Rayleigh-Schrödinger-Goldstone variational perturbation theory for many Fermion systems

We present a Rayleigh-Schrödinger-Goldstone perturbation formalism for many Fermion systems. Based on this formalism, variational perturbation scheme which goes beyond the Gaussian approximation is developed. In order to go beyond the Gaussian approximation, we identify a parent Hamiltonian which has an effective Gaussian vacuum as a variational solution and carry out further perturbation with respect to the renormalized interaction using Goldstones expansion. Perturbation rules for the ground state wavefunctional and energy are found, thus, opening a way for general use of the Schrödinger picture method for many Fermion systems. Useful commuting relations between operators and the Gaussian wavefunctional are also found, which could reduce the calculational efforts substantially. As examples, we calculate the first order correction to the Gaussian wavefunctional and the second order correction to the ground state of an electron gas system with the Yukawa-type interaction.     

Variational Sparse Bayesian Learning for Estimation of Gaussian Mixture Distributed Wireless Channels

In this paper, variational sparse Bayesian learning is utilized to estimate the multipath parameters for wireless channels. Due to its flexibility to fit any probability density function (PDF), the Gaussian mixture model (GMM) is introduced to represent the complicated fading phenomena in various communication scenarios. First, the expectation-maximization (EM) algorithm is applied to the parameter initialization. Then, the variational update scheme is proposed and implemented for the channel parameters’ posterior PDF approximation. Finally, in order to prevent the derived channel model from overfitting, an effective pruning criterion is designed to eliminate the virtual multipath components. The numerical results show that the proposed method outperforms the variational Bayesian scheme with Gaussian prior in terms of root mean squared error (RMSE) and selection accuracy of model order.     

A Note on Cactus Trees: Variational vs. Recursive Approach

In this article we deal with the variational approach to cactus trees (Husimi trees) and the more common recursive approach, that are in principle equivalent for finite systems. We discuss in detail the conditions under which the two methods are equivalent also in the analysis of infinite (self-similar) cactus trees, usually investigated to the purpose of approximating ordinary lattice systems. Such issue is hardly ever considered in the literature. We show (on significant test models) that the phase diagram and the thermodynamic quantities computed by the variational method, when they deviates from the exact bulk properties of the cactus system, generally provide a better approximation to the behavior of a corresponding ordinary system. Generalizing a property proved by Kikuchi, we also show that the numerical algorithm usually employed to perform the free energy minimization in the variational approach is always convergent.     

Compressed and stabilized bright solitons of a derivative Gross–Pitaevskii model

We discuss bright soliton compression in Bose–Einstein condensates described by a cubic-quintic derivative Gross–Pitaevskii model which takes into account the delayed nonlinear response of condensates confined in a complex potential. The external potential consists of an attractive parabolic background, a linear potential that may represent the gravitational field, and a complex part (introduced phenomenologically) which characterizes the rate of injection of atoms into the condensate. In our approach, the bright soliton evolution is described analytically by applying the variational approximation. The influences of the soliton width and the rate of injection of atoms are considered because of practical applications. To ensure the validity of the variational approximation, all analytical results are compared with the numerical data obtained with the split-step Fourier algorithm. The roles of the linear potential on the evolution and stabilization of compressed solitons are unveiled.     

High order numerical approximation of minimal surfaces

We present an algorithm for finding high order numerical approximations of minimal surfaces with a fixed boundary. The algorithm employs parametrization by high order polynomials and a linearization of the weak formulation of the Laplace–Beltrami operator to arrive at an iterative procedure to evolve from a given initial surface to the final minimal surface. For the steady state solution we measure the approximation error in a few cases where the exact solution is known. In the framework of parametric interpolation, the choice of interpolation points (mesh nodes) is directly affecting the approximation error, and we discuss how to best update the mesh on the evolutionary surface such that the parametrization remains smooth. In our test cases we may achieve exponential convergence in the approximation of the minimal surface as the polynomial degree increases, but the rate of convergence greatly differs with different choices of mesh update algorithms. The present work is also of relevance to high order numerical approximation of fluid flow problems involving free surfaces.     

Sampling the Variational Posterior with Local Refinement

Variational inference is an optimization-based method for approximating the posterior distribution of the parameters in Bayesian probabilistic models. A key challenge of variational inference is to approximate the posterior with a distribution that is computationally tractable yet sufficiently expressive. We propose a novel method for generating samples from a highly flexible variational approximation. The method starts with a coarse initial approximation and generates samples by refining it in selected, local regions. This allows the samples to capture dependencies and multi-modality in the posterior, even when these are absent from the initial approximation. We demonstrate theoretically that our method always improves the quality of the approximation (as measured by the evidence lower bound). In experiments, our method consistently outperforms recent variational inference methods in terms of log-likelihood and ELBO across three example tasks: the Eight-Schools example (an inference task in a hierarchical model), training a ResNet-20 (Bayesian inference in a large neural network), and the Mushroom task (posterior sampling in a contextual bandit problem).     

像素重要性测量特征下的侧扫声呐目标检测

研究了在较低信噪比下,在保证检测概率的前提下尽量降低虚警概率的目标检测,提出了一种针对特定目标的两阶段筛选算法.第一阶段中,首先使用阈值分割出有效点,并定义了一种新的像素重要性测量特征用于初步筛选目标。即通过有效像素点之间的距离来赋以高斯分布的权值,当前像素重要性的值定义为剩余有效点的距离加权和,具有较高的像素重要性值的聚集性强的区域内像素点会被定位出来。第二阶段,使用卷积神经网络分类器排除虚假目标.在实验中,使用近期无人潜器获得的海底数据,召回率与虚警概率分别达到90.39%与2.39%,证明了其相比声呐目标检测主流算法有更好的检测能力。      

Gaussian solitons in nonlocal media: variational analysis

Exact solutions of Gaussian solitons in nonlinear media with a Gaussian nonlocal response are obtained. Using the variational approach, we obtain the approximate solutions of such solitons when the degree of the nonlocality is arbitrary. Specifically, we study the conditions for Gaussian solitons that propagate in weakly and highly nonlocal media. We also compare the variational result with the known exact solutions for weakly and highly nonlocal media.     

New Variational Perturbation Theory Based on q−Deformed Oscillator

A new variational perturbation theory is developed based on the q−deformed oscillator. It is shown that the new variational perturbation method provides 200 and 10 times better accuracy for the ground state energy of anharmonic oscillator than the Gaussian and the post Gaussian approximation, respectively, for weak coupling.     

A finite element dynamical nonlinear subscale approximation for the low Mach number flow equations

In this work we propose a variational multiscale finite element approximation of thermally coupled low speed flows. The physical model is described by the low Mach number equations, which are obtained as a limit of the compressible Navier–Stokes equations in the small Mach number regime. In contrast to the commonly used Boussinesq approximation, this model permits to take volumetric deformation into account. Although the former is more general than the latter, both systems have similar mathematical structure and their numerical approximation can suffer from the same type of instabilities.     

Electronic states of a hydrogenic impurity in a zinc-blende GaN/AlGaN quantum well

Binding energies of ground and a few low lying excited states of a hydrogenic donor confined in a zinc-blende GaN/AlGaN quantum well are investigated. They are computed within the framework of single band effective mass approximation, by means of a variational approach. The donor states are investigated with the various impurity positions as a function of well width. The calculations have been carried out with the inclusion of conduction band non-parabolicity through the energy dependent effective mass. The variational solutions have been improved by using a two-parametric trial wavefunction. The results seem better and good agreement with the other investigators. To support our results, we observe that the values of variational parameters are consistent when two parameter wave function is used. We find that the inclusion of non-parabolic effects leads to more binding for all the values of well width and is significant for narrow wells. The results are compared with the existing available literature.     

Variational calculation of the excited states in the sinh-Gordon field theory

The Gaussian wave-functional variational approximation is used to calculate the energy of one-particle state as well as the energy and wave function of two-particle state in the sinh-Gordon model in (1+1) dimensions. We find that there are no bound states in this model. The phase shifts of the scattering states are investigated in detail.     

Study of Dibaryon States Containing Three Different Types of Quarks

Previously we have shown, in a simple chromomagnetic model, that including heavy quarks in the dibaryon sector can lead to favorable configurations for stability against decay into two baryons. In this study we investigate a reduced set of favorable candidates that have emerged from our previous works. We use a non-relativistic quark model with quarks interacting through a QCD-inspired potential, which has been tested previously in meson and baryon spectroscopy. A variational procedure is performed using a great number of Gaussian functions containing all the possibilities for colour, isospin, and spin components. Received February 14, 1997; revised June 12, 1997; accepted for publication June 27, 1997     

Diffraction characteristics of optical vortex passing through an aperture-iris diaphragm

The diffraction of an optical vortex through an iris diaphragm which is a close approximation to a circular aperture has been investigated. The results are compared with those obtained from the diffraction of a Gaussian beam through the same aperture. In our findings the diffraction of an optical vortex and a Gaussian beam produce ball bearing sort of structure of darkness and brightness. The singularity of the vortex beam is found to be consistent even after the diffraction through the aperture. The presence of singularity at the centre of diffraction pattern of an optical vortex has been confirmed by interferometry. There is a good agreement between the experimental and numerical results. We propose that these results may find various applications in optical trapping experiments.     

线性与非线性光晶格中偶极孤立子的稳定性

利用变分法研究了线性和非线性交叉光晶格中偶极玻色-爱因斯坦凝聚(BEC)体系中物质波孤立子的稳定性.选用柱对称高斯型试探波函数,得出参数的Euler-Lagrange方程和体系的有效作用势能,根据有效势能是否具有局域最小值判断体系是否具有稳定孤立子解.结果表明,由于存在接触相互作用的空间调制,在排斥和吸引偶极相互作用下,均能形成稳定的孤立子解.给出了参数空间中存在稳定解的区域和物质波波包宽度随时间的变化曲线.     

Relativistic motion on Gaussian quantum steering for two-mode localized Gaussian states

Realistic quantum systems always exhibit gravitational and relativistic features. In this paper, we investigate the properties of Gaussian steering and its asymmetry by the localized two-mode Gaussian quantum states, instead of the traditional single-mode approximation method in the relativistic setting. We find that the one-side Gaussian quantum steering will monotonically decrease with increasing observers of acceleration. Meanwhile, our results also reveal the interesting behavior of the Gaussian steering asymmetry, which increases for a specific range of accelerated parameter and then gradually approaches to a finite value. Such finding is well consistent and explained by the well-known Unruh effect, which could significantly destroy the one-side Gaussian quantum steering. Finally, our results could also be applied to the dynamical studies of Gaussian steering between the Earth and satellites, since the effects of acceleration are equal to the effects of gravity according to the equivalence principle.     

Modelling the excitation field of an optical resonator

Assuming the paraxial approximation, we derive efficient recursive expressions for the projection coefficients of a Gaussian beam over the Gauss--Hermite transverse electro-magnetic (TEM) modes of an optical cavity. While previous studies considered cavities with cylindrical symmetry, our derivation accounts for “simple” astigmatism and ellipticity, which allows to deal with more realistic optical systems. The resulting expansion of the Gaussian beam over the cavity TEM modes provides accurate simulation of the excitation field distribution inside the cavity, in transmission, and in reflection. In particular, this requires including counter-propagating TEM modes, usually neglected in textbooks. As an illustrative application to a complex case, we simulate reentrant cavity configurations where Herriott spots are obtained at cavity output. We show that the case of an astigmatic cavity is also easily modelled. To our knowledge, such relevant applications are usually treated under the simplified geometrical optics approximation, or using heavier numerical methods.     

Semi-blind image restoration based on Chan-Vese denoising model

A semi-blind image restoration algorithm is proposed based on reduced non-convex approximation of Luminita Vese and Tony Chan's (C-V) denoising model. Compared with C-V denoising model, we modify the fidelity term and add a term on point spread function (PSF). The function depends on two variables: the image function to be restored u and the standard deviation of Gaussian kernel to be estimated σ. Then the problems consist in solving a system with two coupled equations. Compared with the Leah Bar's semi-blind image restoration model which must solve three coupled equations, our method only needs to solve two equations. Furthermore, the estimation of f by our algorithm is superior to Leah Bar's algorithm. The experimental results demonstrate that the proposed method is effective.     

A quasi-discrete Hankel transform for nonlinear beam propagation

We propose and implement a quasi-discrete Hankel transform algorithm based on Dini series expansion (DQDHT) in this paper. By making use of the property that the zero-order Bessel function derivative J' ₀(0)=0, the DQDHT can be used to calculate the values on the symmetry axis directly. In addition, except for the truncated treatment of the input function, no other approximation is made, thus the DQDHT satisfies the discrete Parseval theorem for energy conservation, implying that it has a high numerical accuracy. Further, we have performed several numerical tests. The test results show that the DQDHT has a very high numerical accuracy and keeps energy conservation even after thousands of times of repeating the transform either in a spatial domain or in a frequency domain. Finally, as an example, we have applied the DQDHT to the nonlinear propagation of a Gaussian beam through a Kerr medium system with cylindrical symmetry. The calculated results are found to be in excellent agreement with those based on the conventional 2D-FFT algorithm, while the simulation based on the proposed DQDHT takes much less computing time.     

Non-Gaussian corrections to higgs mass in autonomousλφ 3+1 4

Recent calculations in one-loop and Gaussian approximation, using the so-called autonomous renormalization scheme, indicate a comparatively massive, narrow Higgs excitation at about 2 TeV. Here I show that this result persists in the framework of a post-Gaussian variational approximation for the pureO(N)-symmetric ⁴-theory forN>1. The method is based on nonlinear transformations of path-integral variables, and the optimization amounts to a Schwinger-Dyson-type summation of diagrams. In the case ofO(4), for example, I findM _Higgs=2.3 TeV, compared with 1.9 TeV and 2.1 TeV in one-loop and Gaussian approximation, respectively. My results are also consistent with the Consoli-Stevenson conjecture that the Gaussian and one-loop results forN=1 are exact and may even hold for generalN.