Since August 2002, I have worked at Florida State University (FSU), where I am an Associate in Research at the School for Computational Science (SCS). I am a part of Max Gunzburger's research group.
In October 2002, I was the local SCS coordinator for an MPI workshop broadcast over the AccessGrid. When our feed of the AccessGrid went down only a half-hour into the first class, I had to present the remaining 90 minutes of the class "by hand".
In November 2002, I worked on porting Peter Beerli's MIGRATE program to the SCS supercomputer known as "Teragold". To Peter's shock, the program did not compile and run immediately. In fact, it took a number of weeks before that happened! Since Peter could not log in to the system, I was responsible for setting up the runs, examining the results, and diagnosing as many problems locally as I could before contacting him. I identified a number of difficulties, some involving the compiler invocation, others involving nonstandard MPI usage, such as discrepancies between the amount of data sent from one preocessor and the amount received by another. We discovered that the IBM implementation of MPI is much more strict than other versions. In the time before Peter Beerli arrived here, we worked together to get his program to run satisfactorily, on an "arbitrary" number of processors, on the IBM systems.
In 2003, I worked with Max Gunzburger on a project to implement and compare a number of methods for producing sample points from a unit hypercube, including quasi-Monte Carlo methods, Latin hypercubes, and Max's particular research interest of Centroidal Voronoi Tessellations. Max also devised a number of measures of how well such a set of points was spread out in the hypercube, and I implemented these as well. Some of these implementations were absorbed into the DAKOTA package as Sandia National Laboratory.
From June to July 2003, I participated in the Young Scholars Program.
In October 2003, I was the local coordinator for an IBM Supercomputing Workshop for the users of the SCS/FSU IBM pSeries 690 supercomputers.
In October 2003, I also gave an introductory talk to our research group.
In March 2004, I was one of the organizers for a workshop on Emerging Methodologies and Applications in PDE's.
In May 2004, I was the local coordinator for Individual Help Sessions with Charles Grassl, for research groups using the SCS/FSU IBM pSeries 690 supercomputers. You might be interested in Notes from Charles Grassl's Visit.
In February 2005, many students and postdocs in Max's group attended the SIAM Conference on Computational Science and Engineering held in Orlando.
In July 2005, many students and postdocs in Max's group attended the annual SIAM Meeting in New Orleans, just a few days after one hurricane and just a month before hurricane Katrina.
In September 2005, our research group had a picnic in Maclay Gardens State Park.
In March 2006, I was a coorganizer of one minisymposium at the Joint MAA/SIAM Section Meeting at Auburn University.
Over Summer 2006, I made plans to teach a FORTRAN class in the fall - but this was canceled when another department claimed the exclusive right to teach such a class - while also refusing to teach it!
In September 2006, I gave a presentation to the FSU School of Computational Science Graduate Student Seminar on PDE Model Reduction Using the Singular Value Decomposition.
I spent the month of January 2007 cleaning up and extending a collection of programs for creating, modifying, and analyzing triangulations and tetrahedral meshes. The 2D codes have been very helpful in designing meshes for finite element programs, and I hope that the 3D codes may be used that way eventually as well. One result of this effort was that most of my programs are now available in C++, FORTRAN90 and MATLAB. I spent some time developing graphics routines to display the edges of a tetrahedral mesh, using MATLAB or OpenGL, so that I could check some subdivision algorithms. I also wrote some programs to display a triangulated surface in 3D.
In February 2007, I was a "guest lecturer" in the SCS graduate class on computational science. I presented some algorithms related to the representation and analysis of triangles, and had the students work out implementations of these procedures. You can refer to the lecture notes. The students spent the entire 6 hours of two lab sessions on the triangles, so I never finished or presented the lecture notes on triangulations..
In February 2007, I worked on implementing algorithms for high precision integration in triangles and in tetrahedrons. This was related to the 2D and 3D simplex-based fluid flow calculations that Jeff Borggaard was working on. This brought to my attention the triangle quadrature rules by Dunvant, Fekete, and Wanzura, and the tetrahedral rules by Keast, as well as some families of Newton Cotes rules. To test the resulting packages, I assembled some test integrands over triangles, as described by deDoncker, but have not yet had the time to do the same thing for tetrahedrons.
On 16 February 2007, SCS hosted a hands-on workshop that presented the COMSOL multiphysics software: http://www.comsol.com/events/tallahassee_fl/1632/.
In February 2007 I attended the 7th SIAM Conference on Computational Science and Engineering held in Costa Mesa, California.
Over the spring of 2007, I interacted with Clayton Webster as he finished his dissertation on sparse grid collocation methods for stochastic PDE's. I occasionally helped him by writing codes to compute Clenshaw-Curtis rules, or to display sparse grids in 2D or 3D. As I worked with him, I began to understand the processes by which Smolyak grids were constructed. I computed and stored examples of such grids. One such collection is Newton-Cotes-Closed sparse grid data.
One important test of Smolyak rules involves applying them to families of test integrands. To facilitate this, I installed a version of Genz's testpack code in a C++ version as well as MATLAB and FORTRAN90.
It soon became clear that, to understand a Smolyak grid, I would need to be able to construct multidimensional product grids. I started by writing product_rule, a program which creates an M-dimensional product rule by repeated use of the same 1D rule.
However, a Smolyak grid is made up of product rules that are formed from a variety of distinct factors. So I had to write another program, product_factor, which creates an M-dimensional product rule based on whatever sequence of rules the user provides.
Once I was able to generate a variety of grids, I then became interested in determining the monomial exactness of a given multidimensional quadrature rule, that is, to characterize the terms like xlymzn which would be integrated exactly. This required the construction of the program nint_exactness, which produced some surprisingly interesting information, especially about symmetric rules of odd order, and about Smolyak grids.
I had a program called test_nint which allowed me to easily apply any integration procedure to a series of test integrands. However, I suddenly found myself not with a bunch of integration procedures, but with integration rules, so I wrote another version of the program, quadrature_test, which allowed me to interactively choose a quadrature rule and have it applied to a series of test integrands.
As a side effect, my investigations resulted in a collection of a variety of multidimensional quadrature rules of various types. For convenience and reference, I tossed most of them into a single quadrature rule directory.
At the end of May 2007, Clayton Webster left for Sandia National Laboratory. However, I hope to continue some research projects with him, involving variations of the basic Smolyak algorithm to detect anisotropy in the integrand, to estimate the integration error, and to allow for the integration order to be increased only in particular dimensions.
In June 2007, I gave a presentation to the FSU School of Computational Science on High Dimensional Sparse Grids, in which I outlined the problem of computing integrals in high dimensions, and explained a little about Smolyak quadrature.
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