Finite difference WENO schemes have established themselves as very worthy performers for entire classes of applications that involve hyperbolic conservation laws. In this paper we report on two major advances that make finite difference WENO schemes more efficient.
This paper considers regularized block multiconvex optimization, where the feasible set and objective
function are generally nonconvex but convex in each block of variables. It also accepts nonconvex
blocks and requires these blocks to be updated by proximal minimization. We review some interesting
applications and propose a generalized block coordinate descent method. Under certain
conditions, we show that any limit point satisfies the Nash equilibrium conditions. Furthermore, we
establish global convergence and estimate the asymptotic convergence rate of the method by assuming
a property based on the Kurdyka-Lojasiewicz inequality. The proposed algorithms are tested on
nonnegative matrix and tensor factorization, as well as matrix and tensor recovery from incomplete
observations. The tests include synthetic data and hyperspectral data, as well as image sets from
the CBCL and ORL databases. Compared to the existing state-of-the-art algorithms, the proposed
algorithms demonstrate superior performance in both speed and solution quality. The MATLAB
code of nonnegative matrix/tensor decomposition and completion, along with a few demos, are
accessible from the authors' homepages.
A recently developed nonlocal vector calculus is exploited to provide a variational analysis
for a general class of nonlocal diffusion problems described by a linear integral equation on
bounded domains in Rn. The nonlocal vector calculus also enables striking analogies to be
drawn between the nonlocal model and classical models for diffusion, including a notion
of nonlocal flux. The ubiquity of the nonlocal operator in applications is illustrated by a
number of examples ranging from continuum mechanics to graph theory. In particular, it is
shown that fractional Laplacian and fractional derivative models for anomalous diffusion
are special cases of the nonlocal model for diffusion that we consider. The numerous
applications elucidate different interpretations of the operator and the associated governing
equations. For example, a probabilistic perspective explains that the nonlocal spatial
operator appearing in our model corresponds to the infinitesimal generator for a symmetric
jump process. Sufficient conditions on the kernel of the nonlocal operator and the notion
of volume constraints are shown to lead to a well-posed problem. Volume constraints are
a proxy for boundary conditions that may not be defined for a given kernel. In particular,
we demonstrate for a general class of kernels that the nonlocal operator is a mapping
between a volume constrained subspace of a fractional Sobolev subspace and its dual.
We also demonstrate for other particular kernels that the inverse of the operator does
not smooth but does correspond to diffusion. The impact of our results is that both a
continuum analysis and a numerical method for the modeling of anomalous diffusion on
bounded domains in Rn are provided. The analytical framework allows us to consider
finite-dimensional approximations using discontinuous and continuous Galerkin methods,
both of which are conforming for the nonlocal diffusion equation we consider; error and
condition number estimates are derived.
In this paper, we survey our recent work on designing high order positivity-preserving well-balanced finite difference and finite volume WENO (weighted essentially non-oscillatory) schemes, and discontinuous Galerkin finite element schemes for solving the shallow water equations with a non-flat bottom topography. These schemes are genuinely high order accurate
in smooth regions for general solutions, are essentially non-oscillatory for general solutions with discontinuities, and at the same time they preserve exactly the water at rest or the more general moving water steady state solutions. A simple positivity-preserving limiter, valid under suitable CFL condition, has been introduced in one dimension and reformulated to two dimensions with triangular meshes, and we prove that the resulting schemes guarantee the positivity of the water depth.