Sparse grid

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

Pricing multi-asset options with sparse grids

Abstract: Multi-asset options are based on more than one underlying asset, in contrast to standard vanilla options. A very significant problem within the pricing techniques for multi-asset options is the curse of dimensionality. This curse of dimensionality is the exponential growth of the complexity of the problem when the dimensionality increases, because the number of unknowns to solve simultaneously grows exponentially. Modern computer systems cannot handle this huge amount of data. In order to handle the multi-dimensional option pricing problem, the curse of dimensionality has to be dealt with. The sparse grid solution technique is one of the key techniques to do this. The sparse grid technique divides the original problem into many smaller sized sub-problems, which can be handled efficiently on a modern computer system. Because every sub-problem is independent of all others, this technique is parallelisable at a high efficiency rate. This means, that every sub-problem can be solved simultaneously. However, because of the dimensionality, the size of the sub-problems may remain too large to solve and should be parallelised further. The main restriction to the application of the sparse grid method is that the mixed derivative of the solution of a multi-dimensional option pricing problem has to be bounded. Because of the typical non-differentiability of the final condition of the option pricing problem, this restrictions has to be taken seriously. In the first part of this thesis, it is shown, experimentally, that indeed the sparse grid technique does not lead to a satisfactory accuracy without the use of advanced techniques. If a coordinate transformation is used, the accuracy increases significantly. This transformation aligns the non-differentiability along a grid line. Coordinate transformations are not applicable to any type of multi-asset option, which seriously restricts the sparse grid solution technique for real life financial applications. Sometimes, however, it is not necessary to use it, because the non-differentiability is already aligned with grid line. These types of options are the options based on the best or worst performing underlying asset. The boundary conditions of these contracts are unknown en henceforth these options are computed and analysed with a second alternative method in this thesis. This method arises from the risk-neutral expectation valuation of the final condition which can be written as a multi-dimensional integral over the transition density. By use of a discrete Fourier transform, we can solve this integral efficiently. The fast Fourier transform is a fast algorithm to compute the discrete Fourier transform. This algorithm serves as the basis for a sophisticated algorithm to parallelise the computation of the discrete Fourier transform, by dividing the transform in several parts. In this thesis, a complete parallel algorithm which does not require communication between the sub-problems is developed, which subdivides the problem in a sophisticated way. In combination with the sparse grid technique, the numerical results have a satisfactory accuracy.

The Combination Technique for Parallel Sparse-Grid-Preconditioning or -Solution of PDEs on Workstation Networks

Abstract: In this paper we study the parallel solution of elliptic partial differential equations with the sparse grid combination technique. This new algorithmic concept is based on the independent solution of many problems with reduced size and their linear combination. The resulting algorithm can be used as a solver and within a preconditioner. We will describe the algorithm for two and three-dimensional problems and discuss its parallel implementation on a network of HP720 workstations.

Advances in Software Tools for Scientific Computing

Abstract: A Modern Framework for Portable High-Performance Numerical Linear Algebra.- Blitz++: The Library that Thinks it is a Compiler.- The Design of Sparse Direct Solvers using Object-Oriented Techniques.- A Sparse Grid PDE Solver Discretization, Adaptivity, Software Design and Parallelization.- Java as an Environment for Scientific Computing.- ODE Software that Computes Guaranteed Bounds on the Solution.- The Evolution and Testing of a Medium Sized Numerical Package.- An Object-Oriented Approach to the Finite Element Modeling and Design of Material Processes.- Object-Oriented Field Recovery and Error Estimation in Finite Ele- ment Methods.- Designing an ODE Solving Environment.- Generalized Maps in Geological Modeling: Object-Oriented Design of Topological Kernels.- Author Index.

Adaptive Sparse Grid Techniques for Data Mining

Abstract: It was shown in [GaGTO1] that the task of classification in data mining can be tackled by employing ansatz functions associated to grid points in the (often high dimensional) feature-space rather than using data-centered ansatz functions. To cope with the curse of dimensionality, sparse grids have been used.