Hybrid Monte Carlo on Hilbert spaces
The authors consider the problem of sampling a probability measure \(\pi\) on a Hilbert space defined via the density with respect to a Gaussian measure \(\pi_{0}\): \[ \frac{d\pi}{d\pi_{0}}(q)\propto\exp(-\Phi(q)). \] Any algorithm designed to sample \(\pi\) should be implemented on a finite-dimensional space of dimension \(N\). The number of steps required to explore the target distribution \(\pi\) typically grows with \(N\). The authors propose a generalized hybrid Monte Carlo algorithm which overcomes these shortcomings. They develop the following issues in the infinite-dimensional setting: (i) construction of a probability measure \(\Pi\) in an enlarged phase space having the target \(\pi\) as a marginal together with a Hamiltonian flow that preserves \(\Pi\); (ii) development of a geometric numerical integrator for the Hamiltonian flow; (iii) derivation of an accept/reject rule to ensure preservation of \(\Pi\) when using the above integrator instead of the actual Hamiltonian flow. The standard HMC algorithm was introduced in [\textit{S. Duane, A. D. Kennedy, P. Pendleton} and \textit{D. Roweth}, ``Hybrid Monte Carlo'', Phys. Lett. B 195, No.~2, 216--222 (1987; \url{doi:10.1016/0370-2693(87)91197-X})].
scientific article; zbMATH DE number 5955960
scientific article