Stochastic Reformulations of Linear Systems: Algorithms and Convergence Theory

Stochastic Reformulations of Linear Systems: Algorithms and Convergence Theory

We develop a family of reformulations of an arbitrary consistent linear system into a stochastic problem. The reformulations are governed by two user-defined parameters: a positive definite matrix defining a norm, and an arbitrary discrete or continuous distribution over random matrices. Our reformulation has several equivalent interpretations, allowing for researchers from various communities ...

Convergence Rates and Decoupling in Linear Stochastic Approximation Algorithms

Convergence of Eigenvalues in State-Discretization of Linear Stochastic Systems

The transition operator that describes the time evolution of the state probability distribution for continuousstate linear systems is given by an integral operator. A state-discretization approach is proposed, which consists of a finite rank approximation of this integral operator. As a result of the state-discretization procedure, a Markov chain is obtained, in which case the transition operat...

Convergence Results on Iteration Algorithms to Linear Systems

In order to solve the large scale linear systems, backward and Jacobi iteration algorithms are employed. The convergence is the most important issue. In this paper, a unified backward iterative matrix is proposed. It shows that some well-known iterative algorithms can be deduced with it. The most important result is that the convergence results have been proved. Firstly, the spectral radius of ...

Finite Abstractions of Max-Plus-Linear Systems: Theory and Algorithms

ion function f , i.e. f−1(ŝ) = {s : f (s) = ŝ}. Recall that f−1(ŝ) is a block (or in fact a DBM) and the dynamics in each block are affine. If there exists a transition from an outgoing state s to an incoming state s′ in the concrete transition system, i.e. s γ −−→ s′, then there is a transition from f (s) to f (s′) in the abstract transition system, i.e. f (s) γ −−→ f f (s′) [23, Def. 7.51]. S...