Conjugate gradient methods

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Comput. Appl. Math.  vol.27 número1

Comput. Appl. Math. vol.27 número1

Motivated by [19], we can apply the trust region technique to the conjugate gradient method and propose a new algorithm of nonlinear conjugate gradient methods. This new algorithm has both global convergence and good numerical performance in practical computation. Theoretical analysis and numerical results show that the proposed algorithm is promising and can solve some ill-conditioned minimization problems.

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Comput. Appl. Math.  vol.28 número1

Comput. Appl. Math. vol.28 número1

searches. Moreover, we proved that the proposed HS methods converge glob- ally for strongly convex functions. Two modified schemes are introduced and proved to be globally convergent for general nonconvex functions. These results are also extended to some other conjugate gradient methods. Some results of the paper extend some work of the references [22, 23, 24]. The performance pro- files showed that the proposed methods are also efficient for problems from the CUTE library.

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Comput. Appl. Math.  vol.31 número1

Comput. Appl. Math. vol.31 número1

conjugate gradient methods in practical computation and studied extensively. Now let us simply introduce some results on the PRP method. The global con- vergence of the PRP method with exact line search has been proved by Polak and Ribière [3] under strong convexity assumption of f . However, Powell con- structed an example which showed that the PRP method with exact line searches can cycle infinitely without approaching a solution point [8]. Powell also gave a suggestion to restrict β k = max{β k PRP , 0} to ensure the convergence. Based

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Assessment Of Some Acceleration Schemes In The Solution Of Systems Of Linear Equations.

Assessment Of Some Acceleration Schemes In The Solution Of Systems Of Linear Equations.

Abstract: In this paper, assessment of acceleration schemes in the solution of systems of linear equations has been studied. The iterative methods: Jacobi, Gauss-Seidel and SOR methods were incorporated into the acceleration scheme (Chebyshev extrapolation, Residual smoothing, Accelerated gradient and Richardson Extrapolation) to speed up their convergence. The Conjugate gradient methods of GMRES, BICGSTAB and QMR were also assessed. The research focused on Banded systems, Tridiagonal systems and Dense Symmetric positive definite systems of linear equations for numerical experiments. The experiments were based on the following performance criteria: convergence, number of iterations, speed of convergence and relative residual of each method. Matlab version 7.0.1 was used for the computation of the resulting algorithms. Assessment of the numerical results showed that the accelerated schemes improved the performance of Jacobi, Gauss-Seidel and SOR methods. The Chebyshev and Richardson acceleration methods converged faster than the conjugate gradient methods of GMRES, MINRES, QMR and BICGSTAB in general.
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A STUDY ON CERTAIN NEW NUMERICAL ALGORITHMS FOR MINIMIZATION OF NON LINEAR FUNCTIONS

A STUDY ON CERTAIN NEW NUMERICAL ALGORITHMS FOR MINIMIZATION OF NON LINEAR FUNCTIONS

To solve unconstrained nonlinear minimization problems arising in the diversified field of engineering and technology, we have several methods to get solutions. For instance, multi-step nonlinear conjugate gradient methods [8], a scaled nonlinear conjugate gradient algorithm[2], a method called, ABS-MPVT algorithm [10] are used for solving unconstrained optimization problems. Newton’s method [11] is used for various classes of optimization problems, such as unconstrained minimization problems, equality constrained minimization problems. Chun[6] and Basto[3] have proposed and studied several methods for nonlinear equations with higher order convergence by using the decomposition technique of Adomian[1,13]. Vinay Kanwar et al. [18] introduced new algorithm called, external touch technique for solving the nonlinear equations. Further, they did the comparative study of the new algorithms and Newton’s algorithm. Jishe Feng [9] introduced two step iterative method for solving nonlinear equation with the comparative study of the new iterative method with Newton’s Algorithm, Abbasbandy method and Basto method. Changbum Chun [5] introduced sequence of iterative techniques improving Newton’s Method by the Decomposition Method for solving nonlinear equations. Recently, Rostam K.Saeed et. al. [12] presented a family of new iterative methods for solving nonlinear equations based on Newton’s method. Numerical examples are discussed to illustrate the efficiency of the methods and Behzad Ghanbari [4] introduced new general fourth–order family of methods for finding simple roots of nonlinear equations which is free from second derivative. J.F.Traub [15] introduced several iterative techniques for the solution of equations. C.Chun and Y. Ham[7] proposed some sixth order variants of Ostrowski root finding methods.
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Two New PRP Conjugate Gradient Algorithms for Minimization Optimization Models.

Two New PRP Conjugate Gradient Algorithms for Minimization Optimization Models.

Two new PRP conjugate Algorithms are proposed in this paper based on two modified PRP conjugate gradient methods: the first algorithm is proposed for solving unconstrained opti- mization problems, and the second algorithm is proposed for solving nonlinear equations. The first method contains two aspects of information: function value and gradient value. The two methods both possess some good properties, as follows: 1)β k  0 2) the search direction has the trust region property without the use of any line search method 3) the search direction has sufficient descent property without the use of any line search method. Under some suitable conditions, we establish the global convergence of the two algorithms. We conduct numerical experiments to evaluate our algorithms. The numerical results indi- cate that the first algorithm is effective and competitive for solving unconstrained optimiza- tion problems and that the second algorithm is effective for solving large-scale nonlinear equations.
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Predicting for Sustainable Insurance with Adaptive Gradient Methods

Predicting for Sustainable Insurance with Adaptive Gradient Methods

The main focus of the research is to improve the speed and accuracy of convergence in network training. Convergence of adaptive methods is compared with simple gradient descend and second order methods. It is concluded that adaptive gradient based method are slightly better than simple gradient but still there performance is not comparable to second order techniques. Adaptive methods took lesser training time than simple gradient but they were not able to converge even in 1000 epochs toward error gradient of the order of 10 -4 . While on the other hand, second order techniques were able to reach an accuracy level of 10 -4 and 10 -5 and prove far better in terms of training time and accuracy. Simple gradient (GD) with constant learning rate has shown the poor convergence, conjugate and scaled conjugate gradient methods (CGM, SCGM) show the fastest convergence and adaptive methods (GDA, GDM) fall in between, in terms of convergence towards the minimum error gradient.
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Bridging Small Molecules to Modified Bacterial Microparticles Using a Disulphide Linkage: MIS416 as a Cargo Delivery System.

Bridging Small Molecules to Modified Bacterial Microparticles Using a Disulphide Linkage: MIS416 as a Cargo Delivery System.

The incorporation of a disulfide linkage between MIS416 and the peptide cargo was used to investigate whether rapid release of the attached peptide would occur in the cytoplasm of DCs [4]. To demonstrate that the disulfide bond was potentially cleavable we monitored release of fluorescently labelled peptide from the MIS416 conjugates upon cleavage of the disulfide bond in reducing environments in vitro. The two MIS416 conjugates were prepared as in Figure A in S1 File; one that included a disulfide bond between the MIS416 microparti- cle and attached fluorescent peptide (conjugate A, see Fig 1), and a conjugate without the disulfide bond (conjugate B, Fig 1). Cleavage of the disulfide bond was carried out by incu- bating the conjugates for 30 min in 50 mM tris(2-carboxyethyl)phosphine (TCEP), pH 7 at room temperature (RT) (Fig 3A). The results showed that TCEP treatment of conjugate A (containing a disulfide bond in the linker) was associated with release of the fluorescent pep- tide into solution, as compared to little or no release from conjugate B, or when the conju- gates were treated with phosphate buffered saline (PBS). In addition, treatment of the conjugates with glutathione as a reducing agent for 30, 60 or 120 minutes, to simulate the reducing environment in the cytoplasm of cells, gave an almost identical outcome as treat- ment with TCEP (Fig 3B and 3C)
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A simple ESR identification methodology for Electrolytic Capacitors conditon monitoring

A simple ESR identification methodology for Electrolytic Capacitors conditon monitoring

When experimental data is used for ESR estimation in view of electrolytic capacitor conditioning monitoring as describe in the introduction section, there are two aspects to be considered. First, the convergence time of the algorithms becomes much longer and, second, the estimation using the RLS algorithm with a forgetting factor less than 1 is too noisy, even with a high value of the forgetting factor as shown in figure 5. The data was acquired with an additional resistance (ESR) of 50 mΩ. Due to this, and since the forgetting factor approach with λ=1 corresponds to Kalman filter algorithm as shown previously, only the estimations obtained with gradient and Kalman filter algorithms are presented with experimental data. Hence, in figure 6, the estimation of the ESR with an additional resistance of 50 mΩ, using the Kalman filter algorithm, is shown for different starting instants of computation. Both algorithms, KF and UG have similar behaviour. However, even though the UG is simpler it requires an initial gain higher to have similar convergence to the KF.
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Enhanced membrane pore formation through high-affinity targeted antimicrobial peptides.

Enhanced membrane pore formation through high-affinity targeted antimicrobial peptides.

In order to test if vancomycin was indeed a good targeting module for the peptide, the leakage of carboxyfluorescein was measured from LUVs with a lipid composition that represented the outer leaflet of the eukaryotic cell (60% DOPC:SM:DOPE (46:42:12), 40% cholesterol), and compared this to similar vesicles containing 0.1% lipid II. This lipid composition would give a clear indication of the targeting ability of the vancomycin portion since the cationic peptide would not be strongly attracted to the zwitterionic head groups (phosphatidylcholine, and sphingomy- elin). In this system, nisin caused leakage in vesicles that contained lipid II at nanomolar concentrations, whereas vesicles that did not contain lipid II required much higher concentrations (Figure 2). We found that the vancomycin-magainin construct 1 caused leakage with a similar potency to nisin, i.e. at low nanomolar concentrations. Similarly, the shorter conjugate 2 caused lipid II dependent leakage, although only a maximum of 20% leakage could be achieved. Control experiments with unfunctionalized magainin peptides tested on vesicles with or without lipid II, or the targeted peptides and nisin tested on vesicles that did not contain lipid II, all caused carboxyfluorescein leakage at the expected micromolar concentrations. Previously reported data showed that magainin 2 does not depend on lipid II content [19]. Our results
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Rev. Esc. Minas  vol.67 número1

Rev. Esc. Minas vol.67 número1

The study of gravity-free surface lows presents dificulties, such as the nonlinearity of the dynamic boundary condition in the free surface, and also the fact that the location of this surface is not known a priori. Traditionally, this phenomenon has been investigated by physical models, but the progress of computer science and numeric methods has allowed more and more the successful use of mathematical models to simulate this type of low. This work presents a boundary element method (BEM) numerical simulation of spillway lows with discontinuous linear elements. The solution procedure involves an iterative process in the determination of the free surface. The Newton-Raphson method is adopted together with the use of pseudo- nodes on the free surface and an empiric step factor (or damping factor) which controls the stability and the rate of convergence. An example of WES standard spillway shape is presented. The obtained results are compared with experimental data and they check the eficiency and good precision of the adopted method.
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Optimization of viral resuspension methods for carbon-rich soils along a permafrost thaw gradient

Optimization of viral resuspension methods for carbon-rich soils along a permafrost thaw gradient

While no significant differences across the methods were detected, we note that there are many variations of each method reported in the literature, which were not tested here. Variants of the sonication method include storing the samples on ice during sonication (Duhamel & Jacquet, 2006; Danovaro & Middelboe, 2010) and varying the sonication times (Danovaro & Middelboe, 2010). Sonication has been the most widely used method across sediment and soil environments and has performed better when combined with a well-paired resuspension buffer such as the KC buffer examined in Experiment 1A (Williamson, Wommack & Radosevich, 2003; Duhamel & Jacquet, 2006; Danovaro & Middelboe, 2010). Bead-beating methods have also been used frequently (Ashelford, Day & Fry, 2003; Williamson et al., 2013), but there is no evidence for its increased efficacy over other methods. However, not all types and sizes of beads have been tested. In our case, vortexing was performed using the same equipment and method as bead-beating, but without beads. Vortexers come with many different speed settings and attachments, and each could potentially influence viral yields. Finally, while vortexing is thought to reduce (by 20%) tail breakage as compared to sonication methods (Williamson, Helton & Wommack, 2012), we did not investigate this further since our goals were largely to obtain VLPs for metagenomic sequencing.
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Numerical Mathematics Livros e Materiais Online  Evandro Marquesone

Numerical Mathematics Livros e Materiais Online Evandro Marquesone

In Figure 4.14 we report the convergence history of several non precondi- tioned iterative methods in the case n = 5 corresponding to a matrix size of 19 × 19. The plots show the Euclidean norms of the residual normalized to the initial residual. The dashed curve refers to the Gauss-Seidel method, the dash-dotted line refers to the gradient method, while the solid and circled lines refer respectively to the conjugate gradient (CG) and SOR method (with an optimal value of the relaxation parameter ω ≃ 1.76 computed according to (4.19) since Y is block tridiagonal symmetric positive defi- nite). The SOR method converges in 109 iterations, while the CG method converges in 10 iterations.
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Real-time 3D rendering of water using CUDA

Real-time 3D rendering of water using CUDA

Looking at Figs. 5.6 and 5.7, the reader may observe several details. The visual quality of both CPU and GPU-based Gauss-Seidel and Jacobi solvers is very similar. The Conjugate Gradient method, in both versions, gives slightly better water shaped effects than in the other two methods. However, the GPU-based version of Conjugate Gradient is only used on the diffusion of velocities and densities. Possibly due to floating point errors, the GPU- version does not give the correct results in the projection step. Therefore, instead of using the Conjugate Gradient, we use the Gauss-Seidel or Jacobi solvers in the projection step. The visual aspect of both the GPU and CPU-based versions differ mostly from the usage of different grid sizes. The rendering technique used for grid sizes smaller then 128x64x64 does not produce an uniform volume; consequently, we cannot distinguish the individual elements (i.e. we see the dots individually and not a continuous water volume). For a grid size of 128x64x64, the rendering technique resulted in a uniform water volume, where the individual water elements could not be distinguished from each other. However, none of the versions allowed such grid size because all water elements, and not only the water surface elements, were rendered. The reader will notice also that the textures of the GPU- based version look better. This results from the fact that the texture dimensions used are equivalent to the biggest grid dimension allowed. Therefore, the CPU-based version uses 32 × 32 textures, while the GPU-based version uses 64 × 64 textures. Another
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Progress towards meningitis prevention in the conjugate vaccines era

Progress towards meningitis prevention in the conjugate vaccines era

The success achieved by Hib immunization has stimulated the development of a pneumococcal conjugate vaccine. In February 2000 the Food and Drug Administration approved the use of a seven- valent conjugate vaccine for American children under 2 years old. Among the 90 S. pneumoniae serotypes identified so far [40], the formulation of this vaccine contains only 7 serotypes – 4, 6B, 9V, 14, 18C, 19F and 23F, but these are responsible for 70% to 90% of the invasive disease cases in children in the USA, Canada, Africa and Europe and are 70% of the pneumococci that cause acute otitis media in USA, Canada and Europe [41,42]. Clinical trials carried out with the 7-valent vaccine in USA have revealed high levels of protection against invasive pneumococcal disease, and another review found a 50% to 60% protection range against serotype specific pneumococcal otitis in Finland [43,44]. In Brazil, the heptavalent vaccine was licensed in February 2001 [45], but it has still not been added to the PNI (national program of immunizations) schedule. The 9-valent conjugate vaccine contains the 7-valent serotypes, as well as serotypes 1 and 5, and it has been evaluated by large-scale field trials in South Africa and Gambia
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Rev. bras. epidemiol.  vol.18 número4

Rev. bras. epidemiol. vol.18 número4

ABSTRACT: Objectives: Vaccines represent an important advancement for improving the general health of a population. The efective recording of vaccine data is a factor for the deinition of its supply chain. This study investigated vaccine data recording relatively to data collected from vaccination rooms and data obtained from a government-developed Internet platform. Methods: The monthly recorded total number of diphtheria and tetanus toxoids and pertussis vaccine (alone or in combination with the Haemophilus inluenzae type b conjugate vaccine) doses administered in a medium-sized city of the Southeast region of Brazil was collected for the period January/2006 through December/2010 from two sources: City level (directly from vaccination rooms, the study “gold standard”), and Federal level (from an Internet platform developed by the country government). Data from these sources were compared using descriptive statistics and the Percentage error. Results: The data values made available by the Internet platform difered from those obtained from the vaccination rooms, with a Percentage error relatively to the actual values in the range [-0.48; 0.39]. Concordant values were observed only in one among the sixty analyzed months (1.66%). Conclusions: A frequent and large diference between the number of diphtheria and tetanus toxoids and pertussis vaccine doses administered in the two levels was detected.
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Analysis of Genetic Diversity of Trypanosoma cruzi: an Application of Riboprinting and Gradient Gel Electrophoresis Methods

Analysis of Genetic Diversity of Trypanosoma cruzi: an Application of Riboprinting and Gradient Gel Electrophoresis Methods

Typically, a double stranded DNA fragment has a characteristic melting profile upon the disasso- ciation of this homoduplex into its respective single stranded chains (Cotton 1997). This melting pro- file is reversible, giving rise to a sigmoid transi- tion curve (Fig. 2A). Double stranded DNA mo- bility is greater than that of single stranded or par- tially melted DNA. This profile is also sequence dependent, such that differing DNA sequences have different melting profiles despite being iden- tical in length (Fig. 2B). Gradient gels can utilize either a chemical (DGGE) or temperature (TGGE) method. Gels can also be perpendicular (the gradi- ent is at right angles to the direction of electro- phoresis; Fig. 2A, B) or parallel (the gradient is in the same direction of electrophoresis; Fig. 2C). The precise gradient is usually determined empirically but computer programs, such as MELT 87 (Lerman & Silverstein 1987), can predict the behaviour of a specified sequence and an optimal gradient (Cot- ton 1997). Gradient gel electrophoresis can detect single point mutations between test and reference DNA. It is also a good method to separate com- plex mixtures of DNA that would otherwise co- migrate during conventional electrophoresis. As such, gradient gel methods are ideally suited to study the variation in gene families such as the ri- bosomal RNA complex (Schlötterer 1995).
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Gradient evolution for potential vorticity flows

Gradient evolution for potential vorticity flows

right is nonlinear, in general, since the PV is an active scalar, and thus, S would depend on λ in some nontrivial fashion. It is clear that none of the Statements 1–6 are automatically ap- plicable for this diffusive case; when λ is zero at a point on a trajectory, there are not any nice implications. A fluid parcel placed at a critical point would distance itself from it with time t, such that |λ(t )| approximately goes as Dt for small times. Thus, the magnitude of (horizontal) eddy diffusivity may be roughly approximated by simply releasing a float at a (centre of an) eddy, and observing how its PV gradient devi- ates from zero, with time. The slope of this graph near time zero would estimate the eddy diffusivity. The usage of this idea for the particular example of a (diffusive) Rossby wave is presented in Sect. 6.
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J. Braz. Chem. Soc.  vol.27 número1

J. Braz. Chem. Soc. vol.27 número1

spectroscopy (gCOSY), J-resolved, gradient-selected heteronuclear multiple quantum coherence (gHMQC), gradient-selected heteronuclear multiple bond coherence (gHMBC) and nuclear Overhauser effect spectroscopy (NOESY) experiments. All hydrogen coupling constants were measured, clarifying all the hydrogen signals multiplicities. Computational methods were also used to simulate the 1 H and 13 C chemical shifts and showed good agreement with the trans configuration

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Opinion formation models on a gradient.

Opinion formation models on a gradient.

Statistical physicists have become interested in models of collective social behavior such as opinion formation, where individuals change their inherently preferred opinion if their friends disagree. Real preferences often depend on regional cultural differences, which we model here as a spatial gradient g in the initial opinion. The gradient does not only add reality to the model. It can also reveal that opinion clusters in two dimensions are typically in the standard (i.e., independent) percolation universality class, thus settling a recent controversy about a non- consensus model. However, using analytical and numerical tools, we also present a model where the width of the transition between opinions scales !g { 1=4
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