Accelerating diffusions

Abstract

Let U be a given function defined on R^d and \pi(x) be a density function proportional to \exp -U(x). The following diffusion X(t) is often used to sample from \pi(x), dX(t)=-\nabla U(X(t)) dt+\sqrt2 dW(t),\qquad X(0)=x_0. To accelerate the convergence, a family of diffusions with \pi(x) as their common equilibrium is considered, dX(t)=\bigl(-\nabla U(X(t))+C(X(t))\bigr) dt+\sqrt2 dW(t),\qquad X(0)=x_0. Let L_C be the corresponding infinitesimal generator. The spectral gap of L_C in L^2(\pi) (\lambda (C)), and the convergence exponent of X(t) to \pi in variational norm (\rho(C)), are used to describe the convergence rate, where \lambda(C)= Sup{real part of \mu\dvtx\mu is in the spectrum of L_C, \mu is not zero}, {-2.8cm}\rho(C) = Inf\biggl{\rho\dvtx\int | p(t,x,y) -\pi(y)| dy \le g(x) e^{\rho t}\biggr}.Roughly speaking, L_C is a perturbation of the self-adjoint L_0 by an antisymmetric operator C\cdot\nabla, where C is weighted divergence free. We prove that \lambda (C)\le \lambda (0) and equality holds only in some rare situations. Furthermore, \rho(C)\le \lambda (C) and equality holds for C=0. In other words, adding an extra drift, C(x), accelerates convergence. Related problems are also discussed.Comment: Published at http://dx.doi.org/10.1214/105051605000000025 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Optimal Non-reversible Linear Drift for the Convergence to Equilibrium of a Diffusion

“…Then, by considering J = −J opt , where J opt ∈ A N (R) refers to the matrix considered in Theorem 2 to get (24). Then the inequality…”

“…The identity t(ε + L)e −t(ε+L) − Le −tL = tεe −εt e −tL + (e −εt − 1)tLe −tL with t > 0 fixed and e −tL , tLe −tL ∈ L(H) implies lim ε→0 t(ε+L)e −t(ε+L) − tLe −tL L(H) = 0 , which yields the result in the general case. ⊓ ⊔ In view of (29), Proposition 12 yields the estimate (24) in Theorem 2 with a constant…”

“…As noticed in [23,24], one way to accelerate the convergence to equilibrium is to depart from reversible dynamics (see also [10] for related discussions for Markov Chains). Let us recall that the dynamics (2) is reversible in the sense that if X 0 is distributed according to ψ ∞ (x) dx, then (X t ) 0≤t≤T and (X T −t ) 0≤t≤T have the same law.…”

“…This problem was studied in [23] for a linear drift (namely V is quadratic and b is linear) and in [24] for the general case. It was shown in these works that the addition of a drift function b satisfying (9) helps to speed up convergence to equilibrium.…”

Using Perturbed Underdamped Langevin Dynamics to Efficiently Sample from Probability Distributions

“…The asymptotic behaviour of this process was considered for Gaussian diffusions in [20], where the rate of convergence of the covariance to equilibrium was quantified in terms of the choice of γ . This work was extended to the case of non-Gaussian target densities, and consequently for nonlinear SDEs of the form (7) in [21]. The problem of constructing the optimal nonreversible perturbation, in terms of the L 2 (π) spectral gap for Gaussian target densities was studied in [34] see also [55].…”