2,166 research outputs found
Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications
In this paper, we propose accurate and efficient finite difference methods to
discretize the two- and three-dimensional fractional Laplacian
() in hypersingular integral
form. The proposed finite difference methods provide a fractional analogue of
the central difference schemes to the fractional Laplacian, and as , they collapse to the central difference schemes of the classical Laplace
operator . We prove that our methods are consistent if ,
and the local truncation error is , with a small constant and denoting the floor function. If
, they can achieve the
second order of accuracy for any . These results hold for
any dimension and thus improve the existing error estimates for the
finite difference method of the one-dimensional fractional Laplacian. Extensive
numerical experiments are provided and confirm our analytical results. We then
apply our method to solve the fractional Poisson problems and the fractional
Allen-Cahn equations. Numerical simulations suggest that to achieve the second
order of accuracy, the solution of the fractional Poisson problem should {\it
at most} satisfy . One merit of our methods is
that they yield a multilevel Toeplitz stiffness matrix, an appealing property
for the development of fast algorithms via the fast Fourier transform (FFT).
Our studies of the two- and three-dimensional fractional Allen-Cahn equations
demonstrate the efficiency of our methods in solving the high-dimensional
fractional problems.Comment: 24 pages, 6 figures, and 6 table
LSTM with Working Memory
Previous RNN architectures have largely been superseded by LSTM, or "Long
Short-Term Memory". Since its introduction, there have been many variations on
this simple design. However, it is still widely used and we are not aware of a
gated-RNN architecture that outperforms LSTM in a broad sense while still being
as simple and efficient. In this paper we propose a modified LSTM-like
architecture. Our architecture is still simple and achieves better performance
on the tasks that we tested on. We also introduce a new RNN performance
benchmark that uses the handwritten digits and stresses several important
network capabilities.Comment: Accepted at IJCNN 201
- …