Nawaf BouRabee
Assistant Professor of Mathematics
nawaf.bourabee@rutgers.edu
I completed my PhD in 2007 from the California Institute of Technology under the supervision of the late Jerrold E. Marsden. From 20082011, I was an NSF Mathematical Sciences Postdoctoral Fellow at New York University. Since 2011, I have been an Assistant Professor at the Mathematical Sciences Department of Rutgers University in Camden, New Jersey. I am interested in the numerical solution of stochastic partial differential equations and Markov Chain Monte Carlo methods.
SPECTRWM: Spectral Random Walk Method for the Numerical Solution of Stochastic Partial Differential Equations, arXiv:1608.00885 [math.PR].  
Randomized Hamiltonian Monte Carlo, with Jesús María SanzSerna, arXiv:1511.09382 [math.PR], To be published in Annals of Applied Probability.  
Continuous Time Random Walks for the Numerical Solution of Stochastic Differential Equations, with Eric VandenEijnden, arXiv:1502.05034 [math.PR], To be published in Memoirs of the American Mathematical Society.  
Metropolis Integration Schemes for SelfAdjoint Diffusions, with Aleksandar Donev and Eric VandenEijnden, Multiscale Modeling & Simulation, Vol. 12, No. 2, pp. 781831, 2014. 

Time Integrators for Molecular Dynamics, Entropy, 16, 138162, 2014. 

Nonasymptotic mixing of the MALA algorithm, with Martin Hairer, IMA J Numer Anal, 33, 80110, 2013. 

A patch that imparts unconditional stability to explicit integrators for Langevinlike equations, with Eric VandenEijnden, J Comput Phys, 231, 25652580, 2012. 

Pathwise accuracy and ergodicity of Metropolized integrators for
SDEs, with Eric VandenEijnden, Commun Pure Appl Math, 63, 655696, 2010. 

Longrun accuracy of variational integrators in the stochastic context, with Houman Owhadi, SIAM J of Numer Anal, 48, 278297, 2010. 

A Comparison of GHMC with and without Momentum Flips, with Elena Akhmatskaya and Sebastian Reich, J Comput Phys, 228, 2256  2265, 2009. 

Stochastic variational integrators, with Houman Owhadi, IMA J of Numer Anal, 2, 421443, 2009. 

HamiltonPontryagin integrators on Lie groups, with Jerrold Marsden, Found Comput Math, 9, 197219, 2009. 
Good/Bad News on Hamiltonian MonteCarlo Oberseminar on Stochastic Analysis, Institute for Applied Mathematics, Bonn University, Bonn, Germany, January, 2017 
MiniCourse at Gene Golub SIAM Summer School 2016 entitled:
SPECTRWM: Spectral Random Walk Method for the Numerical Solution of Stochastic Partial Differential Equations. Drexel University, Philadelphia, PA, July 25August 5, 2016 q.v. SPECTRWM paper 
Generalized Markov Chain Approximation Methods Mathematics Colloquium, UC3M, Madrid, Spain, May, 2016 
Markov Chain Approximation Methods for Sampling Quasistationary Distributions Oberseminar on Stochastic Analysis, Institute for Applied Mathematics, Bonn University, Bonn, Germany, September, 2015 
MiniCourse at New Perspectives in Markov Chain Monte Carlo entitled: MCMCbased integrators for Stochastic Differential Equations Mathematics Research Institute at University of Valladolid, Valladolid, Spain, June 812, 2015 
How to simulate stochastic differential equations without discretizing time? Stochastic Computation Workshop, Montevideo, Uruguay, December 2014 
Structurepreserving algorithms for selfadjoint diffusions Scientific and Statistical Computing Seminar, U. of Chicago, IL, 2014 
This video animates a random walk approximation of Brownian motion on the interval shown with periodic boundary conditions.
The state space of the walker is an evenly spaced grid. The walker jumps left or right with equal probability.
Note that the jump size of the walker is fixed. The amount of time the walker spends in any given state is an exponentially
distributed random variable with mean given by the jump size squared  as required by local accuracy.
The spectral random walk method (SPECTRWM) generalizes this simple concept to stochastic partial differential equations (SPDE).
SPECTRWM makes jumps of fixed size forward or backward along the leading eigenfunctions of the linear part of the drift of the SPDE
with probabilities that depend only on its current state. Moreover, like the 1D random walk, the holding time of SPECTRWM is
an exponentially distributed random variable whose mean is determined by local accuracy.
Aside from its mathematical interest, this generalization allows the following benefits:
I would like to gratefully acknowledge the extensive support I have received from Rutgers University and the National Science Foundation.