Struct VariationalProblem 

pub struct VariationalProblem;

Variational problem solver: Euler-Lagrange equations and constrained minimization with Lagrange multipliers.

Implementations

impl VariationalProblem

pub fn euler_lagrange_residual<L>(lagrangian: L, u: &[f64], dx: f64) -> Vec<f64>

where L: Fn(f64, f64, f64) -> f64,

Compute the Euler-Lagrange residual for the functional J\[u\] = ∫ L(x, u, u') dx.

The Lagrangian L takes (x, u, u_prime). Returns the residual ∂L/∂u - d/dx(∂L/∂u') evaluated at each interior grid point using finite differences.

pub fn augmented_lagrangian<J, G>( j: J, g: G, u0: &[f64], rho: f64, step: f64, max_iter: usize, ) -> (Vec<f64>, f64)

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

Minimize the functional J\[u\] subject to equality constraint G\[u\] = 0 using the augmented Lagrangian method.

Returns (u, lambda) where u is the minimizer and lambda is the Lagrange multiplier estimate.

pub fn first_variation(u: &[f64], v: &[f64], f_prime: &[f64], dx: f64) -> f64

Compute the first variation of J\[u\] = ∫ f(u(x)) dx at u in direction v.

f_prime is the derivative of the integrand w.r.t. u.

pub fn steepest_descent<J>( j: J, u0: &[f64], step: f64, tol: f64, max_iter: usize, ) -> (Vec<f64>, Vec<f64>)

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

Find a stationary point via steepest descent on the functional gradient.

Returns the stationary point and the history of functional values.