An Abs Algorithm for a Class of Systems of Stochastic Linear Equations

TL;DR

This paper introduces an ABS algorithm tailored for solving systems of stochastic linear equations where the right-hand side is normally distributed, analyzing the probabilistic distribution of stepsizes and solutions.

Contribution

It extends the ABS algorithm framework to stochastic systems, providing a probabilistic analysis of stepsizes and solutions for such models.

Findings

Stepsize $oldsymbol{\alpha_i}$ follows a normal distribution $N(u_i, \sigma_i)$.

Solution approximation $oldsymbol{\xi_i}$ follows a multivariate normal distribution $N_n(U_i, oldsymbol{\Sigma_i})$.

The model characterizes the distributional properties of the algorithm's steps and solutions.

Abstract

This paper is to explore a model of the ABS Algorithms dealing with the solution of a class of systems of linear stochastic equations $A\xi=\eta$ when $\eta$ is a $m$-dimensional normal distribution. It is shown that the stepsize $\alpha_i$ is distributed as $N(u_i,\sigma_i)$ (being $u_i$ the expected value of $\alpha_i$ and $\sigma_i$ its variance) and the approximation to the solutions $\xi_{i}$ is distributed as $N_n(U_i,\Sigma_i)$ (being $U_i$ the expected value of $\xi_i$ and $\Sigma_i$ its variance), for this algorithm model.