3 research outputs found
Numerical solution of a one-dimensional nonlocal Helmholtz equation by Perfectly Matched Layers
We consider the computation of a nonlocal Helmholtz equation by using
Perfectly Matched Layer (PML). We first derive the nonlocal PML equation by
extending PML modifications from the local operator to the nonlocal operator of
integral form. We then give stability estimates of some weighted average value
of the nonlocal Helmholtz solution and prove that (i) the weighted average
value of the nonlocal PML solution decays exponentially in PML layers in one
case; (ii) in the other case, the weighted average value of the nonlocal
Helmholtz solution itself decays exponentially outside some domain.
Particularly for a typical kernel function , we
obtain the Green's function of the nonlocal Helmholtz equation, and use the
Green's function to further prove that (i) the nonlocal PML solution decays
exponentially in PML layers in one case; (ii) in the other case, the nonlocal
Helmholtz solution itself decays exponentially outside some domain. Based on
our theoretical analysis, the truncated nonlocal problems are discussed and an
asymptotic compatibility scheme is also introduced to solve the resulting
truncated problems. Finally, numerical examples are provided to verify the
effectiveness and validation of our nonlocal PML strategy and theoretical
findings.Comment: 22 pages, 7 figure
A Novel Evolution Strategy with Directional Gaussian Smoothing for Blackbox Optimization
We propose an improved evolution strategy (ES) using a novel nonlocal
gradient operator for high-dimensional black-box optimization. Standard ES
methods with -dimensional Gaussian smoothing suffer from the curse of
dimensionality due to the high variance of Monte Carlo (MC) based gradient
estimators. To control the variance, Gaussian smoothing is usually limited in a
small region, so existing ES methods lack nonlocal exploration ability required
for escaping from local minima. We develop a nonlocal gradient operator with
directional Gaussian smoothing (DGS) to address this challenge. The DGS
conducts 1D nonlocal explorations along orthogonal directions in
, each of which defines a nonlocal directional derivative as a 1D
integral. We then use Gauss-Hermite quadrature, instead of MC sampling, to
estimate the 1D integrals to ensure high accuracy (i.e., small variance).
Our method enables effective nonlocal exploration to facilitate the global
search in high-dimensional optimization. We demonstrate the superior
performance of our method in three sets of examples, including benchmark
functions for global optimization, and real-world science and engineering
applications
On a nonlocal Cahn-Hilliard model permitting sharp interfaces
A nonlocal Cahn-Hilliard model with a nonsmooth potential of double-well
obstacle type that promotes sharp interfaces in the solution is presented. To
capture long-range interactions between particles, a nonlocal Ginzburg-Landau
energy functional is defined which recovers the classical (local) model for
vanishing nonlocal interactions. In contrast to the local Cahn-Hilliard problem
that always leads to diffuse interfaces, the proposed nonlocal model can lead
to a strict separation into pure phases of the substance. Here, the lack of
smoothness of the potential is essential to guarantee the aforementioned
sharp-interface property. Mathematically, this introduces additional inequality
constraints that, in a weak form, lead to a coupled system of variational
inequalities which at each time instance can be restated as a constrained
optimization problem. We prove the well-posedness and regularity of the
semi-discrete and continuous in time weak solutions, and derive the conditions
under which pure phases are admitted. Moreover, we develop discretizations of
the problem based on finite elements and implicit-explicit time stepping
methods that can be realized efficiently. Finally, we illustrate our
theoretical findings through several numerical experiments in one and two
spatial dimensions that highlight the differences in features of local and
nonlocal solutions and also the sharp interface properties of the nonlocal
model