In Proposition 4.1 a characterization is given of Hessian Rieman-nian structures in terms of a natural connection in the general linear group GL(n; R) + , which is viewed as a principal SO(n)-bundle over the space of positive deenite symmetric n n-matrices. For n = 2, Proposition 5.3 contains an interpretation of the curvature of a Hessian Riemannian structure at a given point, in terms of an u...

The well-known symmetric rank-one trust-region method—where the Hessian approximation is generated by the symmetric rank-one update—is generalized to the problem of minimizing a real-valued function over a d-dimensional Riemannian manifold. The generalization relies on basic differential-geometric concepts, such as tangent spaces, Riemannian metrics, and the Riemannian gradient, as well as on t...

We propose a new algorithm for minimizing regularized empirical loss: Stochastic Dual Newton Ascent (SDNA). Our method is dual in nature: in each iteration we update a random subset of the dual variables. However, unlike existing methods such as stochastic dual coordinate ascent, SDNA is capable of utilizing all local curvature information contained in the examples, which leads to striking impr...

The score and hessian functions The score vector ∇L(β, x) and hessian matrix ∇ 2 L(β, x) used in the Newton-Raphson

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 ...

We expand and update the software tool SMCSolver, presented at the SMCTools workshop in 2006, for the numerical solution of structured Markov chains encountered in queuing models. In particular the new version of the package implements different transformation techniques and different shift strategies which are combined in order to speed up and optimize the solution of structured Markov chains....

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...