Gradient-based Iteration Research Articles

A gradient-based method for atmospheric tomography

In Adaptive Optics (AO) systems for ground-based telescopes, one aims at mechanically correcting for atmospheric aberrations by means of quickly moving deformable mirrors. In complex AO systems, which are using several light sources and aim at a good reconstruction in a large field of view, the derivation of optimal mirror commands from measured light typically includes the problem of atmospheric tomography. As the computational effort for such a limited-angle tomography problem is strongly increasing for growing telescope sizes, fast algorithms are needed. We present a novel algorithm for atmospheric tomography that takes real-life effects such as tip/tilt indetermination, cone effect and spot elongation into account. Furthermore, we discuss two models for the tip and tilt components of an incoming wavefront and incorporate them into the reconstruction. We find a fast step size choice for our Gradient-based iteration and compare it with different existing step size choices. Numerical results are demonstrated for two different AO systems on a 42 m telescope, using the European Southern Observatory's end-to-end simulation tool, OCTOPUS.

GRADIENT BASED ITERATIVE ALGORITHM TO SOLVE GENERAL COUPLED DISCRETE-TIME PERIODIC MATRIX EQUATIONS OVER GENERALIZED REFLEXIVE MATRICES

The discrete-time periodic matrix equations are encountered in periodic state feedback problems and model reduction of periodic descriptor systems. The aim of this paper is to compute the generalized reflexive solutions of the general coupled discrete-time periodic matrix equations. We introduce a gradient-based iterative (GI) algorithm for finding the generalized reflexive solutions of the general coupled discretetime periodic matrix equations. It is shown that the introduced GI algorithm always converges to the generalized reflexive solutions for any initial generalized reflexive matrices. Finally, two numerical examples are investigated to confirm the efficiency of GI algorithm.

Minimum-energy bandwidth management for QoS live migration of virtual machines

Live virtual machine (VM) migration aims at enabling the dynamic balanced use of the networking/computing physical resources of virtualized datacenters, so to lead to reduced energy consumption. However, the bandwidth consumption and latency of current state-of-the-art live VM migration techniques still reduce the experienced benefits to much less than their potential. Motivated by this consideration, in this paper, we analytically characterize, prototype in software and test through field trials the optimal bandwidth manager for intra-datacenter live migration of VMs. The goal is the minimization of the migration-induced communication energy under service level agreement (SLA)-induced hard constraints on the total migration time, downtime, slowdown of the migrating applications and overall available bandwidth. For this purpose, after recognizing that the resulting (nonconvex) optimization problem is an instance of Geometric Programming, we solve it by resorting to suitably developed adaptive version of the so-called primal–dual gradient-based iterations and, then, we analytically characterize its feasibility conditions. Hence, we prototype the resulting bandwidth manager atop an intra-datacenter wired test-bed, and, then, test and compare its energy performance through extensive field trials. The carried out field trials point out that: (i) the energy savings attained by the proposed bandwidth manager over the state-of-the-art ones currently utilized by Xen, KVM and VMware hypervisors are over 40% and approach 66% under strict QoS constraints; (ii) the proposed bandwidth manager is capable to quickly adapt to the abrupt changes possibly experienced by the dirty rates of the running applications and/or the round trip times of the utilized (possibly, congested) TCP/IP connections; and, (iii) its actual implementation may be carried out in a distributed and scalable way, and it consumes less than 1.5% of the CPU computing power per migrated VM.

The scaling conjugate gradient iterative method for two types of linear matrix equations

In this paper, we propose a new iteration method which is based on the conjugate gradient method for solving the linear matrix equations of the form AiXBi=Fi,(i=1,2,…,N) and the generalized Sylvester matrix equation AXB+CXD=E. This method is compared with some existing methods in detail, such as gradient based iterative (GI) method and least squares iterative (LSI) method (Ding et al. 2010) proposed by Ding et al., also cyclic and simultaneous methods (Tang et al. 2014) given by Tang et al. Some numerical experiments demonstrate that the introduced iterative method is more efficient than the four existing methods.

Fast and accurate solution of inverse problem in optical scatterometry using heuristic search and robust correction

Library search is one of the most commonly used methods in optical scatterometry, which consists of the beforehand construction of a signature library and the grid search. The efficiency of existing search algorithms such as k-dimensional tree method and locality-sensitive hashing heavily depends on the size of the signature library and usually is inversely proportional to the library scale. Additionally, since the two-norm based objective function is quite sensitive to the outliers, the abnormally distributed measurement errors will bias the solution of the traditional chi-square or maximum likelihood function. In the present paper, the authors propose a heuristic search algorithm and a robust correction method to realize the fast library search and to achieve the more accurate results, respectively. Instead of searching in the signature library, the authors perform the search procedure in an extra constructed Jacobian library using the principle of gradient-based iteration algorithms, by which the fast search speed can be achieved for an arbitrary scale library. After the search, a robust correction procedure is performed on the basis of the searched optimal parameter set to obtain the more accurate results. Simulations and experiments performed on an etched silicon grating have demonstrated the feasibility of the proposed heuristic search algorithm and robust correction method.

WITHDRAWN: Rank deficiency gradient-based iterations for generalized coupled Sylvester matrix equations

WITHDRAWN: Rank deficiency gradient-based iterations for generalized coupled Sylvester matrix equations

Gradient‐based iterative algorithm for a class of the coupled matrix equations related to control systems

By constructing an objective function and using the gradient search, a gradient-based iteration is established for solving the coupled matrix equations A i XB i = F i , i = 1, 2, …, p. The authors prove that the gradient solution is convergent for any initial values. By analysing the spectral radius of the iterative matrix, the authors obtain an optimal convergence factor. An example is provided to illustrate the effectiveness of the proposed algorithm and to testify the conclusions established in this study.

A Human-Computer Cooperation Fuzzy c-Means Clustering with Interval-Valued Weights

In this paper, a fuzzy c-means clustering algorithm based on interval-valued weights is proposed for improving clustering performance. In the proposed algorithm, the interval-valued weights are first constructed by synergy of the ReliefF algorithm and the analytic hierarchy process AHP method, and then they are transformed into a constraint condition associating with each weight variable in the weighted clustering objective function. In the sequence, the weighted clustering objective function is solved by combining the Lagrange multiplier method with the gradient-based iteration computation. In the whole process of algorithm iteration, a compulsion strategy with human-computer cooperation is adopted to ensure each weight variable satisfies interval constraint itself. Three well-known data set are used to perform profound experiments. Experimental results clearly show that the proposed algorithm has better clustering performance than other the weighted fuzzy c-means clustering algorithm.

Several gradient-based iterative estimation algorithms for a class of nonlinear systems using the filtering technique

This paper considers iterative identification problems for a class of nonlinear systems with colored noises, which can be described by a linear-in-parameters output error autoregressive model. A gradient-based iterative (GI) algorithm, a filtered GI algorithm, and a filtered three-stage GI algorithm are developed using the decomposition technique and filtering technique, and their computational efficiencies are analyzed and compared. The simulation results indicate that the proposed algorithms can estimate effectively the parameters of nonlinear systems.

The reflexive and Hermitian reflexive solutions of the generalized Sylvester-conjugate matrix equation

The main purpose of this correspondence is to establish two gradient based iterative (GI) methods extending the Jacobi and Gauss-Seidel iterations for solving the generalized Sylvester-conjugate matrix equation \begin{equation*} A_1XB_1+A_2\overline{X}B_2+C_1YD_1+C_2\overline{Y}D_2=E, \end{equation*} over reflexive and Hermitian reflexive matrices. It is shown that the iterative methods, respectively, converge to the reflexive and Hermitian reflexive solutions for any initial reflexive and Hermitian reflexive matrices. We report numerical tests to show the effectiveness of the proposed approaches.

A gradient based iterative algorithm for solving structural dynamics model updating problems

The procedure of updating an existing but inaccurate model is an essential step toward establishing an effective model. Updating damping and stiffness matrices simultaneously with measured modal data can be mathematically formulated as following two problems. Problem 1: Let Ma∈SRn×n be the analytical mass matrix, and Λ=diag{λ1,…,λp}∈Cp×p, X=[x1,…,xp]∈Cn×p be the measured eigenvalue and eigenvector matrices, where rank(X)=p, p<n and both Λ and X are closed under complex conjugation in the sense that \(\lambda_{2j} = \bar{\lambda}_{2j-1} \in\nobreak{\mathbf{C}} \), \(x_{2j} = \bar{x}_{2j-1} \in{\mathbf{C}}^{n} \) for j=1,…,l, and λk∈R, xk∈Rn for k=2l+1,…,p. Find real-valued symmetric matrices D and K such that MaXΛ2+DXΛ+KX=0. Problem 2: Let Da,Ka∈SRn×n be the analytical damping and stiffness matrices. Find \((\hat{D}, \hat{K}) \in\mathbf{S}_{\mathbf{E}}\) such that \(\| \hat{D}-D_{a} \|^{2}+\| \hat{K}-K_{a} \|^{2}= \min_{(D,K) \in \mathbf{S}_{\mathbf{E}}}(\| D-D_{a} \|^{2} +\|K-K_{a} \|^{2})\), where SE is the solution set of Problem 1 and ∥⋅∥ is the Frobenius norm. In this paper, a gradient based iterative (GI) algorithm is constructed to solve Problems 1 and 2. A sufficient condition for the convergence of the iterative method is derived and the range of the convergence factor is given to guarantee that the iterative solutions consistently converge to the unique minimum Frobenius norm symmetric solution of Problem 2 when a suitable initial symmetric matrix pair is chosen. The algorithm proposed requires less storage capacity than the existing numerical ones and is numerically reliable as only matrix manipulation is required. Two numerical examples show that the introduced iterative algorithm is quite efficient.

Power Control With Imperfect Exchanges and Applications to Spectrum Sharing

In various applications, the effect of errors in gradient-based iterations is of particular importance when seeking saddle points of the Lagrangian function associated with constrained convex optimization problems. Of particular interest here are problems arising in power control applications, where network utility is maximized subject to minimum signal-to-interference-plus-noise ratio (SINR) constraints, maximum interference constraints, maximum received power constraints, or simultaneous minimum and maximum SINR constraints. Especially when the gradient iterations are executed in a disributed fashion, imperfect exchanges among the link nodes may result in erroneous gradient vectors. In order to assess and cope with such errors, two running averages (ergodic sequences) are formed from the iterates generated by the perturbed saddle point method, each with complementary strengths. Under the assumptions of problem convexity and error boundedness, bounds on the constraint violation and the suboptimality per iteration index are derived. The two types of running averages are tested on a spectrum sharing problem with minimum and maximum SINR constraints, as well as maximum interference constraints.

A Numerical Algorithm for Finding Solution of Cross-Coupled Algebraic Riccati Equations

In this letter, a computational approach for solving cross-coupled algebraic Riccati equations (CAREs) is investigated. The main purpose of this letter is to propose a new algorithm that combines Newton's method with a gradient-based iterative (GI) algorithm for solving CAREs. In particular, it is noteworthy that both a quadratic convergence under an appropriate initial condition and reduction in dimensions for matrix computation are both achieved. A numerical example is provided to demonstrate the efficiency of this proposed algorithm.

Numerical computation for H∞ output feedback control for strongly coupled large-scale systems

In this paper, H ∞ output feedback control for strongly coupled large-scale systems are discussed. When the positive coupling parameter ε which connect the other subsystems is large, a successive algorithm for solving the algebraic Riccati equations (ARE) is developed for the first time. Since the proposed algorithm is derived using Newton’s method, it is noteworthy that the quadratic convergence and uniqueness of the obtained solution are both guaranteed for strongly coupled parameter ε. Moreover, in order to reduce the computation in the resulting Newton’s method, the gradient-based iterative (GI) algorithm is combined. As a result, it is shown that the reduced-order computation is attained. Finally, in order to demonstrate the efficiency of the proposed algorithms, computational examples are provided.