Research & Students

My research is focused on the design and analysis of efficient and accurate numerical methods for nonlinear and multiscale partial differential equations and eigenvalue problems.

My research on accelerating low-memory iterative methods for eigenvalue problems is funded by a CAREER award from the National Science Foundation, DMS 2045059: Extrapolation methods for matrix and tensor eigenvalue problems.

My ongoing work on nonlinear PDE includes:

  • Design and analysis of extrapolation methods and acceleration for nonlinear problems.
  • Quantifying the behavior of numerical algorithms in the preasymptotic regime.
  • Developing global convergence theory for methods that start far from the asymptotic regime.
  • Proving comparison theorems and developing uniqueness theory for finite element solutions to nonlinear elliptic equations of nonmonotone type.
  • Design of efficient regularized methods for computing solutions to quasilinear PDE, especially those of nonmonotone type.
  • Goal-oriented adaptivity for multiscale methods.

This research is supported by a collaborative grant from the National Science Foundation, DMS 2011519, with Leo Rebholz from Clemson University, and a single PI grant from the NSF, DMS-1852876, 2018-2020 (previously DMS-1719849, 2017-2018). My work on multiscale problems is focused on adaptive enrichment methods, particularly goal-oriented adaptivity.

I also have a continuing interest in computational geometry, which started from my undergraduate research project. I have worked on developing robust algorithms for inverse-kinematic loop closure, and on characterizing the solution space and singularities of both the geometrical and computational problems.

keywords:
nonlinear partial differential equations, eigenvalue problems, extrapolation, Anderson acceleration, finite element analysis, nonmonotone problems, multiscale methods, discrete comparison principles, uniqueness of discrete solutions, adaptive methods, regularization, goal oriented methods, inverse-kinematics.

Students:

Graduate (current):

Michelle Baker
Christian Austin
Rhea Shroff, Ph.D. 2024: Extrapolation Methods for Tensor Z-eigenvalue Solvers
Matt Dallas, Ph.D. 2024: Convergence of Newton-Anderson at Singular Points and Applications to Fluid Problems. Now at University of Dallas.

Undergraduate:

Simon Kato, USP Scholar, Honors thesis, 2022: Extrapolated Restarted Arnoldi for Solving the PageRank Problem. Now at University of Illinois Urbana-Champaign (Computer Science).

Parker Knight, Honors thesis, 2020: Data-driven adaptive penalties for high-dimensional regression. Now at Harvard University (Biostatistics); awarded fellowship from NSF-GRFP.