[Paper Review] Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems

Markos A. Katsoulakis, Petr Plecháč+2 more|ArXiv.org|Aug 1, 2006

Advanced Mathematical Modeling in Engineering22 references4 citations

TL;DR

This paper develops adaptive coarse-graining schemes for stochastic lattice systems using cluster expansion around an initial Monte Carlo-computable approximation, with a posteriori error estimates derived from relative entropy. The method improves predictions of critical behavior and hysteresis in systems with intermediate and long-range interactions, and significantly enhances accuracy over prior schemes for short-range interactions by incorporating higher-order corrections to the effective Hamiltonian.

ABSTRACT

The primary objective of this work is to develop coarse-graining schemes for stochastic many-body microscopic models and quantify their effectiveness in terms of a priori and a posteriori error analysis. In this paper we focus on stochastic lattice systems of interacting particles at equilibrium. %such as Ising-type models. The proposed algorithms are derived from an initial coarse-grained approximation that is directly computable by Monte Carlo simulations, and the corresponding numerical error is calculated using the specific relative entropy between the exact and approximate coarse-grained equilibrium measures. Subsequently we carry out a cluster expansion around this first-and often inadequate-approximation and obtain more accurate coarse-graining schemes. The cluster expansions yield also sharp a posteriori error estimates for the coarse-grained approximations that can be used for the construction of adaptive coarse-graining methods. We present a number of numerical examples that demonstrate that the coarse-graining schemes developed here allow for accurate predictions of critical behavior and hysteresis in systems with intermediate and long-range interactions. We also present examples where they substantially improve predictions of earlier coarse-graining schemes for short-range interactions.

Motivation & Objective

• To develop computationally efficient coarse-graining schemes for stochastic many-body systems at equilibrium.
• To quantify the accuracy of coarse-grained approximations using a posteriori error estimates based on relative entropy.
• To improve upon first-order coarse-grained approximations by applying cluster expansion to higher-order corrections.
• To enable adaptive coarse-graining by providing sharp, computable error bounds for refinement decisions.
• To accurately predict critical phenomena such as phase transitions and hysteresis in systems with long- and intermediate-range interactions.

Proposed method

• Start with a first-order coarse-grained Hamiltonian, \bar{H}^{(0)}_M(\eta), computable via Monte Carlo simulations by averaging over microscopic configurations given coarse-cell magnetization \eta.
• Use the conditional probability measure PN(dσ|η) to reformulate the exact coarse-grained Hamiltonian \bar{H}_M(\eta) as a perturbation of \bar{H}^{(0)}_M(\eta).
• Apply cluster expansion techniques (based on polymer models) to derive a series expansion: \bar{H}_M(\eta) = \bar{H}^{(0)}_M(\eta) + \bar{H}^{(1)}_M(\eta) + \cdots + \bar{H}^{(p)}_M(\eta) + O(\epsilon^{p+1}).
• Treat the small parameter \epsilon (dependent on coarse-graining ratio q, interaction range L, and inverse temperature β) as a control for convergence and error estimation.
• Derive sharp a posteriori error estimates using relative entropy between the exact and approximate coarse-grained Gibbs measures.
• Use the error estimates to guide adaptive refinement of the coarse-graining scheme, particularly in regions of high error or near criticality.

Experimental results

Research questions

• RQ1 How can one systematically improve a first-order coarse-grained approximation for stochastic lattice systems using perturbative corrections?
• RQ2 What is the role of cluster expansion in deriving higher-order corrections to the effective Hamiltonian in the context of coarse-graining?
• RQ3 Can a posteriori error estimates based on relative entropy be used to construct adaptive coarse-graining methods with guaranteed error tolerance?
• RQ4 How do higher-order corrections affect the prediction of critical behavior and hysteresis in systems with short-, intermediate-, and long-range interactions?
• RQ5 To what extent do the derived schemes outperform existing coarse-graining methods in capturing phase transitions and metastable states?

Key findings

• The inclusion of third-order corrections in the effective Hamiltonian eliminates spurious hysteresis in the Ising model with nearest-neighbor interactions, even at high coarse-graining ratios.
• For intermediate-range interactions (e.g., L = 8, q = 8), the 2nd-order approximation fails to capture the correct transition between phases, but the 3rd-order correction restores accurate prediction of hysteresis and critical behavior.
• In long-range interaction regimes (e.g., L = 32, q = 32), the 2nd-order approximation already provides good agreement with the mean-field solution, and the 3rd-order scheme yields nearly identical results.
• Even when the small parameter \epsilon = q/L is not small (e.g., q = 32, L = 8), the higher-order corrections significantly improve accuracy, indicating robustness beyond asymptotic limits.
• The a posteriori error estimates derived from relative entropy are computable and effective in identifying regions where coarse-graining error is high, enabling adaptive refinement strategies.
• Numerical simulations show that the exact solution and fully resolved microscopic simulations are visually indistinguishable (q = 1), validating the accuracy of the coarse-grained approximations at all tested levels of coarsening.