Abstract
Sampling from probability distributions in high dimensional spaces is generally impractical. Diffusion processes with invariant equilibrium distributions can be used as a means to generate approximations. An important task in such an endeavor is to design an equilibrium-preserving drift to accelerate the convergence. Starting from a reversible diffusion, it is desirable to depart for non-reversible dynamics via a perturbed drift so that the convergence rate is maximized with the common equilibrium. In the Gaussian diffusion acceleration, this problem can be cast as perturbing the inverse of a given covariance matrix by skew-symmetric matrices so that all resulting eigenvalues have identical real part. This paper describes two approaches to obtain the optimal rate of Gaussian diffusion. The asymptotical approach works universally for arbitrary Ornstein–Uhlenbeck processes, whereas the direct approach can be implemented as a fast divide-and-conquer algorithm. A comparison with recently proposed Lelièvre–Nier–Pavliotis algorithm is made.
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Notes
Complex eigenvalues cannot be ordered. The continuous dependence of one particular eigenvalue and its associated eigenvector among other eigenpairs therefore has to be carefully discerned.
We remark that fitting these data with polynomials of higher degrees would be of little avail. It can easily be checked numerically that the coefficients associated with the higher degree terms are nearly zero.
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This research was supported in part by the National Science Foundation under Grant DMS-1014666.
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Wu, SJ., Hwang, CR. & Chu, M.T. Attaining the Optimal Gaussian Diffusion Acceleration. J Stat Phys 155, 571–590 (2014). https://doi.org/10.1007/s10955-014-0963-5
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DOI: https://doi.org/10.1007/s10955-014-0963-5