Computational Mathematics: Numerical Analysis and Scientific Computing

The direct numerical simulation of many scientific processes remains impractical, even with modern supercomputers.

The Center for Computational Mathematics is home to a well-established group of researchers in numerical analysis, modeling, simulation, and high-performance scientific computing. Our solvers have applications to fluid dynamics (at both micro- and geo-scale), waves (acoustic or electromagnetic), diffusion and transport, quantum mechanics, and plasmas.

We also build efficient, widely-used software libraries for fundamental tasks in signal processing and scientific computing.

We have active collaborations with all Flatiron Institute centers, the Scientific Computing Core, and national and international partners.

Areas of interest include:

• Analysis-based high-order accurate solvers for boundary-value problems involving elliptic, parabolic and hyperbolic PDE
• Fast (quasi-linear scaling) algorithms, potential theory, boundary integral equations, and quadratures
• Efficient schemes for complicated geometries (multi-scale, periodic, singular, unbounded, etc)
• Adaptive mesh refinement, finite volume and discontinuous Galerkin methods
• Modeling and simulation of geophysical flows, including tsunami and asteroid impact
• Computational physics, including quantum systems and long-range electrostatics in molecular dynamics
• Computational biology, including Stokes and vesicle flows, cell-scale simulations, and electron microscopy
• Numerical linear algebra, including hierarchical direct solvers and randomized algorithms
• Signal processing tools, including non-uniform Fourier transforms and neural data
• Stellarator design/optimization for fusion energy
• Open-source software libraries, high performance computing, and scientific visualization

Our group is led jointly by Marsha Berger, Alex Barnett, and Leslie Greengard. The team is highly collaborative, and bridges disciplines to work with other computational mathematics researchers in areas such as SciML, computational statistics, and biophysics.

Research scientists are responsible for the long-term research vision of CCM. They also serve as mentors for postdocs and summer interns. Other CCM research scientists whose work strongly overlaps with ours include: Jiequn Han (machine learning for science, high dimensional and inverse problems), and Gilles Francfort (calculus of variations, solid mechanics and applied analysis, joint with the Center for Computational Biology).

Software engineers develop, optimize and maintain the group’s open-source libraries. We also receive help from, and collaborate with, other software engineers in the Center for Computational Mathematics and Scientific Computing Core, including Brian Ward and Robert Blackwell.

Postdocs are appointed as Flatiron Research Fellows with three-year terms. They are free to pursue their own intellectual agenda, and they are encouraged to collaborate with researchers across the Flatiron Institute.

Visiting scholars have short-term or part-time appointments, while guest researchers visit frequently on a more informal basis. Both work closely with our postdocs and research scientists.

September 2026: Welcome to newly arriving postdocs Zhen Huang (joint with CCQ), David Krantz, and Peter Nekrasov (joint with NYU).

November 2025: Misha Padidar, Teresa Huang, Andrew Giuliani and Marina Spivak presented Diffusion for Fusion: Designing Stellarators with Generative AI at the NeurIPS-2025 AI for Science workshop. Their conditional diffusion model, trained on the QUASR database, generates new quasisymmetric stellarator designs in under a second.

October 2025: Gaussian process regression is accelerated by orders of magnitude using NUFFT methods, enabling 2D spatial regression from a billion datapoints in a couple of minutes on a workstation; see the paper by Philip Greengard (CCN), Manas Rachh (now at IIT Bombay), and Alex Barnett. This was recently applied to spectra of nebulae by Thomas Hilder, Andy Casey, Julianne Dalcanton, and coworkers at CCA.

September 2025: Welcome to newly arriving postdocs Xuanzhao Gao and Owen Melia, both experts in scientific computing.

September 2025: A paper by Barry Smith, Marsha Berger and their Argonne National Laboratory collaborators, “Towards Seamless MPI/OpenMP Software Interoperability”, appeared in ACM Transactions on Mathematical Software 51(3), 2025. Smith’s implementation in PETSc lets the OpenMP-based GeoClaw solve the linear systems needed for dispersive equations.

July 2025: Marsha Berger delivered the John von Neumann Prize lecture, SIAM’s highest honor, at the SIAM Annual Meeting.

June 2025: Xuanzhao Gao, Shidong Jiang, Jiuyang Liang and Qi Zhou have released DMK-MC, an O(log N) Monte Carlo method for periodic Coulomb systems that consistently outperforms the state-of-the-art FMM-based baseline.

May 2025: Misha Padidar hosted the inaugural SIMSOPT “Fix-it week”, a pause on new feature development devoted to bugs, testing and documentation.

2025: Charles Epstein, Manas Rachh and U Chicago graduate student Yuguan Wang completed an analysis of static solutions of the London equations for Type-I superconductors in the limit of vanishing London penetration depth (J. Geometric Analysis 35:254, 2025).

June 2024: CCM hosted Computational Tools for PDEs in Complicated Geometries and Interfaces, a 5-day workshop including hands-on tutorials on boundary integral equations, other high-order solvers, and application areas in acoustics, electromagnetics/optics, and biophysics.