The Application of Toolkit of Optimization in Designs of Accelerator Magnets

The traditional design of accelerator magnet usually involves many time consuming iterations of the manual analysis process. A software platform to do these iterations automatically is proposed in this paper. In this platform, we use DAKOTA (a open source software developed by Sandia National Laboratories) as the optimizing routine, which provides a variety of optimization methods and algorithms, and OPERA (software from Vector Fields) is selected as the electromagnetic simulating routine. In this paper, two examples of designs of accelerator magnets are used to illustrate how an optimization algorithm is chosen and the platform works.

基于广义Lippmann-Schwinger方程的地震波场模拟算法与应用

In the last several decades, due to the fast development of computer, numerical simulation has been an indispensable tool in scientific research. Numerical simulation methods which based on partial difference operators such as Finite Difference Method (FDM) and Finite Element Method (FEM) have been widely used. However, in the realm of seismology and seismic prospecting, one usually meets with geological models which have piece-wise heterogeneous structures as well as volume heterogeneities between layers, the continuity of displacement and stress across the irregular layers and seismic wave scattering induced by the perturbation of the volume usually bring in error when using conventional methods based on difference operators. The method discussed in this paper is based on elastic theory and integral theory. Seismic wave equation in the frequency domain is transformed into a generalized Lippmann-Schwinger equation, in which the seismic wavefield contributed by the background is expressed by the boundary integral equation and the scattering by the volume heterogeneities is considered. Boundary element-volume integral method based on this equation has advantages of Boundary Element Method (BEM), such as reducing one dimension of the model, explicit use the displacement and stress continuity across irregular interfaces, high precision, satisfying the boundary at infinite, etc. Also, this method could accurately simulate the seismic scattering by the volume heterogeneities. In this paper, the concrete Lippmann-Schwinger equation is specifically given according to the real geological models. Also, the complete coefficients of the non-smooth point for the integral equation are introduced. Because Boundary Element-Volume integral equation method uses fundamental solutions which are singular when the source point and the field are very close，both in the two dimensional and the three dimensional case, the treatment of the singular kernel affects the precision of this method. The method based on integral transform and integration by parts could treat the points on the boundary and inside the domain. It could transform the singular integral into an analytical one both in two dimensional and in three dimensional cases and thus it could eliminate the singularity. In order to analyze the elastic seismic wave scattering due to regional irregular topographies, the analytical solution for problems of this type is discussed and the analytical solution of P waves by multiple canyons is given. For the boundary reflection, the method used here is infinite boundary element absorbing boundary developed by a pervious researcher. The comparison between the analytical solutions and concrete numerical examples validate the efficiency of this method. We thoroughly discussed the sampling frequency in elastic wave simulation and find that, for a general case, three elements per wavelength is sufficient, however, when the problem is too complex, more elements per wavelength are necessary. Also, the seismic response in the frequency domain of the canyons with different types of random heterogeneities is illustrated. We analyzed the model of the random media, the horizontal and vertical correlation length, the standard deviation, and the dimensionless frequency how to affect the seismic wave amplification on the ground, and thus provide a basis for the choice of the parameter of random media during numerical simulation.

地震波阻抗反演的迭代正则化算法

Post-stack seismic impedance inversion is the key technology of reservoir prediction and identification. Geophysicists have done a lot of research for the problem, but the developed methods still cannot satisfy practical requirements completely. The results of different inversion methods are different and the results of one method used by different people are different too. The reasons are due to the quality of seismic data, inaccurate wavelet extraction, errors between normal incidence assumption and real situation, and so on. In addition, there are two main influence factors: one is the band-limited property of seismic data; the other is the ill-posed property of impedance inversion. Thus far, the most effective way to solve the band-limited problem is the constrained inversion. And the most effective way to solve ill-posed problems is the regularization method assisted with proper optimization techniques. This thesis systematically introduces the iterative regularization methods and numerical optimization methods for impedance inversion. A regularized restarted conjugate gradient method for solving ill-posed problems in impedance inversion is proposed. Theoretic simulations are made and field data applications are performed. It reveals that the proposed algorithm possesses the superiority to conventional conjugate gradient method. Finally, non-smooth optimization is proposed as the further research direction in seismic impedance inversion according to practical situation.

Green Theorems and Qualitative Properties of the Optical Flow

How can one compute qualitative properties of the optical flow, such as expansion or rotation, in a way which is robust and invariant to the position of the focus of expansion or the center of rotation? We suggest a particularly simple algorithm, well-suited to VLSI implementations, that exploits well-known relations between the integral and differential properties of vector fields and their linear behaviour near singularities.

Large-Amplitude nonlinear stability results for atmospheric circulations

Douglas, Robert; Cullen, M.J.P., (2002) 'Large-Amplitude nonlinear stability results for atmospheric circulations', The Quarterly Journal of the Royal Meteorological Society 129 pp.1969-1988 RAE2008

Finite difference time domain modeling of phase grating diffusion

In this paper, a method for modeling diffusion caused by non-smooth boundary surfaces in simulations of room acoustics using finite difference time domain (FDTD) technique is investigated. The proposed approach adopts the well-known theory of phase grating diffusers to efficiently model sound scattering from rough surfaces. The variation of diffuser well-depths is attained by nesting allpass filters within the reflection filters from which the digital impedance filters used in the boundary implementation are obtained. The presented technique is appropriate for modeling diffusion at high frequencies caused by small surface roughness and generally diffusers that have narrow wells and infinitely thin separators. The diffusion coefficient was measured with numerical experiments for a range of fractional Brownian diffusers.

Transferring entanglement to the steady state of flying qubits

The transfer of entanglement from optical fields to qubits provides a viable approach to entangling remote qubits in a quantum network. In cavity quantum electrodynamics, the scheme relies on the interaction between a photonic resource and two stationary intracavity atomic qubits. However, it might be hard in practice to trap two atoms simultaneously and synchronize their coupling to the cavities. To address this point, we propose and study entanglement transfer from cavities driven by an entangled external field to controlled flying qubits. We consider two exemplary non-Gaussian driving fields: NOON and entangled coherent states. We show that in the limit of long coherence time of the cavity fields, when the dynamics is approximately unitary, entanglement is transferred from the driving field to two atomic qubits that cross the cavities. On the other hand, a dissipation-dominated dynamics leads to very weakly quantum-correlated atomic systems, as witnessed by vanishing quantum discord.

Numerical Simulation of Tanpura String Vibrations

A numerical model of a tanpura string is presented, based on a recently developed, stability-preserving way of incorporating the non-smooth forces involved in the impactive distributed contact between the string and the bridge. By defining and modelling the string-bridge contact over the full length of the bridge, the simulated vibrations can be monitored through the force signals at both the bridge and the nut. As such it offers a reference model for both measurements and sound synthesis. Simulations starting from different types of initial conditions demonstrate that the model reproduces the main characteristic feature of the tanpura, namely the sustained appearance of a precursor in the force waveforms, carrying a band of overtones which decrease in frequency as the string vibrations decay. Results obtained with the numerical model are used to examine, through comparison, the effect of the bridge and of the thread on the vibrations.

A self-learning teaching-learning based optimization for dynamic economic/environmental dispatch considering multiple plug-in electric vehicle loads

Economic and environmental load dispatch aims to determine the amount of electricity generated from power plants to meet load demand while minimizing fossil fuel costs and air pollution emissions subject to operational and licensing requirements. These two scheduling problems are commonly formulated with non-smooth cost functions respectively considering various effects and constraints, such as the valve point effect, power balance and ramp rate limits. The expected increase in plug-in electric vehicles is likely to see a significant impact on the power system due to high charging power consumption and significant uncertainty in charging times. In this paper, multiple electric vehicle charging profiles are comparatively integrated into a 24-hour load demand in an economic and environment dispatch model. Self-learning teaching-learning based optimization (TLBO) is employed to solve the non-convex non-linear dispatch problems. Numerical results on well-known benchmark functions, as well as test systems with different scales of generation units show the significance of the new scheduling method.

Besov spaces and the Laplacian on fractal H-sets

Nesta tese são estudados espaços de Besov de suavidade generalizada em espaços euclidianos, numa classe de fractais designados conjuntos-h e em estruturas abstractas designadas por espaços-h. Foram obtidas caracterizações e propriedades para estes espaços de funções. Em particular, no caso de espaços de Besov em espaços euclidianos, foram obtidas caracterizações por diferenças e por decomposições em átomos não suaves, foi provada uma propriedade de homogeneidade e foram estudados multiplicadores pontuais. Para espaços de Besov em conjuntos-h foi obtida uma caracterização por decomposições em átomos não suaves e foi construído um operador extensão. Com o recurso a cartas, os resultados obtidos para estes espaços de funções em fractais foram aplicados para definir e trabalhar com espaços de Besov de suavidade generalizada em estruturas abstractas. Nesta tese foi também estudado o laplaciano fractal, considerado a actuar em espaços de Besov de suavidade generalizada em domínios que contêm um conjunto-h fractal. Foram obtidos resultados no contexto de teoria espectral para este operador e foi estudado, à custa deste operador, um problema de Dirichlet fractal no contexto de conjuntos-h.

Regularization and Bang-bang conjugate times in optimal control

Consideramos o problema de controlo óptimo de tempo mínimo para sistemas de controlo mono-entrada e controlo afim num espaço de dimensão finita com condições inicial e final fixas, onde o controlo escalar toma valores num intervalo fechado. Quando aplicamos o método de tiro a este problema, vários obstáculos podem surgir uma vez que a função de tiro não é diferenciável quando o controlo é bang-bang. No caso bang-bang os tempos conjugados são teoricamente bem definidos para este tipo de sistemas de controlo, contudo os algoritmos computacionais directos disponíveis são de difícil aplicação. Por outro lado, no caso suave o conceito teórico e prático de tempos conjugados é bem conhecido, e ferramentas computacionais eficazes estão disponíveis. Propomos um procedimento de regularização para o qual as soluções do problema de tempo mínimo correspondente dependem de um parâmetro real positivo suficientemente pequeno e são definidas por funções suaves em relação à variável tempo, facilitando a aplicação do método de tiro simples. Provamos, sob hipóteses convenientes, a convergência forte das soluções do problema regularizado para a solução do problema inicial, quando o parâmetro real tende para zero. A determinação de tempos conjugados das trajectórias localmente óptimas do problema regularizado enquadra-se na teoria suave conhecida. Provamos, sob hipóteses adequadas, a convergência do primeiro tempo conjugado do problema regularizado para o primeiro tempo conjugado do problema inicial bang-bang, quando o parâmetro real tende para zero. Consequentemente, obtemos um algoritmo eficiente para a computação de tempos conjugados no caso bang-bang.

A Java expression evaluator for nonlinear programming

Finding the optimal value for a problem is usual in many areas of knowledge where in many cases it is needed to solve Nonlinear Optimization Problems. For some of those problems it is not possible to determine the expression for its objective function and/or its constraints, they are the result of experimental procedures, might be non-smooth, among other reasons. To solve such problems it was implemented an API contained methods to solve both constrained and unconstrained problems. This API was developed to be used either locally on the computer where the application is being executed or remotely on a server. To obtain the maximum flexibility both from the programmers’ and users’ points of view, problems can be defined as a Java class (because this API was developed in Java) or as a simple text input that is sent to the API. For this last one to be possible it was also implemented on the API an expression evaluator. One of the drawbacks of this expression evaluator is that it is slower than the Java native code. In this paper it is presented a solution that combines both options: the problem can be expressed at run-time as a string of chars that are converted to Java code, compiled and loaded dynamically. To wide the target audience of the API, this new expression evaluator is also compatible with the AMPL format.

Direct-search penalty/barrier methods

In Nonlinear Optimization Penalty and Barrier Methods are normally used to solve Constrained Problems. There are several Penalty/Barrier Methods and they are used in several areas from Engineering to Economy, through Biology, Chemistry, Physics among others. In these areas it often appears Optimization Problems in which the involved functions (objective and constraints) are non-smooth and/or their derivatives are not know. In this work some Penalty/Barrier functions are tested and compared, using in the internal process, Derivative-free, namely Direct Search, methods. This work is a part of a bigger project involving the development of an Application Programming Interface, that implements several Optimization Methods, to be used in applications that need to solve constrained and/or unconstrained Nonlinear Optimization Problems. Besides the use of it in applied mathematics research it is also to be used in engineering software packages.

Calculation of fractional derivatives of noisy data with genetic algorithms

This paper addresses the calculation of derivatives of fractional order for non-smooth data. The noise is avoided by adopting an optimization formulation using genetic algorithms (GA). Given the flexibility of the evolutionary schemes, a hierarchical GA composed by a series of two GAs, each one with a distinct fitness function, is established.

Adaptive Penalty and Barrier function based on Fuzzy Logic

Optimization methods have been used in many areas of knowledge, such as Engineering, Statistics, Chemistry, among others, to solve optimization problems. In many cases it is not possible to use derivative methods, due to the characteristics of the problem to be solved and/or its constraints, for example if the involved functions are non-smooth and/or their derivatives are not know. To solve this type of problems a Java based API has been implemented, which includes only derivative-free optimization methods, and that can be used to solve both constrained and unconstrained problems. For solving constrained problems, the classic Penalty and Barrier functions were included in the API. In this paper a new approach to Penalty and Barrier functions, based on Fuzzy Logic, is proposed. Two penalty functions, that impose a progressive penalization to solutions that violate the constraints, are discussed. The implemented functions impose a low penalization when the violation of the constraints is low and a heavy penalty when the violation is high. Numerical results, obtained using twenty-eight test problems, comparing the proposed Fuzzy Logic based functions to six of the classic Penalty and Barrier functions are presented. Considering the achieved results, it can be concluded that the proposed penalty functions besides being very robust also have a very good performance.