FSU_2006.TEX
PDE Model Reduction Using the SVD

I gave a talk on 21 September 2006, to the Graduate Student Seminar at the Florida State University School of Computational Science. The talk was titled

PDE Model Reduction Using the SVD

A plain text abstract of the talk is available as fsu_2006_abstract.txt.

A PDF version of the talk is available in ../../presentations/fsu_2006.pdf.

The problem of analyzing data from a fluid flow simulation and then reusing it as a reduced model for further calculations is presented in

John Burkardt, Max Gunzburger, Hyung Chun Lee,
Centroidal Voronoi Tessellation-Based Reduced-Order Modeling of Complex Systems,
SIAM Journal on Scientific Computing,
Volume 28, Number 2, 2006, pages 459-484.

Please note that the "face" images are used for comparison and illustration, and come from the work of an entirely separate research group. That research is described in

Neil Muller, Lourenco Magaia, Ben Herbst,
Singular Value Decomposition, Eigenfaces, and 3D Reconstructions,
SIAM Review,
Volume 46, Number 3, 2004, pages 518-545.

More information about the Singular Value Decomposition (SVD) is available in most textbooks on linear algebra:

Lloyd Trefethen, David Bau,,
Numerical Linear Algebra,
SIAM, 1997,
ISBN: 0-89871-361-7,
LC: QA184.T74.

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

• fsu_2006.tex, the main LaTeX file;
• multimedia.sty, a LaTeX style file;
• accelerator.png, an image of the aftermath of a collision in a particle accelerator.
• faces_data.png, an image of facial width versus height data, from Muller, Magaia, and Herbst.
• faces_eigen.png, the first eight "eigenfaces" extracted from a dataset, from Muller, Magaia, and Herbst.
• faces_energies.png, a plot of the 600 singular values of a facial dataset, from Muller, Magaia, and Herbst.
• faces_new.png, shows how a face not in the dataset was approximated using eigenfaces, from Muller, Magaia, and Herbst.
• faces_three.png, shows the kind of facial images used in the dataset, and in its testing, from Muller, Magaia, and Herbst.
• fingerprint.png, an image of a fingerprint.
• fsu_logo_ybkgrd.pdf, a little FSU logo that appears on each slide.
• inout_flow_elements.png, shows the decomposition of the flow region into elements.
• inout_flow_energies.png, a plot of the first 16 singular values from the INOUT dataset.
• inout_flow_hat.png, shows the forcing of the boundary term by a hat function, when using the reduced model to solve new problems. (This plot was not used)
• inout_flow_pod001_dir.eps, the direction field for the IN/OUT flow, for POD basis vector 1;
• inout_flow_pod002_dir.eps, the direction field for the IN/OUT flow, for POD basis vector 2;
• inout_flow_pod003_dir.eps, the direction field for the IN/OUT flow, for POD basis vector 3;
• inout_flow_pod004_dir.eps, the direction field for the IN/OUT flow, for POD basis vector 4;
• inout_flow_region.png, shows the flow region.
• inout_flow_sinusoid_error.png, plots of the error between an IN/OUT simulation and a CVT approximation using 4, 8 and 16 basis vectors, with a sinusoidal inflow parameter ALPHA;
• inout_flow_sinusoid.png, shows the forcing of the boundary term by a sinusoid function, when using the reduced model to solve new problems.
• inout_flow_steady_dir.png, the velocity direction field for the IN/OUT flow field at steady state;
• inout_flow_step001_dir.png, the velocity direction field for the IN/OUT flow field at time step 1;
• inout_flow_step100_dir.png, the velocity direction field for the IN/OUT flow field at time step 100;
• satellite.png, an image of an earth-orbiting satellite.
• svd.png, an illustration from Muller, Magaia, and Herbst showing the back-rotation, dilation, and roation effects of the SVD.
• thematrix2.png, an image of "data rain" from The Matrix.