共查询到20条相似文献,搜索用时 31 毫秒
1.
A Uniformly Convergent Numerical Method for Singularly Perturbed Nonlinear Eigenvalue Problems
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In this paper we propose a uniformly convergent numerical method for discretizing singularly perturbed nonlinear eigenvalue problems under constraints with applications in Bose-Einstein condensation 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 presented to locate the positions and characterize the widths of boundary layers and/or interior layers in the solution. A uniformly convergent numerical method is proposed by using the normalized gradient flow and piecewise uniform mesh techniques based on the asymptotic approximations for the problem. Extensive numerical results are reported to demonstrate the effectiveness of our numerical method for the problems. Finally, the method is applied to compute ground and excited states of Bose-Einstein condensation in the semiclassical regime and some conclusive findings are reported. 相似文献
2.
A New Family of Nonconforming Elements with $H$(curl)-Continuity for the 3D Quad-Curl Problem
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Baiju Zhang & Zhimin Zhang 《Communications In Computational Physics》2023,33(4):1069-1089
We propose and analyze a new family of nonconforming finite elements
for the three-dimensional quad-curl problem. The proposed finite element spaces are
subspaces of $\boldsymbol{H}$(curl), but not of $\boldsymbol{H}$(grad curl), which are different from the existing
nonconforming ones [10,12,13]. The well-posedness of the discrete problem is proved
and optimal error estimates in discrete $\boldsymbol{H}$(grad curl) norm, $\boldsymbol{H}$(curl) norm and $L^2$ norm
are derived. Numerical experiments are provided to illustrate the good performance
of the method and confirm our theoretical predictions. 相似文献
3.
Renato Spigler 《Communications In Computational Physics》2022,31(5):1341-1361
The quantum lattice Boltzmann (qlB) algorithm solves the 1D Dirac equations and has been used to solve approximately the classical (i.e., non-relativistic)
Schrödinger equation. We point out that the qlB method actually approximates the
hyperbolic version of the non-relativistic Schrödinger equation, whose solution is thus
obtained at the price of an additional small error. Such an error is of order of $(ω_c\tau)^{−1},$ where $ω_c:=\frac{mc^2}{h}$ is the Compton frequency, $ħ$ being the reduced Planck constant, $m$ the rest mass of the electrons, $c$ the speed of light, and $\tau$ a chosen reference time (i.e.,
1 s), and hence it vanishes in the non-relativistic limit $c → +∞.$ This asymptotic result comes from a singular perturbation process which does not require any boundary
layer and, consequently, the approximation holds uniformly, which fact is relevant in
view of numerical approximations. We also discuss this occurrence more generally, for
some classes of linear singularly perturbed partial differential equations. 相似文献
4.
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
5.
Xiaoqiang Yue Shi Shu Xiaowen Xu & Zhiyang Zhou 《Communications In Computational Physics》2015,18(5):1313-1335
The paper aims to develop an effective preconditioner and conduct the convergence
analysis of the corresponding preconditioned GMRES for the solution of discrete
problems originating from multi-group radiation diffusion equations. We firstly
investigate the performances of the most widely used preconditioners (ILU(k) and
AMG) and their combinations ($B_{co}$ and$\widetilde{B}_{co}$), and provide drawbacks on their feasibilities.
Secondly, we reveal the underlying complementarity of ILU(k) and AMG by analyzing
the features suitable for AMG using more detailed measurements on multiscale
nature of matrices and the effect of ILU(k) on multiscale nature. Moreover, we present
an adaptive combined preconditioner $B^α_{co}$ involving an improved ILU(0) along with
its convergence constraints. Numerical results demonstrate that $B^α_{co}$-GMRES holds the
best robustness and efficiency. At last, we analyze the convergence of GMRES with
combined preconditioning which not only provides a persuasive support for our proposed
algorithms, but also updates the existing estimation theory on condition numbers
of combined preconditioned systems. 相似文献
6.
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
7.
A Well-Conditioned Hypersingular Boundary Element Method for Electrostatic Potentials in the Presence of Inhomogeneities within Layered Media
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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
8.
Directional $\mathcal{H}^2$ Compression Algorithm: Optimisations and Application to a Discontinuous Galerkin BEM for the Helmholtz Equation
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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
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.
An Adaptive Grid Method for Singularly Perturbed Time-Dependent Convection-Diffusion Problems
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In this paper, we study the numerical solution of singularly perturbed time-dependent
convection-diffusion problems. To solve these problems, the backward Euler
method is first applied to discretize the time derivative on a uniform mesh, and the
classical upwind finite difference scheme is used to approximate the spatial derivative
on an arbitrary nonuniform grid. Then, in order to obtain an adaptive grid for all
temporal levels, we construct a positive monitor function, which is similar to the arc-length
monitor function. Furthermore, the ε-uniform convergence of the fully discrete
scheme is derived for the numerical solution. Finally, some numerical results are given
to support our theoretical results. 相似文献
11.
Marco Caliari & Simone Zuccher 《Communications In Computational Physics》2021,29(5):1336-1364
We propose an idea to solve the Gross–Pitaevskii equation for dark structures inside an infinite constant background density $ρ_∞$=${|ψ_∞|}^2$, without the introduction of artificial boundary conditions. We map the unbounded physical domain $\mathbb{R}^3$ into the bounded domain ${(−1,1)}^3$ and discretize the rescaled equation by equispaced
4th-order finite differences. This results in a free boundary approach, which can be
solved in time by the Strang splitting method. The linear part is solved by a new, fast
approximation of the action of the matrix exponential at machine precision accuracy,
while the nonlinear part can be solved exactly. Numerical results confirm existing
ones based on the Fourier pseudospectral method and point out some weaknesses of
the latter such as the need of a quite large computational domain, and thus a consequent critical computational effort, in order to provide reliable time evolution of the
vortical structures, of their reconnections, and of integral quantities like mass, energy,
and momentum. The free boundary approach reproduces them correctly, also in finite
subdomains, at low computational cost. We show the versatility of this method by
carrying out one- and three-dimensional simulations and by using it also in the case of
Bose–Einstein condensates, for which $ψ$→0 as the spatial variables tend to infinity. 相似文献
12.
A Comparison of Higher-Order Weak Numerical Schemes for Stopped Stochastic Differential Equations
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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
13.
An Efficient Finite Element Method with Exponential Mesh Refinement for the Solution of the Allen-Cahn Equation in Non-Convex Polygons
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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
14.
McHoney MC Corizia L Eaton S Wade A Spitz L Drake DP Kiely EM Tan HL Pierro A 《Surgical endoscopy》2006,20(3):452-457
Background Laparoscopic surgery is thought to be associated with a reduced metabolic response compared to open surgery. Oxygen consumption
(
) and energy metabolism during laparoscopic surgery have not been characterized in children.
Methods We measured respiratory gas exchange intraoperatively in children undergoing 19 open and 20 laparoscopic procedures. Premature
infants and patients with metabolic, renal, and cardiac abnormalities were excluded. Anesthesia was standardized. Unheated
carbon dioxide was used for insufflation.
was measured by indirect calorimetry. Core temperature was measured using an esophageal temperature probe.
Results We found a steady increase in
during laparoscopy. The increase in
was more marked in younger children and was associated with a significant rise in core temperature. Open surgery was not
associated with significant changes in core temperature or
.
Conclusions Laparoscopy in children is associated with an intraoperative hypermetabolic response characterized by increased oxygen consumption
and core temperature. These changes are more marked in younger children.
M. C. McHoney and L. Corizia contributed equally to the study, analysis, and writing of the paper 相似文献
15.
On the Effect of Ghost Force in the Quasicontinuum Method: Dynamic Problems in One Dimension
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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
16.
G. J. I. M. van der Werf A. J. Tonino 《Archives of orthopaedic and trauma surgery》1984,103(2):128-130
Summary Eleven posterior tibial tendon transfers were performed in eight patients with spastic equinovarus with an average follow-up of 3 years (range
years). The main goals—to eliminate the need for braces and to improve gait—were achieved in all patients. An adequate operation technique kept the well-known postoperative complications, e.g., valgus and calcaneal deformity, to a minimum.
Zusammenfassung Die Transposition der Musc. tibialis posterior-Sehne als operative Behandlung des spastischen Equinovarus wurde elfmal durchgeführt bei acht Patienten. Die Ergebnisse wurden Jahre nach der Operation kontrolliert. Das Hauptziel: Unabhängigkeit von Orthesen and Verbesserung der Gehfähigkeit, konnte bei allen Patienten erreicht werden. Eine adequate Operationstechnik konnte die bekannten postoperativen Komplikationen, wie eine Valgus- und Hackenfuß-Deformität bis auf ein Minimum beschränken.相似文献
17.
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
18.
Efficient Algorithm for Many-Electron Angular Momentum and Spin Diagonalization on Atomic Subshells
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Christian B. Mendl 《Communications In Computational Physics》2016,19(1):192-204
We devise an efficient algorithm for the symbolic calculation of irreducible
angular momentum and spin (LS) eigenspaces within the $n$-fold antisymmetrized tensor
product $Λ^n$$V_u$, where n is the number of electrons and $u$ = s,p,d,··· denotes the
atomic subshell. This is an essential step for dimension reduction in configuration-interaction
(CI) methods applied to atomic many-electron quantum systems. The algorithm
relies on the observation that each $L_z$ eigenstate with maximal eigenvalue is
also an $L^2$ eigenstate (equivalently for $S_z$ and $S^2$ ), as well as the traversal of LS eigenstates
using the lowering operators $L_−$ and $S_−$. Iterative application to the remaining
states in $Λ^n$$V_u$ leads to an implicit simultaneous diagonalization. A detailed complexity
analysis for fixed $n$ and increasing subshell number $u$ yields run time$\mathcal{O}$($u^{3n−2}$). A
symbolic computer algebra implementation is available online. 相似文献
19.
Correlation Functions,Universal Ratios and Goldstone Mode Singularities in $n$-Vector Models
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J. Kaupužs R. V. N. Melnik & J. Rim&scaron āns 《Communications In Computational Physics》2014,15(5):1407-1430
Correlation functions in the $\mathcal{O}$$(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
20.
The validity of using mechanical measures of work to indicate the metabolic energy consumption during normal gait was examined. These mechanical measures were (a) mechanical work done on the center of mass per kilogram body mass per second (\documentclass{article}\pagestyle{empty}\begin{document}$ \dot W_{{\rm cm}} $\end{document}), calculated by integration of ground reaction forces measured by force platforms; (b) total body segmental work per kilogram body mass per second (\documentclass{article}\pagestyle{empty}\begin{document}$ \dot W_{{\rm seg}} $\end{document}), calculated from individual body segment energies measured by motion analysis; and (c) the sum of the normalized absolute moment impulses per second acting on the joints of the lower extremities (\documentclass{article}\pagestyle{empty}\begin{document}$ \dot M $\end{document}), calculated from both force and motion data. The metabolic energy consumption, determined by analysis of expired air, and the three mechanical measures of work were calculated for six normal subjects walking at five speeds. Each measure of mechanical work per second walked was highly correlated with metabolic energy consumption/kg · s (r = 0.89 for Wcm, r = 0.79 for \documentclass{article}\pagestyle{empty}\begin{document}$ \dot W_{{\rm seg}} $\end{document}, and r = 0.85 for M), but a poorer correlation was found between each measure of mechanical work per meter walked and net metabolic energy consumption/kg. m (r = 0.54 for Wcm, r = 0.28 for Wseg, and r = 0.03 for M). These mechanical parameters, particularly when measured per time, may be useful in comparing metabolic energy consumption between individuals or between different walking conditions for the same individual. 相似文献