Ordered Line Integral Methods (OLIM)

• OLIM is a family of numerical solvers that minimizes local geometric action via quadrature updates and Dijkstra-like label-setting to solve Hamilton–Jacobi equations.
• They achieve high accuracy and efficiency in computing quasi-potentials, rare event transition pathways, and front propagation in systems with anisotropic diffusion.
• OLIM employs advanced local variational updates and discretization strategies, ensuring provable convergence and reduced computational errors.

Ordered Line Integral Methods (OLIM) constitute a family of numerical solvers for Hamilton–Jacobi equations that arise in the computation of the quasi-potential in small-noise stochastic differential equations (SDEs) and in front propagation problems, such as the eikonal equation. OLIM combines the causality-driven, label-setting approach of Dijkstra-like algorithms with the direct minimization of geometric action functionals via local quadrature-based variational updates. This methodology yields provably convergent, efficient, and high-accuracy solvers for capturing rare event transition pathways, invariant measures, and wavefront arrival times in systems with complex dynamics, including those with variable anisotropic diffusion.

1. Mathematical Foundation and Quasi-potential Formulation

The prototypical setting for OLIM is the analysis of perturbed dynamical systems

$$dx = b(x)\,dt + \sigma(x)\sqrt{\epsilon}\,dW, \quad x\in\Omega\subset\mathbb{R}^2, \quad \epsilon\ll1,$$

where $b$ is a smooth drift, $\sigma(x)$ is a nondegenerate diffusion matrix, and $W$ is standard Brownian motion. In Freidlin–Wentzell large deviation theory, the quasi-potential $U_E(x)$ associated with an attractor $E$ is the minimal action

$$U_E(x) = \inf_{\substack{T>0,\, \phi(0)\in E,\, \phi(T) = x}} S_T[\phi],$$

where

$$S_T[\phi] = \frac{1}{2} \int_0^T \|\dot{\phi}(t) - b(\phi(t))\|_{A(\phi(t))}^2 dt, \qquad A(x) = [\sigma(x)\sigma(x)^T]^{-1}.$$

Minimizing also over $T$, one obtains a geometric action

$$S[\psi] = \int_0^L \left( \|b(\psi(s))\|_{A(\psi)} \|\psi'(s)\|_{A(\psi)} - \langle \psi'(s), b(\psi(s)) \rangle_{A(\psi)} \right)\,ds,$$

where $b$0 is arc-length and all metric operations are with respect to $b$1. The quasi-potential $b$2 satisfies the Hamilton–Jacobi equation

$b$3

For the isotropic eikonal equation—another principal application—OLIM solves

$b$4

and its factored variant via analogous geometric action principles (Dahiya et al., 2017, Dahiya et al., 2018, Potter et al., 2019).

2. Discretization Strategy and Label-setting Framework

OLIM employs a regular mesh, typically rectangular in $b$5 or $b$6, where each grid point is assigned a label:

• Far (unknown, not yet tentatively updated),
• Considered (tentative value, neighbor to at least one AcceptedFront point),
• AcceptedFront (value finalized; at least one considered neighbor),
• Accepted (finalized; no considered neighbors).

Propagation proceeds by always accepting the Considered point with minimal $b$7, updating its neighbors within a fixed update radius $b$8 $b$9 mesh size$\sigma(x)$0, and strictly advancing in order of increasing $\sigma(x)$1, which guarantees causality and monotonicity. This is analogous to the Fast Marching/Ordered Upwind Method, but OLIM replaces upwind finite difference stencils with direct local minimization of the geometric action functional (Dahiya et al., 2017, Dahiya et al., 2018, Potter et al., 2019).

3. Local Variational Updates and Quadrature Formulations

At the core of OLIM is the local update, formulated as an action/minimization along straight segments or triangles:

• One-point (line) update: For a pair $\sigma(x)$2, the cost is

$\sigma(x)$3

Common quadrature choices include: - Midpoint: Employs $\sigma(x)$4 and $\sigma(x)$5 at $\sigma(x)$6; order $\sigma(x)$7 for segment length $\sigma(x)$8. - Right-hand: Uses $\sigma(x)$9; order $W$0. - Trapezoid, Simpson: Higher-order, yielding further improved accuracy for smooth fields.

• Two-point (triangle) update: Given $W$1, the update solves

$W$2

with $W$3. The minimization is performed either analytically or by a root-finding method for the nonlinear equation. This triangle-update increases directional coverage and accuracy.

For the eikonal case and general Hamilton–Jacobi class, OLIM generalizes using local cost functionals $W$4 over $W$5-simplexes $W$6, minimizing

$W$7

with prescribed quadrature rule (Potter et al., 2019).

4. Algorithmic Complexity, Parameter Selection, and Error Analysis

The global complexity of OLIM on an $W$8 grid is $W$9 in two dimensions. Each mesh point may require $U_E(x)$0 neighbor updates due to the update radius $U_E(x)$1. Efficient hierarchical update strategies, such as prioritizing one-point over two-point (triangle) updates and applying top-down/bottom-up simplex enumeration in 3D, yield significant constant-factor savings (Potter et al., 2019, Dahiya et al., 2018). Memory usage is $U_E(x)$2, dominated by storage for $U_E(x)$3, state labels, and the heap.

Selection of the update factor $U_E(x)$4 is guided by error analysis: too small $U_E(x)$5 leads to missed nonlocal characteristics; too large increases computational cost and may reduce efficiency due to excessive quadrature/curvature error. Rules-of-thumb, empirically validated, are:

• For OLIM-MID (midpoint quadrature) or higher: $U_E(x)$6 for $U_E(x)$7, effective for moderate anisotropy ($U_E(x)$8). Higher anisotropy may require up to $U_E(x)$9 increase in $E$0.
• For OLIM-R (right-hand): $E$1 (Dahiya et al., 2017, Dahiya et al., 2018).

Theoretical and empirical error rates: OLIM-MID, OLIM-TR, OLIM-SIM achieve superlinear convergence in practice (exponents $E$2) with error constants mildly dependent on anisotropy; OLIM-R and OUM are typically first-order or slightly sublinear (Dahiya et al., 2017, Dahiya et al., 2018).

5. Extension to Variable and Anisotropic Diffusion

OLIM extends to SDEs with general position-dependent and anisotropic diffusion via the matrix $E$3, entering all norms and inner products in the geometric action. For constant anisotropy models $E$4, the induced metric affects both numerical errors and the geometry of transition paths. The local metric unit ball becomes an ellipse, and OLIM’s error and the structure of quasi-potential level sets depend sensitively on the anisotropy ratio and orientation.

OLIM has been demonstrated in nonlinear, variable-diffusion models using closed-form or numerically constructed action-minimizers, and supports both global quasi-potential surface computation and pathwise transition analysis (instantons), even in biologically realistic models (Dahiya et al., 2018).

6. Benchmarks, Applications, and Extensions

Key benchmarks for OLIM include:

• Linear SDEs with analytic $E$5 under constant anisotropic diffusion (error exponents $E$6; optimal $E$7 mildly increasing with anisotropy) (Dahiya et al., 2018, Dahiya et al., 2017).
• Nonlinear SDEs with variable diffusion, designed for analytic comparison.
• Complex systems:
  • Maier–Stein model: OLIM computes quasi-potentials and instantons on fine grids (e.g., $E$8), revealing strong deformation of level sets and transition paths under anisotropy.
  • Genetic toggle switch in λ-phage: OLIM captures position-dependent transitions and quantitative rate constants, revealing up to $E$9 change due to diffusion structure (Dahiya et al., 2018).

In eikonal and front propagation contexts, OLIM offers alternatives to fast marching, providing higher accuracy per CPU and supporting additively factored equations for singularity handling (Potter et al., 2019).

Recent and ongoing extensions include:

• Three-dimensional OLIM, employing efficient simplex pruning strategies (top-down, bottom-up, KKT-driven) for computational scalability (Potter et al., 2019).
• Higher-order quadrature rules (Simpson, Gaussian) for smoother fields.
• Adaptive mesh refinement near critical regions.
• Coupling with path-based solvers for multiscale approaches.

7. Relationship to Previous Methods and Theoretical Properties

A key theoretical property is the equivalence of OLIM-R (right-hand quadrature) and the Ordered Upwind Method for local updates under appropriate interpolation, establishing continuity with classical Dijkstra-like and upwind PDE schemes. OLIM’s variational structure directly exploits the dynamic programming principle, replacing finite-difference Hamilton–Jacobi updates by more accurate minimizations of the local geometric action (Dahiya et al., 2017, Dahiya et al., 2018).

Causality and monotonicity of the updates are mathematically guaranteed under acuteness conditions on simplices (all pairwise directions make angles ≤ 90°); theoretical error bounds are rigorously linked to quadrature rule order, curvature of minimizers, and update radius (Dahiya et al., 2017, Potter et al., 2019). Convexity and strict minima for the local action minimization are proven under mild regularity for the Lagrangian.

OLIM yields viscosity solutions to the underlying Hamilton–Jacobi equations, and Dijkstra-like label-setting ensures global causal ordering. In practice, higher-order OLIM solvers achieve orders-of-magnitude smaller error for fixed computational budget compared to upwind finite-difference methods.

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For detailed proofs, code listings, and comprehensive numerical data, see the original works: (Dahiya et al., 2017, Dahiya et al., 2018), and (Potter et al., 2019).