Sparse Grid Quadrature

Holtz, Markus

doi:10.1007/978-3-642-16004-2_4

Cited by 8 publications

(7 citation statements)

References 24 publications

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“…Their results yield a convergence rate of order −1/2 for the related two-particle correlation functions, compared to a rate of order −1/4 when applying a linear approximation scheme. Moreover, our scheme might be extended to varying weights and thereby subspaces in an adaptive way (particle-wise adaptivity) similar to dimension-wise adaptive approaches for high-dimensional quadrature [34,35]. Also, an adaptive scheme which applies multiscale frames including ridgelet-like two-particle functions [36], the application of two-particle functions for the electron-electron cusps similar to the R12/F12 methods [37] and the extended geminal model [38] could be promising for the future.…”

Section: Discussionmentioning

confidence: 99%

Tensor Product Multiscale Many-Particle Spaces with Finite-Order Weights for the Electronic Schrödinger Equation

Griebel¹,

Hamaekers²

2010

Zeitschrift Für Physikalische Chemie

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In this article we combine the favorable properties of efficient Gaussian type orbitals basis sets, which are applied with good success in conventional electronic structure methods, and tensor product multiscale bases, which provide guaranteed convergence rates and allow for adaptive resolution. To this end, we develop and study a new approach for the treatment of the electronic Schrödinger equation based on a modified adaptive sparse grid technique and a certain particle-wise decomposition with respect to oneparticle functions obtained by a nonlinear rank-1 approximation. Here, we employ a multiscale Gaussian frame for the sparse grid spaces and we use Gaussian type orbitals to represent the rank-1 approximation. With this approach we are able to treat small atoms and molecules with up to six electrons.2

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Section: Discussionmentioning

confidence: 99%

Tensor Product Multiscale Many-Particle Spaces with Finite-Order Weights for the Electronic Schrödinger Equation

Griebel¹,

Hamaekers²

2010

Zeitschrift Für Physikalische Chemie

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“…Besides basic science and engineering, high-dimensional integrals are abundant in financial mathematics. The price of financial derivatives is evaluated as an expectation value over a multi-dimensional space of hundreds of random variables, which are the sources of uncertainty captured by a financial model [26,52,63]. Other relevant areas include dynamic reinforcement learning [14,28] and data mining techniques [18].…”

Section: Introductionmentioning

confidence: 99%

“…This is similar to the technique used in atomic ensembles with a short-range interaction, where the total energy of a system can be written as a sum of the energies of individual atoms, which depend on the neighbors of the atoms [67]. Other attempts to efficiently compute high-dimensional integrals involve the use of sparse grids [21,26] in the phase space and moment approaches [6], which show the asymptotic convergence of the upper and lower bounds of the integral with increasing order of moments. Finally, works in the late 1990's [13,27] started suggesting that probability distributions in high dimensions have random points clustered around a lower-dimensional geometric shape, and hence constructing such data points for machine learning is a simple alternative [20].…”

Section: Introductionmentioning

confidence: 99%

GNN-Assisted Phase Space Integration with Application to Atomistics

Saxena¹,

Bastek²,

Miguel³

et al. 2023

Preprint

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Overcoming the time scale limitations of atomistics can be achieved by switching from the statespace representation of Molecular Dynamics (MD) to a statistical-mechanics-based representation in phase space, where approximations such as maximum-entropy or Gaussian phase packets (GPP) evolve the atomistic ensemble in a time-coarsened fashion. In practice, this requires the computation of expensive high-dimensional integrals over all of phase space of an atomistic ensemble. This, in turn, is commonly accomplished efficiently by low-order numerical quadrature. We show that numerical quadrature in this context, unfortunately, comes with a set of inherent problems, which corrupt the accuracy of simulations-especially when dealing with crystal lattices with imperfections. As a remedy, we demonstrate that Graph Neural Networks, trained on Monte-Carlo data, can serve as a replacement for commonly used numerical quadrature rules, overcoming their deficiencies and significantly improving the accuracy. This is showcased by three benchmarks: the thermal expansion of copper, the martensitic phase transition of iron, and the energy of grain boundaries. We illustrate the benefits of the proposed technique over classically used third-and fifth-order Gaussian quadrature, we highlight the impact on time-coarsened atomistic predictions, and we discuss the computational efficiency. The latter is of general importance when performing frequent evaluation of phase space or other high-dimensional integrals, which is why the proposed framework promises applications beyond the scope of atomistics.

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“…Either by focusing on qualitative results obtained from extremely simplified models with little heterogene- [3] analyze the welfare implications of social security reform in a model where one period corresponds to six years, thereby reducing the number of adult cohorts and thus the dimensionality of the problem by a factor of six. 20 Similarly, international real business cycle (IRBC) models often include only a very small number of countries or regions. Bengui et al [4], for example, analyze crosscountry risk-sharing at the business cycle frequency using a two country model -one 'focus' country versus the rest of the world.…”

Section: Introductionmentioning

confidence: 99%

“…Sparse grids reduce the number of grid points needed from the order O N d to O N · (log N) d−1 , while the accuracy of the interpolation only slightly deteriorates in the case of sufficiently 60 smooth functions [8]. Sparse grids go back to Smolyak [14] and have been applied to a whole range of different research fields such as physics, visualization, data mining, Hamilton-Jacobi Bellman (HJB) equations, mathematical finance, insurance, and econometrics [15,16,8,17,18,19,20,21,22]. Krueger and Kubler [3] and Judd et al [23] solve dynamic economic models using sparse grids with global polynomials 65 as basis functions.…”

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confidence: 99%

Scalable high-dimensional dynamic stochastic economic modeling

Brumm

Mikushin

Scheidegger

et al. 2015

Journal of Computational Science

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We present a highly parallelizable and flexible computational method to solve high-dimensional stochastic dynamic economic models. Solving such models often requires the use of iterative methods, like time iteration or dynamic programming. By exploiting the generic iterative structure of this broad class of economic problems, we propose a parallelization scheme that favors hybrid massively parallel computer architectures. Within a parallel nonlinear time iteration framework, we interpolate policy functions partially on GPUs using an adaptive sparse grid algorithm with piecewise linear hierarchical basis functions. GPUs accelerate this part of the computation one order of magnitude thus reducing overall computation time by 50%. The developments in this paper include the use of a fully adaptive sparse grid algorithm and the use of a mixed MPI-Intel TBB-CUDA/Thrust implementation to improve the interprocess communication strategy on massively parallel architectures. Numerical experiments on "Piz Daint" (Cray XC30) at the Swiss National Supercomputing Centre show that high-dimensional international real business cycle models can be efficiently solved in parallel. To our knowledge, this performance on a massively parallel petascale architecture for such nonlinear high-dimensional economic models has not been possible prior to present work. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractWe present a highly parallelizable and flexible computational method to solve highdimensional stochastic dynamic economic models. Solving such models often requires the use of iterative methods, like time iteration or dynamic programming. By exploiting the generic iterative structure of this broad class of economic problems, we propose a parallelization scheme that favors hybrid massively parallel computer architectures.Within a parallel nonlinear time iteration framework, we interpolate policy functions partially on GPUs using an adaptive sparse grid algorithm with piecewise linear hierarchical basis functions. GPUs accelerate this part of the computation one order of magnitude thus reducing overall computation time by 50%. The developments in this paper include the use of a fully adaptive sparse grid algorithm and the use of a mixed MPI-Intel TBB-CUDA/Thrust implementation to improve the interprocess communication strategy on massively parallel architectures. Numerical experiments on "Piz Daint" (Cray XC30) at the Swiss National Supercomputing Centre show that highdimensional international real business cycle models can be efficiently solved in parallel. To our knowledge, this performance on a massively ...

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Sparse Grid Quadrature

Cited by 8 publications

References 24 publications

Tensor Product Multiscale Many-Particle Spaces with Finite-Order Weights for the Electronic Schrödinger Equation

Tensor Product Multiscale Many-Particle Spaces with Finite-Order Weights for the Electronic Schrödinger Equation

GNN-Assisted Phase Space Integration with Application to Atomistics

Scalable high-dimensional dynamic stochastic economic modeling

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