Research

My research is in applied mathematics, classical mathematical modeling, analysis of ODE and PDE, and applied optimal control theory. I have been lucky to work on a variety of projects along these lines as described below. My chronological publication list can be found on my CV.

Level Set Inspired Optimal Path Planning and Applications

I am generally interested in level set methods, Hamilton-Jacobi equations, applied optimal control theory and the corresponding numerical methods. My recent work is in designing efficient and interpretable methods for optimal path planning in high dimensions using Hopf-Lax type formulas.

A Hopf-Lax Type Formula for Multi-Agent Path Planning with Pattern Coordination, C. Parkinson and A. Baca, to appear at the 2025 IEEE Conference on Decision and Control. (ArXiv Version)
Efficient and Scalable Path-Planning Algorithms for Curvature Constrained Motion in the Hamilton-Jacobi Formulation, C. Parkinson and I. Boyle. Journal of Computational Physics, 509, 2024, 113050. (ArXiv Version; Journal link)
An Efficient Semi-Real-Time Algorithm for Path Planning in the Hamilton-Jacobi Formulation, C. Parkinson and K. Polage. IEEE Control Systems Letters, vol. 7, 2023, pp. 3621-3626. (ArXiv version; Journal link)
Time-Optimal Paths for Simple Cars with Moving Obstacles in the Hamilton-Jacobi Formulation, C. Parkinson, M. Ceccia. 2022 American Control Conference (ACC), 2022, pp. 2944-2949. (ArXiv Version; Journal link)
Modeling Illegal Logging in Brazil, B. Chen, K. Peng, C. Parkinson, A. L. Bertozzi, T. L. Slough, J. Urpelainen. Research in the Mathematical Sciences, 8(29), 2021. (Journal link (open access provided by publisher))
A Rotating-Grid Upwind Fast Sweeping Scheme for a Class of Hamilton-Jacobi Equations, C. Parkinson. Journal of Scientific Computing, 88(13), 2021. (ArXiv version; Journal link)
A Hamilton-Jacobi Formulation for Time-Optimal Paths of Rectangular Nonholonomic Vehicles, C. Parkinson, A. L. Bertozzi, S. Osher. 2020 59th IEEE Conference on Decision and Control (CDC), 2020, pp. 4073-4078. (ArXiv version; Journal link)
A Model for Optimal Human Navigation with Stochastic Effects, C. Parkinson, D. Arnold, A. L. Bertozzi, S. Osher. SIAM Journal on Applied Mathematics, 80(4), 2020, pp. 1862-1881. (ArXiv version; Journal link)
Modeling Environmental Crime in Protected Areas Using the Level Set Method, D. Arnold, A. L. Bertozzi, D. Fernandez, R. Jia, S. Osher, C. Parkinson, D. Tonne, Y. Yaniv. SIAM Journal on Applied Mathematics, 79(3), 2019, 802-821. (ArXiv version; Journal link)
Optimal Human Navigation in Steep Terrain: A Hamilton-Jacobi-Bellman Approach, C. Parkinson, D. Arnold, A. L. Bertozzi, Y. T. Chow, S. Osher. Communications in Mathematical Sciences, 17, 2019, pp. 227-242. (ArXiv version; Journal link)
A Scalable Method for Optimal Path Planning on Manifolds via a Hopf-Lax Type Formula, E. Huynh and C. Parkinson, unpublished manuscript (Arxiv Version)

Biological Applications (Mathematical Virology and Epidemiology)

I am interested in population modeling, mostly using ODE and PDE, though I have also worked on network models. Specifically, my current interest is in designing and analyzing epidemic models which account for human behavior.

Optimal Control of an SIR model with Noncompliance as a Social Contagion, C. Ngo, C. Parkinson, and W. Wang. Submitted.
A Compartmental Model for Epidemiology with Human Behavior and Stochastic Effects, C. Parkinson, and W. Wang. Submitted. (ArXiv version)
Optimal Control of a Reaction-Diffusion Epidemic Model with Noncompliance, M. Bongarti, C. Parkinson, and W. Wang, to appear in the European Journal of Applied Mathematics. (ArXiv version; Open Access to Online Publication Provided by Publisher)
Optimal lockdowns under constraints, J.C. Dagher and C. Parkinson, Economic Inquiry, 63(2), 2025, pp. 523–544. (Journal Link)
Analysis of a Reaction-Diffusion SIR Epidemic Model with Noncompliant Behavior, C. Parkinson and W. Wang. SIAM Journal on Applied Mathematics, 83(5), 2023, pp. 1969-2002. (ArXiv version; Journal link)
Alternative SIAR models for infectious diseases and applications in the study of non-compliance, M. Bongarti, L. D. Galvan, L. Hatcher, M. R. Lindstrom, C. Parkinson, C. Wang, A. L. Bertozzi. Mathematical Models and Methods in Applied Sciences, 32(10), 2022, pp. 1987-2015. (Journal link (manuscript can be provided upon request))
A Multilayer Network Model of the Coevolution of the Spread of a Disease and Competing Opinions, K. Peng, Z. Lu, V. Lin, M. R. Lindstrom, C. Parkinson, C. Wang, A. L. Bertozzi, M. A. Porter. Mathematical Models and Methods in Applied Sciences, 31(12), 2021, pp. 2455-2494. (ArXiv version; Journal link)
Mathematical Analysis of an In-Host Model of Viral Dynamics with Spatial Heterogeneity, S. Pankavich and C. Parkinson. Discrete and Continuous Dynamical Systems B., 21(5), 2016, pp. 1237-1257. (ArXiv version; Journal link)

Discrete Differential Geometry for Hyperbolic Surfaces

Thin elastic sheets which bend, buckle, and wrinkle arise ubiquitously from both natural and artificial processes. Some applications where these appear are biological processes like leafy growth or marine flatworm locomotion, the compression of thin steel sheets during automotive collisions, the irregular crenellations at the edges of torn plastic , and photo- and thermo-sensitive hydrogels. I am interested in developing continuous and discrete methods for describing and generating hyperbolic surfaces to model such phenomena. In particular, hyperbolic surfaces are often analyzed via immersions into Euclidean space which are assumed to be minimizers of some elastic energy functional. While smooth immersions of finite hyperbolic surfaces exist, they do not predict the type of buckling and subwrinkling often observed in application. However, one can observe these features mathematically by allowing for piecewise-smooth Lipschitz-continuous immersions. Theory and methods regarding these topics have been developed for surfaces of constant negative curvature. I have worked to generalize to surfaces of non-constant negative curvature, which could account for things like flower petals which are relatively flat toward the middle and only become curvy near the edges. Pictured below: discrete surfaces of non-constant negative curvature. The surfaces are comprised of sectors which are images of smooth immersions, and are patched together so that the normal vector along the surface remains continuous. The curvature is constant in a disc of geodesic radius 1 from the center (black). Outside of this disc, the curvature increases rapidly with both geodesic distance and with the parameter epsilon.

Discrete Differential Geometry for C1,1 Hyperbolic Surfaces of Non-Constant Curvature, C. Parkinson and S. Venkataramani. Submitted. (ArXiv version)

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