Spatially adaptive sparse grids for high-dimensional data-driven problems

Pflüger, Dirk; Peherstorfer, Benjamin; Bungartz, Hans‐Joachim

doi:10.1016/j.jco.2010.04.001

Cited by 140 publications

(204 citation statements)

References 20 publications

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“…Modified basis functions on the boundary. Looking more closely at the number of basis functions used for a regular sparse grid of level n, we observe that the ratio of points on the boundary versus that in the interior grows significantly with increasing dimensionality [25], i.e. more and more grid points are spent on ∂Ω.…”

Section: Sparse Gridsmentioning

confidence: 98%

“…Thus, the curse of dimensionality of full grid methods arises for sparse grids to a much smaller extent. In case the smoothness conditions are not fulfilled, spatially adaptive sparse grids have been used with good success [3,12,25]. There, as in any adaptive grid refinement procedure, the employed hierarchical basis functions are chosen during the actual computation depending on the function to be represented.…”

Section: Introductionmentioning

confidence: 99%

“…Note that the introduced sparse grid scheme is not monotone as the interpolation with sparse grids is not monotone [25]. Thus neither convergence towards the viscosity solutions of (1) nor stability can presently be guaranteed, even for the linear advection equation.…”

Section: Introductionmentioning

confidence: 99%

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An Adaptive Sparse Grid Semi-Lagrangian Scheme for First Order Hamilton-Jacobi Bellman Equations

et al. 2012

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ABSTRACT. We propose a semi-Lagrangian scheme using a spatially adaptive sparse grid to deal with non-linear time-dependent Hamilton-Jacobi Bellman equations. We focus in particular on front propagation models in higher dimensions which are related to control problems. We test the numerical efficiency of the method on several benchmark problems up to space dimension d = 8, and give evidence of convergence towards the exact viscosity solution. In addition, we study how the complexity and precision scale with the dimension of the problem.

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Section: Sparse Gridsmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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An Adaptive Sparse Grid Semi-Lagrangian Scheme for First Order Hamilton-Jacobi Bellman Equations

et al. 2012

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“…In such cases, the classical sparse grid methods A c c e p t e d M a n u s c r i p t outlined so far may fail to provide a good approximation. One effective way to overcome this problem is to adaptively refine the sparse grid in regions with high function variation and spend fewer points in regions of low variation (see, e.g., [8,9,26], and references therein). The working principle of the refinement strategy we use is to monitor the size of the hierarchical surpluses (see (21)), which reflect the local irregularity of the function.…”

Section: Adaptive Sparse Gridsmentioning

confidence: 99%

“…In contrast, we use piecewise-linear local basis functions first introduced by Zenger [24] in the context of sparse grids. The hierarchical structure of these basis functions lends itself for an adaptive refinement strategy as, e.g., in Ma and Zabaras [9], Bungartz and Dirnstorfer [25], or Pflüger [26]. This adaptive grid can better capture the local behavior of functions that have steep gradients or even nondif-70 ferentiabilities.…”

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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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Gradient‐based optimization with B‐splines on sparse grids for solving forward‐dynamics simulations of three‐dimensional, continuum‐mechanical musculoskeletal system models

Valentin

Sprenger

Pflüger

et al. 2018

Numer Methods Biomed Eng

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Investigating the interplay between muscular activity and motion is the basis to improve our understanding of healthy or diseased musculoskeletal systems. To be able to analyze the musculoskeletal systems, computational models are used. Albeit some severe modeling assumptions, almost all existing musculoskeletal system simulations appeal to multibody simulation frameworks. Although continuum-mechanical musculoskeletal system models can compensate for some of these limitations, they are essentially not considered because of their computational complexity and cost. The proposed framework is the first activation-driven musculoskeletal system model, in which the exerted skeletal muscle forces are computed using 3-dimensional, continuum-mechanical skeletal muscle models and in which muscle activations are determined based on a constraint optimization problem. Numerical feasibility is achieved by computing sparse grid surrogates with hierarchical B-splines, and adaptive sparse grid refinement further reduces the computational effort. The choice of B-splines allows the use of all existing gradient-based optimization techniques without further numerical approximation. This paper demonstrates that the resulting surrogates have low relative errors (less than 0.76%) and can be used within forward simulations that are subject to constraint optimization. To demonstrate this, we set up several different test scenarios in which an upper limb model consisting of the elbow joint, the biceps and triceps brachii, and an external load is subjected to different optimization criteria. Even though this novel method has only been demonstrated for a 2-muscle system, it can easily be extended to musculoskeletal systems with 3 or more muscles.

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Spatially adaptive sparse grids for high-dimensional data-driven problems

Cited by 140 publications

References 20 publications

An Adaptive Sparse Grid Semi-Lagrangian Scheme for First Order Hamilton-Jacobi Bellman Equations

An Adaptive Sparse Grid Semi-Lagrangian Scheme for First Order Hamilton-Jacobi Bellman Equations

Scalable high-dimensional dynamic stochastic economic modeling

Gradient‐based optimization with B‐splines on sparse grids for solving forward‐dynamics simulations of three‐dimensional, continuum‐mechanical musculoskeletal system models

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