General Irreducible Markov Chains and Non-Negative Operators

Nummelin, Esa

doi:10.1017/cbo9780511526237

Cited by 640 publications

(740 citation statements)

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“…The foundations of a theory of general state space Markov chains are described in [50], and although the theory is much more refined now, this is still the best source of much basic material. The next generation of results is developed in [51] and more current treatments are contained in [52].…”

Section: Finite Markov Chainsmentioning

confidence: 99%

Integrated logic synthesis using simulated annealing

Färm

Dubrova

Kuehlmann

2011

Proceedings of the 21st Edition of the Great Lakes Symposium on Great Lakes Symposium on VLSI

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KTH School of Information andCommunication Technology SE-164 40 Kista SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen i elektronik fredagen den 26 januari 2007 klockan 9.00 i Sal E, Forumhuset, Kungl Tekniska högskolan, Isafjordsgatan 39, Kista. © Petra Färm, januari 2007Tryck: Universitetsservice US AB iii Abstract A conventional logic synthesis flow is composed of three separate phases: technology independent optimization, technology mapping, and technology dependent optimization. A fundamental problem with such a three-phased approach is that the global logic structure is decided during the first phase without any knowledge of the actual technology parameters considered during later phases. Although technology dependent optimization algorithms perform some limited logic restructuring, they cannot recover from fundamental mistakes made during the first phase, which often results in non-satisfiable solutions.We present a global optimization approach combining technology independent optimization steps with technology dependent objectives in an annealing-based framework. We prove that, for the presented move set and selection distribution, detailed balance is satisfied and thus the annealing process asymptotically converges to an optimal solution. Furthermore, we show that the presented approach can smoothly trade-off complex, multiple-dimensional objective functions and achieve competitive results. The combination of technology independent and technology dependent objectives is handled through dynamic weighting. Dynamic weighting reflects the sensitivity of the local graph structures with respect to the actual technology parameters such as gate sizes, delays, and power levels. The results show that, on average, the presented advanced annealing approach can improve the area and delay of circuits optimized using the Boolean optimization technique provided by SIS with 11.2% and 32.5% respectively.Furthermore, we demonstrate how the developed logic synthesis framework can be applied to two emerging technologies, chemically assembled nanotechnology and molecule cascades. New technologies are emerging because a number of physical and economic factors threaten the continued scaling of CMOS devices. Alternatives to silicon VLSI have been proposed, including techniques based on molecular electronics, quantum mechanics, and biological processes. We are hoping that our research in how to apply our developed logic synthesis framework to two of the emerging technologies might provide useful information for other designers moving in this direction.iv

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Section: Finite Markov Chainsmentioning

confidence: 99%

Integrated logic synthesis using simulated annealing

Färm

Dubrova

Kuehlmann

2011

Proceedings of the 21st Edition of the Great Lakes Symposium on Great Lakes Symposium on VLSI

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“…If (1.2) holds, then whenever (X n−1 , X n−1 ) ∈ C ×C, we can use Nummelin splitting [20,18,24] to jointly update X n and X n in such a way that X n = X n with probability at least . Thus, we have managed to "force" X n = X n with non-zero probability, as desired.…”

Section: Minorisation Conditions (Small Sets)mentioning

confidence: 99%

Markov Chain Monte Carlo Algorithms: Theory and Practice

Rosenthal

2009

Monte Carlo and Quasi-Monte Carlo Methods 2008

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Summary. We describe the importance and widespread use of Markov chain Monte Carlo (MCMC) algorithms, with an emphasis on the ways in which theoretical analysis can help with their practical implementation. In particular, we discuss how to achieve rigorous quantitative bounds on convergence to stationarity using the coupling method together with drift and minorisation conditions. We also discuss recent advances in the field of adaptive MCMC, where the computer iteratively selects from among many different MCMC algorithms. Such adaptive MCMC algorithms may fail to converge if implemented naively, but they will converge correctly if certain conditions such as Diminishing Adaptation are satisfied.

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“…Under (A1), the environment ω is an ergodic sequence (see e.g. [10, p. 338] or [19,Theorem 6.15]). Condition (A3) guarantees, by convexity, the existence of a unique κ in (1.4).…”

Section: (A1) Eithermentioning

confidence: 99%

Limit theorems for one-dimensional transient random walks in Markov environments*1

Mayer-Wolf

Roitershtein

Zeitouni

2004

Annales de l?Institut Henri Poincare (B) Probability and Statis

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We obtain non-Gaussian limit laws for one-dimensional random walk in a random environment in the case that the environment is a function of a stationary Markov process. This is an extension of the work of Kesten, M. Kozlov and Spitzer [13] for random walks in i.i.d. environments. The basic assumption is that the underlying Markov chain is irreducible and either with finite state space or with transition kernel dominated above and below by a probability measure. MSC2000: primary 60K37, 60F05; secondary 60J05, 60J80.

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General Irreducible Markov Chains and Non-Negative Operators

Cited by 640 publications

References 0 publications

Integrated logic synthesis using simulated annealing

Integrated logic synthesis using simulated annealing

Markov Chain Monte Carlo Algorithms: Theory and Practice

Limit theorems for one-dimensional transient random walks in Markov environments*1

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