Sparse grid

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

Fast algorithms for the solution of stochastic partial differential equations

Abstract: Title of dissertation: FAST ALGORITHMS FOR THE SOLUTION OF STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS Christopher W. Miller, Doctor of Philosophy, 2012 Dissertation directed by: Professor Howard Elman Department of Computer Science Institute for Advanced Computer Studies We explore the performance of several algorithms for the solution of stochastic partial differential equations including the stochastic Galerkin method and the stochastic sparse grid collocation method. We also introduce a new method called the adaptive kernel density estimation (KDE) collocation method, which addresses some of the deficiencies present in other stochastic PDE solution methods. This method combines an adaptive sparse grid collocation method with KDE to optimally allocate stochastic degrees of freedom. Several components of this method can be computationally expensive, such as automatic bandwidth selection for the kernel density estimate, evaluation of the kernel density estimate, and computation of the coefficients of the approximate solution. Fortunately all of these operations are easily parallelizable. We present an implementation of adaptive KDE collocation that makes use of NVIDIA’s complete unified device architecture (CUDA) to perform the computations in parallel on graphics processing units (GPUs). FAST ALGORITHMS FOR THE SOLUTION OF STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

Opticom and the Iterative Combination Technique for Convex Minimisation

Abstract: Since “A combination technique for the solution of sparse grid problems” Griebel et al. (1992), the sparse grid combination technique has been successfully employed to approximate sparse grid solutions of multi-dimensional problems. In this paper we study the technique for a minimisation problem coming from statistics. Our methods can be applied to other convex minimisation problems. We improve the combination technique by adapting the “Opticom” method developed in Hegland et al. (Linear Algebra Appl 420:249–275, 2007). We also suggest how the Opticom method can be extended to other numerical problems. Furthermore, we develop a new technique of using the combination technique iteratively. We prove this method yields the true sparse grid solution rather than an approximation. We also present numerical results which illustrate our theory.

High order combination technique for the efficient pricing of basket options

Abstract: In computational finance high dimensional problems typically arise, when pricing basket options, foreign-exchange (FX) options etc. Since the number of grid points grows exponentially with the dimension, the so called curse of dimensionality shows its eect very quickly. Sparse grids and the combination technique have proven their great ability to reduce the computational effort. In this article we introduce a fourth order scheme for the combination technique to solve efficiently high dimensional partial differential equation problems. In order to linearly combine the sub-solutions, we propose a tensor-based interpolation method. We show that our approach can preserve the error splitting structure of the sub-solutions and lead to a highly accurate sparse grid solution.

Non-intrusive Stochastic Approaches for Efficient Quantification of Uncertainty Associated with Reservoir Simulations

Abstract: This paper presents non-intrusive, efficient stochastic approaches for predicting uncertainties associated with petroleum reservoir simulations. The Monte Carlo simulation method, which is the most common and straightforward approach for uncertainty quantification in the industry, requires to perform a large number of reservoir simulations and is thus computationally expensive especially for large-scale problems. We propose an efficient and accurate alternative through the collocation-based stochastic approaches. The reservoirs are considered to exhibit randomly heterogeneous flow properties. The underlying random permeability field can be represented by the Karhunen-Loeve expansion (or principal component analysis), which reduces the dimensionality of random space. Two different collocation-based methods are introduced to propagate uncertainty of the reservoir response. The first one is the probabilistic collocation method that deals with the random reservoir responses by employing the orthogonal polynomial functions as the bases of the random space and utilizing the collocation technique in the random space. The second one is the sparse grid collocation method that is based on the multi-dimensional interpolation and high-dimensional quadrature techniques. They are non-intrusive in that the resulting equations have the exactly the same form as the original equations and can thus be solved with existing reservoir simulators. These methods are efficient since only a small number of simulations are required and the statistical moments and probability density functions of the quantities of interest in the oil reservoirs can be accurately estimated. The proposed approaches are demonstrated with a 3D reservoir model originating from the 9th SPE comparative project. The accuracy, efficiency, and compatibility are compared against Monte Carlo simulations. This study reveals that, compared to traditional Monte Carlo simulations, the collocation-based stochastic approaches can accurately quantify uncertainty in petroleum reservoirs and greatly reduce the computational cost.