Topic
Sparse grid
About: Sparse grid is a research topic. Over the lifetime, 1013 publications have been published within this topic receiving 20664 citations.
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TL;DR: This work proposes the use of oversampled lattice rules known from multivariate numerical integration for the highly efficient and perfectly stable evaluation and reconstruction of trigonometric polynomials using only one ordinary FFT.
82 citations
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01 Jan 2012TL;DR: The technique of sparse grids allows to overcome the curse of dimensionality, which prevents the use of classical numerical discretization schemes in more than three or four dimensions, under suitable regularity assumptions.
Abstract: The technique of sparse grids allows to overcome the curse of dimensionality, which prevents the use of classical numerical discretization schemes in more than three or four dimensions, under suitable regularity assumptions. The approach is obtained from a multi-scale basis by a tensor product construction and subsequent truncation of the resulting multiresolution series expansion. This entry level article gives an introduction to sparse grids and the sparse grid combination technique.
81 citations
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TL;DR: A new computational method for finding feedback optimal control and solving HJB equations which is able to mitigate the curse of dimensionality is presented and an upper bound for the approximation error is proved.
Abstract: We address finding the semi-global solutions to optimal feedback control and the Hamilton–Jacobi–Bellman (HJB) equation. Using the solution of an HJB equation, a feedback optimal control law can be implemented in real-time with minimum computational load. However, except for systems with two or three state variables, using traditional techniques for numerically finding a semi-global solution to an HJB equation for general nonlinear systems is infeasible due to the curse of dimensionality. Here we present a new computational method for finding feedback optimal control and solving HJB equations which is able to mitigate the curse of dimensionality. We do not discretize the HJB equation directly, instead we introduce a sparse grid in the state space and use the Pontryagin’s maximum principle to derive a set of necessary conditions in the form of a boundary value problem, also known as the characteristic equations, for each grid point. Using this approach, the method is spatially causality free, which enjoys the advantage of perfect parallelism on a sparse grid. Compared with dense grids, a sparse grid has a significantly reduced size which is feasible for systems with relatively high dimensions, such as the 6-D system shown in the examples. Once the solution obtained at each grid point, high-order accurate polynomial interpolation is used to approximate the feedback control at arbitrary points. We prove an upper bound for the approximation error and approximate it numerically. This sparse grid characteristics method is demonstrated with three examples of rigid body attitude control using momentum wheels.
80 citations
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TL;DR: This work presents a flexible and scalable method for computing global solutions of high‐dimensional stochastic dynamic models, combining distributed and shared memory parallelization paradigms, and thus permits an efficient use of high-performance computing architectures.
Abstract: We present a flexible and scalable method for computing global solutions of high-dimensional stochastic dynamic models. Within a time iteration or value function iteration setup, we interpolate functions using an adaptive sparse grid algorithm. With increasing dimensions, sparse grids grow much more slowly than standard tensor product grids. Moreover, adaptivity adds a second layer of sparsity, as grid points are added only where they are most needed, for instance in regions with steep gradients or at non-differentiabilities. To further speed up the solution process, our implementation is fully hybrid parallel, combining distributed and shared memory parallelization paradigms, and thus allows for an efficient use of high-performance computing architectures. To demonstrate the broad applicability of our method, we apply it to two very different dynamic models: First, to high-dimensional international real business cycle models with capital adjustment costs and irreversible investment. Second, to multi-good menu-cost models with temporary sales and economies of scope in price setting.
79 citations
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TL;DR: An approach to solve finite time horizon suboptimal feedback control problems for partial differential equations is proposed by solving dynamic programming equations on adaptive sparse grids with semi-discrete optimal control problem and the feedback control is derived from the corresponding value function.
Abstract: An approach to solve finite time horizon suboptimal feedback control problems for partial differential equations is proposed by solving dynamic programming equations on adaptive sparse grids. A semi-discrete optimal control problem is introduced and the feedback control is derived from the corresponding value function. The value function can be characterized as the solution of an evolutionary Hamilton---Jacobi Bellman (HJB) equation which is defined over a state space whose dimension is equal to the dimension of the underlying semi-discrete system. Besides a low dimensional semi-discretization it is important to solve the HJB equation efficiently to address the curse of dimensionality. We propose to apply a semi-Lagrangian scheme using spatially adaptive sparse grids. Sparse grids allow the discretization of the value functions in (higher) space dimensions since the curse of dimensionality of full grid methods arises to a much smaller extent. For additional efficiency an adaptive grid refinement procedure is explored. The approach is illustrated for the wave equation and an extension to equations of Schrodinger type is indicated. We present several numerical examples studying the effect the parameters characterizing the sparse grid have on the accuracy of the value function and the optimal trajectory.
77 citations