In this paper, a detailed description of CG for evaluating eigenvalue problems by minimizing the Rayleigh quotient is presented from both theoretical and computational viewpoints. Three variants of CG together with their asymptotic behaviours and restarted schemes are discussed. In addition, it is shown that with a generally selected preconditioning matrix the actual performance of the PCG sche...

Iteration arithmetic is formally introduced based on iteration multiplication and αaddition which is a special multisplitting. This part focuses on construction of convergent splittings and approximate inverses for Hermitian positive definite matrices by applying stair matrices, their generalizations and iteration arithmetic. Analysis of the splittings and the approximate inverses is also prese...

This paper introduces the theoretical development of a numerical method, named NeAREst, for solving nonnegative linear inverse problems, which arise often from physical or probabilistic models, especially, in image estimation with limited and indirect measurements. The Richardson-Lucy (RL) iteration is omnipresent in conventional methods that are based on probabilistic assumptions, arguments an...

We discuss iterative methods for computing criticality in nuclear reactors. In general this requires the solution of a generalised eigenvalue problem for an unsymmetric integro-differential operator in 6 independent variables, modelling transport, scattering and fission, where the dependent variable is the neutron angular flux. In engineering practice this problem is often solved iteratively, u...

We discuss the close connection between eigenvalue computation and optimization using the Newton method and subspace methods. From the connection we derive a new class of Newton updates. The new update formulation is similar to the well-known Jacobi-Davidson method. This similarity leads to simplified versions of the Jacobi-Davidson method and the inverse iteration generalized Davidson (IIGD) m...

Topology optimization problem, which involves many design variables, is commonly solved by finite element method, a method must recalculate structure-stiffness matrix each time of analysis. OC method is a good way to solve topology optimization problem, nevertheless, it can not solve multiobjective topology optimization problems. This paper introduces an effective solution to Multi-objective to...

Owing to their fast convergence, second-order Newton-type learning methods have recently received attention in the federated (FL) setting. However, current solutions are based on communicating Hessian matrices from devices parameter server, at every iteration, incurring a large number of communication rounds; calling for novel communication-efficient methods. In this article, we propose method ...

We consider inverse iteration-based eigensolvers, which require at each step solving an “inner” linear system. We assume that this linear system is solved by some (preconditioned) Krylov subspace method. In this framework, several approaches are possible, which differ by the linear system to be solved and/or the way the preconditioner is used. This includes methods such as inexact shift-and-inv...

The limited memory BFGS method pioneered by Jorge Nocedal is usually implemented as a line search method where the search direction is computed from a BFGS approximation to the inverse of the Hessian. The advantage of inverse updating is that the search directions are obtained by a matrix{ vector multiplication. Furthermore, experience shows that when the BFGS approximation is appropriately re{...

Several major inverse methods (Stommel-Schott method, Wunsch method, and Bernoulli method) have been successfully developed to quantitatively estimate the geostrophic velocity at the reference level from hydrographic data. No matter the different appearance, they are based on the same dynamical sophistication: geostrophy, hydrostatic, and potential density (ρ) conservation (Davis, 1978). The cu...