Kinetic Computation

A part of our work targets at providing efficient computational tools for solving kinetic equations.

The focus has been placed on capturing the asymptotic limiting system on the numerical level. Two approaches have been taken, including

Exploiting low rank/low dimensional structure of the solution manifold
Develop Asymptotic preserving schemes

Multi-scale structure and low-rank

Many multiscale problems have their corresponding limiting systems. Can we develop a general solver that captures all limits?

We develop a generic PDE solver by exploiting the low rank structure using random sampling. This could be viewed as a numerical counterpart of asymptotic Hilbert expansion.

On optimal bases for multiscale PDEs and Bayesian homogenization. S.~Chen, Z.~Ding, Q.~Li, and S.~Wright, submitted, 2023
A reduced order Schwarz method for nonlinear miltiscale elliptic equations based on two-layer neural networks. S. Chen, Z. Ding, Q. Li and S. Wright. accepted, J. Comp. Math arXiv: 2111.02280, 2021
Low-Dimensional Approximation to PDE Solution Manifold. S. Chen, Q. Li, J. Lu and S. Wright. accepted, SIAM-Multiscale Model. Simul. arXiv: 2011.00568, 2020
Low-rank approximation for multiscale PDEs. K. Chen, S. Chen, Q. Li J. Lu and S. Wright. AMS-notices
A low-rank Schwarz method for radiative transport equation with heterogeneous scattering coefficient. K. Chen, Q. Li, J. Lu and S. Wright. SIAM-Multiscale Model. Simul. 19(2), 2021
Random sampling and efficient algorithms for multiscale PDEs. K. Chen, Q. Li, J. Lu and S. Wright. SIAM J. Sci. Comput., 42(5), 2020
Randomized sampling for basis functions construction in generalized finite element methods. K. Chen, Q. Li, J. Lu and S. Wright. SIAM-Multiscale Model. Simul., 18(2), 2020
Schwarz iteration method for elliptic equation with rough media based on random sampling. K. Chen, Q. Li and S. Wright. invited, Proceeding, ICCM 2018, arXiv:1910.02022, 2019

Along the way, we also worked on similar systems:

A sparse decomposition of the symmetric positive semidefinite matrices. T.Y. Hou, Q. Li and P. Zhang. SIAM-Multiscale Model. Simul. 15(1), 410-444, 2017.
Exploring the locally low dimensional structure in solving random elliptic PDEs. T.Y. Hou, Q. Li and P. Zhang. SIAM-Multiscale Model. Simul. 15(2), 661-695, 2017.
Sparse + low energy decomposition for viscous conservation laws. T.Y. Hou, Q. Li and H. Schaeffer. J. Comput. Physics. 288, 150-166, 2015.
Computing conservation laws with L1 minimization: existence, uniqueness and error analysis. A. Gelb, X. Hou and Q. Li. J. Sci. Comput, 81, 2019

Asymptotic Preserving

Kinetic equations have asymptotic limit in the fluid regime. Can one utilize this structure to design better solvers? Asymptotic preserving is a concept that requires a scheme to automatically preserve the equations' asymptotic limit in the small parameter regime, typically this is done through building in the limiting solver.

Dynamical Low-Rank Integrator for the Linear Boltzmann Equation: Error Analysis in the Diffusion Limit. Z. Ding, L. Einkemmer, and Q. Li. SIAM J. Numer. Anal., 59(4), 2021
Generalized multiscale finite element method for the steady state linear Boltzmann equation. E. Chung, Y. Efendiev, Y. Li and Q. Li. SIAM-Multiscale Model. Simul., 18(1), 2020
An asymptotic preserving method for transport equations with oscillatory scattering coefficients. Q. Li and J. Lu. SIAM-Multiscale Model. Simul.15(4),2017.
Implicit asymptotic preserving method for linear transport equations. Q. Li, L. Wang. Commun. Comput. Phys., 22(1):157-181, 2017.
Asymptotic-preserving exponential methods for the quantum Boltzmann equation with high- order accuracy. J. Hu, Q. Li, and L. Pareschi. J. Sci. Comput. 62(2), 555-574, 2014.
Exponential Runge-Kutta for the inhomogeneous Boltzmann equations with high order of accuracy. Q. Li and L. Pareschi. J. Comput. Phys., 259:402–420, 2014.
Exponential Runge-Kutta methods for the multispecies Boltzmann equation. Q. Li and X. Yang. Commun. Comput. Phys., 15(4): 996-1011, 2014.
A BGK-penalization-based asymptotic-preserving scheme for the multispecies Boltzmann equation. S. Jin and Q. Li. Numer. Methods Partial Differential Equations., 29(3):1056–1080, 2013.
Numerical methods for plasma physics in collisional regimes. G. Dimarco, Q. Li, L. Pareschi and B. Yan. J. Plasma. Phys., 81, 305810106, 2015.
1 Asymptotic-Preserving schemes for multiscale hyperbolic and kinetic equations , Handbook of Numerical Methods for Hyperbolic Problems. J. Hu, S. Jin and Q. Li. (ed. by R. Abgrall and C.-W. Shu), North Holland/Elsevier

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