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检索条件"丛书名=Lecture Notes in Computational Science and Engineering 37"

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Pod-Galerkin modeling and sparse-grid collocation for a natural convection problem with stochastic boundary conditions 1

引用

2nd Workshop on Sparse Grids and Applications, SGA 2012

作者： Ullmann, Sebastian Lang, Jens Department of Mathematics Technische Universität Darmstadt Dolivostraße 15 Darmstadt64293 Germany

ISBN: (数字)9783319045375

ISBN: (纸本)9783319045368

The computationally most expensive part of the stochastic collocation method are usually the numerical solutions of a deterministic equation at the collocation points. We propose a way to reduce the total computation time by replacing the deterministic model with its Galerkin projection on the space spanned by a small number of basis functions. The proper orthogonal decomposition (POD) is used to compute the basis functions from the solutions of the deterministic model at a few collocation points. We consider the computation of the statistics of the Nusselt number for a two-dimensional stationary natural convection problem with a stochastic temperature boundary condition. It turns out that for the estimation of the mean and the probability density, the number of finite element simulations can be significantly reduced by the help of the POD-Galerkin reduced-order model. © Springer International Publishing Switzerland 2014.

关键词： Boundary conditions

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Alternating direction method of multipliers for hierarchical basis approximators 1

引用

2nd Workshop on Sparse Grids and Applications, SGA 2012

作者： Khakhutskyy, Valeriy Pflüger, Dirk Institute for Advanced Study Technische Universität München Lichtenbergstrasse 2a Garching85748 Germany Institute for Parallel and Distributed Systems University of Stuttgart Universitätsstr. 3 Stuttgart70569 Germany

ISBN: (数字)9783319045375

ISBN: (纸本)9783319045368

Sparse grids have been successfully used for the mining of vast datasets with a moderate number of dimensions. Compared to established machine learning techniques like artificial neural networks or support vector machines, sparse grids provide an analytic approximant that is easier to analyze and to interpret. More important, they are based on a high-dimensional discretization of the feature space, are thus less data-dependent than conventional approaches, scale only linearly in the number of data points and are well-suited to deal with huge amounts of data. But with an increasing size of the datasets used for learning, computing times clearly can become prohibitively large for normal use, despite the linear scaling. Thus, efficient parallelization strategies have to be found to exploit the power of modern hardware. We investigate the parallelization opportunities for solving high-dimensional machine learning problems with adaptive sparse grids using the alternating direction method of multipliers (ADMM). ADMM allows us to split the initially large problem into smaller ones. They can then be solved in parallel while their reduced problem sizes can even be small enough for an explicitly assembly of the system matrices. We show the first results of the new approach using a set of problems and discuss the challenges that arise when applying ADMM to a hierarchical basis. © Springer International Publishing Switzerland 2014.

关键词： Neural networks

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Detecting edges in high order methods for hyperbolic conservation laws 1

引用

European Workshop on High Order Nonlinear Numerical Methods for Evolutionary PDEs, HONOM 2013

作者： Wirz, Martina Institut Computational Mathematics Technische Universität Braunschweig Pockelsstr. 14 Braunschweig38106 Germany

ISBN: (数字)9783319054551

ISBN: (纸本)9783319054544

In this paper the edge detection technique based on generalized conjugated Fourier partial sums is extended to the fully twodimensional case using concentration kernels. Several examples with different kernels are provided which show the improved edge detection property. Furthermore, one possibility how to apply this approach to a local method, namely the Spectral-Difference method with modal filtering in the Proriol-Koornwinder-Dubiner basis, is presented and compared with a common coefficient based shock indicator. Some examples which show a better shock indication using the new Fourier based approach are provided. © Springer International Publishing Switzerland 2014.

关键词： Edge detection

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An adaptive wavelet stochastic collocation method for irregular solutions of partial differential equations with random input data 1

引用

2nd Workshop on Sparse Grids and Applications, SGA 2012

作者： Gunzburger, Max Webster, Clayton G. Zhang, Guannan Department of Scientific Computing 400 Dirac Science Library Florida State University TallahasseeFL32306-4120 United States Computer Science and Mathematics Division Oak Ridge National Laboratory One Bethel Valley Road P.O. Box 2008 Oak RidgeTN37831-6164 United States

ISBN: (数字)9783319045375

ISBN: (纸本)9783319045368

A novel multi-dimensional multi-resolution adaptive wavelet stochastic collocation method (AWSCM) for solving partial differential equations with random input data is proposed. The uncertainty in the input data is assumed to depend on a finite number of random variables. In case the dimension of this stochastic domain becomes moderately large, we show that utilizing a hierarchical sparse grid AWSCM (sg-AWSCM) not only combats the curse of dimensionality but, in contrast to the standard sg-SCMs built from global Lagrange-type interpolating polynomials, maintains fast convergence without requiring sufficiently regular stochastic solutions. Instead, our non-intrusive approach extends the sparse-grid adaptive linear stochastic collocation method (sg-ALSCM) by employing a compactly supported wavelet approximation, with the desirable multi-scale stability of the hierarchical coefficients guaranteed as a result of the wavelet basis having the Riesz property. This property provides an additional lower bound estimate for the wavelet coefficients that are used to guide the adaptive grid refinement, resulting in the sg-AWSCM requiring a significantly reduced number of deterministic simulations for both smooth and irregular stochastic solutions. Second-generation wavelets constructed from a lifting scheme allows us to preserve the framework of the multi-resolution analysis, compact support, as well as the necessary interpolatory and Riesz property of the hierarchical basis. Several numerical examples are given to demonstrate the improved convergence of our numerical scheme and show the increased efficiency when compared to the sg-ALSCM method. © Springer International Publishing Switzerland 2014.

关键词： Partial differential equations

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High-order discontinuous Galerkin solution of unsteady flows by using an advanced implicit method 1

引用

European Workshop on High Order Nonlinear Numerical Methods for Evolutionary PDEs, HONOM 2013

作者： Nigro, Alessandra De Bartolo, Carmine Bassi, Francesco Ghidoni, Antonio Department of Mechanical Energetic and Management Engineering University of Calabria Ponte P. Bucci cubo 44/C RendeCS87036 Italy Department of Industrial Engineering University of Bergamo Viale Marconi 5 DalmineBG24044 Italy Department of Mechanical and Industrial Engineering University of Brescia Via Branze 38 BresciaBS25123 Italy

ISBN: (数字)9783319054551

ISBN: (纸本)9783319054544

The aim of this paper is to investigate and evaluate a multi-stage and multi-step method that is an evolution of the more common Backward Differentiation Formulae (BDF). This new class of formulae, called Two Implicit Advanced Step-point (TIAS), has been applied to a high-order Discontinuous Galerkin (DG) discretization of the Navier-Stokes equations, coupling the high temporal accuracy gained by the TIAS scheme with the high space accuracy of the DG method. The performance of the DG-TIAS scheme has been evaluated by means of two test cases: an inviscid isentropic convecting vortex and a laminar vortex shedding behind a circular cylinder. The advantages of the high-order time discretization are illustrated comparing the sixth-order accurate TIAS scheme with the second-order accurate BDF scheme using the same spatial discretization. © Springer International Publishing Switzerland 2014.

关键词： Circular cylinders

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Discontinuous Galerkin method and applications to fluid-structure interaction problems 1

引用

European Workshop on High Order Nonlinear Numerical Methods for Evolutionary PDEs, HONOM 2013

作者： Feistauer, Miloslav Česenek, Jan Hadrava, Martin Horáček, Jaromír Kosík, Adam Department of Numerical Mathematics Charles University in Prague Sokolovská 83 Praha 818675 Czech Republic Institute of Thermomechanics The Academy of Sciences of the Czech Republic v.v.i. Dolejškova 1402/5 Praha 8182 00 Czech Republic Aerospace Research and Test Establishment Beranových 130 Praha-Letňany199 05 Czech Republic

ISBN: (数字)9783319054551

ISBN: (纸本)9783319054544

The subject of the paper is the numerical simulation of viscous compressible flow in time dependent domains. The motion of the boundary of the domain occupied by the fluid is taken into account with the aid of the ALE (Arbitrary Lagrangian-Eulerian) formulation of the Navier-Stokes equations. The flow problem is coupled with the dynamical linear elasticity problem. Both problems are discretized in space by the discontinuous Galerkin (DG) finite element method using piecewise polynomial discontinuous approximations. The time discretization is carried out by the BDF scheme or the DG in time. The developed methods are tested by numerical experiments and applied to the solution of a fluid-structure interaction problem. © Springer International Publishing Switzerland 2014.

关键词： Navier Stokes equations

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Robust solutions to PDEs with multiple grids 1

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2nd Workshop on Sparse Grids and Applications, SGA 2012

作者： Harding, Brendan Hegland, Markus Australian National University CanberraACT0200 Australia

ISBN: (数字)9783319045375

ISBN: (纸本)9783319045368

In this paper we will discuss some approaches to fault-tolerance for solving partial differential equations. In particular we will discuss how one can combine the solution from multiple grids using ideas related to the sparse grid combination technique and multivariate extrapolation. By utilising the redundancy between the solutions on different grids we will demonstrate how this approach can be adapted for fault-tolerance. Much of this will be achieved by assuming error expansions and examining the extrapolation of these when various solutions from different grids are combined. © Springer International Publishing Switzerland 2014.

关键词： Fault tolerance

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Fast discrete fourier transform on generalized sparse grids 1

引用

2nd Workshop on Sparse Grids and Applications, SGA 2012

作者： Griebel, Michael Hamaekers, Jan Institute for Numerical Simulation University of Bonn Wegelerstr. 6 Bonn53115 Germany Fraunhofer Institute for Algorithms and Scientific Computing SCAI Schloss Birlinghoven Sankt Augustin53754 Germany

ISBN: (数字)9783319045375

ISBN: (纸本)9783319045368

In this paper, we present an algorithm for trigonometric interpolation of multivariate functions on generalized sparse grids and study its application for the approximation of functions in periodic Sobolev spaces of dominating mixed smoothness. In particular, we derive estimates for the error and the cost. We construct interpolants with a computational cost complexity which is substantially lower than for the standard full grid case. The associated generalized sparse grid interpolants have the same approximation order as the standard full grid interpolants, provided that certain additional regularity assumptions on the considered functions are fulfilled. Numerical results validate our theoretical findings. © Springer International Publishing Switzerland 2014.

关键词： Sobolev spaces

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On the convergence of the combination technique 1

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2nd Workshop on Sparse Grids and Applications, SGA 2012

作者： Griebel, Michael Harbrecht, Helmut Institut für Numerische Simulation Universität Bonn Wegelerstr. 6 Bonn53115 Germany Mathematisches Institut Universität Basel Rheinsprung 21 Basel4051 Switzerland

ISBN: (数字)9783319045375

ISBN: (纸本)9783319045368

Sparse tensor product spaces provide an efficient tool to discretize higher dimensional operator equations. The direct Galerkin method in such ansatz spaces may employ hierarchical bases, interpolets, wavelets or multilevel frames. Besides, an alternative approach is provided by the so-called combination technique. It properly combines the Galerkin solutions of the underlying problem on certain full (but small) tensor product spaces. So far, however, the combination technique has been analyzed only for special model problems. In the present paper, we provide now the analysis of the combination technique for quite general operator equations in sparse tensor product spaces. We prove that the combination technique produces the same order of convergence as the Galerkin approximation with respect to the sparse tensor product space. Furthermore, the order of the cost complexity is the same as for the Galerkin approach in the sparse tensor product space. Our theoretical findings are validated by numerical experiments. © Springer International Publishing Switzerland 2014.

关键词： Galerkin methods

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High order SFV and mixed SDG/FV methods for the uncertainty quantification in multidimensional conservation laws 1

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European Workshop on High Order Nonlinear Numerical Methods for Evolutionary PDEs, HONOM 2013

作者： Tokareva, Svetlana Schwab, Christoph Mishra, Siddhartha ASCOMP GmbH Zurich Switzerland Seminar for Applied Mathematics ETH Zurich Zurich Switzerland

ISBN: (数字)9783319054551

ISBN: (纸本)9783319054544

We analyze and apply the high order Stochastic Finite Volume (SFV) and mixed Stochastic Discontinuous Galerkin/Finite Volume (SDG/FV) methods used to quantify the uncertainty in hyperbolic conservation laws with random initial data and flux coefficients. We describe incomplete information in the conservation law mathematically as random fields. The SFVM is formulated to solve numerically the system of conservation laws with sources of randomness in both flux coefficients and initial data. We formulate the Stochastic Discontinuous Galerkin method which we primarily use to solve the multidimensional stochastic conservation laws on unstructured grids. Finally, we compare the efficiency of the SFV and SDG methods with of Monte-Carlo type methods. Finally, we introduce an adaptation technique based on the Karhunen-Loève expansion of the random flux and/or initial data and apply it in order to reduce the computational cost of the SFVM. © Springer International Publishing Switzerland 2014.

关键词： Galerkin methods

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