On the link between Markovian trees and tree-structured Markov chains

Hautphenne, Sophie; Houdt, Benny Van

doi:10.1016/j.ejor.2009.03.052

Cited by 13 publications

(17 citation statements)

References 22 publications

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“…Recently Hautphenne and Van Houdt [17] proposed a different version of Newton's method for E1 that has a better convergence rate than the traditional one. Their idea is to apply the Newton method to the equation…”

Section: Modified Newton Methodsmentioning

confidence: 99%

Quadratic vector equations

Poloni¹

2011

Algorithms for Quadratic Matrix and Vector Equations

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We study in a unified fashion several quadratic vector and matrix equations with nonnegativity hypotheses, by seeing them as special cases of the general problem M x = a + b (x, x), where a and the unknown x are componentwise nonnegative vectors, M is a nonsingular M-matrix, and b is a bilinear map from pairs of nonnegative vectors to nonnegative vectors. Specific cases of this equation have been studied extensively in the past by several authors, and include unilateral matrix equations from queuing problems [Bini, Latouche, Meini, 2005], nonsymmetric algebraic Riccati equations [Guo, Laub, 2000], and quadratic matrix equations encountered in neutron transport theory .We present a unified approach which treats the common aspects of their theoretical properties and basic iterative solution algorithms. This has interesting consequences: in some cases, we are able to derive in full generality theorems and proofs appeared in literature only for special cases of the problem; this broader view highlights the role of hypotheses such as the strict positivity of the minimal solution. In an example, we adapt an algorithm derived for one equation of the class to another, with computational advantage with respect to the existing methods. We discuss possible research lines, including the relationship among Newton-type methods and the cyclic reduction algorithm for unilateral quadratic equations.

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Section: Modified Newton Methodsmentioning

confidence: 99%

Quadratic vector equations

Poloni¹

2011

Algorithms for Quadratic Matrix and Vector Equations

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“…L 1 = (−Q 11 ) −1 Q 12 , since, starting from level 1, the probability of reaching level 2 before level 0 is the probability of directly moving to level 2 when leaving level 1. Equation (12) can be rewritten as…”

Section: A Catastrophe Occurs and Killsmentioning

confidence: 99%

“…[3] ). This equation relies on the assumption of independence between individuals and may be solved numerically: linear algorithms have been developed in Bean, Kontoleon, and Taylor [3] and in Hautphenne, Latouche, and Remiche [9] , and quadratic algorithms have been proposed in Hautphenne, Latouche, and Remiche [8] and in Hautphenne and van Houdt [12] . We superimpose on the MBT an independent process of catastrophe.…”

Section: Introductionmentioning

confidence: 99%

Markovian Trees Subject to Catastrophes: Transient Features and Extinction Probability

Hautphenne

Latouche

2011

Stochastic Models

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We study transient features and the extinction probability of a particular class of multitype Markovian branching processes called Markovian binary trees, subject at random epochs to catastrophes, controlled by a Markovian arrival process, and killing random numbers of living individuals. We provide two types of probabilistic methods to numerically compute the extinction probability: the first one is based on an integral equation for the probability generating function of the population size, and the second one is based on the correspondence between Markovian branching processes with catastrophes and structured Markov chains of G/M/1-type.

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“…Several algorithms have been devised for the computation of the minimal nonnegative solution of QVE (1.1) such as the depth and order algorithm [2], the thickness algorithm [7], which are all linearly convergent; the Newton's iteration [6] and the variations of Newton's iteration [5], which are shown to be quadratically convergent; the Perron vector iteration [3,11], which converges faster than Newton's iteration for close to critical problems. We refer the readers to [12] and the references therein for the details of the numerical methods.…”

Section: Introductionmentioning

confidence: 99%

Perturbation analysis of the extinction probability of a Markovian binary tree

Guo

Cai

Qian

et al. 2015

Linear Algebra and its Applications

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The extinction probability of the Markovian Binary Tree (MBT) is the minimal nonnegative solution of a Quadratic Vector Equation (QVE). In this paper, we present a perturbation analysis for the extinction probability of a supercritical MBT. We derive a perturbation bound for the minimal nonnegative solution of the QVE, which is a bound on the difference between the solutions of two nearby equations in terms of the perturbation magnitude. A posteriori error bound is also given, which is a bound on the distance between an approximate solution and the real solution, in terms of the residual of the approximate solution. Numerical experiments show that these bounds are fairly sharp.

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On the link between Markovian trees and tree-structured Markov chains

Cited by 13 publications

References 22 publications

Quadratic vector equations

Quadratic vector equations

Markovian Trees Subject to Catastrophes: Transient Features and Extinction Probability

Perturbation analysis of the extinction probability of a Markovian binary tree

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