Showing papers on "Sparse grid published in 2003"

Dimension-adaptive tensor-product quadrature

Abstract: 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.

Adaptive sparse grids

Abstract: Sparse grids, as studied by Zenger and Griebel in the last 10 years have been very successful in the solution of partial differential equations, integral equations and classification problems. Adaptive sparse grid functions are elements of a function space lattice. Such lattices allow the generalisation of sparse grid techniques to the fitting of very high-dimensional functions with categorical and continuous variables. We have observed in first tests that these general adaptive sparse grids allow the identification of the ANOVA structure and thus provide comprehensible models. This is very important for data mining applications. Perhaps the main advantage of these models is that they do not include any spurious interaction terms and thus can deal with very high dimensional data.

Multivariate quadrature on adaptive sparse grids

Abstract: In this paper, we study the potential of adaptive sparse grids for multivariate numerical quadrature in the moderate or high dimensional case, i.e. for a number of dimensions beyond three and up to several hundreds. There, conventional methods typically suffer from the curse of dimension or are unsatisfactory with respect to accuracy. Our sparse grid approach, based upon a direct higher order discretization on the sparse grid, overcomes this dilemma to some extent, and introduces additional flexibility with respect to both the order of the 1 D quadrature rule applied (in the sense of Smolyak's tensor product decomposition) and the placement of grid points. The presented algorithm is applied to some test problems and compared with other existing methods.

Sparse finite elements for stochastic elliptic problems: higher order moments

Abstract: We define the higher order moments associated to the stochastic solution of an elliptic BVP in D ⊂ Rd with stochastic input data. We prove that the k-th moment solves a deterministic problem in Dk ⊂ Rdk, for which we discuss well-posedness and regularity. We discretize the deterministic k-th moment problem using sparse grids and, exploiting a spline wavelet basis, we propose an efficient algorithm, of logarithmic-linear complexity, for solving the resulting system.

Higher order sparse grid methods for elliptic partial differential equations with variable coefficients

Abstract: 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.

Parallelisation of sparse grids for large scale data analysis

Abstract: Sparse Grids (SG), due to Zenger, are the basis for efficient high dimensional approximation and have recently been applied successfully to predictive modelling They are spanned by a collection of simpler function spaces represented by regular grids The combination technique prescribes how approximations on simple grids can be combined to approximate the high dimensional functions It can be improved by iterative refinement Fitting sparse grids admits the exploitation of parallelism at various stages The fit can be done entirely by fitting partial models on regular grids This allows parallelism over the partial grids In addition, each of the partial grid fits can be parallelised as well, both in the assembly phase where parallelism is done over the data and in the solution stage using traditional parallel solvers for the resulting PDEs A simple timing model confirms that the most effective methods are obtained when both types of parallelism are used

Multiscale Methods for the Simulation of Turbulent Flows

Abstract: In this paper we apply the finite difference method on adaptive sparse grids [9] to the simulation of turbulent flows. This method combines the flexibility and efficiency of finite difference schemes with the advantages of an adaptive approximation by tensor product multiscale bases. We shortly discuss the method. Then, we present numerical results for a simple linear convection problem for a validation of our scheme. Finally, results for three-dimensional turbulent shear layers are shown.

Additive sparse grid fitting

Abstract: We propose an iterative algorithm for high-dimensional sparse grid regression with penalty. The algorithm is of additive Schwarz type and is thus parallel. We show that the convergence of the algorithm is better than additive Schwarz and examples demonstrate that convergence is between that of additive Schwarz and multiplicative Schwarz procedures. Similarly, the method shows improved performance compared to (additive) iterative methods based on the combination technique. §

Multigrid for discrete differential forms on sparse grids

Abstract: Discrete differential forms are a generalization of the common H1 (Ω)-conforming Lagrangian elements. For the latter, Galerkin schemes based on sparse grids are well known, and so are fast iterative multilevel solvers for the discrete Galerkin equations. We extend both the sparse grid idea and the design of multilevel methods to arbitrary discrete differential forms. The focus of this presentation will be on issues of efficient implementation and numerical studies of convergence of multigrid solvers.

Mixed finite elements on sparse grids

Abstract: This paper generalizes the idea of approximation on sparse grids to discrete differential forms that include \(\vec H({\rm div}; \Omega\))- and \(\vec (H{\bf curl}; \Omega)\)-conforming mixed finite element spaces as special cases. We elaborate on the construction of the spaces, introduce suitable nodal interpolation operators on sparse grids and establish their approximation properties. We discuss how nodal interpolation operators can be approximated. The stability of \(\vec H({\rm div}; \Omega)\)-conforming finite elements on sparse grids, when used to approximate second order elliptic problems in mixed formulation, is investigated both theoretically and in numerical experiments.

A new approach to analysing spatial data using sparse grids

Abstract: We describe in this paper a new data mining approach for the analysis of spatial data for environmental modelling. The sparse grids analysis system models the functional relationship between a set of predictor variables and a response variable by using a combination of easily computable functions defined on grids of varying mesh sizes in attribute space. The approach circumvents the so-called “curse of dimensionality” by using, instead of a costly high-dimensional grid a with a fine mesh size in every dimension, a collection of grids that are coarse along some dimensions but fine along others. Adaptive sparse grid regression and classification methods select combinations of grids that suit a particular data set. One advantage of the sparse grids approach from an environmental analysis perspective is that it uses machine learning approaches, and so can deal with correlated data, as are common in environmental problems. One advantage of the sparse grids approach from an environmental analysis perspective is that it uses machine learning approaches, and so can deal with correlated data, as is commonly the case with geographic data. They also require fewer degrees of freedom than do full grid models, allowing them to be applied to more datasets. The parameters defining the adaptive sparse grids can be used to interpret relationships in terms of scale and resolution. For example, the distribution of mesh points used in the set of lattices describes the complexity of the relationships present. It can be used to understand if the system is responding to fine scale variations (many mesh points used) or to gross patterns (few mesh points used). This is valuable information for environmental modelling.