# Multiscale Modeling

Most theoretical and real world systems can be understood at multiple scales of detail. Consider a system of gas particles. This system could be understood in terms of:

• quantum mechanical wavefunctions for the particles comprising each atom,
• trajectories of individual gas molecules,
• probability density functions of particle positions and velocities,
• hydrodynamic quantities like pressure and bulk velocity.

Hierarchies of description linked by assumptions and simplifications exist in biology, chemistry, engineering, physics, and theoretical mathematics systems. However, there exist systems for which the assumptions necessary to model them at one level are not met, but which are also impractical or impossible to analyze and simulate at lower levels. There are also cases in which the different scales of detail cannot be neatly separated.

In both of these cases, a multiscale approach to the problem is necessary. Multiscale techniques leverage the different information made available by viewing the same problem at multiple scales. My research focuses on multiscale techniques as both a practical simulation tool and as a means of understanding the structure of complex systems with multiple scales.

• ### Reduced order modeling with memory

Constructing novel reduced order models with memory that are accurate for long times

Many mathematical systems require a large (or even infinite!) number of variables to fully resolve. Reduced order modeling is the process of constructing a simplified model for only a small number of these variables. Ideally, in this reduced model, the resolved variables should evolve in the same way that they would have in the full system.

The Mori-Zwanzig formalism is a mathematical framework for studying reduced order modeling. It contains a memory term, so called because it encodes the history of interactions between the resolved and the unresolved modes. In order to construct a reduced order model from the Mori-Zwanzig formalism, one must approximate this memory term in some way.

There have been several successful approximation methods for the memory term. I work on developing new and more accurate approximation methods that require fewer assumptions. The nature of these reduced order models reveal impressive and beautiful structure inherited from the parent systems. Now, I am working on fully characterizing the roles memory plays in multiscale modeling.

• ### Multiscale plasma simulation

Combining molecular dynamics and kinetic simulation methods for plasma systems

The heterogeneous multiscale method (HMM) is a popular multiscale simulation scheme for systems with scale separation. Scale separation between two descriptions of the same system means that each description is relevant on widely different scales. For example, in the simulation of a crack propagating through a solid, a detailed molecular understanding may be relevant only at the crack tip, while a continuum description may work elsewhere. Alternatively, in a fast-slow dynamical system, the “fast” variables may evolve on an effectively instantaneous timescale as far as the “slow” variables are concerned.

I worked at Los Alamos National Laboratory to construct an HMM simulation method of plasma physics systems. My collaborators were Jeff Haack and Mathieu Marciante of Los Alamos National Laboratory, Gil Shohet of Stanford University, and Michael Murillo of Michigan State University. We married a kinetic and molecular description of the same system. This required careful mathematical derivation of one from the other, as well as detailed knowledge of numerical techniques relevant for each scale. This method vastly expands the size and timescales of plasma systems we can probe, which may lead to advances in understanding plasma systems with large shocks such as those that occur in inertial confinement fusion reactions.

The figure below depicts multiple simulation methods applied to a temperature relaxation problem between hydrogen ions (dotted lines) and aluminum ions (solid lines). The green curves are the “exact answer” from a full molecular simulation. Our multiscale method gives the blue curves, while the purple and red curves are other non-multiscale methods.

I am always eager to work with scientists and engineers in academia, national laboratories, and industry to construct problem-specific multiscale simulation methods. If you have such a project, I would love to talk with you!

In addition to my primary work with multiscale simulation methods, I have worked on several related projects. Each exists in the world of numerical methods and scientific computing.

• ### Conditional path sampling

Sampling transition paths between metastable states

A dynamical system may possess multiple equilibria, or points where the system no longer changes. In a stochastic system, random noise is introduced to the dynamics. In this case, the system may not stay in a stable equilibrium for all time. A series of “random” jumps may drive the system from one stable equilibrium to another. These transition paths can be extremely rare!

Protein folding is an example of this phenomenon. A protein may stably rest in an unfolded state, and also possesses a stable folded conformation. A sequence of random kicks from water molecules can cause the protein to fold and shift from one to another.

Because these transition paths are rare, it is challenging to simulate them and observe what they look like. One would need to wait an exceptionally long time for a transition to occur by chance, and even that would only give one example. I work on methods of sampling different paths a system might take conditioned upon it beginning in one state and ending in another!

This work was conducted as a project with the Washington Experimental Mathematics Laboratory with undergraduate research assistants Landon Shock, Qingtong Zeng, and Jesse Rivera.

• ### Initializing molecular dynamics simulations

Finding new initializations of particle simulations that can accelerate start-up times

Molecular dynamics simulations are a classical area of computational physics. Particles interact with one another through gravity, electrical attraction and repulsion, or other interparticle forces. The trajectories of many particles are tracked through time and from these trajectories we can compute statistical quantities such as temperature and pressure.

A central issue is the specification of the initial positions of particles. This is a level of detail that is typically not known or of interest. However, if two positively charged particles are placed very near one another at the onset, they will rapidly repel one another and immediately increase the temperature of the system. The lack of knowledge about the initial state causes the simulation to immediately veer away from the desired starting point!

There are solutions: scientists use a temporary equilibration phase to allow the inconsistencies of the initial condition to shake out over time. Sometimes the equilibration phase takes longer than the simulation itself. I work with Michael Murillo of Michigan State University on using mathematical sequences that evenly distribute points through a domain to reduce the need for lengthy equilibration phases.

On the top is a random initialization. Notice that the points cluster! Below is one of our initializations, which distribute the points more evenly. On the right are the respective radial distribution functions, indicating the likelihood of finding particles with a particular distance between them.

• ### Mathematics and music

Characterizing the patterns found in musical compositions

I have an abiding passion for music as well as mathematics. In college, I conducted research with Eric Barth at Kalamazoo College on numerical analysis of musical compositions.

There are countless points of entry for analyzing the mathematical structure of music. All sounds are acoustic waves, and the frequency of those waves determines the pitch. Different pitches sound harmonious together, and are used to construct chords. One can analyze the distribution of different chords in a musical composition or many compositions of a composer. Musical chords and scales also possess rich structure. Constructing informative visualizations of musical concepts can draw from diverse mathematical areas such as topology.

A musical piece consists of pitches arranged in time as well in the form of rhythm. Certain composers might also show an affinity for particular rhythmic patterns. With these ideas, we began exploring the idea that the development of music across the centuries can be understood as composers exploring and expanding these different patterns.