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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

Faculty with secondary interests in Computational Mathematics

» J. Duan » L. Kang » 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 Publications

  • Z. Cheng, J. Duan, and L. Wang. Most Probable Dynamics of Some Non-Linear Systems under Noisy Fluctuations. Communications in Nonlinear Science and Numerical Simulation (2016), Vol. 30, Issue 1-3, p. 108-114.
  • T. Gao and J. Duan. Quantifying Model Uncertainty in Dynamical Systems Driven by Non-Gaussian Lévy Stable Noise with Observations on Mean Exit Time or Escape Probability. Communications in Nonlinear Science and Numerical Simulation (2016), Vol. 39, p. 1-6.
  • T. Gao, J. Duan, and X. Li. Fokker-Planck Equations for Stochastic Dynamical Systems with Symmetric Lévy Motions. Applied Mathematics and Computation (2016), Vol. 278, p. 1-20.
  • T. Wang, J. Duan, and T. Liu. Competition promotes the persistence of populations in ecosystems. Nature - Scientific Reports 6 (2016), Article number 30477.
  • J. Ren, J. Duan, and X. Wang. A Parameter Estimation Method Based on Random Slow Manifolds. Applied Mathematical Modelling (2015), Vol. 39, Issue 13, p. 3721-3732.
  • T. Gao, J. Duan, X. Li, and R. Song. Mean Exit Time and Escape Probability for Dynamical Systems Driven by Lévy Noise. SIAM Journal on Scientific Computing (2014), Vol. 36, No. 3, p. A887-A906.
  • S.-C. T. Choi, Y. Ding, F. J. Hickernell, and X. Tong, Local Adaption for Approximation and Minimization of Univariate Functions, J. Complexity (2017) 40, 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, C. R. Math. Acad. Sci. Paris (2017) 355, 10–14.
  • 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, Springer-Verlag, Berlin, 2016, pp. 367–383.
  • 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, Springer-Verlag, Berlin, 2016, pp. 407–422.
  • 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, Springer-Verlag, Berlin, 2016, pp. 583–598.
  • G. E. Fasshauer, F. J. Hickernell, and Q. Ye, Solving Support Vector Machines in Reproducing Kernel Banach Spaces with Positive Definite Functions, Appl. Comput. Harmon. Anal. (2015) 38, 115–139.
  • Z. Berkaliev, S. Devi, G. E. Fasshauer, F. J. Hickernell, O. Kartal, X. Li, P. McCray, S. Whitney, and J. S. Zawojewski, Initiating a Programmatic Assessment Report, PRIMUS (2014) 24.
  • N. Clancy, Y. Ding, C. Hamilton, F. J. Hickernell, and Y. Zhang, The Cost of Deterministic, Adaptive, Automatic Algorithms: Cones, not Balls, J. Complexity (2014) 30, 21–45.

Recent Research Grants

  • NSF DMS-1642545 (PI J. Duan), CBMS Conference: Nonlocal Dynamics Theory, Computation and Applications, 2016-2017.
  • NSF DMS-1025422 (Lead PI J. Duan). Collaborative Research: Mathematical Modeling by Bridging Primitive and Boussinesq Equations, 2010-2015.
  • NSF-DMS-1522687 (G. E. Fasshauer and F. J. Hickernell (PI)) Stable, Efficient, Adaptive Algorithms for Approximation and Integration, 2015–2018.
  • Fermilab (F. J. Hickernell), Modern Monte Carlo Methods for High Energy Event Simulation, Parts I, II, 2015.
  • NSF-DMS-1115392 (G. E. Fasshauer and F. J. Hickernell (PI)) Kernel Methods for Numerical Computation, 2011–2014.
  • NSF-DMS-0938235 (I. Cialenco, J. Duan (PI), and F. J. Hickernell) NSF/CBMS Regional Conference in the Mathematical Sciences — Recent Advances in the Numerical Approximation of Stochastic Partial Differential Equations, 2010.