Struct GeodesicSRI 

pub struct GeodesicSRI {
    pub eps: f64,
}

Geodesic SRI: strong order 1.5 approximation for manifold SDEs.

Extends GeodesicMilstein with the dZ iterated-integral correction term.

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

where nabla_g(g) is approximated by finite-difference parallel transport: nabla_g g(x) ≈ (1/eps) * [PT_{y->x}(g(y)) - g(x)], y = exp_x(eps*g(x))

Single-FD note

This uses nabla_g g for both the Milstein correction and the dZ term. Full SRI1 would require a second PT-based FD for nabla_g(nabla_g g). This approximation is O(dt^{3/2}) accurate for smooth diffusion fields.

Fields

eps: f64

Finite-difference step size for covariant derivative approximation.

Implementations

impl GeodesicSRI

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, M::Tangent: Add<Output = M::Tangent> + Mul<f64, Output = M::Tangent> + Sub<Output = M::Tangent> + Clone,

Advance x by one SRI step on the manifold.

Computes the Milstein and dZ corrections 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. Milstein correction: 0.5 * nabla_g(g) * (dW^2 - dt).
6. SRI correction: nabla_g(g) * dZ.
7. Apply exp to the full tangent displacement.

If transport fails (cut locus), falls back to Euler step.

Trait Implementations

impl Default for GeodesicSRI

fn default() -> Self

Returns the “default value” for a type.

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

where M: Manifold + ParallelTransport, D: Fn(&M::Point, f64) -> M::Tangent + Send + Sync, G: Fn(&M::Point, f64) -> M::Tangent + Send + Sync, M::Tangent: Add<Output = M::Tangent> + Mul<f64, Output = M::Tangent> + Sub<Output = M::Tangent> + Clone,

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.