The self-scaling quasi-Newton method solves an unconstrained optimization problem by scaling the Hessian approximation matrix before it is updated at each iteration to avoid the possible large eigenvalues in the Hessian approximation matrices of the objective function. It has been proved in the literature that this method has the global and superlinear convergence when the objective function is...

Techniques for obtaining safely positive definite Hessian approximations with selfscaling and modified quasi-Newton updates are combined to obtain ‘better’ curvature approximations in line search methods for unconstrained optimization. It is shown that this class of methods, like the BFGS method has global and superlinear convergence for convex functions. Numerical experiments with this class, ...

A new Newton-Raphson method based preconditioner for Krylov type linear equation solvers for GPGPU is developed, and the performance is investigated. Conventional preconditioners improve the convergence of Krylov type solvers, and perform well on CPUs. However, they do not perform well on GPGPUs, because of the complexity of implementing powerful preconditioners. The developed preconditioner is...

SUMMARY Full-waveform inversion has become an essential technique for mapping geophysical subsurface structures. However, proper uncertainty quantification is often lacking in current applications. In theory, related to the inverse Hessian (or posterior covariance matrix). Even common problems its calculation beyond computational and storage capacities of largest high-performance computing syst...

In this paper, a trust-region algorithm combining with the limited memory BFGS (L-BFGS) update is proposed for solving nonlinear equations, where the super relaxation technique(SRT) is used. We choose the next iteration point by SRT. The global convergence without the nondegeneracy assumption is obtained under suitable conditions. Numerical results show that this method is very effective for la...

Doubly stochastic matrix plays an essential role in several areas such as statistics and machine learning. In this paper we consider the optimal approximation of a square set doubly matrices. A structured BFGS method is proposed to solve dual primal problem. The resulting algorithm builds curvature information into diagonal components true Hessian, so that it takes only additional linear cost o...

We present a line search algorithm for minimizing nonconvex and/or nonsmooth objective functions. The algorithm is a hybrid between a standard Broyden-Fletcher-Goldfarb-Shanno (BFGS) and an adaptive gradient sampling (GS) method. The BFGS strategy is employed as it typically yields fast convergence to the vicinity of a stationary point, and along with the adaptive GS strategy the algorithm ensu...

The purpose of this paper is an optimal control problem of temperature using the Newton based method and the finite element method. This method is based on the first and second order adjoint technique allowing to obtained a better approximation to the Newton line search direction. The formulation is based on the optimal control theory in which the performance function is expressed by the comput...

In this paper, a modified BFGS algorithm is proposed for unconstrained optimization. The proposed algorithm has the following properties: (i) a nonmonotone line search technique is used to obtain the step size [Formula: see text] to improve the effectiveness of the algorithm; (ii) the algorithm possesses not only global convergence but also superlinear convergence for generally convex functions...