Some Applications of the Generalized Multiscale Finite Element Method

Many materials in nature are highly heterogeneous and their properties can vary at different scales. Direct numerical simulations in such multiscale media are prohibitively expensive and some types of model reduction are needed. Typical model reduction techniques include upscaling and multiscale methods. In upscaling methods, one upscales the multiscale media properties so that the problem can be solved on a coarse grid. In multiscale method, one constructs multiscale basis functions that capture media information and solves the problem on the coarse grid. Generalized Multiscale Finite Element Method (GMsFEM) is a recently proposed model reduction technique and has been used for various practical applications. This method has no assumption about the media properties, which can have any type of complicated structure. In GMsFEM, we first create a snapshot space, and then solve a carefully chosen eigenvalue problem to form the offline space. One can also construct online space for the parameter dependent problems. It is shown theoretically and numerically that the GMsFEM is very efficient for the heterogeneous problems involving high-contrast, no-scale separation. In this dissertation, we apply the GMsFEM to perform model reduction for the steady state elasticity equations in highly heterogeneous media though some of our applications are motivated by elastic wave propagation in subsurface. We will consider three kinds of coupling mechanism for different situations. For more practical purposes, we will also study the applications of the GMsFEM for the frequency domain acoustic wave equation and the Reverse Time Migration (RTM) based on the time domain acoustic wave equation

Uncertainty Quantification of Nonlinear Lagrangian Data Assimilation Using Linear Stochastic Forecast Models

Lagrangian data assimilation exploits the trajectories of moving tracers as observations to recover the underlying flow field. One major challenge in Lagrangian data assimilation is the intrinsic nonlinearity that impedes using exact Bayesian formulae for the state estimation of high-dimensional systems. In this paper, an analytically tractable mathematical framework for continuous-in-time Lagrangian data assimilation is developed. It preserves the nonlinearity in the observational processes while approximating the forecast model of the underlying flow field using linear stochastic models (LSMs). A critical feature of the framework is that closed analytic formulae are available for solving the posterior distribution, which facilitates mathematical analysis and numerical simulations. First, an efficient iterative algorithm is developed in light of the analytically tractable statistics. It accurately estimates the parameters in the LSMs using only a small number of the observed tracer trajectories. Next, the framework facilitates the development of several computationally efficient approximate filters and the quantification of the associated uncertainties. A cheap approximate filter with a diagonal posterior covariance derived from the asymptotic analysis of the posterior estimate is shown to be skillful in recovering incompressible flows. It is also demonstrated that randomly selecting a small number of tracers at each time step as observations can reduce the computational cost while retaining the data assimilation accuracy. Finally, based on a prototype model in geophysics, the framework with LSMs is shown to be skillful in filtering nonlinear turbulent flow fields with strong non-Gaussian features