We examine the evolution of internal gravity waves in the ionosphere, the region in the upper atmosphere where there is a high concentration of ions and electrons. The waves are influenced by electromagnetic forces that fluctuate randomly in time and the problem is thus represented in terms of stochastic partial differential equations. We discuss the implementation of a numerical method based on Weiner chaos expansions and present the results of the numerical simulations.

Biochemical systems have important practical applications, in particular to understanding key intra-cellular processes. Often biochemical kinetic models represent cellular processes as systems of chemical reactions, traditionally modelled by the deterministic reaction rate equations. In the cellular environment, many biological processes are inherently stochastic. The stochastic fluctuations due to the presence of some low molecular populations may have a great impact on the biochemical system behavior. Then, stochastic models are required for an accurate description of the system dynamics. Biochemically reacting systems often evolve on multiple time-scales, thus their mathematical models manifest stiffness. Stochastic models which, in addition, are stiff are computationally challenging, therefore the need for developing effective and accurate numerical methods for approximating their solution. We present an adaptive stepsize strategy for the numerical solution of stochastic models of biochemical systems. Numerical experiments on models of practical interest show the advantages of the adaptive strategies over the fixed-step methods.

Despite recent advances in supercomputing, current general circulation models (GCMs) represent poorly the variability associated with convection and cloud processes in the tropics. The reason for this failure is believed to be due to the inadequate treatment of organized convection by the underlying cumulus parameterizations. Most of these parameterizations are based on the quasi-equilibrium theory which assumes that convection has an instantaneous or rapid response to large scale instability and thus fail to capture the intermittent and sluggish nature of deep convection that are believed to be key for its tremendous capability to organize itself into mesoscale to planetary scale convective systems including synoptic scale convectively coupled waves and the Madden-Julian oscillation (MJO). In this talk I will discuss a new stochastic lattice-gas model with three order parameters to represent the sugrid-scale variability due to the random interactions between the convective activity and the environment as well as the mutual interactions between the three main cloud types, congestus, deep, and stratiform that, according to recent satellite and in situ observations, are the dominant cloud features in organized tropical convective systems (TCS) of all scales.

In particular, I will present a new coarse graining technique for multi-particle lattice-gas models that permits to derive a multi-dimensional birth death process for the particles area coverage (cloud area fraction) that approximate the lattice model so the stochastic dynamics can be integrated with very little computational cost. Some tropical climate simulations will be presented for the case when the stochastic model is coupled to a simplified/toy climate model to demonstrate the multiple advantages of using such a stochastic parameterization for organized tropical convection.

Such particle interacting systems are widely used in science and engineering (Material Science, Pedestrian traffic, Forest fires, etc.) but because of the sheer computational burden many scientists and engineers rely on the corresponding deterministic mean field limit approximation thus eliminating all the benefits of the stochastic dynamics. The coarse-graining strategy presented here can be easily adopted to such applications.

We propose a framework for modelling stochastic systems with variable diffusion rates which satisfy detailed balance (or in other terminology, time-reversibility). Rather than specifying the dynamics through a state-dependent drift and diffusion coefficients, we specify an equilibrium probability density and a state-dependent diffusion coefficient. We argue that our framework is more natural from the modelling point of view and has a distinct advantage in situations where either the equilibrium probability density or the diffusion coefficient is discontinuous. We introduce a numerical method for simulating dynamics in our framework that samples from the equilibrium probability density exactly and elegantly handles discontinuities in the coefficients. This is joint work with Xin Yang.

The S-transform (a multi-scale Fourier transform) reveals the temporal variation of frequency content in a signal. It has been successfully used in various fields from geophysics to biomedicine. The S-transform and its variations in the current literature are defined in the context of finite-length signals. However, in many applications such as audio or real-time signal processing, the size of the signal is large or infinite. Therefore, in this paper, we present new formula of the discrete S-transforms for infinite-length signals. Our new computational scheme makes it possible to process a long or infinite-length signal at low computational cost without boundary effects and are suitable to be realized in real-time systems.

This is a joint work with Yusong Yan.