Numerical analysis of saddle point problems with random data
Mathematical models with data uncertainty are ubiquitous in engineering applications. These models may come, in particular, as PDE problems with correlated random data (i.e., coefficients, sources, boundary conditions, geometry etc.).
Hence, reliable numerical methods that evaluate the propagation of uncertainty to the output variables (solutions to the PDE problems) are of special importance.
Whilst there exists a large body of research work on numerical approximation of primal formulations for elliptic PDEs with random data, the case of mixed variational problems is not so well developed.
In this talk we will consider two representative model problems:
i) a mixed formulation of a second-order elliptic problem with random diffusion coefficient and ii) the steady-state Navier-Stokes equations with random viscosity.
We will focus on a particular numerical method for solving these problems, namely on the stochastic Galerkin finite element method, which combines conventional Galerkin finite elements on the (spatial) computational domain with spectral approximations in the stochastic variables.
We will discuss the issues involved in the error analysis of stochastic Galerkin approximations as well as the associated linear algebra.
Joint work with Catherine Powell and David Silvester (University of Manchester).
Deviations for processes with averaging phenomena
Systems with coupled fast-slow dynamics are introduced. Large deviations (in the limit of fast dynamics) are discussed for finite-dimensional diffusion processes. The infinite-dimensional case (or the case of jump processes) will be treated depending on time and audience.
Stochastic modeling in single-molecule biophysics
Recent advances in nanotechnology allow scientists to follow a biological process on the individual molecule basis. These advances also raise many challenging stochastic modeling problems, because the experimental capability of zooming in on single molecules reveal that many classical models derived from oversimplified assumptions are no longer valid. In this talk we will look at two particular experimental discoveries: single-molecule Michaelis-Menten equations in enzymatic reactions, and subdiffusion in a single protein's conformational fluctuation; both strongly contradict the classical models. We will introduce models to explain these experimental findings, and explore the connection between the two phenomena.
Growth rate identification in the crystallization of polymers
Nucleation and growth mechanisms are important kinetics of the phase transformation model which arises in the crystallization of the polymer materials. In each stage, the nucleation rate and the growth rate have been the crucial coefficients describing the kinetics of the process as well as the properties of the specimens. Moreover, identification of these physical parameters describing the nucleation or the growth mechanisms is essential for controlling the crystallizati! on of polymers and so is a significant subject also from industrial viewpoints.
In this talk, we will revisit the time cone approach in Cahn 1996 where a hyperbolic governing equation is derived for the heterogeneous nucleation rate and spatially homogeneous growth rate. As for the inverse problem, by utilizing the eigenfunction expansion method, we investigate the identification of the growth rate for an isothermal one dimension specimen. Such a problem can be seem as an inverse coefficient problem for a hyperbolic equation which is highly nonlinear with respect to the observation data. A two-step Tikhonov type regularization method is proposed to reconstruct the growth rate provided with the final noisy observation data. Numerical prototype examples are presented to illustrate the validity and effectiveness of the proposed scheme.