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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 ..."
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Cited by 67 (5 self)
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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 ..."
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Cited by 40 (16 self)
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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 ..."
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Cited by 24 (12 self)
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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.
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. ..."
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Cited by 2 (2 self)
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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.
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 ..."
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Cited by 2 (1 self)
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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.
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 ..."
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Cited by 1 (1 self)
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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.
AN ARTIFICIAL NEURAL NETWORK METHOD FOR SOLVING BOUNDARY VALUE PROBLEMS WITH ARBITRARY IRREGULAR BOUNDARIES Approved by:
, 2006
"... Acknowledgements This dissertation would never have come about without the support and direction of Dr. J. Robert Mahan. No one could hope for a more engaged advisor and better role model. The author owes a debt of gratitude to the Conseil Régional de Lorraine for its generous financial support of t ..."
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Cited by 1 (0 self)
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Acknowledgements This dissertation would never have come about without the support and direction of Dr. J. Robert Mahan. No one could hope for a more engaged advisor and better role model. The author owes a debt of gratitude to the Conseil Régional de Lorraine for its generous financial support of this research at the European Campus of the Georgia Institute of Technology, located in Metz, France. And although never directly involved in this work, my family – including a certain dangerous redhead – has perhaps made the largest contribution through years of constant and unwavering love and support. iii
unknown title
, 1993
"... A new set of benchmarks has been developed for the performance evaluation of highly parallel supercomputers. These benchmarks consist of a set of kernels, the "Parallel Kernels, " and a simulated application benchmark. Together they mimic the computation and data movement characteristics o ..."
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A new set of benchmarks has been developed for the performance evaluation of highly parallel supercomputers. These benchmarks consist of a set of kernels, the "Parallel Kernels, " and a simulated application benchmark. Together they mimic the computation and data movement characteristics of large scale computational fluid dynamics (CFD) applications. The principal distinguishing feature of these benchmarks is their "pencil and paper " specificationall details of these benchmarks are specified only algorithmically. In this way many of the difficulties associated with conventional benchmarking approaches on highly parallel systems are avoided.