Struct GeodesicMilstein 

pub struct GeodesicMilstein {
    pub eps: f64,
}

Geodesic Milstein: strong order 1.0 scheme via finite-difference covariant derivative.

Correction via finite-difference approximation of nabla_g(g): nabla_g g(x) ≈ (1/eps) * [PT_{y->x}(g(y)) - g(x)] where y = exp_x(eps * g(x))

Full step: v = f(x)*dt + g(x)*dW + 0.5 * nabla_g(g) * (dW^2 - dt) x_new = exp_x(v)

Requires M: ParallelTransport to compute the covariant derivative via transporting g(y) back to T_x(M) along the geodesic from y to x.

References:

• Milstein (1974), Platen & Wagner (1982) for the scalar correction.
• Said & Manton (2012) for geodesic extension to Lie groups.

Fields

eps: f64

Finite-difference step size for covariant derivative approximation.

Implementations

impl GeodesicMilstein

pub fn new() -> Self

Create with default eps = 1e-4.

pub fn step<M, D, G>( &self, sde: &ManifoldSDE<M, D, G>, x: &M::Point, t: f64, dt: f64, inc: &Increment<f64>, ) -> M::Point

where M: Manifold + ParallelTransport, D: Fn(&M::Point, f64) -> M::Tangent + Send + Sync, G: Fn(&M::Point, f64) -> M::Tangent + Send + Sync,

Advance x by one Milstein step on the manifold.

Computes the Milstein correction via finite-difference parallel transport:

1. Walk eps along g(x) to get y = exp_x(eps * g(x)).
2. Evaluate g at y.
3. Transport g(y) back from y to x via ParallelTransport.
4. Approx covariant deriv: nabla_g g ≈ (PT(g(y)) - g(x)) / eps.
5. Add Milstein correction: 0.5 * nabla_g(g) * (dW^2 - dt).
6. Apply exp to the full tangent displacement.

If transport fails (cut locus), falls back to Euler step (no correction).

Trait Implementations

impl Default for GeodesicMilstein

fn default() -> Self

Returns the “default value” for a type.

impl<M, D, G> GeoScheme<M, D, G> for GeodesicMilstein

where M: Manifold + ParallelTransport, D: Fn(&M::Point, f64) -> M::Tangent + Send + Sync, G: Fn(&M::Point, f64) -> M::Tangent + Send + Sync,

fn step_geo( &self, sde: &ManifoldSDE<M, D, G>, x: &M::Point, t: f64, dt: f64, inc: &Increment<f64>, ) -> M::Point

Advance x by one scheme step.