Sparse grid

About: Sparse grid is a research topic. Over the lifetime, 1013 publications have been published within this topic receiving 20664 citations.

Accelerating Stochastic Collocation Methods for Partial Differential Equations with Random Input Data

Abstract: This work proposes and analyzes a generalized technique for decreasing the computational complexity of stochastic collocation (SC) methods to solve partial differential equations (PDEs) with random input data. Specifically, we predict the solution of the parametrized PDE at each collocation point using a previously assembled lower fidelity interpolant and use this prediction to provide deterministic (linear/nonlinear) iterative solvers with initial approximations which continue to improve as the algorithm progresses through the levels of the interpolant. With nested collocation points, these coarse predictions can be assembled as a substep in the construction of the high-fidelity interpolant. As a concrete example, we develop our approach in the context of SC approaches employing sparse tensor products of globally defined Lagrange polynomials on nested one-dimensional Clenshaw--Curtis abscissas, providing a rigorous computational complexity analysis of the resulting fully discrete sparse grid SC approxima...

Efficient Regular Sparse Grid Hierarchization by a Dynamic Memory Layout

Abstract: We consider a new hierarchization algorithm for sparse grids of high dimension and low level. The algorithm is inspired by the theory of memory efficient algorithms. It is based on a cache-friendly layout of a compact data storage, and the idea of rearranging the data for the different phases of the algorithm. The core steps of the algorithm can be phrased as multiplying the input vector with two sparse matrices. A generalized counting makes it possible to create (or apply) the matrices in constant time per row. The algorithm is implemented as a proof of concept and first experiments show that it performs well in comparison with the previous implementation SG++, in particular for the case of high dimensions and low level.

Sparse Grids and Applications - Munich 2012

Abstract: Sparse grids have gained increasing interest in recent years for the numerical treatment of high-dimensional problems. Whereas classical numerical discretization schemes fail in more than three or four dimensions, sparse grids make it possible to overcome the curse of dimensionality to some degree, extending the number of dimensions that can be dealt with. This volume of LNCSE collects the papers from the proceedings of the second workshop on sparse grids and applications, demonstrating once again the importance of this numerical discretization scheme. The selected articles present recent advances on the numerical analysis of sparse grids as well as efficient data structures, and the range of applications extends to uncertainty quantification settings and clustering, to name but a few examples.

Use of scaling information for stochastic atmospheric absolute phase screen retrieval

Abstract: Scaling information is an important tool for the description of natural processes. Many applications of SAR (differential) interferometry lead to a set of sparse phase measurements, e.g. the monitoring of permanent scatterers. In this case, the atmospheric phase screen component of a given SAR image can be estimated over the PS sparse grid. Usually such data have to be unwrapped and then interpolated on a regular grid. We investigate the utility of the scaling information, valid for atmospheric phase screen data, in the process of unwrapping a set of sparse measurements. We show how the power-law behaviour of the data variogram can be used as an a priori constraint for optimization through techniques such as simulated annealing. The results are interpreted in view of operational applications to real data.