共查询到20条相似文献,搜索用时 31 毫秒
1.
Zuowei Shen Haizhao Yang & Shijun Zhang 《Communications In Computational Physics》2020,28(5):1768-1811
This paper quantitatively characterizes the approximation power of deep
feed-forward neural networks (FNNs) in terms of the number of neurons. It is shown
by construction that ReLU FNNs with width$\mathcal{O}$(max{$d⌊N^{1/d}⌋$,$N$+1}) and depth $\mathcal{O}(L)$ can approximate an arbitrary Hölder continuous function of order $α∈(0,1]$ on $[0,1]^d$ with a nearly tight approximation rate $\mathcal{O}(\sqrt{d}N^{−2α/d}L^{−2α/d})$ measured in $L^p$ -norm for
any $N,L∈\mathbb{N}^+$ and $p∈[1,∞]$. More generally for an arbitrary continuous function $f$ on $[0,1]^d$ with a modulus of continuity $ω_f
(·)$, the constructive approximation rate
is $\mathcal{O}(\sqrt{d}ω_f(N^{−2α/d}L^{−2α/d}))$. We also extend our analysis to $f$ on irregular domains or
those localized in an ε-neighborhood of a $d_\mathcal{M}$-dimensional smooth manifold $\mathcal{M}⊆[0,1]^d$ with $d_\mathcal{M}≪d$. Especially, in the case of an essentially low-dimensional domain, we
show an approximation rate $\mathcal{O}(ω_f(\frac{ε}{1−δ}\sqrt{\frac{d}{d_δ}}+ε)+\sqrt{d}ω_f(\frac{\sqrt{d}}{1−δ\sqrt{d_δ}}N^{−2α/d_δ}L^{−2α/d_δ})$ for
ReLU FNNs to approximate $f$ in the $ε$-neighborhood, where $d_δ=\mathcal{O}(d_\mathcal{M}\frac{\rm{ln}(d/δ)}{δ^2})$ for any $δ∈(0,1)$ as a relative error for a projection to approximate an isometry when projecting $\mathcal{M}$ to a $d_δ$-dimensional domain. 相似文献
2.
Directional $\mathcal{H}^2$ Compression Algorithm: Optimisations and Application to a Discontinuous Galerkin BEM for the Helmholtz Equation 下载免费PDF全文
Nadir-Alexandre Messaï Sebastien Pernet & Abdesselam Bouguerra 《Communications In Computational Physics》2022,31(5):1585-1635
This study aimed to specialise a directional $\mathcal{H}^2
(\mathcal{D}\mathcal{H}^2)$ compression to matrices arising from the discontinuous Galerkin (DG) discretisation of the hypersingular
equation in acoustics. The significant finding is an algorithm that takes a DG stiffness matrix and finds a near-optimal $\mathcal{D}\mathcal{H}^2$ approximation for low and high-frequency
problems. We introduced the necessary special optimisations to make this algorithm
more efficient in the case of a DG stiffness matrix. Moreover, an automatic parameter
tuning strategy makes it easy to use and versatile. Numerical comparisons with a classical Boundary Element Method (BEM) show that a DG scheme combined with a $\mathcal{D}\mathcal{H}^2$ gives better computational efficiency than a classical BEM in the case of high-order finite elements and $hp$ heterogeneous meshes. The results indicate that DG is suitable
for an auto-adaptive context in integral equations. 相似文献
3.
Ziqing Xie Jiangxing Wang Bo Wang & Chuanmiao Chen 《Communications In Computational Physics》2016,19(5):1242-1264
In this paper, an approach combining the DG method in space with CG
method in time (CG-DG method) is developed to solve time-dependent Maxwell's
equations when meta-materials are involved. Both the unconditional $L^2$-stability and
error estimate of order $\mathcal{O}$($τ^ {r+1}$+$h^{k+\frac{1}{2}}$) are obtained when polynomials of degree at
most r is used for the temporal discretization and at most k for the spatial discretization.
Numerical results in 3D are given to validate the theoretical results. 相似文献
4.
On the Effect of Ghost Force in the Quasicontinuum Method: Dynamic Problems in One Dimension 下载免费PDF全文
Numerical error caused by "ghost forces" in a quasicontinuum method is
studied in the context of dynamic problems. The error in the discrete W1,∞ norm is analyzed for the time scale $\mathcal{O}$($ε$) and the time scale $\mathcal{O}$(1) with ε being the lattice spacing. 相似文献
5.
Lei Wang Robert Krasny & Svetlana Tlupova 《Communications In Computational Physics》2020,28(4):1415-1436
A kernel-independent treecode (KITC) is presented for fast summation of
particle interactions. The method employs barycentric Lagrange interpolation at
Chebyshev points to approximate well-separated particle-cluster interactions. The
KITC requires only kernel evaluations, is suitable for non-oscillatory kernels, and relies on the scale-invariance property of barycentric Lagrange interpolation. For a given
level of accuracy, the treecode reduces the operation count for pairwise interactions
from $\mathcal{O}$($N^2$) to $\mathcal{O}$($N$log$N$), where $N$ is the number of particles in the system. The algorithm is demonstrated for systems of regularized Stokeslets and rotlets in 3D, and
numerical results show the treecode performance in terms of error, CPU time, and
memory consumption. The KITC is a relatively simple algorithm with low memory
consumption, and this enables a straightforward OpenMP parallelization. 相似文献
6.
An Efficient Finite Element Method with Exponential Mesh Refinement for the Solution of the Allen-Cahn Equation in Non-Convex Polygons 下载免费PDF全文
Emine Celiker & Ping Lin 《Communications In Computational Physics》2020,28(4):1536-1560
In this paper we consider the numerical solution of the Allen-Cahn type
diffuse interface model in a polygonal domain. The intersection of the interface with
the re-entrant corners of the polygon causes strong corner singularities in the solution.
To overcome the effect of these singularities on the accuracy of the approximate solution, for the spatial discretization we develop an efficient finite element method with
exponential mesh refinement in the vicinity of the singular corners, that is based on
($k$−1)-th order Lagrange elements, $k$≥2 an integer. The problem is fully discretized by
employing a first-order, semi-implicit time stepping scheme with the Invariant Energy
Quadratization approach in time, which is an unconditionally energy stable method.
It is shown that for the error between the exact and the approximate solution, an accuracy of $\mathcal{O}$($h^k$+$τ$) is attained in the $L^2$-norm for the number of $\mathcal{O}$($h^{−2}$ln$h^{−1}$) spatial
elements, where $h$ and $τ$ are the mesh and time steps, respectively. The numerical
results obtained support the analysis made. 相似文献
7.
A fully discrete discontinuous Galerkin method is introduced for solving
time-dependent Maxwell's equations. Distinguished from the Runge-Kutta discontinuous Galerkin method (RKDG) and the finite element time domain method (FETD), in
our scheme, discontinuous Galerkin methods are used to discretize not only the spatial
domain but also the temporal domain. The proposed numerical scheme is proved to be
unconditionally stable, and a convergent rate $\mathcal{O}((∆t)^{r+1}+h^{k+1/2})$ is established under the $L^2$ -norm when polynomials of degree at most $r$ and $k$ are used for temporal and
spatial approximation, respectively. Numerical results in both 2-D and 3-D are provided to validate the theoretical prediction. An ultra-convergence of order $(∆t)^{2r+1}$ in
time step is observed numerically for the numerical fluxes w.r.t. temporal variable at
the grid points. 相似文献
8.
Some Random Batch Particle Methods for the Poisson-Nernst-Planck and Poisson-Boltzmann Equations 下载免费PDF全文
We consider in this paper random batch interacting particle methods for
solving the Poisson-Nernst-Planck (PNP) equations, and thus the Poisson-Boltzmann
(PB) equation as the equilibrium, in the external unbounded domain. To justify the
simulation in a truncated domain, an error estimate of the truncation is proved in
the symmetric cases for the PB equation. Then, the random batch interacting particle methods are introduced which are $\mathcal{O}(N)$ per time step. The particle methods can
not only be considered as a numerical method for solving the PNP and PB equations,
but also can be used as a direct simulation approach for the dynamics of the charged
particles in solution. The particle methods are preferable due to their simplicity and
adaptivity to complicated geometry, and may be interesting in describing the dynamics of the physical process. Moreover, it is feasible to incorporate more physical effects
and interactions in the particle methods and to describe phenomena beyond the scope
of the mean-field equations. 相似文献
9.
We describe our implementation of a parallel fast multipole method for evaluating
potentials for discrete and continuous source distributions. The first requires
summation over the source points and the second requiring integration over a continuous
source density. Both problems require$\mathcal{O}$($N^2$) complexity when computed directly;
however, can be accelerated to $\mathcal{O}$($N$) time using FMM. In our PVFMM software
library, we use kernel independent FMM and this allows us to compute potentials for
a wide range of elliptic kernels. Our method is high order, adaptive and scalable. In
this paper, we discuss several algorithmic improvements and performance optimizations
including cache locality, vectorization, shared memory parallelism and use of
coprocessors. Our distributed memory implementation uses space-filling curve for
partitioning data and a hypercube communication scheme. We present convergence
results for Laplace, Stokes and Helmholtz (low wavenumber) kernels for both particle
and volume FMM. We measure efficiency of our method in terms of CPU cycles per
unknown for different accuracies and different kernels. We also demonstrate scalability
of our implementation up to several thousand processor cores on the Stampede
platform at the Texas Advanced Computing Center. 相似文献
10.
Fast One-Dimensional Convolution with General Kernels Using Sum-of-Exponential Approximation 下载免费PDF全文
Yong Zhang Chijie Zhuang & Shidong Jiang 《Communications In Computational Physics》2021,29(5):1570-1582
Based on the recently-developed sum-of-exponential (SOE) approximation,
in this article, we propose a fast algorithm to evaluate the one-dimensional convolution potential $φ(x)=K∗ρ=∫^1_{0}K(x−y)ρ(y)dy$ at (non)uniformly distributed target grid
points {$x_i$}$^M_{i=1}$, where the kernel $K(x)$ might be singular at the origin and the source
density function $ρ(x)$ is given on a source grid ${{{y_i}}}^N_{j=1}$ which can be different from
the target grid. It achieves an optimal accuracy, inherited from the interpolation of
the density $ρ(x)$, within $\mathcal{O}(M+N)$ operations. Using the kernel's SOE approximation $K_{ES}$, the potential is split into two integrals: the exponential convolution $φ_{ES}$=$K_{ES}∗ρ$ and the local correction integral $φ_{cor}=(K−K_{ES})∗ρ$. The exponential convolution is
evaluated via the recurrence formula that is typical of the exponential function. The
local correction integral is restricted to a small neighborhood of the target point where
the kernel singularity is considered. Rigorous estimates of the optimal accuracy are
provided. The algorithm is ideal for parallelization and favors easy extensions to complicated kernels. Extensive numerical results for different kernels are presented. 相似文献
11.
Jinchao Xu 《Communications In Computational Physics》2020,28(5):1707-1745
We study a family of $H^m$-conforming piecewise polynomials based on the
artificial neural network, referred to as the finite neuron method (FNM), for numerical
solution of $2m$-th-order partial differential equations in$\mathbb{R}^d$ for any $m,d≥1$ and then
provide convergence analysis for this method. Given a general domain Ω$⊂\mathbb{R}^d$ and a
partition$\mathcal{T}_h$ of Ω, it is still an open problem in general how to construct a conforming finite element subspace of $H^m$(Ω) that has adequate approximation properties. By using
techniques from artificial neural networks, we construct a family of $H^m$-conforming
functions consisting of piecewise polynomials of degree $k$ for any $k≥m$ and we further obtain the error estimate when they are applied to solve the elliptic boundary
value problem of any order in any dimension. For example, the error estimates that $‖u−u_N‖_{H^m(\rm{Ω})}=\mathcal{O}(N^{−\frac{1}{2}−\frac{1}{d}})$ is obtained for the error between the exact solution $u$ and
the finite neuron approximation $u_N$. A discussion is also provided on the difference
and relationship between the finite neuron method and finite element methods (FEM).
For example, for the finite neuron method, the underlying finite element grids are not
given a priori and the discrete solution can be obtained by only solving a non-linear
and non-convex optimization problem. Despite the many desirable theoretical properties of the finite neuron method analyzed in the paper, its practical value requires
further investigation as the aforementioned underlying non-linear and non-convex optimization problem can be expensive and challenging to solve. For completeness and
the convenience of the reader, some basic known results and their proofs are introduced. 相似文献
12.
Optimal Error Estimates of Compact Finite Difference Discretizations for the Schrödinger-Poisson System 下载免费PDF全文
Yong Zhang 《Communications In Computational Physics》2013,13(5):1357-1388
We study compact finite difference methods for the Schrödinger-Poisson
equation in a bounded domain and establish their optimal error estimates under proper
regularity assumptions on wave function $ψ$ and external potential $V(x)$. The Crank-Nicolson compact finite difference method and the semi-implicit compact finite difference method are both of order $\mathcal{O}$($h^4$+$τ^2$) in discrete $l^2$, $H^1$ and $l^∞$ norms with mesh
size $h$ and time step $τ$. For the errors of compact finite difference approximation to
the second derivative and Poisson potential are nonlocal, thus besides the standard
energy method and mathematical induction method, the key technique in analysis is
to estimate the nonlocal approximation errors in discrete $l^∞$ and $H^1$ norm by discrete
maximum principle of elliptic equation and properties of some related matrix. Also
some useful inequalities are established in this paper. Finally, extensive numerical results are reported to support our error estimates of the numerical methods. 相似文献
13.
14.
Andrew T. Barker Tyrone Rees & Martin Stoll 《Communications In Computational Physics》2016,19(1):143-167
In this paper we consider PDE-constrained optimization problems which incorporate
an $\mathcal{H}_1$ regularization control term. We focus on a time-dependent PDE, and
consider both distributed and boundary control. The problems we consider include
bound constraints on the state, and we use a Moreau-Yosida penalty function to handle
this. We propose Krylov solvers and Schur complement preconditioning strategies
for the different problems and illustrate their performance with numerical examples. 相似文献
15.
An Adaptive Finite Element Method with Hybrid Basis for Singularly Perturbed Nonlinear Eigenvalue Problems 下载免费PDF全文
Ye Li 《Communications In Computational Physics》2016,19(2):442-472
In this paper, we propose a uniformly convergent adaptive finite element
method with hybrid basis (AFEM-HB) for the discretization of singularly perturbed
nonlinear eigenvalue problems under constraints with applications in Bose-Einstein
condensation (BEC) and quantum chemistry. We begin with the time-independent
Gross-Pitaevskii equation and show how to reformulate it into a singularly perturbed
nonlinear eigenvalue problem under a constraint. Matched asymptotic approximations
for the problem are reviewed to confirm the asymptotic behaviors of the solutions
in the boundary/interior layer regions. By using the normalized gradient flow, we
propose an adaptive finite element with hybrid basis to solve the singularly perturbed
nonlinear eigenvalue problem. Our basis functions and the mesh are chosen adaptively
to the small parameter ε. Extensive numerical results are reported to show the
uniform convergence property of our method. We also apply the AFEM-HB to compute
the ground and excited states of BEC with box/harmonic/optical lattice potential
in the semiclassical regime (0<ε≪1). In addition, we give a detailed error analysis of
our AFEM-HB to a simpler singularly perturbed two point boundary value problem,
show that our method has a minimum uniform convergence order $\mathcal{O}$(1/$(NlnN)^\frac{2}{3}$). 相似文献
16.
Kei Mizukoshi Kengo Suzuki Kihei Yoneyama Ryo Kamijima Seisyou Kou Manabu Takai Masaki Izumo Akio Hayashi Eiji Ohtaki Yoshihiro J. Akashi Naohiko Osada Kazuto Omiya Tomoo Harada Sachihiko Nobuoka Fumihiko Miyake 《Journal of Echocardiography》2013,11(1):9-17
Background
Hypertrophic cardiomyopathy (HCM) patients with preserved left ventricular ejection fraction (LVEF) often develop dyspnea and exercise intolerance. Diastolic dysfunction may contribute to exercise intolerance in these patients. This study aimed to clarify our hypothesis as to whether diastolic function rather than systolic function would be associated with exercise intolerance in HCM using two-dimensional (2D) speckle tracking echocardiography during exercise.Methods
Thirty-three HCM patients (mean age 59.3 ± 15.7 years) underwent 2D speckle tracking echocardiography at rest and during submaximal semi-supine bicycle exercise. Global longitudinal strain (LS), LS rate during systole (LSRs), early diastole (LSRe), and late diastole (LSRa) were measured. The symptom-limited cardiopulmonary exercise testing was performed using a cycle ergometer for measuring the peak oxygen consumption (peak $ \dot{V}_{{{\text{O}}_{2} }} $ ).Results
In the multivariate linear regression analysis, peak $ \dot{V}_{{{\text{O}}_{2} }} $ did not associate with strain or strain rate at rest. However, peak $ \dot{V}_{{{\text{O}}_{2} }} $ correlated with LS (β = ?0.403, p = 0.007), LSRe (β = 6.041, p = 0.001), and LSRa (β = 5.117, p = 0.021) during exercise after adjustment for age, gender, and heart rate. The first quartile peak $ \dot{V}_{{{\text{O}}_{2} }} $ (14.2 mL/min/kg) was assessed to predict exercise intolerance. The C-statistic of delta LSRe was 0.74, which was relatively greater than that of delta LS (0.70) and delta LSRa (0.58), indicating that early diastolic function rather than systolic and late diastolic function affects exercise intolerance.Conclusions
LSRe during exercise is closely associated with the peak $ \dot{V}_{{{\text{O}}_{2} }} $ . Early diastolic function during exercise is an important determinant of exercise capacity in patients with HCM. 相似文献17.
A Comparison of Higher-Order Weak Numerical Schemes for Stopped Stochastic Differential Equations 下载免费PDF全文
Francisco Bernal & Juan A. Acebró n 《Communications In Computational Physics》2016,20(3):703-732
We review, implement, and compare numerical integration schemes for spatially
bounded diffusions stopped at the boundary which possess a convergence rate of
the discretization error with respect to the time step $h$ higher than $\mathcal{O}$$(√h)$. We address
specific implementation issues of the most general-purpose of such schemes. They
have been coded into a single Matlab program and compared, according to their accuracy
and computational cost, on a wide range of problems in up to R48. The paper is
self-contained and the code will be made freely downloadable. 相似文献
18.
Correlation Functions,Universal Ratios and Goldstone Mode Singularities in $n$-Vector Models 下载免费PDF全文
J. Kaupužs R. V. N. Melnik & J. Rim&scaron āns 《Communications In Computational Physics》2014,15(5):1407-1430
Correlation functions in the $\mathcal{O}$$(n)$ models below the critical temperature are
considered. Based on Monte Carlo (MC) data, we confirm the fact stated earlier by Engels and Vogt, that the transverse two-plane correlation function of the $\mathcal{O}$$(4)$ model for
lattice sizes about $L=120$ and small external fields $h$ is very well described by a Gaussian approximation. However, we show that fits of not lower quality are provided by
certain non-Gaussian approximation. We have also tested larger lattice sizes, up to $L=512$. The Fourier-transformed transverse and longitudinal two-point correlation
functions have Goldstone mode singularities in the thermodynamic limit at $k→0$ and $h=+0$, i.e., $G_⊥$(k)$≃ak^{−λ_⊥}$ and$G_{||}($k$)≃bk^{−λ_{||}}$, respectively. Here $a$ and $b$ are the amplitudes, $k$=|k| is the magnitude of the wave vector k. The exponents $λ_⊥$, $λ_{||}$ and the
ratio $bM^2/a^2$, where $M$ is the spontaneous magnetization, are universal according to
the GFD (grouping of Feynman diagrams) approach. Here we find that the universality follows also from the standard (Gaussian) theory, yielding $bM^2/a^2$=$(n−1)/16$. Our
MC estimates of this ratio are $0.06±0.01$ for $n=2$, $0.17±0.01$ for $n=4$ and $0.498±0.010$ for $n=10$. According to these and our earlier MC results, the asymptotic behavior and
Goldstone mode singularities are not exactly described by the standard theory. This is
expected from the GFD theory. We have found appropriate analytic approximations
for $G_⊥$(k) and $G_{||}$(k), well fitting the simulation data for small $k$. We have used them
to test the Patashinski-Pokrovski relation and have found that it holds approximately. 相似文献
19.
Shi Jin & Xiantao Li 《Communications In Computational Physics》2020,28(5):1907-1936
Random batch algorithms are constructed for quantum Monte Carlo simulations. The main objective is to alleviate the computational cost associated with the
calculations of two-body interactions, including the pairwise interactions in the potential energy, and the two-body terms in the Jastrow factor. In the framework of variational Monte Carlo methods, the random batch algorithm is constructed based on the
over-damped Langevin dynamics, so that updating the position of each particle in an $N$-particle system only requires$\mathcal{O}(1)$ operations, thus for each time step the computational cost for $N$ particles is reduced from$\mathcal{O}(N^2)$ to$\mathcal{O}(N)$. For diffusion Monte Carlo
methods, the random batch algorithm uses an energy decomposition to avoid the computation of the total energy in the branching step. The effectiveness of the random
batch method is demonstrated using a system of liquid $^4$He atoms interacting with a
graphite surface. 相似文献
20.
A Well-Conditioned Hypersingular Boundary Element Method for Electrostatic Potentials in the Presence of Inhomogeneities within Layered Media 下载免费PDF全文
In this paper, we will present a high-order, well-conditioned boundary element
method (BEM) based on Müller's hypersingular second kind integral equation
formulation to accurately compute electrostatic potentials in the presence of inhomogeneity
embedded within layered media. We consider two types of inhomogeneities:
the first one is a simple model of an ion channel which consists of a finite height cylindrical
cavity embedded in a layered electrolytes/membrane environment, and the second
one is a Janus particle made of two different semi-spherical dielectric materials.
Both types of inhomogeneities have relevant applications in biology and colloidal material,
respectively. The proposed BEM gives$\mathcal{O}$(1) condition numbers, allowing fast
convergence of iterative solvers compared to previous work using first kind of integral
equations. We also show that the second order basis converges faster and is more
accurate than the first order basis for the BEM. 相似文献