Sparse grid

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

On coordinate transformation and grid stretching for sparse grid pricing of basket options

Abstract: We evaluate two coordinate transformation techniques in combination with a coordinate stretching for pricing basket options in a sparse grid setting. The sparse grid technique is a basic technique for solving a high-dimensional partial differential equation. By creating a small hypercube sub-grid in the 'composite' sparse grid we can also determine hedge parameters accurately. We evaluate these techniques for multi-asset examples with up to five underlying assets in the basket.

GPU based sparse grid technique for solving multidimensional options pricing PDEs

Abstract: It has been shown that the sparse grid combination technique can be a practical tool to solve high dimensional PDEs arising in multidimensional option pricing problems in finance. Hierarchical approximation of these problems leads to linear systems that are smaller in size compared to those arising from standard finite element or finite difference discretizations. However, these systems are still excessively demanding in terms of memory for direct methods and challenging to solve by iterative methods. In this paper we address iterative solutions via preconditioned Krylov subspace based methods, such as Stabilized BiConjugate Gradient (BiCGStab) and CG Squared (CGS), with the main focus on the design of such iterative solvers to harness massive parallelism of general purpose Graphics Processing Units (GPGPU)s. We discuss data structures and efficient implementation of iterative solvers. We also present a number of performance results to demonstrate the scalability of these solvers on the NVIDIA's CUDA platform.

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.