Thomas Hogancamp

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Quantum Computing

My Postdoctoral work at the Naval Research Laboratory is focused on investigating applications of Quantum Computing to Partial Differential Equations arising in fluid dynamics. More specifically, I investigate novel methods to implement numerical schemes in the quantum framework for PDEs such as Poisson problems, the viscous Burgers’ equation, and the Discrete Velocity Boltzmann equation (or the closely related Lattice Boltzmann Equation). Some key aspects of these works include matrix decomposition, block-encoding strategies for both unitary and non-unitary operators, quantum circuit design, complexity analyses/resource estimates for both NISQ and fault tolerant implementations, and accuracy/convergence/stability analyses for the relevant numerical schemes.

Thesis Research

The focus of my PhD research was on applications of bifurcation theory to nonlinear PDEs. The equations I studied have the following general structure:

\[\nabla \cdot (\mathcal{W}'(|\nabla u|^{2})\nabla u)=0.\]

In practice, one would specify suitable boundary conditions in addition to the PDE given above. The function $\mathcal{W}$ is interpreted as the strain energy density and $u$ as the displacement in the context of nonlinear elasticity theory (when studying gas dynamics, these function represent instead the density and velocity of a gas).

I am also interested in adapting tools from classical elliptic theory to degenerate elliptic PDEs. The Maximum Principle, the Hopf Lemma, and the standard machinery of elliptic regularity either fail outright or become vastly more complex in this case. Consequently, far less is known about the qualitative and quantitative properties of solutions to such problems. Insights from transonic gas dynamics suggest that stability and smoothness can’t be expected generally.

In my research article titled Broadening Global Families of Anti-Plane Shear Equilibria, families of equilibrium solutions are constructed for several classes of materials. One class is shown to exhibit broadening. This is characterized by displacements which become increasingly wide and flat. Numerical simulations predict related behavior for internal solitary water waves, but it remains a challenging open question to rigourously verifiy their existence. Another class of materials are shown to generate solutions that lose ellipticity; a phenomena that is that is intimately linked to shock formation.

I also conducted follow-up research to help shed light on the consequences of degenerate ellipticity in the governing equations of anti-plane shear. Singularities are shown to develop as the strain reaches a critical state and ellipticity is lost. Degenerate elliptic PDEs, and more generally mixed elliptic-hyperbolic equations, continue to present serious mathematical challenges for many physically relevant problems.

Publications and Preprints Pre-prints can be found on the arXiv or through my Researchgate account.

Conference Presentations/Talks