On the problem of computational vector gravimetry

It is suggested that the vector problem of terrestrial gravimetry to be solved in the framework of the inertial method be provided with an uncontrollable hardware platform in order that the problem be solved by purely computational means. Estimates of solvability of and accuracy of a solution to the problem made in the course of the numerical experiment are given.     

The Pomraning-Eddington approximation to diffusion of light in turbid materials

The source-free diffusion problem of light in turbid media with generalized boundary conditions is considered. The intensity of light is considered as a sum of collimated and diffused radiance. In this way the problem is transformed to a source problem with a collimated source (problem 1). This problem is solved in terms of the corresponding source-free problem of simple boundary conditions (problem 2). The Pomraning-Eddington method is used to solve problem 2. Two coupled first-order differential equations are obtained involving the energy density and the radiation net flux. Weight functions are introduced in order to force the boundary conditions to be fulfilled. Numerical results are given and compared with previous calculations. The calculations show that the accuracy depends on the choice of the weight function.     

Riemann-Hilbert problems for the Ernst equation and fibre bundles

Riemann-Hilbert techniques are used in the theory of completely integrable differential equations to generate solutions that contain a free function which can be used at least in principle to solve initial or boundary-value problems. The solution of a boundary-value problem is thus reduced to the identification of the jump data of the Riemann-Hilbert problem from the boundary data. But even if this can be achieved, it is very difficult to get explicit solutions since the matrix Riemann-Hilbert problem is equivalent to an integral equation. In the case of the Ernst equation (the stationary axisymmetric Einstein equations in vacuum), it was shown in a previous work that the matrix problem is gauge equivalent to a scalar problem on a Riemann surface. If the jump data of the original problem are rational functions, this surface will be compact which makes it possible to give explicit solutions in terms of hyperelliptic theta functions. In the present work, we discuss Riemann-Hilbert problems on Riemann surfaces in the framework of fibre bundles. This makes it possible to treat the compact and the non-compact case in the same setting and to apply general existence theorems.     

一维声波方程声阻抗反演问题的稳定性分析

讨论了一维声波方程的声阻抗反演问题。对于脉冲激发的情况,得到反问题适定的充分、必要条件,适定条件的形式非常简单,在实际计算中易于判定。对于子波激发的情况,由子波响应反演声阻抗通常是不适定的,不适定程度与子波的奇性有关,子波越光滑,相应的反问题的不适定程度越高。同时给出了定量的说明,为一维声波方程声阻抗反演问题的数值计算提供了理论。     

Phase retrieval

The problem of phase retrieval arises in experimental uses of diffraction to determine intrinsic structure because the modulus of a Fourier transform is all that can usually be measured after diffraction occurs. For finite distributions, the phase retrieval problem can be solved by methods of factorization in suitable rings of polynomials; for continuous distributions with compact support, the methods of complex analysis are needed to solve the phase retrieval problem. These methods are discussed and examples are given for illustration.Partially supported by NSF Grant MCS 8218800     

动态随机最短路径算法研究

静态最短路径问题已经得到很好解决, 然而现实中的网络大多具有动态性和随机性. 网络弧和节点的状态及耗费不仅具有不确定性且相互关联, 弧和节点的耗费都服从一定的概率分布, 因此把最短路径问题看作是一个动态随机优化问题更具有一般性. 文中分析了网络弧和节点的动态随机特性及其相互关系, 定义了动态随机最短路径; 给出了动态随机最短路径优化数学模型, 提出了一种动态随机最短路径遗传算法; 针对网络的拓扑特性设计了高效合理的遗传算子. 实验结果表明, 文中提出的模型和算法能有效地解决动态随机最短路径问题, 可以运用到交通、通信等网络的网络流随机优化问题中.     

Joint power allocation and user assignment for licensed/unlicensed band users in massive MIMO cellular systems

In this paper, we study the joint user assignment and power allocation for the defined utility function (central cell throughput) maximization in massive Multiple Input-Multiple Output (MIMO) cellular system coexistence with Wireless Fidelity (WiFi) network. Firstly, the power allocation of problem is formulated as a convex optimization. Unfortunately, the formulated problem has not a closed-form solution. For solving the mentioned problem, it is converted to three sub-problem based on the number of lemmas that are expressed. Due to two of these problems remain difficult to solve, this two sub-problem are relaxed. The Ellipsoid algorithm is an iterative algorithm that used for solving of the relaxed problems. In the following, joint user assignment and power allocation will be addressed, in which two approaches are proposed for solving. In the first approach, we propose an iterative algorithm that user assignment problem and power allocation problem are solved in each iteration. In the second approach, at first, users are assigned to licensed and unlicensed bands, then for the obtained arrangement, the power allocation problem is solved. The simulation results showed that the proposed algorithms are significantly close to the benchmark methods.     

Hartree-Fock theory with constrained deformation

We study the variational Hartree-Fock problem with constrained deformation. If the deformation is defined by prescribed expectation values of the conventional multipole moments, we show that a necessary condition for the problem to have physically meaningful solutions is that the multipole operators are negligible outside the nucleus, so that they must be state dependent. In this way, a further non-linearity is introduced into the problem. An alternative formulation is to impose a given shape to the nuclear surface by means of two inequality constraints.     

A Compton scattering image reconstruction algorithm based on total variation minimization

Compton scattering imaging is a novel radiation imaging method using scattered photons.Its main characteristics are detectors that do not have to be on the opposite side of the source,so avoiding the rotation process.The reconstruction problem of Compton scattering imaging is the inverse problem to solve electron densities from nonlinear equations,which is ill-posed.This means the solution exhibits instability and sensitivity to noise or erroneous measurements.Using the theory for reconstruction of sparse images,a reconstruction algorithm based on total variation minimization is proposed.The reconstruction problem is described as an optimization problem with nonlinear data-consistency constraint.The simulated results show that the proposed algorithm could reduce reconstruction error and improve image quality,especially when there are not enough measurements.     

Disturbance rejection and H∞ pinning control of linear complex dynamical networks

This paper concerns the disturbance rejection problem of a linear complex dynamical network subject to external disturbances.A dynamical network is said to be robust to disturbance,if the H ∞ norm of its transfer function matrix from the disturbance to the performance variable is satisfactorily small.It is shown that the disturbance rejection problem of a dynamical network can be solved by analysing the H ∞ control problem of a set of independent systems whose dimensions are equal to that of a single node.A counter-intuitive result is that the disturbance rejection level of the whole network with a diffusive coupling will never be better than that of an isolated node.To improve this,local feedback injections are applied to a small fraction of the nodes in the network.Some criteria for possible performance improvement are derived in terms of linear matrix inequalities.It is further demonstrated via a simulation example that one can indeed improve the disturbance rejection level of the network by pinning the nodes with higher degrees than pinning those with lower degrees.     

一类时变时滞混沌系统的参数辨识方法

时变的未知时滞参数普遍存在于混沌系统中,它使得混沌系统同步控制变得非常困难. 针对时滞混沌系统中参数时变且未知的问题, 提出了一种新颖的辨识方法. 该方法首先将未知时变参数用分段常数函数来近似, 把求解非线性函数的问题转化为参数向量选择问题, 其中分段常数函数的高度向量成为待求解参数向量; 然后推导了目标函数对分段常数高度向量的梯度信息, 结合序列二次规划法求解得最优分段函数; 随着分段数的增加, 最优分段函数将逼近原非线性时变函数. 数值实例结果验证了该方法的有效性.     

变系数瞬态热传导问题边界元格式的特征正交分解降阶方法

提出了一种基于边界元法求解变系数瞬态热传导问题的特征正交分解(POD)降阶方法,重组并推导出变系数瞬态热传导问题适合降阶的边界元离散积分方程,建立了变系数瞬态热传导问题边界元格式的POD降阶模型,并用常数边界条件下建立的瞬态热传导问题的POD降阶模态,对光滑时变边界条件瞬态热传导问题进行降阶分析.首先,对一个变系数瞬态热传导问题,建立其边界域积分方程,并将域积分转换成边界积分;其次,离散并重组积分方程,获得可用于降阶分析的矩阵形式的时间微分方程组;最后,用POD模态矩阵对该时间微分方程组进行降阶处理,建立降阶模型并对其求解.数值算例验证了本文方法的正确性和有效性.研究表明:1)常数边界条件下建立的低阶POD模态矩阵,能够用来准确预测复杂光滑时变边界条件下的温度场结果;2)低阶模型的建立,解决了边界元法中采用时间差分推进技术求解大型时间微分方程组时求解速度慢、算法稳定性差的问题.     

On asymptotics of the solution of the moving oscillator problem

Asymptotic behavior of the solution of the moving oscillator problem is examined for large and small values of the spring stiffness for the general case of non-zero beam initial conditions. In the limiting case of infinite spring stiffness, it is shown that the moving oscillator problem for a simply supported beam is not equivalent, in a strict sense, to the moving mass problem. The two problems are shown to be equivalent in terms of the beam displacements but are not equivalent in terms of stresses (the higher order derivatives of the two solutions differ). In the general case, the force acting on the beam from the oscillator is shown to contain a high-frequency component , which does not vanish and can even grow when the spring coefficient tends to infinity. The magnitude of this force and its dependence on the oscillator parameters can be estimated by considering the asymptotics of the solution for the initial stage of the oscillator motion. It is shown that, for the case of a simply supported beam, the magnitude of the high-frequency force depends linearly on the oscillator eigenfrequency and velocity. The deficiency of the moving mass model is principally that it fails to predict stresses in the supporting structure. For small values of the spring stiffness, the moving oscillator problem is shown to be equivalent to the moving force problem. The discussion is amply illustrated by results of numerical experiments.     

Dynamics of topological multipoles: I. High-order nonparaxial singular beams

Approximate boundary conditions for matching the high-order nonparaxial and paraxial singular beams propagating through free space or through a homogeneous isotropic medium are formulated. It is found that the nonparaxial and paraxial beams have almost the same structure both in the vicinity of the focal caustic and away from it. In the intermediate zone, the beam profiles are distorted, and a phase mismatch arises and increases with increasing mode order. The combined nonparaxial singular beams are devised, and their dynamics is studied. It is shown that the problem of matching the paraxial and nonparaxial wave functions may be reformulated as a problem of the phase matching for nonparaxial wave function. Such a problem is similar to the mode-dispersion problem and is unsolvable in terms of a linear problem for the entire optical axis. Thus, it makes sense to discuss either the precise matching of the paraxial and nonparaxial beams within small spatial zones or an approximate matching away from the focal plane but at long distances along the optical axis.     

Nonlinear non-extensive approach for identification of structured information

The problem of separating structured information representing phenomena of differing natures is considered. A structure is assumed to be independent of the others if can be represented in a complementary subspace. When the concomitant subspaces are well separated the problem is readily solvable by a linear technique. Otherwise, the linear approach fails to correctly discriminate the required information. Hence, a non-extensive approach is proposed. The resulting nonlinear technique is shown to be suitable for dealing with cases that cannot be tackled by the linear one.     

Sampling the posterior: An approach to non-Gaussian data assimilation

The viewpoint taken in this paper is that data assimilation is fundamentally a statistical problem and that this problem should be cast in a Bayesian framework. In the absence of model error, the correct solution to the data assimilation problem is to find the posterior distribution implied by this Bayesian setting. Methods for dealing with data assimilation should then be judged by their ability to probe this distribution. In this paper we propose a range of techniques for probing the posterior distribution, based around the Langevin equation; and we compare these new techniques with existing methods.

When the underlying dynamics is deterministic, the posterior distribution is on the space of initial conditions leading to a sampling problem over this space. When the underlying dynamics is stochastic the posterior distribution is on the space of continuous time paths. By writing down a density, and conditioning on observations, it is possible to define a range of Markov Chain Monte Carlo (MCMC) methods which sample from the desired posterior distribution, and thereby solve the data assimilation problem. The basic building-blocks for the MCMC methods that we concentrate on in this paper are Langevin equations which are ergodic and whose invariant measures give the desired distribution; in the case of path space sampling these are stochastic partial differential equations (SPDEs).

Two examples are given to show how data assimilation can be formulated in a Bayesian fashion. The first is weather prediction, and the second is Lagrangian data assimilation for oceanic velocity fields. Furthermore the relationship between the Bayesian approach outlined here and the commonly used Kalman filter based techniques, prevalent in practice, is discussed. Two simple pedagogical examples are studied to illustrate the application of Bayesian sampling to data assimilation concretely. Finally a range of open mathematical and computational issues, arising from the Bayesian approach, are outlined.     

Darboux Transformation and Explicit Solutions for Discretized Modified Korteweg-de Vries Lattice Equation

The modified Korteweg-de Vries （mKdV） typed equations can be used to describe certain nonlinear phenomena in fluids, plasmas, and optics. In this paper, the discretized mKdV lattice equation is investigated. With the aid of symbolic computation, the discrete matrix spectral problem for that system is constructed. Darboux transformation for that system is established based on the resulting spectral problem. Explicit solutions are derived via the Darboux transformation. Structures of those solutions are shown graphically, which might be helpful to understand some physical processes in fluids, plasmas, and optics.     

Analytic Hessian derivation for the quasi-one-dimensional Euler equations

The Hessian for the quasi-one-dimensional Euler equations is derived. A pressure minimization problem and a pressure matching inverse problem are considered. The flow sensitivity, adjoint sensitivity, gradient and Hessian are calculated analytically using a direct approach that is specific to the model problems. For the pressure minimization problem we find that the Hessian exists and it contains elements with significantly larger values around the shock location. For the pressure matching inverse problem we find at least one case for which the gradient as well as the Hessian do not exist. In addition, two formulations for calculating the Hessian are proposed and implemented for the given problems. Both methods can be implemented in industrial applications such as large scale aerodynamic optimization.     

Propagation of electromagnetic dipole waves through dielectric interfaces

The problem of divergent electromagnetic dipole waves propagating through parallel dielectric interfaces is solved. The solution is obtained in an analytic form that can be readily evaluated numerically. The result is obtained as a solution to a boundary-value problem. Applications of the solution are described.