In this work, we present an upscaled model for mixed dimensional coupled flow problem in fractured porous media. We consider both embedded and discre…

It is important to develop fast yet accurate numerical methods for seismic wave propagation to characterize complex geological structures and oil and…

In this paper, we consider multiscale methods for nonlinear elasticity. In particular, we investigate the Generalized Multiscale Finite Element Metho…

In this paper, we present a staggered discontinuous Galerkin (SDG) method for a class of nonlinear elliptic equations in two dimensions. The SDG meth…

In this paper, we present a new multiscale model reduction technique for the Stokes flows in heterogeneous perforated domains. The challenge in the n…

In this paper, we construct an adaptive multiscale method for solving H(curl)-elliptic problems in highly heterogeneous media. Our method is based on…

Local multiscale methods often construct multiscale basis functions in the offline stage without taking into account input parameters, such as source…

The main goal of this paper is to design multiscale basis functions within GMsFEM framework such that the convergence of method is independent of the…

We show that two Dehn surgeries on a knot $K$ never yield manifolds that are homeomorphic as oriented manifolds if $V_K''(1)\neq 0$ or $V_K'''(1)\neq…

We present a new stabilization technique for multiscale convection diffusion problems. Stabilization for these problems has been a challenging task, …

For a toric Calabi-Yau (CY) orbifold $\mathcal{X}$ whose underlying toric variety is semi-projective, we construct and study a non-toric Lagrangian t…

Traditional two level upscaling techniques suffer from a high offline cost when the coarse grid size is much larger than the fine grid size. Thus, mu…

In \cite{LuLa13}, two of the authors initiated a study of Lipschitz equivalence of self-similar sets through the augmented trees, a class of hyperbol…

Surface parameterizations have been widely used in computer graphics and geometry processing. In particular, as simply-connected open surfaces are co…

We investigate the perturbative aspects of Rozansky-Witten's 3d $\sigma$-model using Costello's approach to the Batalin-Vilkovisky (BV) formalism. We…

The piecewise constant Mumford-Shah (PCMS) model and the Rudin-Osher-Fatemi (ROF) model are two of the most famous variational models in image segmen…

We shall derive and propose several efficient overlapping domain decomposition methods for solving some typical linear inverse problems, including th…

We study the Levenberg-Marquardt (L-M) method for solving the highly nonlinear and ill-posed inverse problem of identifying the Robin coefficients in…

In this paper, we prove uniform curvature estimates for immersed stable free boundary minimal hypersurfaces which satisfy a uniform area bound. Our r…

For any contractive iterated function system (IFS, including the Moran systems), we show that there is a natural hyperbolic graph on the symbolic spa…

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