312 research outputs found
Experiments with Conjugate Gradient Algorithms for Homotopy Curve Tracking
There are algorithms for finding zeros or fixed points of nonlinear systems of equations that are globally convergent for almost all starting points, i.e., with probability one. The essence of all such algorithms is the construction of an appropriate homotopy map and then tracking some smooth curve in the zero set of this homotopy map. HOMPACK is a mathematical software package implementing globally convergent homotopy algorithms with three different techniques for tracking a homotopy zero curve, and has separate routines for dense and sparse Jacobian matrices. The HOMPACK algorithms for sparse Jacobian matrices use a preconditioned conjugate gradient algorithm for the computation of the kernal of the homotopy Jacobian matrix, a required linear algebra step for homotopy curve tracking. Here variants of the conjugate gradient algorithm are implemented in the context of homotopy curve tracking and compared with Craig's preconditioned conjugate gradient method used in HOMPACK. The test problems used include actual large scale, sparse structural mechanics problems
Quantum Control Landscapes
Numerous lines of experimental, numerical and analytical evidence indicate
that it is surprisingly easy to locate optimal controls steering quantum
dynamical systems to desired objectives. This has enabled the control of
complex quantum systems despite the expense of solving the Schrodinger equation
in simulations and the complicating effects of environmental decoherence in the
laboratory. Recent work indicates that this simplicity originates in universal
properties of the solution sets to quantum control problems that are
fundamentally different from their classical counterparts. Here, we review
studies that aim to systematically characterize these properties, enabling the
classification of quantum control mechanisms and the design of globally
efficient quantum control algorithms.Comment: 45 pages, 15 figures; International Reviews in Physical Chemistry,
Vol. 26, Iss. 4, pp. 671-735 (2007
Development of homotopy algorithms for fixed-order mixed H2/H(infinity) controller synthesis
A major difficulty associated with H-infinity and mu-synthesis methods is the order of the resulting compensator. Whereas model and/or controller reduction techniques are sometimes applied, performance and robustness properties are not preserved. By directly constraining compensator order during the optimization process, these properties are better preserved, albeit at the expense of computational complexity. This paper presents a novel homotopy algorithm to synthesize fixed-order mixed H2/H-infinity compensators. Numerical results are presented for a four-disk flexible structure to evaluate the efficiency of the algorithm
Taylor-newton homotopy method for computing the depth of flow rate for a channel
Homotopy approximation methods (HAM) can be considered as one of the new methods belong to the general classification of the computational methods which can be used to find the numerical solution of many types of the problems in science and engineering. The general problem relates to the flow and the depth of water in open channels such as rivers and canals is a nonlinear algebraic equation which is known as continuity equation. The solution of this equation is the
depth of the water. This paper represents attempt to solve the equation of depth and flow using Newton homotopy based on Taylor series. Numerical example is given to show the
effectiveness of the purposed method using MATLAB language
Recommended from our members
Parallel software for nonlinear systems of equations. Final report, February 28, 1995--June 30, 1997
The nonlinear systems of equations arising in circuit simulation, structural optimization, closed loop optimal control, chemical engineering of distillation systems, combustion chemistry, CAD/CAM modeling, robotics, computer vision, and orbital mechanics have several properties that make them especially amenable to homotopy methods. Even so, the homotopy zero curves are not trivial to track, and sophisticated curve tracking techniques are sometimes required. The size of typical engineering problems also presents some interesting numerical linear algebra challenges, and the supported work has been geared toward developing parallel sparse matrix techniques specifically tailored to the sparsity structures corresponding to the mentioned problem areas, in the context of homotopy algorithms. There are many different algorithms for tracking the zero curve {gamma}; the previous proposal discussed three such algorithms: ordinary differential equation based, normal flow, and augmented Jacobian matrix. The descriptions of these algorithms are now in the literature for the software package HOMPACK, so will not be repeated here. The development of sparse homotopy algorithms within HOMPACK specifically tailored for various parallel machines (e.g., distributed memory, shared memory, and vector) and problem areas (e.g., circuit simulation, structural optimization, optimal control, and combustion chemistry) was the central theme of this research
Efficient Optimization Algorithms for Nonlinear Data Analysis
Identification of low-dimensional structures and main sources of variation from multivariate data are fundamental tasks in data analysis. Many methods aimed at these tasks involve solution of an optimization problem. Thus, the objective of this thesis is to develop computationally efficient and theoretically justified methods for solving such problems.
Most of the thesis is based on a statistical model, where ridges of the density estimated from the data are considered as relevant features. Finding ridges, that are generalized maxima, necessitates development of advanced optimization methods. An efficient and convergent trust region Newton method for projecting a point onto a ridge of the underlying density is developed for this purpose. The method is utilized in a differential equation-based approach for tracing ridges and computing projection coordinates along them. The density estimation is done nonparametrically by using Gaussian kernels. This allows application of ridge-based methods with only mild assumptions on the underlying structure of the data.
The statistical model and the ridge finding methods are adapted to two different applications. The first one is extraction of curvilinear structures from noisy data mixed with background clutter. The second one is a novel nonlinear generalization of principal component analysis (PCA) and its extension to time series data. The methods have a wide range of potential applications, where most
of the earlier approaches are inadequate. Examples include identification of faults from seismic data and identification of filaments from cosmological data. Applicability of the nonlinear PCA to climate analysis and reconstruction of periodic patterns from noisy time series data are also demonstrated.
Other contributions of the thesis include development of an efficient semidefinite optimization method for embedding graphs into the Euclidean space. The method produces structure-preserving embeddings that maximize interpoint distances. It is primarily developed for dimensionality reduction, but has also potential applications in graph theory and various areas of physics, chemistry
and engineering. Asymptotic behaviour of ridges and maxima of Gaussian kernel densities is also investigated when the kernel bandwidth approaches infinity. The results are applied to the nonlinear PCA and to finding significant maxima of such densities, which is a typical problem in visual object tracking.Siirretty Doriast
Singular Continuation: Generating Piece-wise Linear Approximations to Pareto Sets via Global Analysis
We propose a strategy for approximating Pareto optimal sets based on the
global analysis framework proposed by Smale (Dynamical systems, New York, 1973,
pp. 531-544). The method highlights and exploits the underlying manifold
structure of the Pareto sets, approximating Pareto optima by means of
simplicial complexes. The method distinguishes the hierarchy between singular
set, Pareto critical set and stable Pareto critical set, and can handle the
problem of superposition of local Pareto fronts, occurring in the general
nonconvex case. Furthermore, a quadratic convergence result in a suitable
set-wise sense is proven and tested in a number of numerical examples.Comment: 29 pages, 12 figure
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