Multiscale Approximation

Error Analysis of Diffusion Approximation Methods for Multiscale Systems in Reaction Kinetics

Several different methods exist for efficient approximation of paths in multiscale stochastic chemical systems. Another approach is to use bursts of stochastic simulation to estimate the parameters of a stochastic differential equation approximation of the paths. In this paper, multiscale methods for approximating paths are used to formulate different strategies for estimating the dynamics by d...

Redheffer Products and Numerical Approximation of Currents in One-Dimensional Semiconductor Kinetic Models

A Systematic Coarse-Scale Model Reduction Technique for Parameter-Dependent Flows in Highly Heterogeneous Media and Its Applications

In this paper, we propose a multiscale approach for solving the parameter-dependent elliptic equation with highly heterogeneous coefficients. In particular, we assume that the coefficients have both small scales and high contrast (where the high contrast refers to the large variations in the coefficients). The main idea of our approach is to construct local basis functions that encode the local...

Efficient algorithms for multiscale modeling in porous media

We describe multiscale mortar mixed finite element discretizations for second order elliptic and nonlinear parabolic equations modeling Darcy flow in porous media. The continuity of flux is imposed via a mortar finite element space on a coarse grid scale, while the equations in the coarse elements (or subdomains) are discretized on a fine grid scale. We discuss the construction of multiscale mo...

Multiscale mass conservative domain decomposition preconditioners for elliptic problems on irregular grids

Multiscale methods can in many cases be viewed as special types of domain decomposition preconditioners. The localisation approximations introduced within the multiscale framework are dependent upon both the heterogeneity of the reservoir and the structure of the computational grid. While previous works on multiscale control volume methods have focused on heterogeneous elliptic problems on regu...

Sub-sampling and Parametric Estimation for Multiscale Dynamics

We study the problem of adequate data sub-sampling for consistent parametric estimation of unobservable stochastic differential equations (SDEs), when the data are generated by multiscale dynamic systems approximating these SDEs in some suitable sense. The challenge is that the approximation accuracy is scale dependent, and degrades at very small temporal scales. Therefore, maximum likelihood p...