sparse_intro_2009_ajou

sparse_intro_2009_ajou, "An Introduction to High Dimensional Sparse Grids", a talk to the Mathematics Department at Ajou University, Suwon, Korea, on 11 May 2009.

• sparse_intro_2009_ajou.tex
• sparse_intro_2009_ajou.pdf

Some information on the sparse grid research I participated in is included in these references:

1. John Burkardt, Max Gunzburger, Clayton Webster,
   Reduced Order Modeling of Some Nonlinear Stochastic Partial Differential Equations,
   International Journal of Numerical Analysis and Modeling,
   Volume 4, Number 3-4, 2007, pages 368-391.
2. Michael Eldred, John Burkardt,
   Comparison of Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Quantification,
   American Institute of Aeronautics and Astronautics,
   Paper 2009-0976
3. Clayton Webster,
   Sparse Grid Stochastic Collocation Techniques for the Numerical Solution of Partial Differential Equations with Random Input Data,
   PhD Dissertation,
   Mathematics Department,
   Florida State University, May 2007.

The following files were used:

• art_owen.png, a PNG image of Art Owen, a Stanford researcher in the area of high dimensional quadrature.
• cc_1d.png, a PNG image of the arrangement of Clenshaw Curtis abscissas in 1D.
• cc_d2_o17x17.png, a PNG image of a 2D product grid made from two order 17 Clenshaw Curtis rules.
• cc_d2_o9x5.png, a PNG image of a 2D product grid made from order 9 and order 5 Clenshaw Curtis rules.
• cc_d3_level5.png, a PNG image of a 3D Smolyak sparse grid of level 5, based on Clenshaw Curtis rules.
• cc_sparse_2d.png, a PNG image of a 2D Smolyak sparse grid of level 4.
• explorer.png, a PNG image of an explorer.
• fsu_logo.pdf, a logo.
• integral_rectangles.m, a MATLAB program for displaying an approximation to an integral.
• integral_rectangles.png, a PNG image of suggesting how a 1D integral can be approximated by rectangles;
• interp.m, a MATLAB program for displaying stages in interpolatory quadrature.
• interp1.png, a PNG image of of the stage 1 in interpolatory quadrature;
• interp2.png, a PNG image of of the stage 2 in interpolatory quadrature;
• interp3.png, a PNG image of of the stage 3 in interpolatory quadrature;
• interp4.png, a PNG image of of the stage 4 in interpolatory quadrature;
• lhs_2d.png, a PNG image of 100 points in a 2D Latin Square.
• level4_o1x17.png, a PNG image of the 1x17 component of a 2D level 4 sparse grid.
• level4_o3x9.png, a PNG image of the 3x9 component of a 2D level 4 sparse grid.
• level4_o5x5.png, a PNG image of the 5x5 component of a 2D level 4 sparse grid.
• level4_o9x3.png, a PNG image of the 17x1 component of a 2D level 4 sparse grid.
• level4_o17x1.png, a PNG image of the 17x1 component of a 2D level 4 sparse grid.
• lhs_2d.png, a PNG image of a Latin Hypercube sample in 2D.
• mc5_p28.png, a PNG image of the results of 5 distinct Monte Carlo approximations to integrand 28.
• mc5_versus_smolyak_p28.png, a PNG image of a comparison of 5 distinct Monte Carlo approximations versus the Smolyak approach on integrand 28.
• mc_versus_smolyak_p28.png, a PNG image of a comparison of a Monte Carlo approximation versus the Smolyak approach on integrand 28.
• monte_carlo_1d.png, a PNG image of a possible arrangement of the first 100 sample points in a 1D Monte Carlo procedure.
• monte_carlo_2d.png, a PNG image of a possible arrangement of the first 1000 sample points in a 2D Monte Carlo procedure.
• monte_carlo_2d.txt, the coordinates of 1000 Monte Carlo points in the unit square.
• monte_carlo_p28.f90, a FORTRAN90 program, for use with TEST_NINT, for applying Monte Carlo to integrand 28.
• monte_carlo_p28.out, output from a run of the monte_carlo_p28.f90 program.
• monte_carlo_p28.png, a PNG image of the results of a Monte Carlo approximation to integrand 28.
• monte_carlo_p28_one.txt, integration error data for one Monte Carlo approximation to integrate problem 28.
• monte_carlo_p28_five.txt, integration error data for 5 Monte Carlo approximations to integrate problem 28.
• pool.png, a PNG image of a swimming pool.
• pool_depth.png, a PNG image of the variation in depth of a swimming pool;
• quasi_monte_carlo_1d.png, a PNG image of a possible arrangement of the first 100 sample points in a 1D Quasi Monte Carlo procedure.
• quasi_monte_carlo_2d.png, a PNG image of a possible arrangement of the first 1000 sample points in a 2D Quasi Monte Carlo procedure.
• quasi_monte_carlo_2d.txt, the coordinates of 1000 Quasi Monte Carlo points in the unit square.
• roulette_wheel.png, a PNG image of a roulette wheel;
• sergey_smolyak.png, a PNG image of Sergey Smolyak, who devised the Smolyak sparse grid technique.
• smolyak_p28.png, a PNG image of the results of a series of Smolyak approximations to integrand 28.
• smolyak_p28.txt, integration error data for Smolyak approximations to integrate problem 28.
• stochastic_mc.pdf, a PDF image of the results of 5 Monte Carlo approximations to the stochastic diffusion problem, in 11D, with varying values of the correlation length L.
• stochastic_mc_and_sm.pdf, a PDF image of a comparison of the Smolyak procedure and 5 Monte Carlo approximations to the stochastic diffusion problem, in 11D, with varying values of the correlation length L.