211 research outputs found
Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains
Many PDEs involving fractional Laplacian are naturally set in unbounded
domains with underlying solutions decay very slowly, subject to certain power
laws. Their numerical solutions are under-explored. This paper aims at
developing accurate spectral methods using rational basis (or modified mapped
Gegenbauer functions) for such models in unbounded domains. The main building
block of the spectral algorithms is the explicit representations for the
Fourier transform and fractional Laplacian of the rational basis, derived from
some useful integral identites related to modified Bessel functions. With these
at our disposal, we can construct rational spectral-Galerkin and direct
collocation schemes by pre-computing the associated fractional differentiation
matrices. We obtain optimal error estimates of rational spectral approximation
in the fractional Sobolev spaces, and analyze the optimal convergence of the
proposed Galerkin scheme. We also provide ample numerical results to show that
the rational method outperforms the Hermite function approach
Spectral Method for Solving the Nonlinear Thomas-Fermi Equation Based on Exponential Functions
We present an efficient spectral methods solver for the Thomas-Fermi equation for neutral atoms in a semi-infinite domain. The ordinary differential equation has been solved by applying a spectral method using an exponential basis set. One of the main advantages of this approach, when compared to other relevant applications of spectral methods, is that the underlying integrals can be solved analytically and numerical integration can be avoided. The nonlinear algebraic system of equations that is derived using this method is solved using a minimization approach. The presented method has shown robustness in the sense that it can find high precision solution for a wide range of parameters that define the basis set. In our test, we show that the new approach can achieve a very high rate of convergence using a small number of bases elements. We also present a comparison of recently published results for this problem using spectral methods based on several different basis sets. The comparison shows that our method is highly competitive and in many aspects outperforms the previous work
ChebNet: Efficient and Stable Constructions of Deep Neural Networks with Rectified Power Units using Chebyshev Approximations
In a recent paper [B. Li, S. Tang and H. Yu, arXiv:1903.05858], it was shown
that deep neural networks built with rectified power units (RePU) can give
better approximation for sufficient smooth functions than those with rectified
linear units, by converting polynomial approximation given in power series into
deep neural networks with optimal complexity and no approximation error.
However, in practice, power series are not easy to compute. In this paper, we
propose a new and more stable way to construct deep RePU neural networks based
on Chebyshev polynomial approximations. By using a hierarchical structure of
Chebyshev polynomial approximation in frequency domain, we build efficient and
stable deep neural network constructions. In theory, ChebNets and the deep RePU
nets based on Power series have the same upper error bounds for general
function approximations. But numerically, ChebNets are much more stable.
Numerical results show that the constructed ChebNets can be further trained and
obtain much better results than those obtained by training deep RePU nets
constructed basing on power series.Comment: 18 pages, 6 figures, 2 table
Fast Fourier-like Mapped Chebyshev Spectral-Galerkin Methods for PDEs with Integral Fractional Laplacian in Unbounded Domains
In this paper, we propose a fast spectral-Galerkin method for solving PDEs
involving integral fractional Laplacian in , which is built upon
two essential components: (i) the Dunford-Taylor formulation of the fractional
Laplacian; and (ii) Fourier-like bi-orthogonal mapped Chebyshev functions
(MCFs) as basis functions. As a result, the fractional Laplacian can be fully
diagonalised, and the complexity of solving an elliptic fractional PDE is
quasi-optimal, i.e., with being the number of modes in
each spatial direction. Ample numerical tests for various decaying exact
solutions show that the convergence of the fast solver perfectly matches the
order of theoretical error estimates. With a suitable time-discretization, the
fast solver can be directly applied to a large class of nonlinear fractional
PDEs. As an example, we solve the fractional nonlinear Schr{\"o}dinger equation
by using the fourth-order time-splitting method together with the proposed
MCF-spectral-Galerkin method.Comment: This article has a total of 24 pages and including 22 figure
The rational SPDE approach for Gaussian random fields with general smoothness
A popular approach for modeling and inference in spatial statistics is to
represent Gaussian random fields as solutions to stochastic partial
differential equations (SPDEs) of the form , where
is Gaussian white noise, is a second-order differential
operator, and is a parameter that determines the smoothness of .
However, this approach has been limited to the case ,
which excludes several important models and makes it necessary to keep
fixed during inference.
We propose a new method, the rational SPDE approach, which in spatial
dimension is applicable for any , and thus remedies
the mentioned limitation. The presented scheme combines a finite element
discretization with a rational approximation of the function to
approximate . For the resulting approximation, an explicit rate of
convergence to in mean-square sense is derived. Furthermore, we show that
our method has the same computational benefits as in the restricted case
. Several numerical experiments and a statistical
application are used to illustrate the accuracy of the method, and to show that
it facilitates likelihood-based inference for all model parameters including
.Comment: 28 pages, 4 figure
On the numerical stability of Fourier extensions
An effective means to approximate an analytic, nonperiodic function on a
bounded interval is by using a Fourier series on a larger domain. When
constructed appropriately, this so-called Fourier extension is known to
converge geometrically fast in the truncation parameter. Unfortunately,
computing a Fourier extension requires solving an ill-conditioned linear
system, and hence one might expect such rapid convergence to be destroyed when
carrying out computations in finite precision. The purpose of this paper is to
show that this is not the case. Specifically, we show that Fourier extensions
are actually numerically stable when implemented in finite arithmetic, and
achieve a convergence rate that is at least superalgebraic. Thus, in this
instance, ill-conditioning of the linear system does not prohibit a good
approximation.
In the second part of this paper we consider the issue of computing Fourier
extensions from equispaced data. A result of Platte, Trefethen & Kuijlaars
states that no method for this problem can be both numerically stable and
exponentially convergent. We explain how Fourier extensions relate to this
theoretical barrier, and demonstrate that they are particularly well suited for
this problem: namely, they obtain at least superalgebraic convergence in a
numerically stable manner
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