AJOU_2009_SPARSE_INTRO.TEX
An Introduction
to High Dimensional Sparse Grids

I gave a talk on 11 May 2009 to the Mathematics Department at Ajou University, Suwon, Korea. The talk was titled

An Introduction to High Dimensional Sparse Grids

A PDF version of the talk is available in ../../presentations/ajou_2009_sparse_intro.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,
   bgw_ijnam_2007.pdf.
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,
   eb_aiaa_asm_2009.pdf.
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,
   webster_thesis.pdf.

The following files constitute the LaTeX file used to create the slides:

• ajou_2009_sparse_intro.tex, the main LaTeX file;
• multimedia.sty, a LaTeX style file;
• vt_logo.pdf, a PDF image of a logo.

The following graphics 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.
• integral_rectangles.png, a PNG image suggesting how a 1D integral can be approximated by rectangles;
• interp1.png, a PNG image of the stage 1 in interpolatory quadrature;
• interp2.png, a PNG image of the stage 2 in interpolatory quadrature;
• interp3.png, a PNG image of the stage 3 in interpolatory quadrature;
• interp4.png, a PNG image 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_p28.png, a PNG image of the results of a Monte Carlo approximation to integrand 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.
• 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.
• 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.

The following graphics files were used:

• integral_rectangles.m, a MATLAB program for displaying an approximation to an integral.
• interp.m, a MATLAB program for displaying stages in interpolatory quadrature.
• 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_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.
• quasi_monte_carlo_2d.txt, the coordinates of 1000 Quasi Monte Carlo points in the unit square.
• smolyak_p28.txt, integration error data for Smolyak approximations to integrate problem 28.

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