Showing papers on "Sparse grid published in 1996"

The solution of multidimensional real Helmholtz equations on Sparse Grids

Abstract: Sparse grids provide a very efficient method for the multilinear approximation of functions, especially in higher-dimensional spaces. In the $d$-dimensional space, the nodal multilinear basis on a grid with mesh size $h = 2^{-n}$ consists of $O(2^{nd})$ basis functions and leads to an $L\_2$-error of order $O(4^{-n})$ and an $H\_1$-error of order $O(2^{-n})$. With sparse grids we get an $L\_2$-error of order $O(4^{-n}n^{d-1})$ and an $H\_1$-error of order $O(2^{-n})$ with only $O(2^n n^{d-1})$ basis functions, if the function $u$ fulfills the condition ${\partial^{2d} \over {\partial x_1^2}{\partial x_2^2} \ldots {\partial x_d^2}} u<\infty$. Therefore, we can achieve much more accurate approximations with the same amount of storage. A data structure for the sparse grid representation of functions defined on cubes of arbitrary dimension and a finite element approach for the Helmholtz equation with sparse grid functions are introduced. Special emphasis is taken in the development of an efficient algorithm for the multiplication with the stiffness matrix. With an appropriate preconditioned conjugate gradient method (cg-method), the linear systems can be solved efficiently. Numerical experiments are presented for Helmholtz equations and eigenvalue problems for the Laplacian in two and three dimensions, and for a six-dimensional Poisson problem. The results support the assertion that the $L\_2$-error bounds for the sparse-grid approximation are also valid for sparse grid finite element solutions of elliptic differential equations. Problems with nonsmooth solutions are treated with adaptive sparse grids.

Numerical Turbulence Simulation on a Parallel Computer Using the Combination Method

Abstract: The parallel numerical solution of the Navier-Stokes equations with the sparse grid combination method was studied. This algorithmic concept is based on the independent solution of many problems with reduced size and their linear combination. The algorithm for three-dimensional problems is described and its application to turbulence simulation is reported. Statistical results on a pipe flow for Reynolds number Re cl = 6950 are presented and compared with results obtained from other numerical simulations and physical experiments. Its parallel implementation on an IBM SP2 computer is also discussed.

Discretization of elliptic differential equations on curvilinear bounded domains with sparse grids

Abstract: Elliptic differential equations can be discretized with bilinear finite elements. Using sparse grids instead of full grids, the dimension of the finite element space for the 2D problem reduces from O(N 2 ) to O (N log N) while the approximation properties are nearly the same for smooth functions. A method is presented which discretizes elliptic differential equations on curvilinear bounded domains with adaptive sparse grids. The grid is generated by a transformation of the domain. This method has the same behaviour of convergence like the sparse grid discretization on the unit square.

An Abstract Data Type for Parallel Simulations Based on Sparse Grids

Abstract: In this paper we introduce an abstract data type for the distributed representation and efficient handling of sparse grids on parallel architectures. The new data layout, implemented by means of the PVM message passing library, provides partitioning and dynamic load balancing transparent to the application making use of it. This enables parallelization of sequential partial differential equations (PDE) solvers based on sparse grids with hardly any source code modifications.

Numerical Turbulence Simulation on Different Parallel Computers Using the Sparse Grid Combination Method

Abstract: The parallel numerical solution of the Navier-Stokes equations with the sparse grid combination method was studied. This algorithmic concept is based on the independent solution of many problems with reduced size and their linear combination. The algorithm for 3-dimensional problems is described. Its parallel implementation on an IBM SP2 and a cluster of 16 HP workstations is discussed.

An Adaptive Algorithm for Solving the Biharmonic Equation on Sparse Grids

Abstract: This paper presents a sparse grid algorithm with higher order elements for solving generalized problems of the biharmonic equation. Furthermore, an adaptive version of the algorithm together with numerical results will be discussed.

Approximation on partially ordered sets of regular grids

Abstract: In this paper we analyse the approximation of functions on partially ordered sequences of regular grids. We start with the formulation of minimal requirements for useful grid transfer operators in such a partially ordered context, and we continue with the introduction of hierarchical decompositions and the identification of piecewise constant and piecewise linear approximations as special instances of the tensor product case. In the second part of the paper we derive error estimates for approximation in different norms on more-dimensional dyadic sequences of regular and sparse grids. We give special attention to a convenient notation.

Parallel Distributed Representation of Sparse Grids Using Process Arrays

Abstract: In this paper a new data structure for the distributed representation and efficient handling of sparse grids on parallel architectures is introduced. The new data layout is making use of the message passing paradigm and provides dynamic partioning and load balancing transparent to the application making use of it. This way sequential partial differential equations (PDE) solvers based on sparse grids can be data parallelized with hardly any source code modifications.