ACCELERATING MONTE CARLO MARKOV PROCESSES

Let π be a probability density proportional to exp - U(x) in S. A convergent Markov process to π(x) may be regarded as a "conceptual" algorithm. Assume that S is a finite set. Let X0,X1,…,Xn,… be a Markov chain with transition matrix P and invariant probability π. Under suitable condition on P, it is known that converges to π(f) and the corresponding asymptotic variance v(f, P) depends only on f and P. It is natural to consider criteria vw(P) and va(P), defined respectively by maximizing and averaging v(f, P) over f. Two families of transition matrices are considered. There are four problems to be investigated. Some results and conjectures are given. As for the continuum case, to accelerate the convergence a family of diffusions with drift ∇U(x) + C(x) with div(C(x)exp - U(x)) = 0 is considered.