Perturbation Bounds Research Articles

The Arsenal of Perturbation Bounds for Finite Continuous-Time Markov Chains: A Perspective

Perturbation bounds are powerful tools for investigating the phenomenon of insensitivity to perturbations, also referred to as stability, for stochastic and deterministic systems. This perspective article presents a focused account of some of the main concepts and results in inequality-based perturbation theory for finite state-space, time-homogeneous, continuous-time Markov chains. The diversity of perturbation bounds and the logical relationships between them highlight the essential stability properties and factors for this class of stochastic processes. We discuss the linear time dependence of general perturbation bounds for Markov chains, as well as time-independent (i.e., time-uniform) perturbation bounds for chains whose stationary distribution is unique. Moreover, we prove some new results characterizing the absolute and relative tightness of time-uniform perturbation bounds. Specifically, we show that, in some of them, an equality is achieved. Furthermore, we analytically compare two types of time-uniform bounds known from the literature. Possibilities for generalizing Markov-chain stability results, as well as connections with stability analysis for other systems and processes, are also discussed.

Perturbation Bounds for the Group Inverse and its Oblique Projection

Perturbation Bounds for the Group Inverse and its Oblique Projection

The Perturbation Bound of the Extended Vertical Linear Complementarity Problem

In this paper, we discuss the perturbation analysis of the extended vertical linear complementarity problem (EVLCP). Under the assumption of the row $${\mathcal {W}}$$ -property, we derive several absolute and relative perturbation bounds of EVLCP, which extend some existing results. Several numerical examples are given to show the proposed bounds.

Strong Perturbation Bounds for the Stationary Distribution of the Main Multiserver Retrial Queue Model

Strong Perturbation Bounds for the Stationary Distribution of the Main Multiserver Retrial Queue Model

Two Approaches to the Construction of Perturbation Bounds for Continuous-Time Markov Chains

This paper is largely a review. It considers two main methods used to study stability and to obtain appropriate quantitative estimates of perturbations of (inhomogeneous) Markov chains with continuous time and a finite or countable state space. An approach is described to the construction of perturbation estimates for the main five classes of such chains associated with queuing models. Several specific models are considered for which the limit characteristics and perturbation bounds for admissible “perturbed” processes are calculated.

Stationary Distribution and Perturbation Bounds for a Stochastic Inventory Model

In this paper, we use the theory of generalized inverses to compute the stationary distribution of the (s, S) inventory model directly from the transition matrix of the underlying Markov chain and to provide perturbation bounds. Indeed, approximations are employed to build a tractable model, and statistical methods are used to estimate the unknown parameters and distributions. Hence, the system is subject to perturbations that may cause deviation in the characteristics. The proposed perturbation bound provides a means to estimate the impact of the perturbations on the performance measures of the considered inventory system.

Perturbation Bounds for Markov Chains with General State Space

The aim of this paper is to investigate the stability of Markov chains with general state space. We present new conditions for the strong stability of Markov chains after a small perturbation of their transition kernels. Also, we obtain perturbation bounds with respect to different quantities.

Asymptotic Perturbation Bounds for Probabilistic Model Checking with Empirically Determined Probability Parameters

Probabilistic model checking is a verification technique that has been the focus of intensive research for over a decade. One important issue with probabilistic model checking, which is crucial for its practical significance but is overlooked by the state-of-the-art largely, is the potential discrepancy between a stochastic model and the real-world system it represents when the model is built from statistical data. In the worst case, a tiny but nontrivial change to some model quantities might lead to misleading or even invalid verification results. To address this issue, in this paper, we present a mathematical characterization of the consequences of model perturbations on the verification distance. The formal model that we adopt is a parametric variant of discrete-time Markov chains equipped with a vector norm to measure the perturbation. Our main technical contributions include a closed-form formulation of asymptotic perturbation bounds , and computational methods for two arguably most useful forms of those bounds, namely linear bounds and quadratic bounds . We focus on verification of reachability properties but also address automata-based verification of omega-regular properties. We present the results of a selection of case studies that demonstrate that asymptotic perturbation bounds can accurately estimate maximum variations of verification results induced by model perturbations.

Perturbation Bounds and Condition Numbers for a Complex Indefinite Linear Algebraic System

AbstractWe consider perturbation bounds and condition numbers for a complex indefinite linear algebraic system, which is of interest in science and engineering. Some existing results are improved, and illustrative numerical examples are provided.

Ergodicity and Perturbation Bounds for Inhomogeneous Birth and Death Processes with Additional Transitions from and to the Origin

Abstract Service life of many real-life systems cannot be considered infinite, and thus the systems will be eventually stopped or will break down. Some of them may be re-launched after possible maintenance under likely new initial conditions. In such systems, which are often modelled by birth and death processes, the assumption of stationarity may be too strong and performance characteristics obtained under this assumption may not make much sense. In such circumstances, time-dependent analysis is more meaningful. In this paper, transient analysis of one class of Markov processes defined on non-negative integers, specifically, inhomogeneous birth and death processes allowing special transitions from and to the origin, is carried out. Whenever the process is at the origin, transition can occur to any state, not necessarily a neighbouring one. Being in any other state, besides ordinary transitions to neighbouring states, a transition to the origin can occur. All possible transition intensities are assumed to be non-random functions of time and may depend (except for transition to the origin) on the process state. To the best of our knowledge, first ergodicity and perturbation bounds for this class of processes are obtained. Extensive numerical results are also provided.

Some Refined Eigenvalue Perturbation Bounds for Two-by-Two Block Hermitian Matrices

AbstractWe consider eigenvalue perturbation bounds for Hermitian matrices, which are associated with problems arising in various computational science and engineering applications. New bounds are discussed that are sharper than some existing ones, including the well-known Weyl bound. Two numerical examples are investigated, to illustrate our theoretical presentation.

On Perturbation Bounds for the Joint Stationary Distribution of Multivariate Markov Chain Models

AbstractMultivariate Markov chain models have previously been proposed in for studying dependent multiple categorical data sequences. For a given multivariate Markov chain model, an important problem is to study its joint stationary distribution. In this paper, we use two techniques to present some perturbation bounds for the joint stationary distribution vector of a multivariate Markov chain with s categorical sequences. Numerical examples demonstrate the stability of the model and the effectiveness of our perturbation bounds.

Perturbation Bounds for the Stationary Distributions of Markov Chains

In this paper, we are interested in investigating the perturbation bounds for the stationary distributions for discrete-time or continuous-time Markov chains on a countable state space. For discrete-time Markov chains, two new normwise bounds are obtained. The first bound is rather easy to obtain since the needed condition, equivalent to uniform ergodicity, is imposed on the transition matrix directly. The second bound, which holds for a general (possibly periodic) Markov chain, involves finding a drift function. This drift function is closely related to the mean first hitting times. Some $V$-normwise bounds are also derived based on the results in [N. V. Kartashov, J. Soviet Math., 34 (1986), pp. 1493--1498]. Moreover, we show how the bounds developed in this paper and one bound given in [E. Seneta, Adv. Appl. Probab., 20 (1988), pp. 228--230] can be extended to continuous-time Markov chains. Several examples are shown to illustrate our results or to compare our bounds with the known ones in the literature.

On the Perturbation Bounds of Projected Generalized Continuous‐Time Sylvester Equations

This paper is devoted to the perturbation analysis for a projected generalized continuous‐time Sylvester equation. Perturbation bounds of the solution based on the Euclidean norm are presented.

Rigorous Perturbation Bounds of Some Matrix Factorizations

This article presents rigorous normwise perturbation bounds for the Cholesky, LU, and QR factorizations with normwise or componentwise perturbations in the given matrix. The considered componentwise perturbations have the form of backward rounding errors for the standard factorization algorithms. The used approach is a combination of the classic and refined matrix equation approaches. Each of the new rigorous perturbation bounds is a small constant multiple of the corresponding first-order perturbation bound obtained by the refined matrix equation approach in the literature and can be estimated efficiently. These new bounds can be much tighter than the existing rigorous bounds obtained by the classic matrix equation approach, while the conditions for the former to hold are almost as moderate as the conditions for the latter to hold.

An LMI Approach to Computing Delayed Perturbation Bounds for Stabilizing Receding Horizon H Controls

This letter presents new delayed perturbation bounds (DPBs) for stabilizing receding horizon H∞ control (RHHC). The linear matrix inequality (LMI) approach to determination of DPBs for the RHHC is proposed. We show through a numerical example that the RHHC can guarantee an H∞ norm bound for a larger class of systems with delayed perturbations than conventional infinite horizon H∞ control (IHHC).

Perturbation Bounds for Determinants and Characteristic Polynomials

We derive absolute perturbation bounds for the coefficients of the characteristic polynomial of a $n\times n$ complex matrix. The bounds consist of elementary symmetric functions of singular values, and suggest that coefficients of normal matrices are better conditioned with regard to absolute perturbations than those of general matrices. When the matrix is Hermitian positive-definite, the bounds can be expressed in terms of the coefficients themselves. We also improve absolute and relative perturbation bounds for determinants. The basis for all bounds is an expansion of the determinant of a perturbed diagonal matrix.

Delayed Perturbation Bounds for Receding Horizon Controls

This letter presents delayed perturbation bounds (DPBs) for receding horizon controls (RHCs) of continuous-time systems. The proposed DPBs are obtained easily by solving convex problems represented by linear matrix inequalities (LMIs). We show, by numerical examples, that the RHCs have larger DPBs than conventional linear quadratic regulators (LQRs).

Some New Perturbation Bounds for Subunitary Polar Factors

In this paper, we present some new perturbation bounds for subunitary polar factors in a special unitarily invariant norm called a Q–norm. Some recent results in the Frobenius norm and the spectral norm are extended to the Q–norm on one hand. On the other hand we also present some relative perturbation bounds for subunitary polar factors.

A Perturbation Bound of the Drazin Inverse of a Matrix by Separation of Simple Invariant Subspaces

A constructive perturbation bound of the Drazin inverse of a square matrix is derived using a technique proposed by G. Stewart and based on perturbation theory for invariant subspaces. This is an improvement of the result published by the authors Wei and Li [Numer. Linear Algebra Appl., 10 (2003), pp. 563--575]. It is a totally new approach to developing perturbation bounds for the Drazin inverse of a matrix. A numerical example which indicates the sharpness of the perturbation bound is presented.