A Non-static Data Layout Enhancing Parallelism and Vectorization in Sparse Grid Algorithms

Global communication schemes for the numerical solution of high-dimensional PDEs

“…Furthermore, the combination coefficients of the combination technique can be chosen in a problem-dependent, optimal manner [26,27]. Sparse grids and the sparse grid combination technique have been used to solve a variety of high-dimensional numerical problems including PDEs from mechanics, physics and financial mathematics [9,28,29,30], real-time visualization [31,32], machine learning problems [33], data mining [34,35], etc.…”

“…Hence, hierarchization and dehierarchization, the transformation from the full grid basis into the hierarchical basis and vice versa, are essential pre-and postprocessing steps for the presented communication schemes. Many implementations of the hierarchization and dehierarchization algorithms exist [33,48,49,32,50,51], including versions that are tuned for GPUs [48,32]. All these implementations take advantage of the so-called unidirectional principle which decomposes d-dimensional operators into multiple applications of 1-dimensional operators.…”

Hierarchization for the Sparse Grid Combination Technique

“…As SGpp has been designed to account for spatially adaptive sparse grids it is necessary to benchmark the derived code against [8] which has been optimized for performance.…”

“…While several software packages to hierarchize spare grids have been developed, few of these can handle the regular grids of the combination technique [7,8]. We use SGpp [7], the current standard for sparse grids, as baseline against which we benchmark our code.…”

“…This makes navigating on the data layout much easier and should result in significant performance gains. Besides SGpp, [8] provides a reportedly fast implementation of the hierarchization algorithm which can handle combination grids. Time did not allow to benchmark against this code.…”