Struct FunctionalDerivative 

pub struct FunctionalDerivative;

Gâteaux and Fréchet derivatives of functionals on function spaces.

Enables gradient descent in function space and Newton’s method for operator equations.

Implementations

impl FunctionalDerivative

pub fn gateaux<F>(j: F, f: &[f64], h: &[f64], eps: f64) -> f64

where F: Fn(&[f64]) -> f64,

Compute the Gâteaux derivative of functional J at f in direction h.

dJ/dε J(f + εh)|_{ε=0} approximated by finite difference.

pub fn frechet_gradient<F>(j: F, f: &[f64], eps: f64) -> Vec<f64>

where F: Fn(&[f64]) -> f64,

Compute the Fréchet derivative (gradient) of functional J at f.

Returns the gradient vector δJ/δf using component-wise Gâteaux derivatives with standard basis directions.

pub fn gradient_descent<F>( j: F, f0: &[f64], step_size: f64, max_iter: usize, eps: f64, ) -> (Vec<f64>, f64)

where F: Fn(&[f64]) -> f64,

Perform gradient descent in function space to minimize functional J.

Starting from f0, takes max_iter steps of size step_size. Returns the minimizer and the final functional value.

pub fn newton_step<F>(j: F, f: &[f64], eps: f64) -> Vec<f64>

where F: Fn(&[f64]) -> f64,

Newton step in function space: f_new = f - \[D²J(f)\]⁻¹ DJ(f).

Approximates the Hessian-vector product via a second finite difference and solves the linear system by a simple diagonal preconditioned gradient step (one step of Jacobi iteration).

pub fn second_gateaux<F>(j: F, f: &[f64], h: &[f64], k: &[f64], eps: f64) -> f64

where F: Fn(&[f64]) -> f64,

Compute the second Gâteaux derivative (bilinear form) at f in directions h and k.