Research interests

My research touches upon Partial Differential Equations (PDEs) and Stochastic PDEs, Deterministic/Random Dynamical Systems, Model Reductions, and Scientific Computing.

The main focus of my current research is on developing a new small scale parameterization approach to derive low-dimensional reduced systems for both deterministic and stochastic dissipative PDEs. The approach is analytic in nature. It consists of building new objects called parameterizing manifolds that provide approximation — in a mean-square sense — of the small scales, in a possibly noisy environment. The approach has already been successfully applied in a variety of applications such as bifurcation analysis, phase transitions, optimal control, and the study of noise-induced phenomena.

Publications

Monographs

  1. Mickaël D. Chekroun, Honghu Liu & Shouhong Wang. Stochastic Parameterizing Manifolds and Non-Markovian Reduced Equations: Stochastic Manifolds for Nonlinear SPDEs II. SpringerBriefs in Mathematics. Springer, New York, xvii+129 pp., 2015.
  2. Mickaël D. Chekroun, Honghu Liu & Shouhong Wang. Approximation of Stochastic Invariant Manifolds: Stochastic Manifolds for Nonlinear SPDEs I. SpringerBriefs in Mathematics. Springer, New York, xv+127 pp., 2015.
    Note: These two monographs originate from the Preprint [1] listed below.

Research articles

  1. Mickaël D. Chekroun, Axel Kröner & Honghu Liu. Galerkin approximations of nonlinear optimal control problems in Hilbert spaces. Electronic Journal of Differential Equations, 2017, No. 189, 1–40, 2017. [Journal link] [Preprint]
  2. Mickaël D. Chekroun & Honghu Liu. Post-processing finite-horizon parameterizing manifolds for optimal control of nonlinear parabolic PDEs. Proceedings of 55th IEEE Conference on Decision and Control (CDC), 2016. [Journal link] [Preprint]
  3. Mickaël D. Chekroun, Honghu Liu & James C. McWilliams. The emergence of fast oscillations in a reduced Primitive Equation model and its implications for closure theories. Computers and Fluids, 151, 3–22, 2017. [Journal link] [Preprint]
  4. Mickaël D. Chekroun, Michael Ghil, Honghu Liu & Shouhong Wang. Low-dimensional Galerkin approximations of nonlinear delay differential equations. Disc. Cont. Dyn. Sys. A, 36, 4133–4177, 2016. [Journal link]
  5. Mickaël D. Chekroun & Honghu Liu. Finite-horizon parameterizing manifolds, and applications to suboptimal control of nonlinear parabolic PDEs. Acta Appl. Math., 135, pp 81–144, 2015. [Journal link]
  6. Honghu Liu, Taylan Sengul, Shouhong Wang & Pingwen Zhang. Dynamic transitions and pattern formations for a Cahn-Hilliard model with long-range repulsive interactions. Comm. Math. Sci., 13, pp 1289–1315, 2015. [Journal link]
  7. Honghu Liu, Taylan Sengul & Shouhong Wang. Dynamic transitions for quasilinear systems and Cahn-Hilliard equation with Onsager mobility, J. Math. Phys., 53:023518, 33 pp., 2012. [Journal link]
  8. Honghu Liu. Phase transitions of a phase field model, Disc. Cont. Dyn. Sys. B, 16, 883–894, 2011. [Journal link]

Preprints and Articles under review

  1. Mickaël D. Chekroun, Axel Kröner & Honghu Liu. Galerkin approximations for the optimal control of nonlinear delay differential equations. Submitted, 2017.
  2. Niklas Boers, Mickaël D. Chekroun, Honghu Liu, Dmitri Kondrashov, Denis-Didier Rousseau, Anders Svensson, Matthias Bigler, and Michael Ghil, Inverse stochastic-dynamic models for high resolution greenland ice-core records, Earth System Dynamics Discussions, 28 pp., 2017.
  3. Traian Iliescu, Honghu Liu & Xuping Xie. Regularized Reduced Order Models for a Stochastic Burgers Equation. Submitted, 2016.
  4. Mickaël D. Chekroun, Honghu Liu & Shouhong Wang. Non-Markovian reduced systems for stochastic partial differential equations: The additive noise case. Preprint, arXiv:1311.3069, 11 pp., 2014.
  5. Mickaël D. Chekroun, Honghu Liu & Shouhong Wang. On stochastic parameterizing manifolds: Pullback characterization and Non-Markovian reduced equations. Preprint, arXiv:1310.3896, 143 pp., 2013.

Articles in preparation

  1. Mickaël D. Chekroun, Honghu Liu & Shouhong Wang. Stochastic parameterizing manifolds: Application to stochastic transitions in SPDEs.

 

 

 


Last modified on 08/22/2017.