Sparse grid

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

Parallel Iterative Linear Solvers on GPU: A Financial Engineering Case

Abstract: In many numerical applications resulting from computational science and engineering problems, the solution of sparse linear systems is the most prohibitively compute intensive task. Consequently, the linear solvers need to be carefully chosen and efficiently implemented in order to harness the available computing resources. Krylov subspace based iterative solvers have been widely used for solving large systems of linear equations. In this paper, we focus on the design of such iterative solvers to take advantage of massive parallelism of general purpose Graphics Processing Units (GPU)s. We will consider Stabilized BiConjugate Gradient (BiCGStab) and Conjugate Gradient Squared (CGS) methods for the solutions of sparse linear systems with unsymmetric coefficient matrices. We discuss data structures and efficient implementation of these solvers on the NVIDIA's CUDA platform. We evaluate scalability and performance of our implementations in the context of a financial engineering problem of solving multidimensional option pricing PDEs using sparse grid combination technique.

Multi- and many-core data mining with adaptive sparse grids

Abstract: Gaining knowledge out of vast datasets is a main challenge in data-driven applications nowadays. Sparse grids provide a numerical method for both classification and regression in data mining which scales only linearly in the number of data points and is thus well-suited for huge amounts of data. Due to the recursive nature of sparse grid algorithms, they impose a challenge for the parallelization on modern hardware architectures such as accelerators. In this paper, we present the parallelization on several current task- and data-parallel platforms, covering multi-core CPUs with vector units, GPUs, and hybrid systems. Furthermore, we analyze the suitability of parallel programming languages for the implementation.Considering hardware, we restrict ourselves to the x86 platform with SSE and AVX vector extensions and to NVIDIA's Fermi architecture for GPUs. We consider both multi-core CPU and GPU architectures independently, as well as hybrid systems with up to 12 cores and 2 Fermi GPUs. With respect to parallel programming, we examine both the open standard OpenCL and Intel Array Building Blocks, a recently introduced high-level programming approach. As the baseline, we use the best results obtained with classically parallelized sparse grid algorithms and their OpenMP-parallelized intrinsics counterpart (SSE and AVX instructions), reporting both single and double precision measurements. The huge data sets we use are a real-life dataset stemming from astrophysics and an artificial one which exhibits challenging properties. In all settings, we achieve excellent results, obtaining speedups of more than 60 using single precision on a hybrid system.

Asymmetric shuffle keyboard

Abstract: An asymmetric keyboard with a QWERTY style layout comprising a plurality of sparse grids and a plurality of dense grids is provided. A sparse grid is substantially large in size containing large keys with large labels, whereas a dense grid is substantially small in size containing small keys with small labels. All keys are functional but the larger keys in the sparse grid offer greater visibility and operability than the smaller keys in the dense grid. The user makes use of the sparse grid as the primary grid to input data. A swipe across a designated boundary interchanges the key labels between corresponding pairs of keys in the designated sparse and dense grids. On the software-based version, a swipe across another designated boundary compresses or decompresses a corresponding grid. On the hardware-based version, a bi-axial hinge allows the display and the keyboard to rotate around two axes.

Sparse-Grid Finite-Volume Multigrid for 3D-Problems

Abstract: We introduce a multigrid algorithm for the solution of a second order elliptic equation in three dimensions. For the approximation of the solution we use a partially ordered hierarchy of finite-volume discretisations. We show that there is a relation with semicoarsening and approximation by more-dimensional Haar wavelets. By taking a proper subset of all possible meshes in the hierarchy, a sparse grid finite-volume discretisation can be constructed. The multigrid algorithm consists of a simple damped point-Jacobi relaxation as the smoothing procedure, while the coarse grid correction is made by interpolation from several coarser grid levels. The combination of sparse grids and multigrid with semi-coarsening leads to a relatively small number of degrees of freedom, N, to obtain an accurate approximation, together with an O(N) method for the solution. The algorithm is symmetric with respect to the three coordinate directions and it is fit for combination with adaptive techniques. To analyse the convergence of the multigrid algorithm we develop the necessary Fourier analysis tools. All techniques, designed for 3D-problems, can also be applied for the 2D case, and - for simplicity - we apply the tools to study the convergence behaviour for the anisotropic Poisson equation for this 2D case.