ON SEMILOCAL CONVERGENCE OF INEXACT NEWTON METHODS

Inexact Newton methods are constructed by combining Newton＇s method with another iterative method that is used to solve the Newton equations inexactly. In this paper, we establish two semilocal convergence theorems for the inexact Newton methods. When these two theorems are specified to Newton＇s method, we obtain a different Newton-Kantorovich theorem about Newton＇s method. When the iterative method for solving the Newton equations is specified to be the splitting method, we get two estimates about the iteration steps for the special inexact Newton methods.

Inexact Newton method via Lanczos decomposed technique for solving box-constrained nonlinear systems

This paper proposes an inexact Newton method via the Lanczos decomposed technique for solving the box-constrained nonlinear systems. An iterative direction is obtained by solving an affine scaling quadratic model with the Lanczos decomposed technique. By using the interior backtracking line search technique, an acceptable trial step length is found along this direction. The global convergence and the fast local convergence rate of the proposed algorithm are established under some reasonable conditions. Furthermore, the results of the numerical experiments show the effectiveness of the pro- posed algorithm.

GLOBALLY CONVERGENT INEXACT GENERALIZED NEWTON METHODS WITH DECREASING NORM OF THE GRADIENT

Presents information on a study which proposed a type of globally convergent inexact generalized Newton methods to solve unconstrained optimization problems. Theorems on inexact generalized Newton algorithm with decreasing gradient norms; Discussion on the assumption given; Applications of algorithms and numerical tests.

SOLVERS FOR SYSTEMS OF LARGE SPARSE LINEAR AND NONLINEAR EQUATIONS BASED ON MULTI-GPUS

Numerical treatment of engineering application problems often eventually results in a solution of systems of linear or nonlinear equations.The solution process using digital computational devices usually takes tremendous time due to the extremely large size encountered in most real-world engineering applications.So,practical solvers for systems of linear and nonlinear equations based on multi graphic process units（GPUs）are proposed in order to accelerate the solving process.In the linear and nonlinear solvers,the preconditioned bi-conjugate gradient stable（PBi-CGstab）method and the Inexact Newton method are used to achieve the fast and stable convergence behavior.Multi-GPUs are utilized to obtain more data storage that large size problems need.

Improved hybrid iterative optimization method for seismic full waveform inversion

In full waveform inversion （FWI）, Hessian information of the misfit function is of vital importance for accelerating the convergence of the inversion; however, it usually is not feasible to directly calculate the Hessian matrix and its inverse. Although the limited memory Broyden-Fletcher-Goldfarb-Shanno （L-BFGS） or Hessian-free inexact Newton （HFN） methods are able to use approximate Hessian information, the information they collect is limited. The two methods can be interlaced because they are able to provide Hessian information for each other; however, the performance of the hybrid iterative method is dependent on the effective switch between the two methods. We have designed a new scheme to realize the dynamic switch between the two methods based on the decrease ratio （DR） of the misfit function （objective function）, and we propose a modified hybrid iterative optimization method. In the new scheme, we compare the DR of the two methods for a given computational cost, and choose the method with a faster DR. Using these steps, the modified method always implements the most efficient method. The results of Marmousi and overthrust model testings indicate that the convergence with our modified method is significantly faster than that in the L-BFGS method with no loss of inversion quality. Moreover, our modified outperforms the enriched method by a little speedup of the convergence. It also exhibits better efficiency than the HFN method.

A Filter Line Search Algorithm Based on an Inexact Newton Method for Nonconvex Equality Constrained Optimization

We propose an inexact Newton method with a filter line search algorithm for nonconvex equality constrained optimization. Inexact Newton＇s methods are needed for large-scale applications which the iteration matrix cannot be explicitly formed or factored. We incorporate inexact Newton strategies in filter line search, yielding algorithm that can ensure global convergence. An analysis of the global behavior of the algorithm and numerical results on a collection of test problems are presented.

Fully-Coupled Multi-Physical Simulation with Physics-Based Nonlinearity-Elimination Preconditioned Inexact Newton Method for Enhanced Oil Recovery

In this paper,we introduce a physics-based nonlinear preconditioned Inexact Newton Method(INB)for the multiphysical simulation of fractured reservoirs.Instead of solving the partial differential equations(PDE)exactly,Inexact Newton method finds a direction for the iteration and solves the equations inexactly with fewer iterations.However,when the equations are not smooth enough,especially when lo-cal discontinuities exits,and when proper preconditioning operations are not adopted,the Inexact Newton method may be slow or even stagnant.As pointed out by Keyes et al.[1],multi-physical numerical simulation faces several challenges,one of which is the local-scale nonlinearity and discontinuity.In this work,we have proposed and studied a nonlinear preconditioner to improve the performance of Inexact Newton Method.The nonlinear preconditioner is essentially a physics-based strategy to adaptively identify and eliminate the highly nonlinear zones.The proposed algorithm has been implemented into our fully coupled,fully implicit THM reservoir simulator(Wang et al.[2,3])to study the effects of cold water injection on fractured petroleum reservoirs.The results of this work show that after the implementation of this nonlinear preconditioner,the iterative solver has become significantly more robust and efficient.

Numerical Study of Switching Behavior in Finite Media Subject to 3D Ferroelectric-Paraelectric Interactions and Inspection of Calibration Effects

We study numerically the switching behavior aspects and calibration effects relative to finite media embedding fully a three-dimensional ferroelectric layer in a paraelectric environment.Our approach makes use of the Ginzburg-Landau formalism in combination with the electrostatics equations.The associated discrete nonlinear system,which arises from finite element approximations,is solved by an inexact Newton method.The resulting numerical experiments highlight the effects of a balance between the physical and geometrical parameters.In particular,the same state switchings can be retrieved from different ferroelectric layer sizes by acting upon the physical characteristic of the paraelectric environment.Ferroelectric platelet samples are in parallelepipedic and cylindrical configurations involved in these experiments.