also described. In addition, the validity of these preconditioners is illustrated with some numerical experiments arising from the second order elliptic partial differential equations and oil reservoir simulations. 相似文献
19.
We consider the quadrature method developed by Kravanja and Van Barel (Computing 63(1):69–91, 1999) for computing all the zeros of a holomorphic function that lie inside the unit circle. The algorithm uses only the function values and no (first or higher order) derivatives. Information about the location of the zeros is obtained from certain integrals along the unit circle. In numerical computations these are replaced by their trapezoidal rule approximations. We investigate the resulting quadrature error. Our error analysis shows that the zeros located inside the unit circle do not affect the accuracy of the computed approximations whereas the quadrature error related to the zeros located outside the unit circle tends to zero exponentially as the number of quadrature points tends to infinity. 相似文献
20.
The application of Powell-Sabin’s or Clough-Tocher’s schemes to scattered data problems, as known requires the knowledge of
the partial derivatives of first order at the vertices of an underlying triangulation. We study a local method for generating partial derivatives based on the minimization of the energy functional on the star of triangles sharing a
node that we called a cell. The functional is associated to some piecewise polynomial function interpolating the points. The proposed method combines
the global Method II by Renka and Cline (cf. [16, pp. 230–231]) with the variational approach suggested by Alfeld (cf. [2]) with care to efficiency
in the computations. The locality together with some implementation strategies produces a method well suited for the treatment
of a big amount of data. An improvement of the estimates is also proposed. 相似文献
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