Below are some images and animations from some of my research projects. Check back for updates.

The parallel replica dynamics (ParRep) algorithm is a mechanism for accelerating the computation of a sequence of infrequent events corresponding to the transition from one state to another; see A.F. Voter, 1998.

When the system, modeled by the Langevin or overdamped Langevin, enters a new potential well, ParRep applies the following steps to more rapidly escape:

- The system evolves without alteration for \(t_{\rm corr}\) amount of time, the decorrelation time. If the system leaves this well, we follow it to the new state.
- An ensemble of trajectories is generated by, for instance, running \(N\) copies independently for \(t_{\rm phase}\) amount of time, the dephasing time. If a replica leaves the well during this time, it is replaced.
- The \(N\) replicas are now permitted to evolve and we follow the first one to exit.

The following is a simulation of the overdamped Langevin equation, \[ dX_t = - \nabla V(X_t) dt + \sqrt{2\beta^{-1}}dB_t \] for the potential appearing in Cérou et al., 2011. This potential features three local minima. The simulation was performed with \(\beta = 4\), \(t_{\rm corr} =t_{\rm phase}= 1\) and 1000 replicas. Furthermore the dephasing step employs a Fleming-Viot particle algorithm. ParRep permits us to access much longer time scales than would be accessible with a direct serial simulation.

Semilinear elliptic equations of the form \[ -\Delta u + u -|u|^{p-1} u = 0, \quad {\bf x} \in \Omega \subset \mathbb{R}^d, \quad u|_{\partial \Omega} = 0 \] appear in a variety of contexts, particularly in nonlinear wave equations. Solutions to such equations may be solitons, which play an important role in the time dynamics of such problems. However, computing \(u\) is nontrivial since the zero function is a solution. Thus, it is essential to have an algorithm which avoids converging to the zero solution.

One such method is Petviashvilli's method, an iterative scheme for solving the equation. With D. Olson, S. Shukla, and D. Spirn, we were able to prove that the algorithm could converge, globally, to a non-trivial solution, provided the data was nonzero.

- Computation of the positive bound state on the unit square with cubic nonlinearity[mp4]

These animations are from research with M.I. Weinstein on the nonlinear Maxwell equations with a periodic index of refraction. We contend they show evidence of coherent structures, shock formation, and shock inhibition.

- Prepared Data with a Periodic Index [mp4]
- Same Data as above, but with a Constant Index. Note that it delocalizes; periodicity induces localization. [mp4]
- Same Data as above, with a Periodic Index, but zoomed in to show evidence of shocks [mp4]

In the preceding simulations, the variation of the index of refraction was "small." If we increase the variations, making them O(1), the problem appears to regularize and inhibit shocks. The following animations are different visualizations of the same simulation

These simulations were computed using a mixture of CLAWPACK and the GSL. A subtlety to this problem is that the flux function is nonconvex, requiring care in solving the Riemann problem.

Broadband solitons are localized solutions to systems of coupled NLS equations that appeared in research with D. Pelinovsky and M.I. Weinstein. The term broadband refers to the property that there are multiple carrier frequencies, each with its own slowly varying envelope. They have also been called polychromatic solitons. These visualizations show the surface generated by such a multicomponent soliton.

These are simulations of these structures in systems of nonlinear coupled mode equations, where they appear in a small amplitude limit. As the parameter mu is reduced, the perturbations become smaller.

- Two Mode Truncation:
- Four Mode Truncation:

The solitons themselves were computed by continuation using MATLAB's bvp4c/5c algorithm, together with artificial boundary conditions. The time dependent simulations were computed with a straightforward pseudo-spectral/RK4 integrator.

These are 3D FEM solutions to the Stokes equations used in computing the permeability of a medium via homogenization. These were part of my thesis and the JGR papers with M.I. Weinstein and M. Spiegelman. These were computing using an early version of FEniCS, using PETSc for the numerical linear algebra. Though both of those libraries are free and open source, we used CUBIT for generating the meshes. The volume averages of the velocity fields yield the permeability of the medium.

- The magnitude of the velocity on a small porosity domain [png]
- The magnitude of the velocity in a slightly larger domain [png]
- The magnitude of the velocity in a much larger domain [png]

These solutions were computed using FEniCS with PETSc as the linear algebra
backend. We used MINRES and the pressure mass matrix for preconditioning, as
discussed in *Finite
Elements and Fast Iterative Solvers*
by Elman, Silvester and Wathen. A weakness of this approach was
that we used the BoomerAMG (part of Hypre) for preconditioning the vector
Laplacian term. I have been told that ML (part of Trillinos ) is superior at
treating vector Laplacians.

These are simulations from my work on scattering in the energy supercritical NLS equation, conducted with J. Colliander and C. Sulem. The pointwise decay in space as we evolve in time is evidence of scattering. Seperately, we see saturation of the critical Sobolev norm.

- Gaussian Data [mp4]
- Ring Data [mp4]
- Gaussian Data with a Complex Phase [mp4]
- Gaussian Data with a Complex Phase under the linear flow (compare with the previous simulation) [mp4]

These simulations were computed using fourth order finite differences and fourth order Runge-Kutta time stepping. This was my first foray in Fortran programming, and it allowed me to learn Fortran 77/90/95.