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 research studies common error indicators compared to adjoint based error estimates extended to the stochastic framework and shows that such estimates are able to detect the true error very accurately and suggest to replace the expensive adjoint problem by a reduced model.
Abstract: We consider Partial Differential Equations (PDEs) with random parameters, solved by adaptive stochastic collocation. Our research studies common error indicators compared to adjoint based error estimates extended to the stochastic framework. We show that such estimates are able to detect the true error very accurately and suggest to replace the expensive adjoint problem by a reduced model. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)
8 citations
14 Aug 2016
TL;DR: In this article, an adaptive version of the Multi-Index Monte Carlo method, introduced by Haji-Ali, Nobile and Tempone (2016), is presented for simulating PDEs with coefficients that are random fields.
Abstract: We present an adaptive version of the Multi-Index Monte Carlo method, introduced by Haji-Ali, Nobile and Tempone (2016), for simulating PDEs with coefficients that are random fields. A classical technique for sampling from these random fields is the Karhunen–Loeve expansion. Our adaptive algorithm is based on the adaptive algorithm used in sparse grid cubature as introduced by Gerstner and Griebel (2003), and automatically chooses the number of terms needed in this expansion, as well as the required spatial discretizations of the PDE model. We apply the method to a simplified model of a heat exchanger with random insulator material, where the stochastic characteristics are modeled as a lognormal random field, and we show consistent computational savings.
8 citations
01 Jan 2016
TL;DR: Two algorithms to detect and filter silent data corruption when solving time-dependent PDEs with the Sparse Grid Combination Technique (SGCT) are described and it is shown that the SGCT can be used with minor modifications to filter corrupted data and obtain good results.
Abstract: We describe two algorithms to detect and filter silent data corruption (SDC) when solving time-dependent PDEs with the Sparse Grid Combination Technique (SGCT). The SGCT solves a PDE on many regular full grids of different resolutions, which are then combined to obtain a high quality solution. The algorithm can be parallelized and run on large HPC systems. We investigate silent data corruption and show that the SGCT can be used with minor modifications to filter corrupted data and obtain good results. We apply sanity checks before combining the solution fields to make sure that the data is not corrupted. These sanity checks are derived from well-known error bounds of the classical theory of the SGCT and do not rely on checksums or data replication. We apply our algorithms on a 2D advection equation and discuss the main advantages and drawbacks.
8 citations
01 Jan 2016
TL;DR: This paper extends the study of (dimension) adaptive sparse grids by building a lattice framework around projections onto hierarchical surpluses and derive formulas for the explicit calculation of combination coefficients, in particular providing a simple formula for the coefficient update used in the adaptive sparse grid algorithm.
Abstract: In this paper we extend the study of (dimension) adaptive sparse grids by building a lattice framework around projections onto hierarchical surpluses. Using this we derive formulas for the explicit calculation of combination coefficients, in particular providing a simple formula for the coefficient update used in the adaptive sparse grids algorithm. Further, we are able to extend error estimates for classical sparse grids to adaptive sparse grids. Multi-variate extrapolation has been well studied in the context of sparse grids. This too can be studied within the adaptive sparse grids framework and doing so leads to an adaptive extrapolation algorithm.
8 citations
Journal Article•
TL;DR: In this paper, the error estimates for periodic interpolation on equidistant and sparse grids for functions from certain Besov spaces were derived using periodic Strang-Fix conditions. But their approach is limited.
Abstract: Using periodic Strang--Fix conditions, we can give an approach to error estimates for periodic interpolation on equidistant and sparse grids for functions from certain Besov spaces.
8 citations