Research Interest< My primary research interests are in the study of nonlinear friction laws associated with earthquake dynamics. This interdisciplinary research project has been conducted at UCSB under the supervision of my two advisors, Bjorn Birnir (Department of Mathematics, UC Santa Barbara), and Daniel Lavallee (Institute for Crustal Studies, UC Santa Barbara). For the past several years we have studied several types of dynamic models under the nonlinear Dieterich-Ruina (DR) friction law by attaching it to a Burridge-Knopoff model of spring connected blocks. My studies in the papers listed on this page explore the equations proposed by R. Madariaga of single block subject to the DR friction law. I found period doubling bifurcations and the presence of chaos when a specific parameter was increased. In the past, the Dieterich-Ruina friction term has been altered because of the difficulties imposed by the nonlinear term. Either this alteration of the nonlinear term, or the difficulty in numerically integrating the equations of motion may explain why chaotic regimes have rarely been observed. Taking this study a step further, I derived the equations of motion for a system of many blocks subject to DR friction, and from there took the continuum limit, resulting in a coupled nonlinear wave equation. Although the numerical challenges still present in both the discrete and the continuous systems require much more computation time than in the case of a single block, I have been able to obtain results using several factors about the equations themselves. Naive methods to numerically integrate these equations result in very long run times needed to obtain correct results. Part of my current research has been to solve these equations on distributed memory parallel machines. Parallelizing the equations themselves, as well as the solvers has greatly increased speed-up in the numerical integration and has allowed me to explore the equations more deeply. A summary of this parallel project can be found in my paper with Long Nguyen (listed below). Preliminary investigation into both the discrete system and the continuous model shows that for both systems, a rich phenomenology of dynamics exist even in 1 spatial dimension. We have seen transitions to chaos in the discrete system by varying the system size, and bifurcations into chaos occur in the continuum case when a specific parameter is increased (and this is a smaller parameter value needed than in the case for a single block).

Many areas of applied math intrigue me, but I'm mostly interested in stochastic PDEs, Levy distributions in advanced probability theory,
and numerical analysis.

Papers and Projects:

--Parallel Project for Dieterich-Ruina PDE

--with B. Birnir, D. Lavallee: "Dynamic Modeling with Dieterich-Ruina Friction". Research article in preparation, June 2009.

--with L. Nguyen: "High Performance Computing Techniques in Nonlinear Spring Block Models" June 2009.

--With B. Birnir, D. Lavallee: "A Model for Aperiodicity in Earthquakes". Nonlinear Processes in Geophysics, January 2008.

--Master's Thesis: "A Source for Aperiodicity in Earthquakes". Filed, December 2006.

--With P. Muchmore, S. Kiefer, A. Gonzalez, E. Ferri: "Black-Scholes Option Valuation with the Method of Lines" and corresponding Numerical Solutions. March 2005.