Recent Developments in the Theory and Application of the Sparse Grid Combination Technique

2016 ◽  
pp. 143-163 ◽  
Author(s):  
Markus Hegland ◽  
Brendan Harding ◽  
Christoph Kowitz ◽  
Dirk Pflüger ◽  
Peter Strazdins
Keyword(s):  
Sparse Grid ◽  
Combination Technique ◽  
Recent Developments

scholarly journals Parallelisation of sparse grids for large scale data analysis

2006 ◽  
Vol 48 (1) ◽  
pp. 11-22 ◽  
Author(s):  
Jochen Garcke ◽  
Markus Hegland ◽  
Ole Nielsen
Keyword(s):  
Large Scale ◽  
Predictive Modelling ◽  
Sparse Grids ◽  
Sparse Grid ◽  
High Dimensional ◽  
Combination Technique ◽  
Parallel Solvers ◽  
Regular Grids ◽  
Timing Model ◽  
Dimensional Approximation

AbstractSparse grids are the basis for efficient high dimensional approximation and have recently been applied successfully to predictive modelling. They are spanned by a collection of simpler function spaces represented by regular grids. The sparse grid combination technique prescribes how approximations on a collection of anisotropic grids can be combined to approximate high dimensional functions.In this paper we study the parallelisation of fitting data onto a sparse grid. The computation can be done entirely by fitting partial models on a collection of regular grids. This allows parallelism over the collection of grids. In addition, each of the partial grid fits can be parallelised as well, both in the assembly phase, where parallelism is done over the data, and in the solution stage using traditional parallel solvers for the resulting PDEs. Using a simple timing model we confirm that the most effective methods are obtained when both types of parallelism are used.

scholarly journals An optimised sparse grid combination technique for eigenproblems

PAMM ◽  
2007 ◽  
Vol 7 (1) ◽  
pp. 1022301-1022302 ◽  
Author(s):  
Jochen Garcke
Keyword(s):  
Sparse Grid ◽  
Combination Technique

Complex scientific applications made fault-tolerant with the sparse grid combination technique

2016 ◽  
Vol 30 (3) ◽  
pp. 335-359 ◽  
Author(s):  
Md Mohsin Ali ◽  
Peter E Strazdins ◽  
Brendan Harding ◽  
Markus Hegland
Keyword(s):  
Fault Tolerant ◽  
Sparse Grid ◽  
Scientific Applications ◽  
Combination Technique

A Generalized Spatially Adaptive Sparse Grid Combination Technique with Dimension-wise Refinement

2021 ◽  
Vol 43 (4) ◽  
pp. A2381-A2403
Author(s):  
Michael Obersteiner ◽  
Hans-Joachim Bungartz
Keyword(s):  
Sparse Grid ◽  
Combination Technique ◽  
Spatially Adaptive

Error Splitting Preservation for High Order Finite Difference Schemes in the Combination Technique

2017 ◽  
Vol 10 (3) ◽  
pp. 689-710 ◽  
Author(s):  
Christian Hendricks ◽  
Matthias Ehrhardt ◽  
Michael Günther
Keyword(s):  
Finite Difference ◽  
High Order ◽  
Sparse Grid ◽  
Finite Difference Schemes ◽  
Combination Technique ◽  
Four Dimensions ◽  
Grid Solution ◽  
Order Four ◽  
High Order Finite Difference ◽  
Theoretical Results

AbstractIn this paper we introduce high dimensional tensor product interpolation for the combination technique. In order to compute the sparse grid solution, the discrete numerical subsolutions have to be extended by interpolation. If unsuitable interpolation techniques are used, the rate of convergence is deteriorated. We derive the necessary framework to preserve the error structure of high order finite difference solutions of elliptic partial differential equations within the combination technique framework. This strategy enables us to obtain high order sparse grid solutions on the full grid. As exemplifications for the case of order four we illustrate our theoretical results by two test examples with up to four dimensions.

scholarly journals Solution of Time-dependent Advection-Diffusion Problems with the Sparse-grid Combination Technique and a Rosenbrock Solver

2001 ◽  
Vol 1 (1) ◽  
pp. 86-98 ◽  
Author(s):  
Boris Lastdrager ◽  
Barry Koren ◽  
Jan Verwer
Keyword(s):  
Time Integration ◽  
Size Control ◽  
Sparse Grid ◽  
Time Dependent ◽  
Step Size ◽  
Combination Technique ◽  
Burgers Equations ◽  
Advection Diffusion ◽  
Grid Approach ◽  
Diffusion Problems

Abstract In the current paper the efficiency of the sparse-grid combination tech- nique applied to time-dependent advection-diffusion problems is investigated. For the time-integration we employ a third-order Rosenbrock scheme implemented with adap- tive step-size control and approximate matrix factorization. Two model problems are considered, a scalar 2D linear, constant-coe±cient problem and a system of 2D non- linear Burgers' equations. In short, the combination technique proved more efficient than a single grid approach for the simpler linear problem. For the Burgers' equations this gain in efficiency was only observed if one of the two solution components was set to zero, which makes the problem more grid-aligned.

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