Sparse Grid and Extrapolation Methods for Parabolic Problems

Balder, Robert; Rüde, Ulrich; Schneider, S.; Zenger, Christoph

doi:10.1007/978-94-010-9204-3_167

Cited by 4 publications

(5 citation statements)

References 2 publications

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“…In the pioneering work of Zenger (1991) and Griebel (1991b), the foundations for adaptive refinement, multilevel solvers, and parallel algorithms for sparse grids were laid. Subsequent studies included the solution of the 3D Poisson equation Bungartz (1992aBungartz ( , 1992b, the generalization to arbitrary dimensionality d (Balder 1994) and to more general equations (the Helmholtz equation (Balder and Zenger 1996), parabolic problems using a time-space discretization (Balder, Rüde, Schneider and Zenger 1994), the biharmonic equation (Störtkuhl 1995), and general linear elliptic operators of second order in 2D (Pflaum 1996, Dornseifer andPflaum 1996). As a next step, the solution of general linear elliptic differential equations and, via mapping techniques, the treatment of more general geometries was implemented Dornseifer 1998, Dornseifer 1997) (see Figure 4.13).…”

Section: Sparse Grid Applications Pde Discretization Techniquesmentioning

confidence: 99%

Sparse grids

Bungartz¹,

Griebel²

2004

Acta Numerica

1,017

825

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We present a survey of the fundamentals and the applications of sparse grids, with a focus on the solution of partial differential equations (PDEs). The sparse grid approach, introduced in Zenger (1991), is based on a higherdimensional multiscale basis, which is derived from a one-dimensional multiscale basis by a tensor product construction. Discretizations on sparse grids involve O(N ·(log N ) d−1 ) degrees of freedom only, where d denotes the underlying problem's dimensionality and where N is the number of grid points in one coordinate direction at the boundary. The accuracy obtained with piecewise linear basis functions, for example, is O(N −2 · (log N ) d−1 ) with respect to the L 2 -and L ∞ -norm, if the solution has bounded second mixed derivatives. This way, the curse of dimensionality, i.e., the exponential dependence O(N d ) of conventional approaches, is overcome to some extent. For the energy norm, only O(N ) degrees of freedom are needed to give an accuracy of O(N −1 ). That is why sparse grids are especially well-suited for problems of very high dimensionality.The sparse grid approach can be extended to nonsmooth solutions by adaptive refinement methods. Furthermore, it can be generalized from piecewise linear to higher-order polynomials. Also, more sophisticated basis functions like interpolets, prewavelets, or wavelets can be used in a straightforward way.We describe the basic features of sparse grids and report the results of various numerical experiments for the solution of elliptic PDEs as well as for other selected problems such as numerical quadrature and data mining. CONTENTS1 Introduction 148 2 Breaking the curse of dimensionality 151 3 Piecewise linear interpolation on sparse grids 154 4 Generalizations, related concepts, applications 188 5 Numerical experiments 219 6 Concluding remarks 255 References 256 Breaking the curse of dimensionalityClassical approximation schemes exhibit the curse of dimensionality (Bellmann 1961) mentioned above. We havewhere r and d denote the isotropic smoothness of the function f and the problem's dimensionality, respectively. This is one of the main obstacles in the treatment of high-dimensional problems. Therefore, the question is whether we can find situations, i.e., either function spaces or error norms, for which the curse of dimensionality can be broken. At first glance, there is an easy way out: if we make a stronger assumption on the smoothness of the function f such thatOf course, such an assumption is completely unrealistic. However, about ten years ago, Barron (1993) found an interesting result. Denote by FL 1 the class of functions with Fourier transforms in L 1 . Then, consider the class of functions of R d with ∇f ∈ FL 1 .We expect an approximation rateMeanwhile, other function classes are known with such properties. These comprise certain radial basis schemes, stochastic sampling techniques, and approaches that work with spaces of functions with bounded mixed derivatives.A better understanding of these results is possible with the help of harmon...

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Section: Sparse Grid Applications Pde Discretization Techniquesmentioning

confidence: 99%

Sparse grids

Bungartz¹,

Griebel²

2004

Acta Numerica

1,017

825

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“…This potential was illustrated in several studies [16][17][18][19][20]. The basis of all sparse grids is the famous Smolyak's method [21], which provides a construction of interpolation functions with a minimum number of points in multi-dimensional space and extends adequately the univariate interpolation formulas to the multivariate case.…”

Section: Problem Descriptionmentioning

confidence: 99%

Multicriteria shape design of a sheet contour in stamping

Oujebbour

Habbal

Ellaia

et al. 2014

Journal of Computational Design and Engineering

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International audienceOne of the hottest challenges in automotive industry is related to weight reduction in sheet metal forming processes, in order to produce a high quality metal part with minimal material cost. Stamping is the most widely used sheet metal forming process; but its implementation comes with several fabrication flaws such as springback and failure. A global and simple approach to circumvent these unwanted process drawbacks consists in optimizing the initial blank shape with innovative methods. The aim of this paper is to introduce an efficient methodology to deal with complex, computationally expensive multicriteria optimization problems. Our approach is based on the combination of methods to capture the Pareto Front, approximate criteria (to save computational costs) and global optimizers. To illustrate the efficiency, we consider the stamping of an industrial workpiece as test-case. Our approach is applied to the springback and failure criteria. To optimize these two criteria, a global optimization algorithm was chosen. It is the Simulated Annealing algorithm hybridized with the Simultaneous Perturbation Stochastic Approximation in order to gain in time and in precision. The multicriteria problems amounts to the capture of the Pareto Front associated to the two criteria. Normal Boundary Intersection and Normalized Normal Constraint Method are considered for generating a set of Pareto-optimal solutions with the characteristic of uniform distribution of front points. The computational results are compared to those obtained with the well-known Non-dominated Sorting Genetic Algorithm II. The results show that our proposed approach is efficient to deal with the multicriteria shape optimization of highly non-linear mechanical systems

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“…This suggests to replace the l-form ω by its interpolant I (1) n ω first [7]. However, in general, for l > 0, it is impossible to determine the exact interpolant, because of the integrals that occur in the definition (4.2) of the degrees of freedom.…”

Section: Approximate Interpolationmentioning

confidence: 99%

“…Given the smooth differential l-form ω by its components u I (which are smooth functions), we aim to compute an appropriate replacement for I (1) n ω with minimal effort, i.e. with a number of operations proportional to the dimension of the sparse grid space.…”

Section: Approximate Interpolationmentioning

confidence: 99%

“…Meanwhile, there are plenty of applications of sparse grids in the discretization of partial differential equations: Poisson's equation [6,7,30], the Helmholtz equation [2], parabolic problems [1], general linear elliptic operator of second order in two dimensions [14] were treated. Adaptive sparse grid techniques were investigated, too [6,17,20,31].…”

Section: Introductionmentioning

confidence: 99%

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Mixed finite elements on sparse grids

Gradinaru

Hiptmair

2003

Numerische Mathematik

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This paper generalizes the idea of approximation on sparse grids to discrete differential forms that include H(div; Ω)-and H(curl; Ω)-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 H(div; Ω)-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. (2000): 65N30 41A10 58A15 Mathematics Subject Classification

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Sparse Grid and Extrapolation Methods for Parabolic Problems

Cited by 4 publications

References 2 publications

Sparse grids

Sparse grids

Multicriteria shape design of a sheet contour in stamping

Mixed finite elements on sparse grids

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