### Research Overview

Broadly, I am interested in applied mathematics, numerical analysis, and scientific computing. I have a variety of active research projects in these areas, summarized below:

My research is presently supported by:

1. Theory and Computation for Mescopic Materials Modeling, DOE (DE-SC0012733) with M. Luskin (Lead PI, UMN)
2. Computational and Analytical Challenges in Nonlinear Dispersive Wave Equations, NSF (DMS-1409018)

### Computational Tools for Materials Science

One of the outstanding challenges in computational materials science is to reach laboratory timescales in the simulation models of materials with atomistic resolution. The challenge is that the systems spend much of their time in so called metastable'' states, trapped by energetic or entropic barriers, before transitioning into another such state.

In a purely atomistic model, one must resolve the femtosecond time scale ($$10^{-15}$$ s), while the laboratory time scale may be of microseconds ($$10^{-6}$$ s) or longer. This vast separation in time scales is due to the metastable states.

One method for overcoming this scale separation is A.F. Voter's Parallel Replica dynamics, which I have worked to analyze and extend in several works, with M. Luskin, T. Lelievre, D. Aristoff, and A. Binder.

Another challenge in materials science is to understand how materials, including metals, crystals, and proteins, change from one arrangement, or conformation, to another. There is no single path but rather an entire distribution of paths with an associated mean and spread. With F.J. Pinski, A.M. Stuart and H. Weber, we have been making use of the relative entropy metric, or Kullback-Leibler divergence, to obtain best fit Gaussian distributions of the true path space distributions, which provide qualitative information which also assists in sampling. This work also has broader application in high dimensional sampling problems including statistical inverse problems.

Publications:
• F. J. Pinski, G. Simpson, A. M. Stuart, H. Weber. Algorithms for Kullback-Leibler Approximation of Probability Measures in Infinite Dimensions [arXiv]
• F. J. Pinski, G. Simpson, A. M. Stuart, H. Weber. Kullback-Leibler approximation for probability measures on infinite dimensional spaces. [arXiv]
• A. Binder, T. Lelievre, G. Simpson. A Generalized Parallel Replica Dynamics. Journal of Computational Physics, 284, 595-616, 2015. [arXiv]
• D. Aristoff, T. Lelievre, G. Simpson. The parallel replica method for simulating long trajectories of Markov chains. AMRX, 2014(2), 332-352, 2014. [arXiv]
• G. Simpson, M. Luskin. Numerical Analysis of Parallel Replica Dynamics. M2AN, 47:1287-1314, 2013. [pdf][arXiv]
See the Parallel Replica Dynamics in the gallery for examples of these simulations.

### Nonlinear Waves

Much of my rigorous work on PDEs has been inspired by solitary waves, localized, persistent solutions to nonlinear wave equations. These distinguished solutions are of interest both as mathematical objects and in applications, where they may be used in the transmission of energy, information, and mass.

But before a solitary wave could be used for such a purpose, we must consider its stability. If we perturb a solitary wave, will it stay close to the initial state, or will it break up into smaller waves? The stability of solitary waves is a topic of great interest to me, and I look to a variety of techniques in studying it, including variational analysis and spectral theory.

Publications:
• R. Cote, C. Munoz, D. Pilod, G. Simpson. Asymptotic Stability of high-dimensional Zakharov-Kuznetsov solitons [arXiv]
• X. Liu, G. Simpson, C. Sulem. Focusing Singularity in a Derivative Nonlinear Schrodinger Equation. Physica D, 262:45-58, 2013. [pdf][arXiv]
• R. Asad, G. Simpson. Embedded Eigenvalues and the Nonlinear Schrodinger Equation. Journal of Mathematical Physics, 52:033511, 2011. [pdf][arXiv][codes]
• J.L. Marzuola, G. Simpson. Spectral Analysis for Matrix Hamiltonian Operators. Nonlinearity, 24:389-429, 2011. [pdf] [arXiv][codes]
• J. Marzuola, S. Raynor, and G. Simpson. A System of ODEs for a Perturbation of a Minimal Mass Soliton. Journal of Nonlinear Science , 20:425--461, 2010. [pdf]
• G. Simpson and M.I. Weinstein. Asymptotic stability of ascending solitary magma waves. SIAM J. Math. Anal., 40:1337--1391, 2008. [pdf]
• G. Simpson, M.I. Weinstein and P. Rosenau. On a Hamiltonian PDE arising in magma dynamics. DCDS-B, 10:903--924, 2008. [pdf]

### Scientific Computing

Much of my work involves the use of numerical algorithms, either as part of a numerical analysis study or to explore the behavior of the problem they approximate by direct numerical simulation. This includes methods fro metastability, algorithms for computing nonlinear waves, and the simulation of stability and blowup in nonlinear wave equations.

Publications:
• D. Olson, S. Shukla, G. Simpson, D. Spirn. Petviashvilli's Method for the Dirichlet Problem [arXiv]
• A. Binder, T. Lelievre, G. Simpson. A Generalized Parallel Replica Dynamics. Journal of Computational Physics, 284, 595-616, 2015. [arXiv]
• D. Aristoff, T. Lelievre, G. Simpson. The parallel replica method for simulating long trajectories of Markov chains. AMRX, 2014(2), 332-352, 2014. [arXiv]
• G. Simpson, M. Luskin. Numerical Analysis of Parallel Replica Dynamics. M2AN, 47:1287-1314, 2013. [pdf][arXiv]
• X. Liu, G. Simpson, C. Sulem. Stability of Solitary Waves for a Generalized Derivative Nonlinear Schrodinger Equation. Journal of Nonlinear Science, 23:557-583, 2013. [pdf][arXiv]
• D.M. Ambrose, G. Simpson, J.D. Wright, D.G. Yang. Ill-Posedness of Degenerate Dispersive Equations. Nonlinearity, 25(9): 2655--2680, 2012. [pdf][arXiv]
• G. Simpson, M.I. Weinstein. Coherent Structures and Carrier Shocks in the Nonlinear Periodic Maxwell Equations. Submitted [pdf][arXiv]
• G. Simpson, M. Spiegelman. Solitary Wave Benchmarks in Magma Dynamics. Journal of Scientific Computing. [pdf][SpringerLink]
• J. Colliander, G. Simpson, and C. Sulem. Numerical simulations of the energy- supercritical Nonlinear Schrodinger equation. JHDE, 7:279--296, 2010. [pdf]
See the Petviashvilli's method in the gallery for an example of a computational algorithm for computing a nonlinear wave. Also see supercritical simulations from the Colliander, Simpson, and Sulem paper and the nonlinear maxwell simulations from the Simpson and Weinstein paper in the gallery.

### Well-Posedness of PDEs

Studying the well-posedness of equations tells us whether they have solutions, in a mathematical sense, and how those solutions behave. Do they always exist? Do they develop singularities? Answering these questions is of interest not just in the pure sense, but also in applications modeled by the PDE. For example, does the appearance of a singularity in an equation reflect a true singularity of the system, or a failure of the assumptions used to model the system?

Some of the problems I have worked on in this context include a nonlinear wave equation arising in Earth science, the Zakharov equations from plasma physics, and the nonlinear Schrodinger equation.

Publications:
• D. M. Ambrose, G. Simpson. Local Existence Theory for Derivative Nonlinear Schrödinger Equations with Non-Integer Power Nonlinearities [arXiv]
• G. Simpson, I. Zwiers. Vortex Collapse for the L2-Critical Nonlinear Schrodinger Equation. Journal of Mathematical Physics, 52(8):083503, 2011. [pdf][arXiv][codes]
• G. Simpson, C. Sulem, and P.L. Sulem. Arrest of Langmuir wave collapse by quantum effects. PRE , 80:5, 056405, 2009. [pdf]
• G. Simpson, M. Spiegelman, and M.I. Weinstein. Degenerate dispersive equations arising in the study of magma dynamics. Nonlinearity, 20:21--49, 2007. [pdf]

### Applications

As a graduate student, I was funded by an NSF IGERT combining Earth science and applied mathematics. This has led me to become quite interested in problems in the solid Earth, particularly the rheological properties of Earth materials. I am particularly interested in magma migration, how molten rock flows in the Earth's interior, as this may be fundamental to fully understanding the interaction between plate tectonics and mantle convection.

The closure problem inherent to systems with multiple temporal and spatial scales is particularly challenging. Macroscopic variations should impact the fine scale, but can the fine scale dynamics influence the large scale? If so how does one model the fine scale effect on the macroscopic scale, without having to resolve the fine scale?

During my time as a PIRE postdoc, I have become active interested in materials science problems, particularly the metastability problem. I am also interested in modeling challenges in fluid mechanics and nonlinear optics.

Publications:
• G. Simpson, M. Spiegelman, and M.I. Weinstein. A Multiscale Model of Partial Melts 2: Numerical Results. Journal of Geophysical Research -- Solid Earth, 115, B04411. [pdf]
• G. Simpson, M. Spiegelman, and M.I. Weinstein. A Multiscale Model of Partial Melts 1: Effective Equations. Journal of Geophysical Research -- Solid Earth, 115, B04410. [pdf]
See the cell problems in the gallery for some example computations from this work.