@article{SharmaZhang2020, author = {Sharma, Upanshu and Zhang, Wei}, title = {Non-reversible sampling schemes on submanifolds}, series = {SIAM Journal on Numerical Analysis}, volume = {59}, journal = {SIAM Journal on Numerical Analysis}, number = {6}, doi = {10.1137/20M1378752}, pages = {2989 -- 3031}, year = {2020}, abstract = {Calculating averages with respect to probability measures on submanifolds is often necessary in various application areas such as molecular dynamics, computational statistical mechanics and Bayesian statistics. In recent years, various numerical schemes have been proposed in the literature to study this problem based on appropriate reversible constrained stochastic dynamics. In this paper we present and analyse a non-reversible generalisation of the projection-based scheme developed by one of the authors [ESAIM: M2AN, 54 (2020), pp. 391-430]. This scheme consists of two steps - starting from a state on the submanifold, we first update the state using a non-reversible stochastic differential equation which takes the state away from the submanifold, and in the second step we project the state back onto the manifold using the long-time limit of a ordinary differential equation. We prove the consistency of this numerical scheme and provide quantitative error estimates for estimators based on finite-time running averages. Furthermore, we present theoretical analysis which shows that this scheme outperforms its reversible counterpart in terms of asymptotic variance. We demonstrate our findings on an illustrative test example.}, language = {en} } @article{MontefuscoSharmaTse, author = {Montefusco, Alberto and Sharma, Upanshu and Tse, Oliver}, title = {Fourier-Cattaneo equation: stochastic origin, variational formulation, and asymptotic limits}, abstract = {We introduce a variational structure for the Fourier-Cattaneo (FC) system which is a second-order hyperbolic system. This variational structure is inspired by the large-deviation rate functional for the Kac process which is closely linked to the FC system. Using this variational formulation we introduce appropriate solution concepts for the FC equation and prove an a priori estimate which connects this variational structure to an appropriate Lyapunov function and Fisher information, the so-called FIR inequality. Finally, we use this formulation and estimate to study the diffusive and hyperbolic limits for the FC system.}, language = {en} }