Numerical Methods with Coordinate Transforms for Efficient Brownian Dynamics Simulations

TL;DR

The paper addresses numerical challenges in Brownian dynamics with multiplicative noise by developing invertible coordinate transforms—the Lamperti transform and time-rescaling—to map variable diffusion to constant diffusion. It derives multivariate extensions, analyzes when these transforms preserve Brownian structure, and demonstrates how the right integrator combined with a transform yields a highly efficient, second-order weak sampler with just one force and one diffusion evaluation per step. Through extensive one- and two-dimensional numerical experiments, including rare-event and enhanced-sampling contexts, the authors quantify convergence, efficiency, and biases, showing that Lamperti-type transforms generally preserve finite-time statistics while time-rescaling can introduce interpolation bias for fixed-time estimates. The results hold promise for large-scale Brownian dynamics, particularly in biophysical and anisotropic diffusion settings, by enabling accurate recovery of invariant measures and dynamic quantities with reduced computational cost.

Abstract

Many stochastic processes in the physical and biological sciences can be modelled as Brownian dynamics with multiplicative noise. However, numerical integrators for these processes can lose accuracy or even fail to converge when the diffusion term is configuration-dependent. One remedy is to construct a transform to a constant-diffusion process and sample the transformed process instead. In this work, we explain how coordinate-based and time-rescaling-based transforms can be used either individually or in combination to map a general class of variable-diffusion Brownian motion processes into constant-diffusion ones. The transforms are invertible, thus allowing recovery of the original dynamics. We motivate our methodology using examples in one dimension before then considering multivariate diffusion processes. We illustrate the benefits of the transforms through numerical simulations, demonstrating how the right combination of integrator and transform can improve computational efficiency and the order of convergence to the invariant distribution. Notably, the transforms that we derive are applicable to a class of multibody, anisotropic Stokes-Einstein diffusion that has applications in biophysical modelling.

Figures (11)

• Figure 1: Comparison of the Lamperti and time-rescaling transforms when applied to the same quadratic potential $V(x)=x^2$ for a variety of different diffusion coefficients. The abscissa axis represents the original $x$ coordinate for the time-rescaled potential and is the transformed $y(x)$ coordinate for the Lamperti-transformed potential.
• Figure 2: (a) Black: the double-well potential $V(x)$. Red: the configuration-dependent diffusion $D(x; \alpha)$ for various $\alpha$ with $kT=1$. (b) Time-rescaled potentials after transforming to constant diffusion $D=1$ with initial diffusion $D(x; \alpha)$. By design, the time-rescaled potentials are independent of $kT$. The transform has the effect of reducing the metastability of the well, removing it entirely when $\alpha = 1$.
• Figure 3: The mean transition counts ($N_T$) vs $\alpha$ for Brownian dynamics trajectories of length $T=1000$ in a double-well potential with a constant step size $h=0.01$. Each line represents an average over $5 \times 10^3$ independent repeats. Results are shown for four different values of $kT$, equally spaced in logarithmic scale. Solid lines represent integrators for the untransformed potential with diffusion function $D(x; \alpha)$, while dotted lines represent integrators for the time-transformed potential with $D(x) = 1$.
• Figure 4: $L_1$ error in the invariant measure as a function of $kT$ for the different integrators. Solid lines correspond to integrators of the original dynamics with $\alpha = 0$ (constant, unit diffusion). Dotted lined correspond to integrators of the transformed dynamics using the $\alpha$ value that gives the minimum $L_1$ error at that temperature. The left panel shows the error for trajectories to final time $10$, where Monte Carlo error generally dominates. The right panel shows the error for trajectories to final time $1000$, where discretisation error generally dominates.
• Figure 5: Comparison of effective potentials with original diffusion $D(x) = 1 + \vert x \vert$, $V(x) = \frac{x^2}{2} + \sin(1+3x)$ and $kT=1$. The original potential is in black, the Lamperti-transformed potential is in red and the time-rescaled potential is in blue. Metastability in the potential makes this a challenging sampling problem. We observe that the Lamperti transform stiffens the potential, while the time-rescaling softens it.

Theorems & Definitions (14)

• Remark
• Example 3.1
• Remark
• Theorem D.1
• Theorem D.2
• Theorem D.3
• Theorem D.4
• Theorem D.5
• Theorem D.6
• Theorem D.7