Group

The group works on high order accurate numerical methods for differential equations. We will have openings for both postdocs and PhD students — please get in touch if you are interested.

Current members

Daniel Appelö

Group leader, Professor, Mathematics and CMDA, Virginia Tech

Jiuhua Hu

Postdoc

With Jiuhua and Yingda Cheng we build arbitrary high order integrators for the Lindblad equation with time-dependent Hamiltonians. The schemes come from nested Picard iteration, take Kraus form, and are completely positive and trace preserving by construction, so the simulated density matrix never drifts away from being a physical state. A follow-up uses Gregory quadrature on equispaced nodes to reach ninth order.

Måns Andersson

Postdoc

Måns works on sub-linear low rank solvers, at the moment a fast Poisson solver built on Cross-DEIM, an adaptive cross approximation that alternates between selecting indices and rebuilding a low rank solution. A second thread of the work is using PyTorch as a general purpose array language, which buys native GPU access without hand-writing device code.

Amit Rotem

PhD candidate, Virginia Tech (expected 2026)

Amit works on the convergence theory behind WaveHoltz. With Olof Runborg we proved that the iteration converges for any stable semi-discretization of the wave equation, and that for some classes of frequency domain problems it needs only O(omega) iterations. More recently we combined WaveHoltz with a heterogeneous multiscale method, so the same machinery handles materials whose coefficients vary rapidly.

Spencer Lee

PhD candidate, Michigan State University (expected 2026)

Spencer works on quantum optimal control. Our High-Order Hermite Optimization method computes exact discrete gradients for continuous, parameterized control pulses while solving Schrödinger’s equation or the Lindblad master equation with arbitrarily high order Hermite Runge-Kutta schemes; it is implemented in the Julia package QuantumGateDesign.jl. We are now adapting Filon quadrature to the highly oscillatory dynamics that make these simulations expensive.

Peter DelMastro

PhD candidate, Virginia Tech (expected 2027)

Peter works on pushing our Lindblad solvers to larger systems. With Yingda Cheng we compress at two levels: the density matrix is factored into tall- skinny matrices, and the columns of those factors are themselves stored as tensor trains, or matrix product states. This fits directly onto our earlier Kraus is King scheme, so the result stays completely positive and trace preserving.

Nishan Gurung

PhD student, Virginia Tech (expected 2030)

Alumni

Shixu Meng

Postdoc (2024-2025)

Co-mentored with Yingda Cheng (primary mentor).

With Shixu and Yingda Cheng we developed a class of preconditioners for low rank GMRES applied to the multiterm matrix equations that come out of implicit time stepping for stiff PDEs. The iteration runs on the low rank factors of the solution rather than on the full matrix, which is where the savings come from, and also what makes preconditioning delicate.

First job: Assistant Professor, University of Texas at Dallas

Zhichao Peng

Postdoc (2020-2023)

Zhichao’s work spans both halves of the group. On the wave side we built EM- WaveHoltz, which extracts time-harmonic solutions of Maxwell’s equations from time-domain simulation and yields a positive definite system, together with an embedded boundary discretization that avoids small cell stiffness altogether. On the quantum side we worked on characterizing superconducting devices from measurement data. Most recently we have been accelerating WaveHoltz by deflating the eigenvectors nearest the driving frequency.

First job: Assistant Professor, Hong Kong University of Science and Technology

Yann-Meing Law

Postdoc (2021-2023)

Yann-Meing worked on making Hermite-Taylor methods usable on real geometry. The method is highly efficient on periodic domains but had no systematic way to impose boundary conditions; our correction function method supplies one, and extends to embedded boundaries and material interfaces for Maxwell’s equations. We also built energy-conserving Hermite methods for dielectric and dispersive media, and a p-adaptive variant for nonlinear dispersive Maxwell.

First job: Assistant Professor, California State University Long Beach

Allen Alvarez Loya

PhD, CU Boulder (2022)

Allen worked on Hermite methods for problems where the solution is not smooth or the geometry is not a box. For Hamilton-Jacobi equations we built a Hermite solver with a discontinuity sensor that keeps high order accuracy in smooth regions while resolving kinks sharply. With Bill Henshaw we then took Hermite methods to curvilinear grids using centered compatibility conditions at the boundary, and he was part of extending WaveHoltz to elastic waves.

High-Order Methods for Wave Phenomena

First job: NSF Postdoc at Los Alamos National Laboratory

Fortino Garcia

PhD, CU Boulder (2021)

Fortino was there for the start of WaveHoltz, the observation that filtering the solution of the wave equation over one period yields a coercive operator, and so a positive definite system for the Helmholtz problem. With Olof Runborg we extended the analysis to impedance boundary conditions and to damped problems, and then to elastic waves in El-WaveHoltz. He also worked on the quantum side of the group, including the characterization of superconducting devices.

Part I: WaveHoltz, a new iterative method for frequency domain problems. Part II: Numerical methods for quantum control

First job: NSF Postdoc at the Courant Institute, New York University

Oleksii Beznosov

PhD, University of New Mexico (2020)

Oleksii’s work spanned two quite different problems. One was combining Hermite methods with discontinuous Galerkin on overset grids, using thin curvilinear grids near boundaries and Cartesian grids in the volume so that the cost approaches that of a structured Hermite method. The other was spin polarization in high energy electron storage rings, where we worked on the Bloch equation for the polarization density with the FCC-ee and CEPC designs in mind.

From Wave Propagation to Spin Dynamics: Mathematical and Computational Aspects

First job: Postdoc at Los Alamos National Laboratory

Adeline Kornelus

PhD, University of New Mexico (2017)

With Adeline we developed flux-conservative Hermite methods for nonlinear conservation laws, keeping the high order spatial accuracy of Hermite methods on smooth solutions while behaving far better where the solution steepens. Shocks are captured with entropy viscosity, and a companion paper examined how the choice of scaling in that viscosity controls how much dissipation you actually get.

High Order Hermite and Discontinuous Galerkin Methods for Hyperbolic Problems

First job: Postdoc at Arizona State University

Juan Pablo Madrigal Cianci

MS, University of New Mexico (2017)

Co-mentored with Mohammad Motamed (primary mentor).

Deterministic and Probabilistic Methods for Seismic Source Inversion

First job: PhD student at EPFL

Evan Dye

MS, University of New Mexico (2014)

GPU Implementations of Hermite Methods

First job: Researcher in the Computer Science Department at the University of New Mexico

Xi Chen

PhD, University of New Mexico (2012)

Numerical and Analytical Studies of Electromagnetic Waves: Hermite Methods, Supercontinuum Generation, and Multiple Poles in the SEM

First job: Postdoc at the University of Arizona