Reversibility and Acyclicity

1975 ◽  
Vol 12 (S1) ◽  
pp. 217-224 ◽  
Author(s):  
P. Whittle
Keyword(s):  
Markov Process ◽  
State Space ◽  
Transition Matrix ◽  
Reversible Markov Process ◽  
Classical Definition ◽  
Dynamic Reversibility ◽  
Converse Assertion

It is well-known that the transition matrix of a reversible Markov process can have only real eigenvalues. An example is constructed which shows that the converse assertion does not hold. A generalised notion of reversibility is proposed, ‘dynamic reversibility’, which has many of the implications for the form of the transition matrix of the classical definition, but which does not exclude ‘circulation in state-space’ or, indeed, periodicity.

scholarly journals Cluster-based reduced-order modelling of a mixing layer

2014 ◽  
Vol 754 ◽  
pp. 365-414 ◽  
Author(s):  
Eurika Kaiser ◽  
Bernd R. Noack ◽  
Laurent Cordier ◽  
Andreas Spohn ◽  
Marc Segond ◽  
...  
Keyword(s):  
Markov Process ◽  
State Space ◽  
Transition Matrix ◽  
Bluff Body ◽  
Mixing Layer ◽  
The State ◽  
Lorenz Attractor ◽  
Reduced Order ◽  
Physical Mechanisms ◽  
Reduced Order Modelling

AbstractWe propose a novel cluster-based reduced-order modelling (CROM) strategy for unsteady flows. CROM combines the cluster analysis pioneered in Gunzburger’s group (Burkardt, Gunzburger & Lee,Comput. Meth. Appl. Mech. Engng, vol. 196, 2006a, pp. 337–355) and transition matrix models introduced in fluid dynamics in Eckhardt’s group (Schneider, Eckhardt & Vollmer,Phys. Rev. E, vol. 75, 2007, art. 066313). CROM constitutes a potential alternative to POD models and generalises the Ulam–Galerkin method classically used in dynamical systems to determine a finite-rank approximation of the Perron–Frobenius operator. The proposed strategy processes a time-resolved sequence of flow snapshots in two steps. First, the snapshot data are clustered into a small number of representative states, called centroids, in the state space. These centroids partition the state space in complementary non-overlapping regions (centroidal Voronoi cells). Departing from the standard algorithm, the probabilities of the clusters are determined, and the states are sorted by analysis of the transition matrix. Second, the transitions between the states are dynamically modelled using a Markov process. Physical mechanisms are then distilled by a refined analysis of the Markov process, e.g. using finite-time Lyapunov exponent (FTLE) and entropic methods. This CROM framework is applied to the Lorenz attractor (as illustrative example), to velocity fields of the spatially evolving incompressible mixing layer and the three-dimensional turbulent wake of a bluff body. For these examples, CROM is shown to identify non-trivial quasi-attractors and transition processes in an unsupervised manner. CROM has numerous potential applications for the systematic identification of physical mechanisms of complex dynamics, for comparison of flow evolution models, for the identification of precursors to desirable and undesirable events, and for flow control applications exploiting nonlinear actuation dynamics.

The Response of the System Incorporating an Observer in the Control Loop1

1999 ◽  
Vol 122 (2) ◽  
pp. 348-353
Author(s):  
Vladimir Polotski
Keyword(s):  
Asymptotic Behavior ◽  
State Space ◽  
Transition Matrix ◽  
Control Loop ◽  
Transition Processes ◽  
State Space Methods ◽  
Peaking Phenomenon ◽  
Siso System

Transition processes in the SISO system incorporating an observer in the control loop are discussed. The symmetry of the transition matrix with respect to the eigenvalues assigned for the controller and the observer is proved using state space methods. The interrelation with the peaking phenomenon is discussed, asymptotic behavior of the system with a fast observer is analyzed, and illustrative examples are presented. [S0022-0434(00)02102-X]

The calculation of limit probabilities for denumerable Markov processes from infinitesimal properties

1973 ◽  
Vol 10 (01) ◽  
pp. 84-99 ◽  
Author(s):  
Richard L. Tweedie
Keyword(s):  
Markov Process ◽  
Continuous Time ◽  
Transition Matrix ◽  
Invariant Measures ◽  
Equilibrium Distribution ◽  
Transition Functions ◽  
The Matrix ◽  
Limit Probability ◽  
Derivatives Of ◽  
Regular Chains

The problem considered is that of estimating the limit probability distribution (equilibrium distribution) πof a denumerable continuous time Markov process using only the matrix Q of derivatives of transition functions at the origin. We utilise relationships between the limit vector πand invariant measures for the jump-chain of the process (whose transition matrix we write P∗), and apply truncation theorems from Tweedie (1971) to P∗. When Q is regular, we derive algorithms for estimating πfrom truncations of Q; these extend results in Tweedie (1971), Section 4, from q-bounded processes to arbitrary regular processes. Finally, we show that this method can be extended even to non-regular chains of a certain type.

The harmonic functions of (At, Bt,)

1993 ◽  
Vol 114 (2) ◽  
pp. 369-377
Author(s):  
L. C. G. Rogers
Keyword(s):  
Markov Process ◽  
State Space ◽  
Harmonic Functions ◽  
Martin Boundary ◽  
The State ◽  
Process Yield ◽  
Reference Measure

The non-negative harmonic functions of a transient Markov process yield a great deal of information about the ‘behaviour at infinity’ of the process, and can be used to h-transform the process to behave in a certain way at infinity. The traditional analytic way of studying the non-negative harmonic functions is to construct the Martin boundary of the process (see, for example, Meyer [4], Kunita and T. Watanabe[3], and Kemeny, Snell & Knapp[2], Williams [7] for the chain case). However, certain conditions on the process need to be satisfied, one of the most basic of which is that there exists a reference measure η such that Uλ (x, ·) ≪ η for all λ > 0, all x ∈ E, the state space of the Markov process. (Here, (Uλ)λ>0 is the resolvent of the process.)

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