My Research
My research interests lie in the intersection between stochastic analysis and numerical simulation, where I study stochastic processes and stochastic partial differential equations. My thesis work develops the existence and uniqueness theory for a class of stochastic PDE models driven by spatial white noise, using Wiener chaos expansion techniques and the Malliavin calculus, as well as derives the error analysis for numerical solutions from a stochastic finite element method. More recently, I am interested in applying the weak convergence approach for proving large deviation results in SPDEs, as well as in applying the large deviation principle to design efficient rare event simulation techniques. In this vein, my recent work at SAMSI focuses on designing optimal importance sampling schemes for simulating rare events, with applications in stochastic differential equations, reflected diffusions, and random graphs.
Currently, I am studying the formation of spatio-temporal patterns in the stochastic Swift-Hohenberg equation with delay, where I investigate how stochasticity and delayed feedback effects the stability and the evolution of the amplitude equations on the slow timescale.
I am also involved in stochastic inverse problems for a seismic imaging project to use stochastic simulation approaches, such as Markov chain Monte Carlo, to estimate distributional properties of model coefficients from observational data.
Other simulation problems I am interested in is the Gillespie's Stochastic Simulation Algorithm for simulating biochemical reactions, where I have developed a modification of the algorithm for enzymatic reactions that are catalyzed by enzymes that fluctuate randomly between its conformers.
Publications & Preprints
- Large deviations for a stochastic Korteweg-de Vries equation. (with L. Setayeshgar.)
Accepted, Markov Processes and Related Fields. - The importance sampling technique for understanding rare events in Erdos-Renyi random graphs. (with J. Nolen, S. Bhamidi, J. Hannig.)
Resubmitted. arXiv:1302.6551, 33 pages. - On Stochastic Navier-Stokes Equation Driven by Stationary White Noise. (with B. Rozovskii.)
Malliavin Calculus and Stochastic Analysis, Eds.: F. Viens et al., Springer Proceedings in Mathematics, 2012, pp. 219-250. PDF - Effective Approximations of Stochastic Partial Differential Equations based on Wiener Chaos expansions and the Malliavin Calculus.
Ph.D Thesis (2011). PDF - A stochastic finite element method for stochastic parabolic equations driven by purely spatial noise. (with B. Rozovskii.)
Communications on Stochastic Analysis, 2010, Vol. 4, No. 2, pp. 271-297. PDF - Randomization of forcing in large systems of PDEs for improvement of energy estimates. (with B. Rozovskii and H. M. Zhou.)
SIAM Journal of Multiscale Modeling and Simulation, 2010, Vol. 8, No. 4, pp. 1419-1438. PDF/ DOI
Current and other work
- Designing importance sampling schemes for reflected SDEs. (with A. Budhiraja)
- Analysis of pattern formation in Swift-Hohenberg equation with delay (with R. Kuske)
- Statistical inverse and Bayesian inference problems for seismic imaging (with F. Hermann, Z. Fang) Abstract
- Signal Detection for Drug Safety. Faculty mentor for this project at the 2012 Industrial Mathematical and Statistical Modeling Workshop for Graduate Students. Report
- Stochastic simulation of biochemical systems with randomly fluctuating rate constants.
Preprint. arXiv:1202.1266, 17 pages.