- I am a
Professor at the Department
of Mathematics, Iowa State University. My general research interests lie in the area of
computational and applied mathematics. I have conducted research in
analysis and numerical approximation of time-dependent partial differential
equations, including hyperbolic balance laws, kinetic
transports, and Schroedinger equations with
applications in fluids, plasma and polymers.

**Current research topcis**

**Deep Learning:**selection dyanmcis for deep neural networks; fast learning algorithms; optimal transport

**Mathematical Biology:**selection dynamics in population evolution; Evolution in biased dispersal in Ecology; aggregation, population balance

**Critical threhsold analysis**: hyperbolic balance laws and related models

**Recovery of high frequency wave fields**: Gaussian beam methods, convergence theory, level set methods for recovery

Bose-Einstein condensation phenomena in the Kompaneets model

Kinetic theory for photon scattering:

fluid equtions, Fokker-Planck equations; dispersive PDEs

Direct discontinuous Galerkin (DDG) methods:

Hamilton-Jacobi equation, fully nonlinear PDEs

Alternating evolution (AE) methods:

**Publicatons:**

Selection dynamics for deep neural networks (arXiv:1905.09076 )

Third order maximum-principle-satisfying DG schemes for convection-diffusion problems with anisotropic diffusivity

Journal Comp. Physics. 391: 14–36, 2019.

A high order positivity preserving DG method for coagulation-fragmentation equations

SIAM J. Sci. Comput., 41(3): B448--B465, 2019.

A mixed discontinuous Galerkin method without interior penalty for time-dependent fourth order problems,

J. of Scientific Computing. 77(1), 467--501, 2018.

Invariant-region-preserving DG methods for multi-dimensional hyperbolic conservation law systems, with an application to compressible Euler equations,

Journal of Comput. Phys. 373(15), 385--409, 2018.

On a nonlocal selection-mutation model with a gradient flow structure, Nonlinearity, 30:4220--4238, 2017.

A free energy satisfying discontinuous Galerkin method for Poisson-Nernst-Planck systems, J. Comput. Phys. 238: 413--437, 2017.

On traveling wave solutions to the Keller-Segel model of mixed type, J. Differential Equations, 261: 7080--7098, 2016.

Sobolev and Max Norm Error Estimates for Gaussian Beam Superpositions, Comm. Math. Sci. 14(7): 2041--2076, 2016.

Global dynamics of Bose-Einstein condensation for a model of the Kompaneets equation, SIAM J. MATH. ANAL. 48(4): 2454--2494, 2016.

An entropy satisfying discontinuous Galerkin method for nonlinear Fokker-Planck equations J Sci Comput., 68:1217--1240, 2016.

A Hamiltonian preserving discontinuous Galerkin method for the generalized Korteweg-de Vries equation, J. Comp. Phys, 15: 776--796, 2016.

Alternating evolution Galerkin methods for convection-diffusion equations, J. Comp. Phys. 307 (2016), 574--592.

Threshold for shock formation in the hyperbolic Keller--Segel model, Appl. Math. Letters. 50 (2015), 56--63.

Entropy satisfying schemes for computing selection dynamics in competitive interactions, SIAM J. Numer. Anal. 53(3) (2015), 1393--1417.

Optimal error estimates of the directdiscontinuous Galerkin method for convection-diffusion equations, Math. Comp. 84 (2015), 2263--2295.

Error estimates of the Bloch band-based Gaussian beam superposition for the Schroedinger equation, Contemporary math. 640 (2015), 87--114.

Maximum-principle-satisfying third order discontinuous Galerkin schemes for Fokker-Planck equations, SIAM J. Sci. Comput. 36(5)(2014), A2296--A2325.

Gaussian beam methods for the Helmholtz equation, SIAM J. Appl. Math. 74(3) (2014), 771--793.

Alternating evolution DG methods for Hamilton-Jacobi equations, J. Comput. Phys. 258 (2014), 32-46.

Entropy/Energy stable schemes for evolutionary dispersal models, J. Comp. Phys. 256 (2014),656--677.

Thresholds in three-dimensional restricted Euler-Poisson equations, Phys. D. 262 (2013), 59--70.

Error estimates for Gaussian beam superpositions, Math Comp. 82 (2013), 919--952.

An entropy satisfying conservative method for the Fokker Planck equation of FENE dumbbell model. SIAM Journal on Numerical Analysis. 2012, Vol. 50, No. 3, pp. 1207-1239.

Global well-posedness for the microscopic FENE model with a sharp boundary condition, Journal Diff. Equ. 252 (2012), 641--662.

Alternating evolution (AE) schemes for hyperbolic conservation laws, SIAM J. on Scientific Computing. 33(6) (2011), 3210--3240.

The Direct Discontinuous Galerkin (DDG) method for diffusion with interface corrections , Commun Comput. Phys. 8(3) (2010), 541--564.