Yanbo Li

Eric T. Chung, Yanbo Li

In this paper, we construct an adaptive multiscale method for solving H(curl)-elliptic problems in highly heterogeneous media. Our method is based on the generalized multiscale finite element method. We will first construct a suitable snapshot space, and a dimensional reduction procedure to identify important modes of the solution. We next develop and analyze an a posteriori error indicator, and the corresponding adaptive algorithm. In addition, we will construct a coupled offline-online adaptive algorithm, which provides an adaptive strategy to the selection of offline and online basis functions. Our theory shows that the convergence is robust with respect to the heterogeneities and contrast of the media. We present several numerical results to illustrate the performance of our method.

Eric T. Chung, Yalchin Efendiev, Yanbo Li

In this paper, we propose a space-time GMsFEM for transport equations. Multiscale transport equations occur in many geoscientific applications, which include subsurface transport, atmospheric pollution transport, and so on. Most of existing multiscale approaches use spatial multiscale basis functions or upscaling, and there are very few works that design space-time multiscale functions to solve the transport equation on a coarse grid. For the time dependent problems, the use of space-time multiscale basis functions offers several advantages as the spatial and temporal scales are intrinsically coupled. By using the GMsFEM idea with a space-time framework, one obtains a better dimension reduction taking into account features of the solutions in both space and time. In addition, the time-stepping can be performed using much coarser time step sizes compared to the case when spatial multiscale basis are used. Our scheme is based on space-time snapshot spaces and model reduction using space-time spectral problems derived from the analysis. We give the analysis for the well-posedness and the spectral convergence of our method. We also present some numerical examples to demonstrate the performance of the method. In all examples, we observe a good accuracy with a few basis functions.

Eric Chung, Yalchin Efendiev, Yanbo Li et al.

The Boltzmann equation, as a model equation in statistical mechanics, is used to describe the statistical behavior of a large number of particles driven by the same physics laws. Depending on the media and the particles to be modeled, the equation has slightly different forms. In this article, we investigate a model Boltzmann equation with highly oscillatory media in the small Knudsen number regime, and study the numerical behavior of the Generalized Multi-scale Finite Element Method (GMsFEM) in the fluid regime when high oscillation in the media presents. The Generalized Multi-scale Finite Element Method (GMsFEM) is a general approach to numerically treat equations with multi-scale structures. The method is divided into the offline and online steps. In the offline step, basis functions are prepared from a snapshot space via a well-designed generalized eigenvalue problem (GEP), and these basis functions are then utilized to patch up for a solution through DG formulation in the online step to incorporate specific boundary and source information. We prove the wellposedness of the method on the Boltzmann equation, and show that the GEP formulation provides a set of optimal basis functions that achieve spectral convergence. Such convergence is independent of the oscillation in the media, or the smallness of the Knudsen number, making it one of the few methods that simultaneously achieve numerical homogenization and asymptotic preserving properties across all scales of oscillations and the Knudsen number.

Yanbo Li, Zakary Littlefield, Kostas E. Bekris

Sampling-based algorithms are viewed as practical solutions for high-dimensional motion planning. Recent progress has taken advantage of random geometric graph theory to show how asymptotic optimality can also be achieved with these methods. Achieving this desirable property for systems with dynamics requires solving a two-point boundary value problem (BVP) in the state space of the underlying dynamical system. It is difficult, however, if not impractical, to generate a BVP solver for a variety of important dynamical models of robots or physically simulated ones. Thus, an open challenge was whether it was even possible to achieve optimality guarantees when planning for systems without access to a BVP solver. This work resolves the above question and describes how to achieve asymptotic optimality for kinodynamic planning using incremental sampling-based planners by introducing a new rigorous framework. Two new methods, Stable Sparse-RRT (SST) and SST*, result from this analysis, which are asymptotically near-optimal and optimal, respectively. The techniques are shown to converge fast to high-quality paths, while they maintain only a sparse set of samples, which makes them computationally efficient. The good performance of the planners is confirmed by experimental results using dynamical systems benchmarks, as well as physically simulated robots.