69 research outputs found
Ghost Force Influence of a Quasicontinuum Method in Two Dimension
We derive an analytical expression for the solution of a two-dimensional
quasicontinuum method with a planar interface. The expression is used to prove
that the ghost force may lead to a finite size error for the gradient of the
solution. We estimate the width of the interfacial layer induced by the ghost
force is of \co(\sqrt{\eps}\,) with \eps the equilibrium bond length, which
is much wider than that of the one-dimensional problem.Comment: 28 page
Analysis of the divide-and-conquer method for electronic structure calculations
We study the accuracy of the divide-and-conquer method for electronic
structure calculations. The analysis is conducted for a prototypical subdomain
problem in the method. We prove that the pointwise difference between electron
densities of the global system and the subsystem decays exponentially as a
function of the distance away from the boundary of the subsystem, under the gap
assumption of both the global system and the subsystem. We show that gap
assumption is crucial for the accuracy of the divide-and-conquer method by
numerical examples. In particular, we show examples with the loss of accuracy
when the gap assumption of the subsystem is invalid
Two-parameter asymptotic expansions for elliptic equations with small geometric perturbation and high contrast ratio
We consider the asymptotic solutions of an interface problem corresponding to
an elliptic partial differential equation with Dirich- let boundary condition
and transmission condition, subject to the small geometric perturbation and the
high contrast ratio of the conductivity. We consider two types of
perturbations: the first corresponds to a thin layer coating a fixed bounded
domain and the second is the per perturbation of the interface. As the
perturbation size tends to zero and the ratio of the conductivities in two
subdomains tends to zero, the two-parameter asymptotic expansions on the fixed
reference domain are derived to any order after the single parameter expansions
are solved be- forehand. Our main tool is the asymptotic analysis based on the
Taylor expansions for the properly extended solutions on fixed domains. The
Neumann boundary condition and Robin boundary condition arise in two-parameter
expansions, depending on the relation of the geometric perturbation size and
the contrast ratio
Solving multiscale elliptic problems by sparse radial basis function neural networks
Machine learning has been successfully applied to various fields of
scientific computing in recent years. In this work, we propose a sparse radial
basis function neural network method to solve elliptic partial differential
equations (PDEs) with multiscale coefficients. Inspired by the deep mixed
residual method, we rewrite the second-order problem into a first-order system
and employ multiple radial basis function neural networks (RBFNNs) to
approximate unknown functions in the system. To aviod the overfitting due to
the simplicity of RBFNN, an additional regularization is introduced in the loss
function. Thus the loss function contains two parts: the loss for the
residual of the first-order system and boundary conditions, and the
regularization term for the weights of radial basis functions (RBFs). An
algorithm for optimizing the specific loss function is introduced to accelerate
the training process. The accuracy and effectiveness of the proposed method are
demonstrated through a collection of multiscale problems with scale separation,
discontinuity and multiple scales from one to three dimensions. Notably, the
regularization can achieve the goal of representing the solution by
fewer RBFs. As a consequence, the total number of RBFs scales like
, where is the smallest scale,
is the dimensionality, and is typically smaller than . It is
worth mentioning that the proposed method not only has the numerical
convergence and thus provides a reliable numerical solution in three dimensions
when a classical method is typically not affordable, but also outperforms most
other available machine learning methods in terms of accuracy and robustness.Comment: 23 pages, 12 figure
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