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.

Department of Applied Mathematics

# Computational Mathematics

## Related Seminars

- Computational Mathematics & Statistics Seminar
- Nonlinear Algebra and Statistics (NLASTATS) Seminar
- Data Science Seminar

## Ph.D. Students

- Yue Cao
- Min-Jhe Lu
- Xiaoxia Tang
- Kan Zhang

## Recent Research Grants

- NSF (Co-PI S. Li): GOALI: Coupled Analytical Field Approach for Optimal Design and Accurate Modeling of Electric Machines, 2019-2022.
- 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

- R. Jagadeeswaran and F. J. Hickernell. Fast Automatic Bayesian Cubature Using Lattice Sampling. Forthcoming in Statistics and Computing, 2019+. arXiv:1809.09803
- J. A. De Loera, S. Petrovic, L. Silverstein, D. Stasi, and D. Wilburne. Random Monomial Ideals. Journal of Algebra (2019), Vol. 519, No. 1, pp. 440-473.
- S. Petrovic, D. Stasi, and D. Wilburne. Random Monomial Ideals: A Macaulay2 Package. Journal of Software for Algebra and Geometry (2019), Vol. 9, pp. 65–70.
- E. Turian, K. Liu, J. Lowengrub, and S. Li. Morphological Stability of an Elastic Tumor-Host Interface. Journal of Computational and Applied Mathematics (2019), Vol. 362, pp. 410-422.
- M. Zhao, G. Salter, V. Voller, and S. Li. Can the Growth of Deltaic Shoreline be Unstable? Earth Surface Dynamics (2019), Vol. 7, pp. 505-513.
- W. Hao, B. Hu, S. Li, and L. Song. Convergence of Boundary Integral Method for a Free Boundary System. Journal of Computational and Applied Mathematics (2018), Vol. 334, pp. 128-157.
- F. J. Hickernell, Ll. A. Jiménez Rugama, and D Li. Adaptive Quasi-Monte Carlo Methods for Cubature. Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan (J. Dick, F. Y. Kuo, and H. Wozniakowski (eds.)), pp. 597-619, 2018.
- D. Li, M. Zhao, and S. Li. On the Transition Probability for the Near or Exact Resonance by RWA. International Journal of Theoretical Physics (2018), Vol. 57, No. 5, pp. 1391-1403.
- Kara Pham, E. Turian, K. Liu, S. Li, and J. Lowengrub. Nonlinear Studies of Tumor Morphological Stability Using a Two-Fluid Flow Model. Journal of Mathematical Biology (2018), Vol. 77, No. 3, pp. 671-709.
- M. Zhao, X. Li, W. Ying, A. Belmonte, J. Lowengrub, and S. Li. Computation of a Shrinking Interface in a Hele-Shaw Cell. Siam Journal on Scientific Computing (2018), Vol. 40, No. 4, pp. B1206-B1228.
- 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.
- C. Deng and C. Liu. Largest Well-Posed Spaces for the General Diffusion System with Nonlocal Interactions. Journal of Functional Analysis (2017), Vol. 272, Issue 10, pp. 4030–4062.
- M. Endo, Y. Giga, D. Götz, and C. Liu. Stability of a Two-Dimensional Poiseuille-Type Flow for a Viscoelastic Fluid. Journal of Mathematical Fluid Mechanics (2017), Vol. 19, Issue 1, pp. 17–45.
- 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.
- L. Ma, X. Li, and C, Liu. Fluctuation-Dissipation Theorem Consistent Approximation of the Langevin Dynamics Model. Communications in Mathematical Sciences (2017), Vol. 15, No. 4, pp. 1171–1181.
- Y. Wang, C. Liu, and Z. Tan. Well-Posedness on a New Hydrodynamic Model of the Fluid with the Dilute Charged Particles. Journal of Differential Equations (2017), Vol. 262, Issue 1, pp. 68–115.
- 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.
- T. Huang, F. Lin, C. Liu, and C. Wang. Finite Time Singularity of the Nematic Liquid Crystal Flow in Dimension Three. Archive for Rational Mechanics and Analysis (2016), Vol. 221, Issue 3, pp. 1223–1254.
- 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.
- C.-C. Lee, H. Lee, Y. Hyon, T.-C. Lin, and C. Liu. Boundary Layer Solutions of Charge Conservation Poisson-Boltzmann Equations: One Dimensional Case. Communications in Mathematical Sciences (2016), Vol. 14, No. 4, pp. 911–940.
- M. Liao, L. Wan, S. Xu, C. Liu, and P. Sheng. The Poisson-Boltzmann Equation and the Charge Separation Phenomenon at the Silica-Water Interface: A Holistic Approach. Annals of Mathematical Sciences and Applications (2016), Vol. 1, Issue 1, pp. 217–249.
- 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.
- L. Ma, X. Li, and C. Liu. From Generalized Langevin Equations to Brownian Dynamics and Enbedded Brownian Dynamics. The Journal of Chemical Physics (2016), Vol. 145, Issue 11, Article Number 114102.
- L. Ma, X. Li, and C. Liu. The Derivation and Approximation of Coarse-Grained Dynamics from Langevin Dynamics. The Journal of Chemical Physics (2016), Vol. 145, Issue 20, Article Number 204117.
- M. S. Metti, J. Xu, and C. Liu. Energetically Stable Discretizations for the Charge Transport and Electrokinetic Models. Journal of Computational Physics (2016), Vol. 306, pp. 1–18.
- Y. Wang, C. Liu, and Z. Tan. A Generalized Poisson-Nernst-Planck-Navier-Stokes Model on the Fluid with the Crowded Charged Particles: Derivation and its Well-Posedness. SIAM Journal on Mathematical Analysis (2016), Vol. 48, No. 5, pp. 3191–3235.
- 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.