Parameterized matrix equations. Polynomial approximation. Uncertainty quantification.
For a list of my projects, codes, and papers, visit my Stanford website.
I work on approximation methods for the solution of a linear system of equations that depend on a set of independent parameters. Such parameterized matrix equations arise in computational methods for models - particularly differential equation models - with uncertain input data.
I am particularly interested in polynomial approximation, i.e. spectral, methods such as spectral Galerkin and spectral collocation techniques.
| Inquisitor | So, Paul, what do you do? |
|---|---|
| Me | I am a Ph.D. student in computational math. |
| Inquisitor | What exactly is computational math? I'm terrible at math. |
| Me | You want the thirty-second elevator spiel that my grandma can understand? |
| Inquisitor | Um, okay! |
| Me | You've heard of physics, right? |
| Inquisitor | Well, sure. |
| Me | And you've heard that physics is described by equations, right? |
| Inquisitor | I suppose so. |
| Me | Most of those equations are extremely difficult - if not impossible - to solve. Therefore, we have to use computers to find approximate solutions to those equations. It turns out that there's an entire science behind approximating the solutions to physics equations on computers. That's what I study. In particular for my research, I study what happens when you add randomness to those equations, and then you want to quantify the uncertainty in the approximate solutions. |
| Inquisitor | Wow, that's way over my head. |
08.08.08
It's Aug. 8, 2008; how can I not write something different?
The goal of this summer was to really dig into my dissertation, but this goal has been difficult to pursue. At the end of June, I went to Venice for a week for the WCCM. I had hoped to present some really exciting work about the equivalence between Galerkin and collocation for a large class of problems. Unfortunately, I discovered - on a rainy Tuesday morning in Venice - that I had made a mistake in my derivation. However, the mistake led to some very interesting insights about the relationship (no longer equivalent for such a large class of problems). For better or worse, my presentation had relatively low attendance. But that was okay, since I had to change it last minute.
The following week, I was in San Diego at the SIAM Annual Meeting with David. It was a very interesting conference, and I met many students and researchers in the field.
After I got back to Stanford, we had a small UQ workshop with some of the top guys in the field. It was really great to hear some of them speak. I was particularly impressed by Boris Rozovsky. It was nice to hear someone asking questions about existence and posedness of these problems.
Immediately after the workshop, I had to finish a paper for a special issue of IJNME, which I just gave to Gianluca and Alireza yesterday.
Maybe I'll finally get a chance to work on some of my research now...