# Sparse Grid Tutorial

@inproceedings{Garcke2007SparseGT, title={Sparse Grid Tutorial}, author={Jochen Garcke}, year={2007} }

The sparse grid method is a special discretization technique, which allows to cope with the curse of dimensionality of grid based approaches to some extent. It is based on a hierarchical basis [Fab09, Yse86, Yse92], a representation of a discrete function space which is equivalent to the conventional nodal basis, and a sparse tensor product construction. The method was originally developed for the solution of partial differential equations [Zen91, Gri91, Bun92, Bal94, Ach03] and is now also… Expand

#### 21 Citations

Towards Non-blocking Combination Schemes in the Sparse Grid Combination Technique

- 2019

Despite the utilization of High Performance Computing (HPC) to solve high-dimensional partial differential equations (PDEs), the extent to which these problems are solvable in a considerable amount… Expand

A note on the convergence analysis of a sparse grid multivariate probability density estimator

- Mathematics
- 2009

With the recent growth in volume and complexity of available data has come a renewed interest in the problem of estimating multivariate probability density functions. However, traditional methods… Expand

Maximum a posteriori density estimation and the sparse grid combination technique

- Mathematics
- 2013

We study a novel method for maximum a posteriori (MAP) estimation of the probability density function of an arbitrary, independent and identically distributed \(d\)-dimensional data set. We give an… Expand

Regression with the optimised combination technique

- Mathematics, Computer Science
- ICML
- 2006

This article applies the recently introduced optimised combination technique for regression, which repairs instabilities of the sparse grid combination technique, resulting in a non-linear approximation method which achieves very competitive results. Expand

Mitigating the curse of dimensionality: sparse grid characteristics method for optimal feedback control and HJB equations

- Mathematics, Computer Science
- Comput. Optim. Appl.
- 2017

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. Expand

Sparse Grid Method in the Libor Market Model. Option Valuation and the 'greeks' Msc in Grid Computing

- 2007

In this project we study numerical approximations of the Libor Market Model by using Sparse Grid techniques. This approach is a promising method to solve high dimensional partial differential… Expand

Stochastic simulations of ocean waves: An uncertainty quantification study

- Mathematics
- 2015

The primary objective of this study is to introduce a stochastic framework based on generalized polynomial chaos (gPC) for uncertainty quantification in numerical ocean wave simulations. The… Expand

An example of solving HJB equations using sparse grid for feedback control

- Mathematics, Computer Science
- 2015 54th IEEE Conference on Decision and Control (CDC)
- 2015

This paper introduces and demonstrates an example of solving the 6-D HJB equation for the optimal attitude control of a rigid body equipped with two pairs of momentum wheels and integrates the solution into a model predictive control for optimal attitude stabilization. Expand

Masters Thesis: Tensor Network B-splines

- 2020

B-splines are basis functions for the spline function space and are extensively used in applications requiring function approximation. The generalization of B-splines to multiple dimensions is done… Expand

Smolyak Method for Solving Dynamic Economic Models: Lagrange Interpolation, Anisotropic Grid and Adaptive Domain

- Mathematics, Computer Science
- 2013

It is argued that in large-scale economic applications, a solution algorithm based on Smolyak interpolation has substantially lower expense when it uses derivative-free fixed-point iteration instead of standard time iteration. Expand

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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… Expand

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A multigrid algorithm for the solution of a second order elliptic equation in three dimensions is introduced and it is shown that there is a relation with semicoarsening and approximation by more-dimensional Haar wavelets. Expand

Pointwise Convergence Of The Combination Technique For Laplace's Equation

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Classification with sparse grids using simplicial basis functions

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The usage of extended Gauss (Patterson) quadrature formulas as the one‐dimensional basis of the construction is suggested and their superiority in comparison to previously used sparse grid approaches based on the trapezoidal, Clenshaw–Curtis and Gauss rules is shown. Expand

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A multilevel approach for the solution of partial differential equations based on a multiscale basis which is constructed from a one-dimensional multiscales basis by the tensor product approach, which is well suited for higher dimensional problems. Expand

Multivariate Quadrature on Adaptive Sparse Grids

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- 2003

The sparse grid approach, based upon a direct higher order discretization on the sparse grid, overcomes this dilemma to some extent, and introduces additional flexibility with respect to both the order of the 1 D quadrature rule applied (in the sense of Smolyak's tensor product decomposition) and the placement of grid points. Expand

The efficient solution of fluid dynamics problems by the combination technique

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The sparse grid combination technique is very economic on both storage requirements and computing time, but achieves almost the same accuracy as the usual full grid solution. Expand

A Combination Technique For The Solution Of Sparse Grid Problems

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It can be seen that the combination approach works not only for suuciently smooth solutions of linear problems, but, to some extent, also for non-smooth solutions and even forNon-linear problems. Expand

A Note on the Complexity of Solving Poisson's Equation for Spaces of Bounded Mixed Derivatives

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A strong tractability result of the order O(e−1) is given and this paper provides a practically usable hierarchical basis finite element method of this complexity O( e−1), i.e., without logarithmic terms growing exponentially in d, at least for the authors' sparse grid setting with its underlying smoothness requirements. Expand