Sparse grid

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

Adaptive Sparse Grid Multilevel Methods for Elliptic PDEs Based on Finite Differences.

Abstract: We present a multilevel approach for the solution of partial differential equations. It is based on a multiscale basis which is constructed from a one-dimensional multiscale basis by the tensor product approach. Together with the use of hash tables as data structure, this allows in a simple way for adaptive refinement and is, due to the tensor product approach, well suited for higher dimensional problems. Also, the adaptive treatment of partial differential equations, the discretization (involving finite differences) and the solution (here by preconditioned BiCG) can be programmed easily. We describe the basic features of the method, discuss the discretization, the solution and the refinement procedures and report on the results of different numerical experiments. — Author's Abstract

Learn to Invert: Surface Wave Inversion with Deep Neural Network

Abstract: Summary We propose a hybrid analytics and machine learning approach for large-scale surface wave inversion (SWI) for shear-wave velocities in the shallow overburden. A sparse grid of 1D velocity models are inverted using analytic optimization. Then, a deep neural network (DNN) with three hidden layers is trained using a spatially sparse subset of the data and non-linear inversion results. Finally, we use the DNN to predict the velocity model for the whole survey. This approach is demonstrated on a real high density land project. In comparison to the purely analytical approach, the hybrid analytic-ML method estimates a more reliable shear velocity model over the whole survey with significant reduction in computing time. We end with a discussion around the potential of this type of method for other geophysical inverse problems and seismic processing.

Grouped Transformations in High-Dimensional Explainable ANOVA Approximation.

Abstract: Many applications are based on the use of efficient Fourier algorithms such as the fast Fourier transform and the nonequispaced fast Fourier transform. In a high-dimensional setting it is also already possible to deal with special sampling sets such as sparse grids or rank-1 lattices. In this paper we propose fast algorithms for high-dimensional scattered data points with corresponding frequency sets that consist of groups along the dimensions in the frequency domain. From there we propose a fast matrix-vector multiplication, the grouped Fourier transform, that finds theoretical foundation in the context of the analysis of variance (ANOVA) decomposition where there is a one-to-one correspondence from the ANOVA terms to our proposed groups. An application can be found in function approximation for high-dimensional functions where the number of the variable interactions is limited. We tested two different approximation approaches: Classical Tikhonov-regularization, namely, regularized least squares, and the technique of group lasso, which promotes sparsity in the groups. As for the latter, there are no explicit solution formulas which is why we applied the fast iterative shrinking-thresholding algorithm to obtain the minimizer. Numerical experiments in under-, overdetermined, and noisy settings indicate the applicability of our algorithms.

Third-order sparse grid generalized spectral elements on hexagonal cells for uniform-speed advection in a plane

Abstract: This paper investigates sparse grids on a hexagonal cell structure using a Local-Galerkin method (LGM) or generalized spectral element method (SEM). Such methods allow sparse grids to be used, known as serendipity grids in square cells. This means that not all points of the full grid are used. Using a high-order polynomial, some points of each cell are eliminated in the discretization, and thus saving Central Processing Unit (CPU) time. Here a sparse SEM scheme is proposed for hexagonal cells. It uses a representation of fields by second-order polynomials and achieves third-order accuracy. As SEM, LGM is strictly local for explicit time integration. This makes LGM more suitable for multiprocessing computers compared with classical Galerkin methods. The computer time depends on the possible timestep and program implementation. Assuming that these do not change when going to a sparse grid, the potential saving of computer time due to sparseness is 1:2. The projected CPU saving in 3-D from sparseness is by a factor of 3:8. A new spectral procedure is used in this paper, called the implied spectral equation (ISE). This procedure allows for some collocation points to use any finite difference scheme of high order and the time derivatives of other spectral coefficients are implied.