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Computational Mathematics

Computer simulation is recognized as the third pillar of science, complementing theory and experiment. The computational mathematics research group designs and analyzes numerical algorithms and answers fundamental questions about the underlying physics. We construct and analyze algorithms for approximating functions and integration in high dimensions, and solving systems of polynomial equations. The emphasis is on meshfree methods, maximizing algorithm efficiency, avoiding catastrophic round-off error, overcoming the curse of dimensionality, and advancing adaptive computations to meet error tolerances. We develop accurate mathematical models and efficient numerical methods to investigate dynamics of interfaces. Our goal is to understand the underlying mechanisms that govern the process of pattern formation, i.e., growth and form. Examples include multiphase flows in complex fluids and vesicle deformation in bio-related applications such as drug delivery. We establish analytical and computational techniques for extracting effective dynamics from multiscale phenomena that are abundant in geophysical and biophysical systems.

Faculty with primary interests in Computational Mathematics

» F. Hickernell » S. Li » X. Li » c. Liu

Faculty with secondary interests in Computational Mathematics

» J. Duan » L. Kang » K. W. Ong » S. Petrović » D. Stasi

Related seminars

» Meshfree Methods Seminar » Stochastic & Multiscale Modeling & Computation Seminar

Ph.D. Students

  • Yue Cao
  • Hansen Ha
  • Francisco Hernandez
  • Julienne Kabre
  • Kan Zhang
  • Yizhi Zhang
  • Meng Zhao

RECENT RESEARCH GRANTS

  • NSF DMS-1720420 (PI S. Li): Collaborative Research: Modeling and Computation of Three-Dimensional Multicomponent Vesicles in Complex Flow Domains, 2017-2020.
  • NSF DMS-1759535 (PI C. Liu): Topics in Complex Fluids and Biophysiology: the Energetic Variational Approaches, 2017-2020.
  • NSF DMS-1759536 (PI C. Liu): Energetic Variational Approaches in Complex Fluids and Electrophysiology, 2017-2018.
  • NSF DMS-1620449 (PI X. Li and Co-PI J. Duan): Theoretical and Numerical Studies of Nonlocal Equations Derived from Stochastic Differential Equations with Lévy Noises, 2016-2020.
  • NSF DMS-1522687 (PI F. J. Hickernell and Co-PI G. E. Fasshauer): Stable, Efficient, Adaptive Algorithms for Approximation and Integration, 2015–2018.
  • NSF ECCS-1307625 (PI S. Li): Collaborative Research: Computationally Efficient Solvers for Power System Simulation, 2013-2017.
  • Fermilab (PI F. J. Hickernell): Modern Monte Carlo Methods for High Energy Event Simulation, Parts I, II, 2015.
  • NSF DMS-1217277 (PI S. Li): Collaborative Research: Reactive Instabilities, Colloids, and Interfacial Flows: Experiments, Modeling and Numerics, 2012-2015.
  • NSF DMS-1115392 (PI F. J. Hickernell and Co-PI G. E. Fasshauer): Kernel Methods for Numerical Computation, 2011–2014.
  • NSF DMS-0914923 (PI S. Li): Collaborative Research: Computational and Theoretical Approaches for the Morphological Control of Material Microstructures, 2009-2013.

Recent Publications

  • F. J. Hickernell, Ll. A. Jiménez Rugama, and D Li. Adaptive Quasi-Monte Carlo Methods for Cubature. Submitted, 2017. arXiv:1702.01491
  • S.-C. T. Choi, Y. Ding, F. J. Hickernell, and X. Tong. Local Adaption for Approximation and Minimization of Univariate Functions. Journal of Complexity (2017), Vol. 40, pp. 17–33.
  • L. Gilquin, Ll. A. Jiménez Rugama, E. Arnaud, F. J. Hickernell, H. Monod, and C. Prieur. Iterative Construction of Replicated Designs Based on Sobol' Sequences. Comptes Rendus Mathematique (2017), Vol. 355, Issue 1, pp. 10–14.
  • K. Liu, Gary Marple, J. Allard, S. Li, Shravan Veerapaneni, and J. Lowengrub. Dynamics of a Multicomponent Vesicle in Shear Flow. Soft Matter (2017), Vol. 13, Issue 19, pp. 3521-3531.
  • M. Zhao, W. Ying, J. Lowengrub, and S. Li. An Efficient Adaptive Rescaling Scheme for Computing Moving Interface Problems. Communications in Computational Physics (2017), Vol. 21, No. 3, pp. 679-691.
  • H. Feng, A. Cordoba, F. Hernandez, T. Indei, S. Li, X. Li, and J. Schieber. A Boundary Integral Method for Motion of Particles in Unsteady Stokes and Linear Viscoelastic Flows. International Journal for Numerical Methods in Fluids (2016), Vol. 82, Issue 4, pp. 198-217.
  • F. J. Hickernell and Ll. A. Jiménez Rugama. Reliable Adaptive Cubature Using Digital Sequences. Monte Carlo and Quasi-Monte Carlo Methods, MCQMC, Leuven, Belgium, April 2014 (R. Cools and D. Nuyens, eds.), Springer Proceedings in Mathematics and Statistics, Vol. 163, pp. 367–383, Springer, 2016.
  • Ll. A. Jiménez Rugama and F. J. Hickernell. Adaptive Multidimensional Integration Based on Rank-1 Lattices. Monte Carlo and Quasi-Monte Carlo Methods, MCQMC, Leuven, Belgium, April 2014 (R. Cools and D. Nuyens, eds.), Springer Proceedings in Mathematics and Statistics, Vol. 163, pp. 407–422, Springer, 2016.
  • K. Liu, C. Hamilton, J. Allard, J. Lowengrub, and S. Li. Wrinkling Dynamics of Fluctuating Vesicles in Time-Dependent Viscous Flow. Soft Matter (2016) (cover page article), Vol. 12, Issue 26, pp. 5663-5675.
  • M. Zhao, A. Belmonte, S. Li, X. Li, and J. Lowengrub. Nonlinear Simulations of Elastic Fingering in a Hele-Shaw Cell. Journal of Computational and Applied Mathematics (2016), Vol. 307, Issue C, pp. 394-407.
  • X. Zhou and F. J. Hickernell. Tractability of the Radial Function Approximation Problem with Kernels of a Product Form. Monte Carlo and Quasi-Monte Carlo Methods, MCQMC, Leuven, Belgium, April 2014 (R. Cools and D. Nuyens, eds.), Springer Proceedings in Mathematics and Statistics, Vol. 163, pp. 583–598, Springer, 2016.
  • G. E. Fasshauer, F. J. Hickernell, and Q. Ye. Solving Support Vector Machines in Reproducing Kernel Banach Spaces with Positive Definite Functions. Applied and Computational Harmonic Analysis (2015), Vol. 38, Issue 1, pp. 115–139.