riemannian metric tensor
The SDE for the underdamped Langevin dynamics is modified to incorporate anisotropic smoothness through a position-dependent metric tensor: $$ dX_t = -G(x)^{-1} \nabla U(x) , dt + \sqrt{2G(x)^{-1}} , dB_t, $$ where $G(x) \in \mathbb{R}^{d \times d}$ is a positive definite metric tensor, such as $G(x) = \nabla^2 U(x) + \epsilon I$ with regularization parameter $\epsilon > 0$.
Discretization of the SDE using the Euler-Maruyama method yields: $$ X_{k+1} = X_k - \eta G(X_k)^{-1} \nabla U(X_k) + \sqrt{2\eta G(X_k)^{-1}} , \xi_k, $$ where $\eta > 0$ is the step size, and $\xi_k \sim \mathcal{N}(0, I)$.
multiscale time-stepping
Let $U(x)$ decompose as $U(x) = U_s(x) + U_f(x)$, where $U_s(x)$ and $U_f(x)$ represent smooth and stiff components, respectively. The dynamics are given by: $$ \begin{aligned} dX_t &= -\nabla U_s(X_t) , dt - \nabla U_f(X_t) , dt + \sqrt{2} , dB_t. \end{aligned} $$
The multiscale scheme alternates between resolving $U_s(x)$ with a larger step size $\eta_s$ and $U_f(x)$ with a smaller step size $\eta_f$, where $\eta_s \gg \eta_f$. The discretization becomes: $$ \begin{aligned} X_{k+1/2} &= X_k - \eta_s \nabla U_s(X_k), \ X_{k+1} &= X_{k+1/2} - \eta_f \nabla U_f(X_{k+1/2}) + \sqrt{2\eta_f} , \xi_k. \end{aligned} $$