FSU_2010_SPARSE.TEX
Integration, Quadrature, and Sparse Grids
I gave a talk on 10 September 2010 to Max Gunzburger's research group
seminar, titled
Integration, Quadrature, and Sparse Grids
A PDF version of the talk is available in
../../presentations/fsu_2010_sparse.pdf.
Some information on the sparse grid research I participated in
is included in these references:
-
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.
-
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:
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.
-
optical_illusion.png,
a PNG image of an optical illusion.
-
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.
You can go up one level to
the LATEX home page.
Last revised on 10 September 2010.
