Conjugate gradient methods are very important ones for solving nonlinear optimization problems,especially for large scale problems. However, unlike quasi-Newton methods, conjugate gradient methods wereusually analyzed individually. In this paper, we propose a class of conjugate gradient methods, which can beregarded as some kind of convex combination of the Fletcher-Reeves method and the method proposed byDai et al. To analyze this class of methods, we introduce some unified tools that concern a general methodwith the scalarβk having the form of φk/φk-1. Consequently, the class of conjugate gradient methods canuniformly be analyzed.
Deconvolution problem is a main topic in signal processing. Many practical applications are re-quired to solve deconvolution problems. An important example is image reconstruction. Usually, researcherslike to use regularization method to deal with this problem. But the cost of computation is high due to thefact that direct methods are used. This paper develops a trust region-cg method, a kind of iterative methodsto solve this kind of problem. The regularity of the method is proved. Based on the special structure of thediscrete matrix, FFT can be used for calculation. Hence combining trust region-cg method with FFT is suitablefor solving large scale problems in signal processing.
Differential-algebraic equations (DAE’s) arise naturally in many applied fields, but numerical and analytical difficulties that have not appeared in ordinary differential equations (ODE’s) occur in DAE’s because it includes algebraic constrained equations. Some efficient numerical methods for ODE’s can not work well for DAE’s. So many eminent numerical analysis scholars are interested in this field recently. But few numerical methods are able to solve all DAE’s because of its essential difficulties. This paper discusses the simulation algorithm character of DAE’s. And we construct an efficient constrained-algebraic algorithm based on the Runge-Kutta methods of order two for the semi-explicit differential-algebraic equations with index two and give the computational experiment results for specific examples. The experiment results indicate that the constrained-algebraic algorithm is high efficient for semi-explicit differential-algebraic equations with index two.
In this paper, we provide a counter example for a successful method, i.e. IMP-BOT method [6], based on ODE for unconstrained optimization. And we obtainthat methods based on BDF and the general trapezoidal metod for unconstrainedoptimization is bad efficient because these methods even if have A stability, not Lstability.
In this paper we solve large scale ill-posed problems, particularly the image restoration problem in atmospheric imaging sciences, by a trust region-CG algorithm. Image restoration involves the removal or minimization of degradation (blur, clutter, noise, etc.) in an image using a priori knowledge about the degradation phenomena. Our basic technique is the so-called trust region method, while the subproblem is solved by the truncated conjugate gradient method, which has been well developed for well-posed problems.The trust region method, due to its robustness in global convergence, seems to be a promising way to deal with ill-posed problems.
