Using Lie group integrators to solve two and higher dimensional variational problems with symmetry

Improved sequential optimization method (SOM) and dimension reduction optimization method (DROM) are presented for high dimensional optimization design problems of electromagnetic devices. Improved SOM can simultaneously optimize the statistical approximate models and optimization algorithms more efficiently compared with SOM. Using DROM, a high dimensional problem can be converted into a low dimensional problem with expert experience or some design of experiment techniques. Then two TEAM benchmark problems (Problem 22 and Problem 25) are investigated to illustrate the efficiency of the proposed methods. From the experimental results, we can see that the presented methods can obviously reduce the computational cost of finite element analysis, while the optimal results also satisfy design specification.

A velocity-based butterfly optimization algorithm for high-dimensional optimization and feature selection

Memetic algorithms are effective algorithms to obtain reliable and accurate solutions for complex continuous optimization problems. Nowadays, high dimensional optimization problems are an interesting field of research. The high dimensionality introduces new problems for the optimization process, requiring more scalable algorithms that, at the same time, could explore better the higher domain space around each solution. In this work, we proposed a memetic algorithm, MA-SW-Chains, for large scale global optimization. This algorithm assigns to each individual a local search intensity that depends on its features, by chaining different local search applications. MA-SW-Chains is an adaptation to large scale optimization of a previous algorithm, MA-CMA-Chains, to improve its performance on high-dimensional problems. Finally, we present the results obtained by our proposal using the benchmark problems defined in the Special Session of Large Scale Global Optimization on the IEEE Congress on Evolutionary Computation in 2010.

Particle Swarm Optimization (PSO) is an evolution-nary computation technique. Separate adjustment to inertia weight and learning factors in PSO undermines the integrity and intelligent characteristic in the evolutionary process of particle swarm to some extent, thus it is not suitable for solving most complicated optimization problems. On the basis of previous researches, the aim of this study was to improve the computational efficiency of PSO and avoid premature convergence for multimodal, higher dimensional complicated optimization problems by considering the mutual influences of inertia weight and learning factors on the updates of particle's veloci-ties. A great number of experiment data provided evidence that the nonlinear adjustment to inertia weight, cognitive learning factor and social learning factor within a certain interval is a good selection. Moreover, the simulation results on four typical functions show that the improved strategy of PSO proposed in the paper is available for solving multimodal and higher dimensional complicated optimization problems, and can accelerate convergence speed, improve optimization quality effectively in comparison to the algorithms of standard PSO and existing relevant improved PSO.

The calculus of variations is concerned with finding extremal paths of functionals in analogy with the way that classical calculus seeks to find critical points of functions. Variational calculus plays a central role in classical mechanics, connecting the “Principle of Least Action” and Lagrange’s equations of motion (also called the Euler– Lagrange equations). In that setting, generalized coordinates are introduced to describe the geometric configuration of a mechanical system. In this chapter, classical variational calculus is reviewed and extended to describe systems on Lie groups. Of course, the introduction of coordinates such as Euler angles to describe the orientation of a rigid body can be used to formulate classical variational problems at the expense of introducing singularities. However, it is possible to formulate variational problems on Lie groups without coordinates. This results in the Euler’Poincaré equations.

Complexity Theory for Lie-Group Solvers

High dimensional global optimization problems arise frequently in process systems engineering. This is a result of the complex mechanistic relationships that describe process systems, and/or their large-scale nature. High dimensional optimization problems can often be more easily solved by instead solving a sequence of reduced-dimension subproblems. Surrogate models can allow the formulation of reduced-dimension subproblems by approximating the key features of the original model. Surrogate-based optimization (SBO) is to use surrogate modeling to solve a sequence of approximate reduced-dimension subproblems, in order to converge to a high quality solution to the original high dimensional problem. Here we review the key characteristics of SBO frameworks and their application to process systems optimization problems.

Efficient operation of cloud computing platforms depends on the optimised virtual resources and faster fault diagnosis system of the virtual machines. This paper proposes an algorithm by introducing a virtual machines based on elastic reservation mechanism, which can improve the availability of cloud resources through the demand analysis taking the help of support vector machines which has advantages resolving nonlinear and high dimensional classification problems. Secondly it adopts the anomaly detection algorithm based on support vector machines for failure analysis. In addition, the dimensionality problem can be sorted out by means of principal component analysis (PCA) algorithm and a kernel function used for distance measurement. It establishes the topological structure for the image set of feature space with Delaunay triangulation and analyses the relationship between kernel parameter and regulator, in order to build an effective model.

Crow search algorithm (CSA) is a recently proposed metaheuristic optimizer inspired by the intelligent behaviour of crows with attributes like simplicity and ease of implementation. CSA is claimed superior and more effective in optimizing a variety of constrained engineering design problems in comparison to other state-of-art algorithms. In the present work, CSA is applied to high dimensional optimization problems and it is found that CSA suffers from premature convergence which leads to lower precision and less accuracy in optimization or sometimes failure. Therefore an improvement in CSA (ICSA) is suggested to solve high-dimensional global optimization problems efficiently. The balance between exploitation and exploration capabilities of CSA is improved by introducing experience factor, adaptive adjustment operator and Lévy flight distribution in position updating mechanism of crows. Lévy flight distribution promotes continuous exploration of search space and prevents premature convergence by escaping from local optimum at any stage. The performance of ICSA is validated on high-dimensional nonlinear scalable benchmark test functions. The proposed improvement in CSA makes it highly competitive and less sensitive to function dimensions. ICSA is also found superior to other well established optimizers.

Condition (P.-S.) may seem rather restrictive. Actually, as Hildebrandt [4; p. 324] records, for quite a while many mathematicians felt convinced that inspite of its success in dealing with one-dimensional variational problems like geodesics (see Birkhoff’s Theorem I.4.4, for example, or Palais’ [3] work on closed geodesics), the Palais-Smale condition could never play a role in the solution of “interesting” variational problems in higher dimensions.

We consider Sobolev problems with nonlocal boundary conditions associated with an action of a compact Lie group. We find a natural conditions of ellipticity of such problems, obtain the corresponding finiteness theorem, and give the index formula.

: A major issue in modeling and computation is how to handle high dimensional problems. We can divide these high dimensional problems into two classes: Moderately high dimensional problems or very high dimensional problems. In the former class, we have problems such as the Boltzmann equation, whose dimensionality is high but they are still amendable to grid-based methods. In the latter class we have problems such as exploration of the configuration space of a large molecule. These problems often involve hundreds of thousands of dimensions, and methods based on fixed grids are far from being adequate. We have explored various ways of handling these problems using the sparse representation or the adaptive sampling.

Introduction. 1. Hilbert's fifth problem and proper actions of Lie groups S. Illman. 2. Equivariant algebraic vector bundles over representations - a survey M. Masuda. 3. G-manifolds and G-vector bundles in algebraic, semialgebraic, and definable categories T. Kawakami. 4. Geometry of Finite Topological and equivariant finite topological spaces S. Kono, F. Ushitaki. 5. On the theory of homotopy representations. A survey I. Nagasaki. 6. Manifolds as fixed point sets of smooth compact Lie group actions K. Pawalowski. 7. Surgery and homotopy theory in study of the transformation groups Y. Kitada. 8. Kervaire's obstructions of free actions of finite cyclic groups on homotopy spheres Y. Kitada. 9. The Burnside ring revisited M. Morimoto. 10. Multiplicative stabilization and transformation groups S. Kwasik, R. Schultz. 11. Symmetries on manifolds, deformations and rational homotopy: a survey M. Raussen. 12. Rigidity of codimension one locally free actions of solvable Lie groups N. Tsuchiya, A. Yamakawa. 13. Smooth actions of non-compact semi-simple Lie groups F. Uchida, K. Mukoyama. 14. Hamiltonian group actions and equivariant indices T. Takakura. 15. Controlled methods in equivariant topology, a survey E. Pedersen. Index.

Estimation is the computational task of recovering a hidden parameter $x$ associated with a distribution $D_x$, given a measurement $y$ sampled from the distribution. High dimensional estimation problems arise naturally in statistics, machine learning, and complexity theory. Many high dimensional estimation problems can be formulated as systems of polynomial equations and inequalities, and thus give rise to natural probability distributions over polynomial systems. Sum-of-squares proofs provide a powerful framework to reason about polynomial systems, and further there exist efficient algorithms to search for low-degree sum-of-squares proofs. Understanding and characterizing the power of sum-of-squares proofs for estimation problems has been a subject of intense study in recent years. On one hand, there is a growing body of work utilizing sum-of-squares proofs for recovering solutions to polynomial systems when the system is feasible. On the other hand, a general technique referred to as pseudocalibration has been developed towards showing lower bounds on the degree of sum-of-squares proofs. Finally, the existence of sum-of-squares refutations of a polynomial system has been shown to be intimately connected to the existence of spectral algorithms. In this article we survey these developments.

In this paper, Statistical Neural Networks have been proven to be an effective classifier method for large sample and high dimensional letter recognition problem. For this purpose, Probabilistic Neural Network (PNN) and General Regression Neural Networks (GRNN) have been applied to classify the 26 capital letters in the English alphabet. Principal Component Analysis (PCA) has been established as a feature extraction and a data compression method to achieve less computational complexity. The low computational complexity obtained by PCA provides a solution for high dimensional letter recognition problem for online operations. Simulation results illustrate that GRNN and PNN are suitable and effective methods for solving classification problems with higher classification accuracy and better generalization performances than their counterparts.