My current research interests are mainly in combining data and mathematical theory to construct stochastic reduced models for complex dynamical systems, and related topics such as data assimilation, sequential Monte Carlo methods, deterministic and stochastic dynamical systems and PDEs, ergodicity theory, and learning theory.

## Data-driven stochastic model reduction

Many problems in science and engineering involve nonlinear dynamical systems that are too complex or computationally expensive for full solutions, but one is interested mainly in a subset of the variables. Such problems appear, for example, in weather and climate modeling, in statistical mechanics, and in the mechanics of turbulent flow. In this setting, it is desirable to construct effective reduced models for the variables of interest using data that come either from observations or from fine numerical simulations. These reduced models are supposed to capture the key statistical and dynamical properties of the original systems, and therefore stochastic reduced model are often preferred.

** Discrete-time stochastic parametrization.** In collaboration with Alexandre J. Chorin, I proposed a discrete-time parametrization framework to infer from discrete-time partial data a reduced model in form of NARMAX (nonlinear autoregression moving average with exogenous input). This provides flexibility in the parametrization of memory effects as suggested by Mori-Zwanzig formalism, simplifies the inference from data and accounts for the discretization errors.

** Parametrization of approximate inertial manifolds. ** The major challenge in NARMAX (and general semi-parametric) inference is to derive a model structure. Together with Kevin K. Lin, we developed a method deriving structures by parametrizing approximate inertial manifolds. This method applies to dissipative systems with inertial manifolds such as the Kuramoto-Sivashinsky equation.

A. J. Chorin and F. Lu. Discrete approach to stochastic parametrization and dimension reduction in nonlinear dynamics. Proc. Natl. Acad. Sci. USA, 112 (2015), no. 32, 9804-9809.
F. Lu, K. K. Lin and A. J. Chorin. Data-based stochastic model reduction for the Kuramoto--Sivashinsky equation. Physica D, 340 (2017), 46-57.
F. Lu, K. K. Lin and A. J. Chorin. Comparison of continuous and discrete-time data-based modeling for hypoelliptic systems. Comm. App. Math. Com. Sc., 11 (2016), no. 2, 187â€“216.

## Data assimilation

F. Lu, X. Tu and A. J. Chorin. Accounting for model error from unresolved scales in ensemble Kalman filters by stochastic parametrization. To appear on Mon. Wea. Rev. 2017

## Malliavan calculus and SPDEs

Y. Hu, F. Lu and D. Nualart: Convergence of Densities of functionals of Gaussian Processes. J. Funct. Anal. 266 (2014), no. 2, 814-875.
Y. Hu, F. Lu and D. Nualart: Non-degeneracy of Sobolev Pseudo-norms of fractional Brownian motions. Electron. Commun. Probab. 18 (2013), no. 84, 1-8.
Y. Hu, F. Lu and D. Nualart: Holder continuity of the solution for a class of nonlinear SPDEs arising from one-dimensional superprocesses. Probab. Theory Related Fields 156 (2013), no. 1-2, 27-49.
Y. Hu, F. Lu and D. Nualart: Feynman-Kac formula for the heat equation driven by fractional noise with Hurst parameter H<1/2. Ann. Probab. 40 (2012), No. 3, 1041-1068.

## Other notes

Dealiasing of Fast Fourier Transform.