Stochastic algorithm corresponding to a general linear iterative process
Stephen L. Adler · Physical Review Letters · 1988
Let u'=Mu+Nf be a general linear iterative process for solving the system Lu=f, with L=${\mathrm{L}}^{\mathrm{T}}$ and with 1=M+NL. Provided that \ensuremath{\Gamma}\ensuremath{\equiv}1/2(${\mathit{L}}^{\mathrm{\ensuremath{-}}1}$-${\mathit{ML}}^{\mathrm{\ensuremath{-}}1}$${\mathit{M}}^{\mathrm{T}}$${)}^{\mathrm{\ensuremath{-}}1}$ is a positive-definite matrix, it is shown that one can explicitly construct a corresponding stochastic algorithm which satisfies the homogeneous-state condition with respect to the probability distribution exp(-\ensuremath{\beta}S), where S=(1/2${\mathrm{u}}^{\mathrm{T}\mathrm{Lu}\mathrm{\ensuremath{-}}{\mathrm{f}}^{\mathrm{T}\mathrm{u}}}$. When ${\mathrm{M}}^{\mathrm{T}\mathrm{L}=\mathrm{LM}}$, the algorithm also satisfies the detailed-balance condition.