Gradient Flow Finite Element Discretisations with Energy-Based Adaptivity for Excited States of Schrödinger's Equation

The purpose of this paper is to verify that the computational scheme from [Heid et al., Gradient flow finite element discretizations with energy-based adaptivity for the Gross--Pitaevskii equation, J. Comput. Phys. 436(2021)] for the numerical approximation of the ground state of the Gross--Pitaevskii equation can equally be applied for the effective approximation of excited states of Schrödinger's equation. That procedure employs an adaptive interplay of a Sobolev gradient flow iteration and a novel local mesh refinement strategy, and yields a guaranteed energy decay in each step of the algorithm. The computational tests in the present work highlight that this strategy is indeed able to approximate excited states, with (almost) optimal convergence rate with respect to the number of degrees of freedom.     

Modeling and simulation of the CLS cryogenic system

This paper presents results pertaining to the numerical modeling of the cryogenic system at the Canadian Light Source. The cryogenic system consists of a cryostat that houses a Radio Frequency (RF) cavity used for boosting the energy of an electron beam. For consistent operation of the RF cavity, it must be kept immersed in liquid helium at a constant level with the pressure in the gas space maintained to an accuracy of ±1 mbar. An improvement to the cryostat model suggested in [3] using control volumes is described. The model and numerical method developed for the liquid helium supply and gaseous helium return lines are validated using two different cases, viz., the liquid helium flow rate from the liquid helium transfer line and the gaseous helium flow rate from the cryostat for various heater power input settings. The numerical method described here is significantly more accurate, efficient, and flexible than that used in [1] based on an iterative bisection method.     

非线性延迟积分微分方程连续Runge-Kutta方法的稳定性分析

本文主要研究了一般形式的延迟积分微分方程,将连续Runge-Kutt,a方法用于求解该类问题,并讨论了方法的稳定性,证明了(k,l)-代数稳定的Runge-Kutta方法当0k1时对应的连续Runge-Kutta方法是渐近稳定的.最后我们通过数值试验验证了方法的有效性及所获结论的正确性.     

Uniformly exponentially stable approximation for the transmission line with variable coefficients and its application

We analyze an ideal transmission line, which is defined by the telegraph equation with variable coefficients, from the perspectives of numerical analysis and control theory in this note. Because the spatially semi-discrete scheme of the original system is insufficient for discussing uniform exponential stability, we apply a similar transform to the continuous system and produce an intermediate system that may be easily analyzed. To begin, we discuss uniform exponential stability for the intermediate system using an so called average central-difference semi-discrete scheme and the direct Lyapunov function approach. The proof is the same as in the continuous case. The Trotter-Kato Theorem is used to demonstrate the stability and consistency of numerical approximation scheme. Finally, we propose a semi-discrete strategy for the original system through an inverse transform. All results on intermediate system are then translated into the original system. The numerical state reconstruction problem is addressed as an essential application of the main results. Furthermore, several numerical simulations are used to validate the effectiveness of the numerical approximating algorithms.     

Domain imbedding methods for the Stokes equations

Summary We study direct and iterative domain imbedding methods for the Stokes equations on certain non-rectangular domains in two space dimensions. We analyze a continuous analog of numerical domain imbedding for bounded, smooth domains, and give an example of a simple numerical algorithm suggested by the continuous analysis. This algorithms is applicable for simply connected domains which can be covered by rectangular grids, with uniformly spaced grid lines in at least one coordinate direction. We also discuss a related FFT-based fast solver for Stokes problems with physical boundary conditions on rectangles, and present some numerical results.     

Performance and numerical behavior of the second‐order scheme of precise time‐step integration for transient dynamic analysis

Spurious high‐frequency responses resulting from spatial discretization in time‐step algorithms for structural dynamic analysis have long been an issue of concern in the framework of traditional finite difference methods. Such algorithms should be not only numerically dissipative in a controllable manner, but also unconditionally stable so that the time‐step size can be governed solely by the accuracy requirement. In this article, the issue is considered in the framework of the second‐order scheme of the precise integration method (PIM). Taking the Newmark‐β method as a reference, the performance and numerical behavior of the second‐order PIM for elasto‐dynamic impact‐response problems are studied in detail. In this analysis, the differential quadrature method is used for spatial discretization. The effects of spatial discretization, numerical damping, and time step on solution accuracy are explored by analyzing longitudinal vibrations of a shock‐excited rod with rectangular, half‐triangular, and Heaviside step impact. Both the analysis and numerical tests show that under the framework of the PIM, the spatial discretization used here can provide a reasonable number of model types for any given error tolerance. In the analysis of dynamic response, an appropriate spatial discretization scheme for a given structure is usually required in order to obtain an accurate and meaningful numerical solution, especially for describing the fine details of traction responses with sharp changes. Under the framework of the PIM, the numerical damping that is often required in traditional integration schemes is found to be unnecessary, and there is no restriction on the size of time steps, because the PIM can usually produce results with machine‐like precision and is an unconditionally stable explicit method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Semiclassical quantization of Bohr orbits in the helium atom

We use the complex WKB-Maslov method to construct the semiclassical spectral series corresponding to the resonance Bohr orbits in the helium atom. The semiclassical energy levels represented as the Rydberg tetra series correspond to the doubly symmetrically excited states of helium-like atoms. This level series contains the Rydberg triple series reported by Richter and Wintgen in 1991, which corresponds to the Z²⁺e⁻e⁻ configuration of electrons observed by Eichmann and his collaborators in experiments on the laser excitation of the barium atom in 1992. The lower-level extrapolation of the formula obtained for the semiclassical spectrum gives the value of the ground state energy, which differs by 6% from the experimental value obtained by Bergeson and his collaborators in 1998. We also calculate the fine structure of the semiclassical spectrum due to the spin-orbit and spin-spin interactions of electrons. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 261–286, May, 2007.     

The piecewise linear-quadratic model for computational convex analysis

A new computational framework for computer-aided convex analysis is proposed and investigated. Existing computational frameworks are reviewed and their limitations pointed out. The class of piecewise linear-quadratic functions is introduced to improve convergence and stability. A stable convex calculus is achieved using symbolic-numeric algorithms to compute all fundamental transforms of convex analysis. Our main result states the existence of efficient (linear time) algorithms for the class of piecewise linear-quadratic functions. We also recall that such class is closed under convex transforms. We illustrate the results with numerical examples, and validate numerically the resulting computational framework.     

Asymptotic properties of excited states in the Thomas-Fermi limit

Excited states are stationary localized solutions of the Gross-Pitaevskii equation with a harmonic potential and a repulsive nonlinear term that have zeros on a real axis. The existence and the asymptotic properties of excited states are considered in the semi-classical (Thomas-Fermi) limit. Using the method of Lyapunov-Schmidt reductions and the known properties of the ground state in the Thomas-Fermi limit, we show that the excited states can be approximated by a product of dark solitons (localized waves of the defocusing nonlinear Schrödinger equation with nonzero boundary conditions) and the ground state. The dark solitons are centered at the equilibrium points where a balance between the actions of the harmonic potential and the tail-to-tail interaction potential is achieved.     

BCS Model Hamiltonian of the Theory of Superconductivity as a Quadratic Form

Bogolyubov proved that the average energies (per unit volume) of the ground states for the BCS Hamiltonian and the approximating Hamiltonian asymptotically coincide in the thermodynamic limit. In the present paper, we show that this result is also true for all excited states. We also establish that, in the thermodynamic limit, the BCS Hamiltonian and the approximating Hamiltonian asymptotically coincide as quadratic forms.     

Some remarks on the efficient algorithms of numerical statistical simulation

Some numerical algorithms for the simulation of random variables and functions, as well as for parametrical numerical statistical analysis are considered. Specifications and explanations of the algorithms important for practical purposes are given.     

Condition number based complexity estimate for solving polynomial systems

By modifying and combining algorithms in symbolic and numerical computation, we propose a real-root-counting based method for deciding the feasibility of systems of polynomial equations. Along with this method, we also use a modified Newton operator to efficiently approximate the real solutions when the systems are feasible. The complexity of our method can be measured by a number of arithmetic operations which is singly exponential in the number of variables.     

A class of methods based on non-polynomial sextic spline functions for the solution of a special fifth-order boundary-value problems

A family of fourth and second-order accurate numerical schemes is presented for the solution of fifth-order boundary-value problems with two-point-boundary conditions. The non-polynomial sextic spline functions are applied to construct the numerical algorithms. This approach generalizes polynomial spline algorithms, and provides solution at every point of range of integration. Convergence of the methods is discussed through standard convergence analysis. A numerical illustration is given to show the pertinent features of the technique.     

Rotational analysis of the1Π-X1Σ system of AsN

The¹Π-X ¹Σ bands of AsN were excited by means of radio-frequency discharge through nitrogen and helium and traces of arsenic. The 0-0 band (2784.25 Å) and the 0-1 band (2868.74 Å) were photographed in the third order of a 6.6 meter concave grating spectrograph at a dispersion of 0.38 Å/mm. and analysed for their rotational structure. Perturbations observed in the¹Π state were studied in detail.     

线形(n,f,k)系统和系统的可靠度计算新方法

本文应用两端同时进行的有限马尔可夫链嵌入法给出了具有复合失效准则的(n,f,k)系统和系统可靠度计算公式及公式中连接矩阵的结构特点。数值计算结果表明,随着构成系统的单元个数的增加,应用两端同时进行的有限马尔可夫链嵌入法比应用传统的有限马尔可夫链嵌入法计算可靠度能节省大量运算时间,特别是对于具有对称结构的(n,f,k)和系统效果更加明显。     

Numerical solution of the nonstationary Stokes system by methods of adjoint-equation theory and optimal control theory

Methods in optimal control and the adjoint-equation theory are applied to the design of iterative algorithms for the numerical solution of the nonstationary Stokes system perturbed by a skew-symmetric operator. A general scheme is presented for constructing algorithms of this kind as applied to a broad class of problems. The scheme is applied to the nonstationary Stokes equations, and the convergence rate of the corresponding iterative algorithm is examined. Some numerical results are given.     

A numerical evaluation of solvers for the periodic Riccati differential equation

Efficient and accurate structure exploiting numerical methods for solving the periodic Riccati differential equation (PRDE) are addressed. Such methods are essential, for example, to design periodic feedback controllers for periodic control systems. Three recently proposed methods for solving the PRDE are presented and evaluated on challenging periodic linear artificial systems with known solutions and applied to the stabilization of periodic motions of mechanical systems. The first two methods are of the type multiple shooting and rely on computing the stable invariant subspace of an associated Hamiltonian system. The stable subspace is determined using either algorithms for computing an ordered periodic real Schur form of a cyclic matrix sequence, or a recently proposed method which implicitly constructs a stable deflating subspace from an associated lifted pencil. The third method reformulates the PRDE as a convex optimization problem where the stabilizing solution is approximated by its truncated Fourier series. As known, this reformulation leads to a semidefinite programming problem with linear matrix inequality constraints admitting an effective numerical realization. The numerical evaluation of the PRDE methods, with focus on the number of states (n) and the length of the period (T) of the periodic systems considered, includes both quantitative and qualitative results.     

The λ-point in helium-4 and nonholonomic clusters

The λ-point for heat capacity in liquid helium is studied in greater detail than in the previous papers of the author. We use the concept of the corresponding Van der Waals states and the new “parastatistical distribution.”     

Structure and properties of four-kink multisolitons of the sine-Gordon equation

The dynamics of nonlinear waves of the sine-Gordon equation with a spatially modulated periodic potential are studied using analytical and numerical methods. The structure and properties of four-kink multisolitons excited on two identical attracting impurities are determined. For small-amplitude oscillations, an analytical spectrum of the oscillations is obtained, which is in qualitatively agreement with the numerical results.     

CVaR鲁棒均值-CVaR投资组合模型与求解

传统的均值-风险(包括方差、VaR、CVaR等)组合选择模型在计算最优投资组合时,常假定均值是已知的常值,但在实际资产配置中,收益的均值估计会有偏差,即存在着估计风险.在利用CVaR测度估计风险的基础上,研究了CVaR鲁棒均值-CVaR投资组合选择模型,给出了另外两种不同的求解方法,即对偶法和光滑优化方法,并探讨了它们的相关性质及特征,数值实验表明在求解大样本或者大规模投资组合选择问题上,对偶法和光滑优化方法在计算上是可行且有效的.