Results 1  10
of
18
Computer Experiments
, 1996
"... Introduction Deterministic computer simulations of physical phenomena are becoming widely used in science and engineering. Computers are used to describe the flow of air over an airplane wing, combustion of gasses in a flame, behavior of a metal structure under stress, safety of a nuclear reactor, a ..."
Abstract

Cited by 119 (6 self)
 Add to MetaCart
Introduction Deterministic computer simulations of physical phenomena are becoming widely used in science and engineering. Computers are used to describe the flow of air over an airplane wing, combustion of gasses in a flame, behavior of a metal structure under stress, safety of a nuclear reactor, and so on. Some of the most widely used computer models, and the ones that lead us to work in this area, arise in the design of the semiconductors used in the computers themselves. A process simulator starts with a data structure representing an unprocessed piece of silicon and simulates the steps such as oxidation, etching and ion injection that produce a semiconductor device such as a transistor. A device simulator takes a description of such a device and simulates the flow of current through it under varying conditions to determine properties of the device such as its switching speed and the critical voltage at which it switches. A circuit simulator takes a list of devices and the
Numerical Integration using Sparse Grids
 NUMER. ALGORITHMS
, 1998
"... We present and review algorithms for the numerical integration of multivariate functions defined over ddimensional cubes using several variants of the sparse grid method first introduced by Smolyak [51]. In this approach, multivariate quadrature formulas are constructed using combinations of tensor ..."
Abstract

Cited by 82 (16 self)
 Add to MetaCart
(Show Context)
We present and review algorithms for the numerical integration of multivariate functions defined over ddimensional cubes using several variants of the sparse grid method first introduced by Smolyak [51]. In this approach, multivariate quadrature formulas are constructed using combinations of tensor products of suited onedimensional 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 onedimensional basis of the construction and show their superiority in comparison to previously used sparse grid approaches based on the trapezoidal, ClenshawCurtis and Gauss rules in several numerical experiments and applications.
Sparse grids and related approximation schemes for higher dimensional problems
"... The efficient numerical treatment of highdimensional problems is hampered by the curse of dimensionality. We review approximation techniques which overcome this problem to some extent. Here, we focus on methods stemming from Kolmogorov’s theorem, the ANOVA decomposition and the sparse grid approach ..."
Abstract

Cited by 45 (12 self)
 Add to MetaCart
(Show Context)
The efficient numerical treatment of highdimensional problems is hampered by the curse of dimensionality. We review approximation techniques which overcome this problem to some extent. Here, we focus on methods stemming from Kolmogorov’s theorem, the ANOVA decomposition and the sparse grid approach and discuss their prerequisites and properties. Moreover, we present energynorm based sparse grids and demonstrate that, for functions with bounded mixed derivatives on the unit hypercube, the associated approximation rate in terms of the involved degrees of freedom shows no dependence on the dimension at all, neither in the approximation order nor in the order constant.
Transfinite Surface Interpolation
 THE MATHEMATICS OF SURFACES VI
, 1996
"... A transfinite surface interpolant is one which matches given curves exactly, in contrast to a finitedimensional one, which matches a finite number of conditions. This paper ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
A transfinite surface interpolant is one which matches given curves exactly, in contrast to a finitedimensional one, which matches a finite number of conditions. This paper
Principal Manifold Learning by Sparse Grids
, 2008
"... In this paper we deal with the construction of lowerdimensional manifolds from highdimensional data which is an important task in data mining, machine learning and statistics. Here, we consider principal manifolds as the minimum of a regularized, nonlinear empirical quantization error functional. ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
(Show Context)
In this paper we deal with the construction of lowerdimensional manifolds from highdimensional data which is an important task in data mining, machine learning and statistics. Here, we consider principal manifolds as the minimum of a regularized, nonlinear empirical quantization error functional. For the discretization we use a sparse grid method in latent parameter space. This approach avoids, to some extent, the curse of dimension of conventional grids like in the GTM approach. The arising nonlinear problem is solved by a descent method which resembles the expectation maximization algorithm. We present our sparse grid principal manifold approach, discuss its properties and report on the results of numerical experiments for one, two and threedimensional model problems.
A rational cubic spline with tension
 Computer Aided Geometric Design
, 1990
"... Abstract. A rational cubic spline curve is described which has tension control parameters for manipulating the shape of the curve. The spline is presented in both interpolatory and rational Bspline forms, and the behaviour of the resulting representations is analysed with respect to ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Abstract. A rational cubic spline curve is described which has tension control parameters for manipulating the shape of the curve. The spline is presented in both interpolatory and rational Bspline forms, and the behaviour of the resulting representations is analysed with respect to
Reconstruction of G 1 Surfaces with Biquartic patches for hp FE Simulations. 13th International Meshing Roundtable 2004
"... We present an efficient G 1 surface reconstruction scheme for complex solid models used in F E simulations. A novel technique based on low geometric degree (biquartic) polynomial interpolation is proposed to construct a smooth surface on arbitrary unstructured(irregular) rectangular meshes. A suitab ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
We present an efficient G 1 surface reconstruction scheme for complex solid models used in F E simulations. A novel technique based on low geometric degree (biquartic) polynomial interpolation is proposed to construct a smooth surface on arbitrary unstructured(irregular) rectangular meshes. A suitable parametric representation of surface as well as local control of individual rectangular patches is achieved via simultaneous surface fitting of a curve network with corresponding cubic normals. Necessary compatibility conditions are formulated, and proved to satisfy the tangent plane continuity and vertex enclosure constraints.
AN ARTIFICIAL NEURAL NETWORK METHOD FOR SOLVING BOUNDARY VALUE PROBLEMS WITH ARBITRARY IRREGULAR BOUNDARIES
, 2006
"... ..."
Variablefree representation of manifolds via transfinite blending with a functional language
 EDS.) SPRINGERVERLAG, LECTURE NOTES IN COMPUTER SCIENCE
"... In this paper a variablefree parametric representation of manifolds is discussed, using transfinite interpolation or approximation, i.e. function blending in some functional space. This is a power ful appr oach to gener ion of cur es,sur aces and solids (and even higher dimensional manifolds) by bl ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
In this paper a variablefree parametric representation of manifolds is discussed, using transfinite interpolation or approximation, i.e. function blending in some functional space. This is a power ful appr oach to gener ion of cur es,sur aces and solids (and even higher dimensional manifolds) by blending lower dimensional vector valued functions. Tr ansfinite blending, e.g. used in Gor donCoons patches, is well known to mathematicians and CAD people. It is pr esented her e in a ver y simple conceptual and computational fr amewor k, which leads such a power ful modeling to be easily handled even by the non mathematically sophisticated user of gr aphics techniques. In particular, transfinite blending is discussed in this paper by making use of a very powerful and simple functional language for geometric design.