Sparse grid

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

Solving High-Dimensional Dynamic Portfolio Choice Models with Hierarchical B-Splines on Sparse Grids

Abstract: Discrete time dynamic programming to solve dynamic portfolio choice models has three immanent issues: firstly, the curse of dimensionality prohibits more than a handful of continuous states. Secondly, in higher dimensions, even regular sparse grid discretizations need too many grid points for sufficiently accurate approximations of the value function. Thirdly, the models usually require continuous control variables, and hence gradient-based optimization with smooth approximations of the value function is necessary to obtain accurate solutions to the optimization problem. For the first time, we enable accurate and fast numerical solutions with gradient-based optimization while still allowing for spatial adaptivity using hierarchical B-splines on sparse grids. When compared to the standard linear bases on sparse grids or finite difference approximations of the gradient, our approach saves an order of magnitude in total computational complexity for a representative dynamic portfolio choice model with varying state space dimensionality, stochastic sample space, and choice variables.

A Sparse Grid Collocation Method For Parabolic PDEs with random domain deformations

Abstract: This work considers the problem of numerically approximating statistical moments of a Quantity of Interest (QoI) that depends on the solution of a time dependent linear parabolic partial differential equation. The geometry is assumed to be random and is parameterized by N random variables. The parabolic problem is remapped to a fixed deterministic domain with random coefficients and shown to admit an extension to a well defined region in C^{N} with respect to the random variables. To compute the stochastic moments of the QoI, a a Smolyak sparse grid stochastic collocation method is used. To confirm the convergence rates, a comparison to numerical experiments is performed.

Advanced Dimension-Adaptive Sparse Grid Integration Method for Structural Reliability Analysis

Abstract: The conventional sparse grid (SG) integration method treats all considered random variables equally for the numerical integration of performance functions. The dimension-equal treatment ine...

A combined collocation and Monte Carlo method for advection-diffusion equation of a solute in random porous media

Abstract: In this work, we present a numerical analysis of a method which combines a deterministic and a probabilistic approaches to quantify the migration of a contaminant, under the presence of uncertainty on the permeability of the porous medium. More precisely, we consider the flow equation in a random porous medium coupled with the advection-diffusion equation. Quantities of interest are the mean spread and the mean dispersion of the solute. The means are approximated by a quadrature rule, based on a sparse grid defined by a truncated Karhunen-Loeve expansion and a stochastic collocation method. For each grid point, the flow model is solved with a mixed finite element method in the physical space and the advection-diffusion equation is solved with a probabilistic Lagrangian method. The spread and the dispersion are expressed as functions of a stochastic process. A priori error estimates are established on the mean of the spread and the dispersion. Keywords: Uncertainty quantification, elliptic PDE with random coefficients, advection-diffusion equation, collocation techniques, anisotropic sparse grids, Monte Carlo method, Euler scheme for SDE.