Stability Analysis Of Continuous Conjugate Gradient Method

Kaedah Conjugate Gradient adalah sangat berguna untuk: menyelesaikan masalah tiada kekangan paling optimum yang berskala besar. Walaubagaimanapun, carlan garis (line search) dalam Kaedah Conjugate Gradient kadang-kadang sukar didapati dan pengiraannya menggunakan komputer adalah sangat mahal. Berdasarkan penyelidikan oleh Sun dan Zhang [J. Sun and J. Zhang (2001), Global convergence of conjugate gradient methods without line search], menyatakan bahawa Kaedah Conjugate Gradient adalah menumpu secara global (globally convergence) dengan menggunakan langkah (stepsize) ak yang ditetapkan berdasarkan formula 8r/ ft. Darlpada keputusan yang didapati, mereka mencadangkan carlan Ilpkll{4 garis (line search) adalah tidak diperlukan untuk mendapatkan penumpuan secara global (globally convergence) oleh Kaedah Conjugate Gradient. Oleh itu, objektif disertasi ini adalah untuk menentukan julat a dan P di mana julat ini akan memastikan kestabilan Kaedah Conjugate Gradient. In order to solve a large-scale unconstrained optimization, Conjugate Gradient Method has been proven to be successful. However, the line search required in Conjugate Gradient Method is sometimes extremely difficult and computationally expensive. Studies conducted by Sun and Zhang [J. Sun and J. Zhang (2001), Global convergence of conjugate gradient methods without line search], claimed that the Conjugate Gradient Method was globally convergence using "fixed" stepsize at determined using formula at = 8rk T fk . The result suggested that for global Ilpkl~ convergence of Conjugate Gradient Method, line search was not compUlsory. Therefore, tlfts dissertation's objective is to determine the range of a and P where this range will ensure the stability of Conjugate Gradient Method

A novel canonical dual computational approach for prion AGAAAAGA amyloid fibril molecular modeling

Many experimental studies have shown that the prion AGAAAAGA palindrome hydrophobic region (113-120) has amyloid fibril forming properties and plays an important role in prion diseases. However, due to the unstable, noncrystalline and insoluble nature of the amyloid fibril, to date structural information on AGAAAAGA region (113-120) has been very limited. This region falls just within the N-terminal unstructured region PrP (1-123) of prion proteins. Traditional X-ray crystallography and nuclear magnetic resonance (NMR) spectroscopy experimental methods cannot be used to get its structural information. Under this background, this paper introduces a novel approach of the canonical dual theory to address the 3D atomic-resolution structure of prion AGAAAAGA amyloid fibrils. The novel and powerful canonical dual computational approach introduced in this paper is for the molecular modeling of prion AGAAAAGA amyloid fibrils, and that the optimal atomic-resolution structures of prion AGAAAAGA amyloid fibils presented in this paper are useful for the drive to find treatments for prion diseases in the field of medicinal chemistry. Overall, this paper presents an important method and provides useful information for treatments of prion diseases. Overall, this paper could be of interest to the general readership of Theoretical Biology

The recent development of non-monotone trust region methods

Abstract: Trust region methods are a class of numerical methods for optimization. They compute a trial step by solving a trust region sub-problem where a model function is minimized within a trust region. In this paper, we review recent results on non-monotone trust region methods for unconstrained optimization problems. Generally, non-monotone trust region algorithms with non-monotone technique are more effective than the traditional ones, especially when coping with some extreme nonlinear optimization problems. Results on trust region sub-problems and regularization methods are also discussed

A dynamical view of nonlinear conjugate gradient methods with applications to FFT-based computational micromechanics

For fast Fourier transform (FFT)-based computational micromechanics, solvers need to be fast, memory-efficient, and independent of tedious parameter calibration. In this work, we investigate the benefits of nonlinear conjugate gradient (CG) methods in the context of FFT-based computational micromechanics. Traditionally, nonlinear CG methods require dedicated line-search procedures to be efficient, rendering them not competitive in the FFT-based context. We contribute to nonlinear CG methods devoid of line searches by exploiting similarities between nonlinear CG methods and accelerated gradient methods. More precisely, by letting the step-size go to zero, we exhibit the Fletcher–Reeves nonlinear CG as a dynamical system with state-dependent nonlinear damping. We show how to implement nonlinear CG methods for FFT-based computational micromechanics, and demonstrate by numerical experiments that the Fletcher–Reeves nonlinear CG represents a competitive, memory-efficient and parameter-choice free solution method for linear and nonlinear homogenization problems, which, in addition, decreases the residual monotonically

A Modified Conjugacy Condition and Related Nonlinear Conjugate Gradient Method

The conjugate gradient (CG) method has played a special role in solving large-scale nonlinear optimization problems due to the simplicity of their very low memory requirements. In this paper, we propose a new conjugacy condition which is similar to Dai-Liao (2001). Based on this condition, the related nonlinear conjugate gradient method is given. With some mild conditions, the given method is globally convergent under the strong Wolfe-Powell line search for general functions. The numerical experiments show that the proposed method is very robust and efficient

Direction set based Algorithms for adaptive least squares problems improvements and innovations.

The main objective of this research is to provide a mathematically tractable solutions to the adaptive filtering problem by formulating the problem as an adaptive least squares problem. This approach follows the work of Chen (1998) in his study of direction-set based CDS) adaptive filtering algorithm. Through the said formulation, we relate the DS algorithm to a class of projection method. Objektif utama penyelidikan ini ialah untuk menyediakan penyelesaian matematik yang mudah runut kepada masalah penurasan adaptif dengan memfonnulasikan masalah tersebut sebagai masalah kuasa dua terkecil adaptif. Pendekatan ini rnengikut hasil kerja oleh Chen (1998) dalam kajian beliau tentang algoritma penurasan adaptif berasaskan 'direction-set' (DS). Melalui fornulasi tersebut, kami menghubungkaitkan algoritma DS kepada satu kelas kaedah unjuran. Secara khususnya, versi rnudah aigoritma itu, iaitu algoritma 'Euclidean direction search' (EDS) ditunjukkan mempunyai hubungkait dengan satu kelas kaedah berlelaran yang dipanggil kaedah 'relaxation'. Penernuan ini rnembolehkan kami menambahbaik algoritma EDS kepada 'accelerated EDS' eli mana satu parameter pemecutan diperkenalkan untuk rnengoptirnumkan saiz langkah sernasa setiap pencarian garis

The hybrid idea of (energy minimization) optimization methods applied to study prion protein structions focusing on the beta2-alpha2 loop

In molecular mechanics, current generation potential energy functions provide a reasonably good compromise between accuracy and effectiveness. This paper firstly reviewed several most commonly used classical potential energy functions and their optimization methods used for energy minimization. To minimize a potential energy function, about 95% efforts are spent on the Lennard-Jones potential of van der Waals interactions; we also give a detailed review on some effective computational optimization methods in the Cambridge Cluster Database to solve the problem of Lennard- Jones clusters. From the reviews, we found the hybrid idea of optimization methods is effective, necessary and efficient for solving the potential energy minimization problem and the Lennard-Jones clusters problem. An application to prion protein structures is then done by the hybrid idea. We focus on th