ANDYMARK: An Analytical Method to Establish Dynamically the Length of the Markov Chain in Simulated Annealing for the Satisfiability Problem

2006 ◽  
pp. 269-276 ◽  
Author(s):  
Juan Frausto-Solís ◽  
Héctor Sanvicente-Sánchez ◽  
Froilán Imperial-Valenzuela
Keyword(s):  
Markov Chain ◽  
Simulated Annealing ◽  
Analytical Method ◽  
Satisfiability Problem

On the Markov Chain Simulation Method for Uniform Combinatorial Distributions and Simulated Annealing

1987 ◽  
Vol 1 (1) ◽  
pp. 33-46 ◽  
Author(s):  
David Aldous
Keyword(s):  
Markov Chain ◽  
Simulated Annealing ◽  
Simulated Annealing Algorithm ◽  
Simulation Method ◽  
Careful Analysis ◽  
Eigenvalue Estimate ◽  
Uniform Distributions ◽  
Markov Chain Method ◽  
Annealing Algorithm ◽  
Second Largest Eigenvalue

Uniform distributions on complicated combinatorial sets can be simulated by the Markov chain method. A condition is given for the simulations to be accurate in polynomial time. Similar analysis of the simulated annealing algorithm remains an open problem. The argument relies on a recent eigenvalue estimate of Alon [4]; the only new mathematical ingredient is a careful analysis of how the accuracy of sample averages of a Markov chain is related to the second-largest eigenvalue.

Velocity log upscaling based on reversible jump Markov chain Monte Carlo simulated annealing

Geophysics ◽  
2016 ◽  
Vol 81 (5) ◽  
pp. R293-R305 ◽  
Author(s):  
Sireesh Dadi ◽  
Richard Gibson ◽  
Kainan Wang
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Simulated Annealing ◽  
Markov Chain Monte Carlo ◽  
Random Perturbation ◽  
Current Model ◽  
Sampling Technique ◽  
Maximum Deviation ◽  
Reversible Jump ◽  
Layer Boundary

Upscaling log measurements acquired at high frequencies and correlating them with corresponding low-frequency values from surface seismic and vertical seismic profile data is a challenging task. We have applied a sampling technique called the reversible jump Markov chain Monte Carlo (RJMCMC) method to this problem. A key property of our approach is that it treats the number of unknowns itself as a parameter to be determined. Specifically, we have considered upscaling as an inverse problem in which we considered the number of coarse layers, layer boundary depths, and material properties as the unknowns. The method applies Bayesian inversion, with RJMCMC sampling and uses simulated annealing to guide the optimization. At each iteration, the algorithm will randomly move a boundary in the current model, add a new boundary, or delete an existing boundary. In each case, a random perturbation is applied to Backus-average values. We have developed examples showing that the mismatch between seismograms computed from the upscaled model and log velocities improves by 89% compared to the case in which the algorithm is allowed to move boundaries only. The layer boundary distributions after running the RJMCMC algorithm can represent sharp and gradual changes in lithology. The maximum deviation of upscaled velocities from Backus-average values is less than 10% with most of the values close to zero.

A Bayesian Approach to Simulated Annealing

1989 ◽  
Vol 3 (4) ◽  
pp. 453-475 ◽  
Author(s):  
P.J.M. Van Laarhoven ◽  
C.G.E. Boender ◽  
E.H.L. Aarts ◽  
A. H. G. Rinnooy Kan
Keyword(s):  
Markov Chain ◽  
Simulated Annealing ◽  
Probability Distribution ◽  
Posterior Distribution ◽  
Prior Distribution ◽  
Numerical Experiments ◽  
Optimization Problems ◽  
Probabilistic Algorithm ◽  
Unknown Parameters ◽  
Combinatorial Optimization Problems

Simulated annealing is a probabilistic algorithm for approximately solving large combinatorial optimization problems. The algorithm can mathematically be described as the generation of a series of Markov chains, in which each Markov chain can be viewed as the outcome of a random experiment with unknown parameters (the probability of sampling a cost function value). Assuming a probability distribution on the values of the unknown parameters (the prior distribution) and given the sequence of configurations resulting from the generation of a Markov chain, we use Bayes's theorem to derive the posterior distribution on the values of the parameters. Numerical experiments are described which show that the posterior distribution can be used to predict accurately the behavior of the algorithm corresponding to the next Markov chain. This information is also used to derive optimal rules for choosing some of the parameters governing the convergence of the algorithm.

scholarly journals Schedules of lectures and Monte Carlo method

2011 ◽  
Vol 52 ◽  
Author(s):  
Nikolaj Grigorjev ◽  
Gediminas Stepanauskas
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Simulated Annealing ◽  
Markov Chain Monte Carlo ◽  
Monte Carlo Method ◽  
Simulated Annealing Algorithm ◽  
Annealing Algorithm

In this paper the problem of the construction of schedule of lectures is considered. The Markov chain Monte Carlo method is used. A particular program based on simulated annealing algorithm was created.  

Convergence and finite-time behavior of simulated annealing

1986 ◽  
Vol 18 (03) ◽  
pp. 747-771 ◽  
Author(s):  
Debasis Mitra ◽  
Fabio Romeo ◽  
Alberto Sangiovanni-Vincentelli
Keyword(s):  
Markov Chain ◽  
Simulated Annealing ◽  
Finite Time ◽  
Randomized Algorithm ◽  
Global Optimum ◽  
Time Behavior ◽  
Annealing Schedule ◽  
Complete Problems ◽  
Inhomogeneous Markov Chain ◽  
Np Complete

Simulated annealing is a randomized algorithm which has been proposed for finding globally optimum least-cost configurations in large NP-complete problems with cost functions which may have many local minima. A theoretical analysis of simulated annealing based on its precise model, a time-inhomogeneous Markov chain, is presented. An annealing schedule is given for which the Markov chain is strongly ergodic and the algorithm converges to a global optimum. The finite-time behavior of simulated annealing is also analyzed and a bound obtained on the departure of the probability distribution of the state at finite time from the optimum. This bound gives an estimate of the rate of convergence and insights into the conditions on the annealing schedule which gives optimum performance.

scholarly journals On the use of Markovian stick-breaking priors

2021 ◽  
pp. 153-174
Author(s):  
William Lippitt ◽  
Sunder Sethuraman
Keyword(s):  
Markov Chain ◽  
Simulated Annealing ◽  
Dirichlet Process ◽  
Test Cases ◽  
Content Type ◽  
Inference Test ◽  
Irreducible Markov Chain ◽  
Class Of Priors ◽  
Distributional Limits

Recently, a ‘Markovian stick-breaking’ process which generalizes the Dirichlet process ( μ , θ ) (\mu , \theta ) with respect to a discrete base space X \mathfrak {X} was introduced. In particular, a sample from from the ‘Markovian stick-breaking’ processs may be represented in stick-breaking form ∑ i ≥ 1 P i δ T i \sum _{i\geq 1} P_i \delta _{T_i} where { T i } \{T_i\} is a stationary, irreducible Markov chain on X \mathfrak {X} with stationary distribution μ \mu , instead of i.i.d. { T i } \{T_i\} each distributed as μ \mu as in the Dirichlet case, and { P i } \{P_i\} is a GEM ( θ ) (\theta ) residual allocation sequence. Although the previous motivation was to relate these Markovian stick-breaking processes to empirical distributional limits of types of simulated annealing chains, these processes may also be thought of as a class of priors in statistical problems. The aim of this work in this context is to identify the posterior distribution and to explore the role of the Markovian structure of { T i } \{T_i\} in some inference test cases.

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