# Research

I work in computational science and engineering, and my research mixes mathematical analysis, application modeling, and software design. This page introduces some of my work -- I've left out older projects, projects where there aren't any results yet, and projects that can't readily be crammed into the listed categories. My CV gives a more complete list of talks and publications (with hyperlinks).

#### Optimizing stellarators

Advancing magnetic confinement fusion through optimization and hidden symmetries.

#### Nonlinear waves in resonant MEMS

Seeking solid-wave solitons in resonant MEMS

#### Kernel methods

Theory and scalable algorithms for kernel-based function approximation.

#### Parallel surrogate optimization

Asynchronous parallel algorithms for finding minima fast by fitting functions to surrogate models.

#### Spectral network analysis

Fast spectral tools for graph structure.

#### Nonlinear eigenvalue bounds

Generalizing classic eigenvalue localization results (e.g. Gershgorin).

#### Spectral topic modeling

Better topic inference by enforcing implied structures.

## Older projects

#### Making grids smarter

New algorithms for diagnosing line failures in a smart grid.

#### Rotation averaging

How hard is it to combine pictures to understand scene geometry?

#### Scattering and resonances

Linear eigenproblems for standing waves; nonlinear eigenproblems for almost-standing waves.

#### Rank-structured PDE solvers

Moving toward pain-free fast direct sparse solvers for 3D PDEs.

#### Solid-wave MEMS gyroscopes

Building simulation tools to understand the challenges in miniaturizing the current gyro-of-choice for space applications. Success means MEMS IN SPACE!

#### Continuing invariant subspaces

If $A(s)$ is a parameter-dependent matrix, how do we compute an invariant subspace basis $V(s)$?

#### Declarative HPC

Optimizing parallel code is hard -- so automate as much as possible.

#### Opinions as game theory

In the game of expressing our opinions, we all want to both follow our beliefs and agree with our friends. Analyzing Nash equilibria yields interesting insights. Eigenvalues are involved!

#### Network tomography

Inferring internal network state by measuring end-to-end paths. It's a CAT scan for the Internet!

#### Model reduction for MEMS

Simulating small devices using small models!