Daniel Appelö

I am a Professor in the Department of Mathematics and the Computational Modeling and Data Analytics (CMDA) program at Virginia Tech. My group develops fast, stable and accurate numerical algorithms for the approximation of differential equations arising in engineering and the natural sciences.

Much of our work is about waves — acoustic, elastic and electromagnetic — in both the time and the frequency domain, and about what it takes to make high order methods provably stable rather than merely accurate on smooth problems. More recently we have also been working on quantum computing.

Our most ambitious current effort, running over the next five to seven years, is Quantum Digital Twins (QDTs): rigorous, self-correcting, bi-directional replicas of quantum computing hardware, together with the structure preserving and scalable numerical methods they rest on. A QDT couples three layers — the quantum dynamics of the qubits (Schrödinger and Lindblad equations), the electromagnetic design of the device (Maxwell and Helmholtz equations), and quantum error correction — into a single multi-fidelity framework. The framework is learned from measurements through Bayesian experimental design and optimal control, and in return it certifies and drives the physical device. Making this work rests on two pillars of numerical analysis that are worth pursuing in their own right: fast, scalable frequency domain wave solvers, and structure preserving low rank and tensor network integrators for high dimensional open quantum systems — mathematical technology whose value extends well beyond quantum computing.

Research

• Wave propagation. High order accurate methods for acoustic, elastic and electromagnetic waves, built so that stability follows from a discrete energy estimate. This includes energy based discontinuous Galerkin methods, Hermite and Hermite-Taylor methods, and summation-by-parts finite differences.
• Frequency domain solvers. The WaveHoltz iteration solves the Helmholtz equation by time domain wave solves, turning an indefinite problem into a positive definite one that parallelizes and scales. Related work covers elastic and electromagnetic waves, overset grids, and eigenvalue computation.
• Quantum computing. Structure preserving methods for the Schrödinger and Lindblad equations — completely positive and trace preserving integrators in particular — together with quantum optimal control, quantum error correction, and the Bayesian characterization that ties a digital twin back to the device it models.
• Low rank and tensor methods. Adaptive low rank time stepping, low rank Anderson acceleration, and tensor network integrators for the high dimensional problems that open quantum systems and matrix differential equations give rise to.
• Unbounded domains. Artificial boundary conditions — perfectly matched layers and local high order radiation conditions — for problems posed on unbounded domains.

See the publications page for the full list.

Openings

We will have openings for both Postdocs and PhD students. Please contact me to learn more, and see the group page for who you would be working with.

Group

The group currently includes two postdocs and four PhD students at Virginia Tech and Michigan State. Former members have gone on to faculty positions at CSU Long Beach, HKUST and UT Dallas, and to postdocs at Los Alamos, the Courant Institute and Arizona State — the group page has the full roster.