We study adaptive meshes which are quasi-uniform in a metric generated by the Hessian of a P1 finite element function. We consider three most efficient methods for recovering this Hessian, one variational method and two projection methods. We compare these methods for problems with anisotropic singularities to show that all Hessian recovery methods result in acceptable adaptive meshes although ...

Inspired by the recent paper (L. Ying, Journal of Scientific Computing, 84, 1–14 (2020), we explore relationship between mirror descent and variable metric method. When in decent is induced a convex function, whose Hessian close to objective this method enjoys both robustness from superlinear convergence for Newton type methods. applied linearly constrained minimization problem, prove global lo...

A modification of Newton’s method for solving multidimensional problems of Keating’s potential optimization is addressed. The proposed approach is based on two main futures: information about the special structure of Hessian matrix and sparse matrix technology. Developed data structures for store sparse Hessian matrix and modification of the linear conjugate method for the Hessian matrices with...

Recently, a derivative-free optimization algorithm was proposed that utilizes minimum Frobenius norm (MFN) Hessian update for estimating the second derivative information, which in turn is used accelerating search. The formula relies only on computed function values and closed-form expression special case of more general approach first published by Powell. This paper analyzes convergence under ...

The BFGS update and the DFP update perhaps are respectively the most successful Hessian and inverse Hessian approximations for unconstrained minimization problems This paper describes a geometric meaning of these updates understood from the Wedderburn rank one reduction formula