Ari Stern

Professor of Mathematics
Washington University in St. Louis

About Me

I received my B.A. and M.A. from the mathematics department at Columbia University. In 2009, I received my Ph.D. in Applied and Computational Mathematics at Caltech, under the direction of the late Jerrold E. Marsden and Mathieu Desbrun. Before arriving at WashU in 2012, I was a postdoc in the mathematics department at UCSD, where I worked with Michael Holst.

Teaching (Fall 2024)

Math 449, Numerical Applied Mathematics.

Previous Courses

Spring 2024: Math 5052, Measure Theory and Functional Analysis II.
Fall 2023: Math 5051, Measure Theory and Functional Analysis I.
Spring 2023: Math 4121, Introduction to Lebesgue Integration.
Fall 2022: Math 4111, Introduction to Analysis.
Spring 2022: Math 5052, Measure Theory and Functional Analysis II.
Fall 2021: Math 5051, Measure Theory and Functional Analysis I.
Spring 2021: Math 204, Honors Mathematics II.
Fall 2020: Math 203, Honors Mathematics I.
Spring 2020: Math 308, Mathematics for the Physical Sciences.
Spring 2020: Math 450, Numerical Methods for Differential Equations.
Spring 2019: Math 233, Calculus III.
Fall 2018: Math 547, Geometric Mechanics.
Spring 2018: Math 450, Numerical Methods for Differential Equations.
Fall 2017: Math 449, Numerical Applied Mathematics.
Fall 2017: Math 456, Topics in Financial Mathematics.
Spring 2017: Math 217, Differential Equations.
Spring 2016: Math 450, Numerical Methods for Differential Equations.
Fall 2015: Math 449, Numerical Applied Mathematics.
Fall 2015: Math 456, Topics in Financial Mathematics.
Spring 2015: Math 131, Calculus I.
Spring 2015: Math 450, Numerical Methods for Differential Equations.
Fall 2014: Math 449, Numerical Applied Mathematics.
Spring 2014: Math 515, Partial Differential Equations.
Fall 2013: Math 456, Topics in Financial Mathematics.
Spring 2013: Math 5052, Measure Theory and Functional Analysis II.
Fall 2012: Math 5051, Measure Theory and Functional Analysis I.

Research Interests

My research lies at the intersection of geometry, applied analysis, and computational mathematics. I am interested in what I call geometric numerical analysis: using geometry as a means to develop novel numerical methods and techniques to analyze them. The driving idea behind this work is the need for numerical methods for differential equations that are accurate globally, not just locally—and, in recent years, it has been shown that these global features have important (and often surprising) connections with modern geometry, particularly differential and symplectic geometry.

Publications and Preprints

 

Stern, A., and M. Viviani (2024), Quadratic projectable Runge-Kutta methods. Preprint. [ bib | arXiv ]

 

Stern, A., and S. Suri (2024), Functional equivariance and modified vector fields. J. Comput. Dyn., 11 (4), 409-426. [ bib | doi | arXiv ]

 

Barker, M., S. Cao, and A. Stern (2024), A nonconforming primal hybrid finite element method for the two-dimensional vector Laplacian. SMAI J. Comput. Math., 10, 85-106. [ bib | doi | arXiv ]

 

McLachlan, R. I., and A. Stern (2024), Functional equivariance and conservation laws in numerical integration. Found. Comput. Math., 24 (1), 149-177. [ bib | doi | arXiv ]

 

Hu, J., and A. Stern (2024), Hamiltonian mechanics and Lie algebroid connections. J. Nonlinear Sci., 34 (1), Paper No. 9, 23 pages. [ bib | doi | arXiv ]

 

Stern, A., and E. Zampa (2023), Multisymplecticity in finite element exterior calculus. Preprint. [ bib | arXiv ]

 

Awanou, G., M. Fabien, J. Guzmán, and A. Stern (2023), Hybridization and postprocessing in finite element exterior calculus. Math. Comp., 92 (339), 79-115. [ bib | doi | arXiv ]

 

Chambers, E., M. Duchin, R. A. C. Edmonds, P. Edwards, J. Matthews, A. E. Pizzimenti, C. Richardson, P. Rule, and A. Stern (2022), Aggregating community maps. In SIGSPATIAL '22: Proceedings of the 30th International Conference on Advances in Geographic Information Systems, Paper No. 27, 12 pages, ACM Press, New York. [ bib | doi ]

 

Olver, P. J., and A. Stern (2021), Dispersive fractalization in linear and nonlinear Fermi-Pasta-Ulam-Tsingou lattices. European J. Appl. Math., 32 (5), 820-845. [ bib | doi | arXiv ]

 

Smith, G., A. Stern, H. Tran, and D. Zhou (2021), On the Morse index of higher-dimensional free boundary minimal catenoids. Calc. Var. Partial Differential Equations, 60 (6), Paper No. 208, 44 pages. [ bib | doi | arXiv ]

 

Berchenko-Kogan, Y., and A. Stern (2021), Constraint-preserving hybrid finite element methods for Maxwell's equations. Found. Comput. Math., 21 (4), 1075-1098. [ bib | doi | arXiv ]

 

Berchenko-Kogan, Y., and A. Stern (2021), Charge-conserving hybrid methods for the Yang-Mills equations. SMAI J. Comput. Math., 7, 97-119. [ bib | doi | arXiv ]

 

Luckett, P. H., A. McCullough, B. A. Gordon, J. Strain, S. Flores, A. Dincer, J. McCarthy, T. Kuffner, A. Stern, K. L. Meeker, S. B. Berman, J. P. Chhatwal, C. Cruchaga, A. M. Fagan, M. R. Farlow, N. C. Fox, M. Jucker, J. Levin, C. L. Masters, H. Mori, J. M. Noble, S. Salloway, P. R. Schofield, A. M. Brickman, W. S. Brooks, D. M. Cash, M. J. Fulham, B. Ghetti, C. R. Jack Jr., J. Vöglein, W. Klunk, R. Koeppe, H. Oh, Y. Su, M. Weiner, Q. Wang, L. Swisher, D. Marcus, D. Koudelis, N. Joseph-Mathurin, L. Cash, R. Hornbeck, C. Xiong, R. J. Perrin, C. M. Karch, J. Hassenstab, E. McDade, J. C. Morris, T. L. Benzinger, R. J. Bateman, and B. M. Ances, for the Dominantly Inherited Alzheimer Network (DIAN) (2021), Modeling autosomal dominant Alzheimer's disease with machine learning. Alzheimer's Dement., 17 (6), 1005-1016. [ bib | doi ]

 

Chen, Z., B. Raman, and A. Stern (2020), Structure-preserving numerical integrators for Hodgkin-Huxley-type systems. SIAM J. Sci. Comput., 42 (1), B273-B298. [ bib | doi | arXiv ]

 

Munthe-Kaas, H. Z., A. Stern, and O. Verdier (2020), Invariant connections, Lie algebra actions, and foundations of numerical integration on manifolds. SIAM J. Appl. Algebra Geom., 4 (1), 49-68. [ bib | doi | arXiv ]

 

McLachlan, R. I., and A. Stern (2020), Multisymplecticity of hybridizable discontinuous Galerkin methods. Found. Comput. Math., 20 (1), 35-69. [ bib | doi | arXiv ]

 

Wallace, M., R. Feres, G. Yablonsky, and A. Stern (2019), Explicit formulas for reaction probability in reaction-diffusion experiments. Comput. Chem. Eng., 125, 612-622. [ bib | doi | arXiv ]

 

Stern, A., and A. Tettenhorst (2019), Hodge decomposition and the Shapley value of a cooperative game. Games Econom. Behav., 113, 186-198. [ bib | doi | arXiv ]

 

Li, S., A. Stern, and X. Tang (2017), Lagrangian mechanics and reduction on fibered manifolds. SIGMA Symmetry Integrability Geom. Methods Appl., 13, Paper No. 019, 26 pages. [ bib | doi | arXiv ]

 

Brier, M. R., B. Gordon, K. Friedrichsen, J. McCarthy, A. Stern, J. Christensen, C. Owen, P. Aldea, Y. Su, J. Hassenstab, N. J. Cairns, D. M. Holtzman, A. M. Fagan, J. C. Morris, T. L. S. Benzinger, and B. M. Ances (2016), Tau and Aβ imaging, CSF measures, and cognition in Alzheimer’s disease. Science Translational Medicine, 8 (338), 338ra66. [ bib | doi ]

 

Brier, M. R., J. E. McCarthy, T. L. S. Benzinger, A. Stern, Y. Su, K. A. Friedrichsen, J. C. Morris, B. M. Ances, and A. G. Vlassenko (2016), Local and distributed PiB accumulation associated with development of preclinical Alzheimer's disease. Neurobiol. Aging, 38, 104-111. [ bib | doi ]

 

Leopardi, P., and A. Stern (2016), The abstract Hodge-Dirac operator and its stable discretization. SIAM J. Numer. Anal., 54 (6), 3258-3279. [ bib | doi | arXiv ]

 

Marrero, J. C., D. Martín de Diego, and A. Stern (2015), Symplectic groupoids and discrete constrained Lagrangian mechanics. Discrete Contin. Dyn. Syst., 35 (1), 367-397. [ bib | doi | arXiv ]

 

Miller, E., and A. Stern (2015), Maximum principles for the relativistic heat equation. [ bib | arXiv ]

 

Norton, R. A., D. I. McLaren, G. R. W. Quispel, A. Stern, and A. Zanna (2015), Projection methods and discrete gradient methods for preserving first integrals of ODEs. Discrete Contin. Dyn. Syst., 35 (5), 2079-2098. [ bib | doi | arXiv ]

 

Stern, A. (2015), Banach space projections and Petrov-Galerkin estimates. Numer. Math., 130 (1), 125-133. [ bib | doi | arXiv ]

 

Stern, A., Y. Tong, M. Desbrun, and J. E. Marsden (2015), Geometric computational electrodynamics with variational integrators and discrete differential forms. In Geometry, mechanics, and dynamics, volume 73 of Fields Institute Communications, pages 437-475, Springer, New York. [ bib | doi | arXiv ]

 

McLachlan, R. I., and A. Stern (2014), Modified trigonometric integrators. SIAM J. Numer. Anal., 52 (3), 1378-1397. [ bib | doi | arXiv ]

 

Stern, A. (2013), L^p change of variables inequalities on manifolds. Math. Inequal. Appl., 16 (1), 55-67. [ bib | doi | arXiv ]

 

Holst, M., and A. Stern (2012), Semilinear mixed problems on Hilbert complexes and their numerical approximation. Found. Comput. Math., 12 (3), 363-387. [ bib | doi | arXiv ]

 

Holst, M., and A. Stern (2012), Geometric variational crimes: Hilbert complexes, finite element exterior calculus, and problems on hypersurfaces. Found. Comput. Math., 12 (3), 263-293. [ bib | doi | arXiv ]

 

Stern, A. (2010), Discrete Hamilton-Pontryagin mechanics and generating functions on Lie groupoids. J. Symplectic Geom., 8 (2), 225-238. [ bib | doi | arXiv ]

 

Stern, A., and E. Grinspun (2009), Implicit-explicit variational integration of highly oscillatory problems. Multiscale Model. Simul., 7 (4), 1779-1794. [ bib | doi | arXiv ]

 

Stern, A. (2009), Geometric discretization of Lagrangian mechanics and field theories. Ph.D. thesis, California Institute of Technology. [ bib | http ]

 

Stern, A., Y. Tong, M. Desbrun, and J. E. Marsden (2008), Variational integrators for Maxwell's equations with sources. PIERS Online, 4 (7), 711-715. [ bib | doi | arXiv ]

 

Stern, A., and M. Desbrun (2006), Discrete geometric mechanics for variational time integrators. In SIGGRAPH '06: ACM SIGGRAPH 2006 Courses, pages 75-80, ACM Press, New York. [ bib | doi ]

Contact

Ari Stern
Department of Mathematics
Washington University in St. Louis
Campus Box 1146
One Brookings Drive
St. Louis, MO 63130-4889
Email: stern@wustl.edu