sparse_2012_scala

sparse_2012_scala, "Sparse Grid Collocation for Uncertainty Quantification", a talk at the SCALA Conference 2012 (Scientific Computing Around Louisiana), held at Louisiana State University in Baton Rouge, on 20/21 January 2012.

This talk will start by mentioning examples of a few stochastic problems. We'll show how integration is used as a means of averaging out the uncertainties, and consider the behavior of product rules and Monte Carlo methods for estimating integrals. We'll suggest that the accuracy of the product rule can be achieved at a much lower cost, by using Smolyak's sparse grid definition. We'll then look at a common form of the sparse grid based on the Clenshaw Curtis rule. We'll finish with some numerical examples that emphasize how nice it is to have something converge quickly in a high dimensional space.

• sparse_2012_scala.tex
• sparse_2012_scala.pdf

The following files were used:

• cc_d3_level2.png, the abscissas of the level 2, dimension 3 Clenshaw Curtis sparse grid.
• cc_level2_closeup.png, a precision plot for the level 2, dimension 2 Clenshaw Curtis sparse grid.
• cc_level3_closeup.png, a precision plot for the level 3, dimension 2 Clenshaw Curtis sparse grid.
• cc_level4_closeup.png, a precision plot for the level 4, dimension 2 Clenshaw Curtis sparse grid.
• cc_sequence.png, displays the nesting behavior of the chosen set of Clenshaw Curtis rules.
• ccs_level4_closeup.png, a precision plot for the level 4, dimension 2 "slow" Clenshaw Curtis sparse grid.
• d10_con.png, integration test with Genz "Continous" function for dimension 10.
• d10_cpeak.png, integration test with Genz "Corner Peak" function for dimension 10.
• d10_discon.png, integration test with Genz "Discontinous" function for dimension 10.
• d10_gauss.png, integration test with Genz "Gaussian" function for dimension 10.
• d10_oscill.png, integration test with Genz "Oscillatory" function for dimension 10.
• d10_plots.m, MATLAB code to produce the Genz plots for dimension 10.
• d10_ppeak.png, integration test with Genz "Product Peak" function for dimension 10.
• d6_con.png, integration test with Genz "Continous" function for dimension 6.
• d6_cpeak.png, integration test with Genz "Corner Peak" function for dimension 6.
• d6_discon.png, integration test with Genz "Discontinous" function for dimension 6.
• d6_gauss.png, integration test with Genz "Gaussian" function for dimension 6.
• d6_oscill.png, integration test with Genz "Oscillatory" function for dimension 6.
• d6_plots.m, MATLAB code to produce the Genz plots for dimension 6.
• d6_ppeak.png, integration test with Genz "Product Peak" function for dimension 6.
• diffusion_mc.png, error plots for the diffusion equation, Monte Carlo estimates.
• diffusion_mc_and_sg.png, error plots for the diffusion equation, Monte Carlo and sparse grid estimates.
• diffusion_plot.m, a MATLAB program to create the error plots for the diffusion equation.
• diffusivity_plots.png, two plots of stochastic diffusivity fields.
• ghxcc_level4.png, a plot of a 2D sparse grid formed from Gauss Hermite rules in X and Clenshaw Curtis rules in Y.
• grid_04.png, the abscissas of the sparse grid A(2,2).
• ncc_5x5x5.png, the abscissas for a 5x5x5 Newton-Cotes rule.
• orion.png, an illustration of the flight of the Orion space vehicle.
• pce_flow.png, velocity field for Boussinesq flow with error bars.
• pce_heat.png, temperature field for Boussinesq flow with error bars.
• ppeak_plot.m, MATLAB code to create a plot of Genz's product peak function.
• ppeak_plot.png, Genz's product peak integrand function.
• product_grids2.png, the product grids used to construct a sparse grid.
• scala_logo.pdf, a logo for SCALA.
• wing_load_test.png, a photograph of an airplane undergoing the wing load test.