Sparse grid

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

The parallelization of helmholtz equation related to breast cancer growth

Abstract: Detecting breast cancer at an early stage will decrease the mortality rate and improve the cancer treatment successfully. This research focuses on the parallelization of the mathematical modeling on breast cancer growth using one and two dimensional Helmholtz equations. Finite difference method (FDM) is chosen to discretize the Helmholtz equation in order to generate a large sparse grid solution. Some numerical iterative methods are used to simulate the grid solution. The numerical methods under consideration are alternating group explicit (AGE), Red Black Gauss Seidel (RBGS), Gauss Seidel (GS) and Jacobi (JB) method. The alternative numerical method can be detected and quantified by comparing and analyzing the numerical methods under consideration in the aspect of run time, number of iterations, maximum error, root mean square error and computational complexity. Domain decomposition technique of the parallel AGE, RBGS and JB can be applied to decompose the full domain solution into subdomains. The message passing among the neighbourhood of subdomain can be done efficiently using MATLAB Distributed Computing Software. This technique is a straight forward implementation on a distributed parallel computer system (DPCS) because of the non-overlapping subdomain feature. The computer system architecture of DPCS is a single instruction multiple data stream (SIMD) and well suited to support the high computational complexity of a large sparse matrix. The development of DPCS is based on the Linux platform with eight processors of Intel® Core™ Duo Processor architecture and MATLAB Distributed Computing Software version R2011a. The visualization of one and two dimensional of breast cancer growth are captured using Comsol Multiphysic version 4.3a. The parallel performance evaluations of parallel AGE, RBGS and JB are measured in terms of run time, speedup, efficiency, effectiveness and temporal performance. As a conclusion, the parallel algorithm of AGE is superior than RBGS, GS and JB for solving one and two dimensional Helmholtz equations for breast cancer growth early detection.

Solving Dynamic Portfolio Choice Models in Discrete Time Using Spatially Adaptive Sparse Grids

Abstract: In this paper, I propose a dynamic programming approach with value function iteration to solve Bellman equations in discrete time using spatially adaptive sparse grids. In doing so, I focus on Bellman equations used in finance, specifically to model dynamic portfolio choice over the life cycle. Since the complexity of the dynamic programming approach—and other approaches—grows exponentially in the dimension of the (continuous) state space, it suffers from the so called curse of dimensionality. Approximation on a spatially adaptive sparse grid can break this curse to some extent. Extending recent approaches proposed in the economics and computer science literature, I employ local linear basis functions to a spatially adaptive sparse grid approximation scheme on the value function. As economists are interested in the optimal choices rather than the value function itself, I discuss how to obtain these optimal choices given a solution to the optimization problem on a sparse grid. I study the numerical properties of the proposed scheme by computing Euler equation errors to an exemplary dynamic portfolio choice model with varying state space dimensionality.

Scientific Visualization on Sparse Grids

Abstract: The ever growing size of data sets resulting from industrial and scientific simulations and measurements have created an enormous need for analysis tools allowing interactive visualization. A promising hierarchical approach in the area of numerical simulation is called sparse grids. We present two major visualization algorithms working directly on the sparse grid representation of the data set. One of them is interactive particle tracing, which continues to be an important utility for evaluating CFD simulations. The other one is volume ray casting, which is of interest in many areas dealing with three-dimensional scalar data. Additionally we have been able to employ texture hardware support for the necessary function interpolation. Hence, we are able to perform volume visualization methods on compressed data sets at interactive frame rates, which is not possible with other methods like wavelets or fractal compression. In particular, we are able to handle sparse grids of level 13, which correspond to regular volumes of 8193^3 voxels.

High-dimensional integration for optimization under uncertainty

Abstract: High-Dimensional Integration for Optimization Under Uncertainty

Pricing American Options under High-Dimensional Models with Recursive Adaptive Sparse Expectations

Abstract: We introduce a novel numerical framework for pricing American options in high dimensions. Our scheme manages to alleviate the problem of dimension scaling through the use of adaptive sparse grids. We approximate the value function with a low number of points and recursively apply fast approximations of the expectation operator from an exercise period to the previous period. Given that available option databases gather several thousands of prices, there is a clear need for fast approaches in empirical work. Our method processes an entire cross section of options in a single execution and offers an immediate solution to the estimation of hedging coefficients through finite differences. It thereby brings valuable advantages over Monte Carlo simulations, which are usually considered to be the tool of choice in high dimensions, and satisfies the need for fast computation in empirical work with current databases containing thousands of prices. We benchmark our algorithm under the canonical model of Black and Scholes and the stochastic volatility model of Heston, the latter in the presence of discrete dividends. We illustrate the massive improvement of complexity scaling over dense grids with a basket option study including up to eight underlying assets. We show how the high degree of parallelism of our scheme makes it suitable for deployment on massively parallel computing units to scale to higher dimensions or further speed up the solution process.