Comparison with Finite Differences

finitediff

Finite difference approximation of parametric derivatives.

Warning

Finite difference approximations are typically computationally expensive and inaccurate. They should only be used for comparison in small test cases.

Functions:

Name	Description
finite_diff_1_forward	Forward finite difference approximation of a first order derivative
finite_diff_1_backward	Backward finite difference approximation of a first order derivative
finite_diff_1_central	Central finite difference approximation of a first order derivative
finite_diff_2	Implement second order finite differences
compute_fd_jacobian	Compute the Jacobian of the Eikonal equation w.r.t. to parameter with finite differences

eikonax.finitediff.finite_diff_1_forward

finite_diff_1_forward(
    func: Callable[[jtReal[npt.NDArray, M]], jtReal[npt.NDArray, M]],
    eval_point: jtReal[npt.NDArray, M],
    step_width: Real,
    index: int,
) -> jtReal[npt.NDArray, N]

Forward finite difference approximation of a first order derivative.

This method expects vector-valued functions \(f: \mathbb{R}^M \to \mathbb{R}^N\), and approximates the first derivative of \(f\) at a given point \(x \in \mathbb{R}^M\) with respect to the \(i\)-th component of \(x\) as

\[ \frac{\partial f(x)}{\partial x_i} \approx \frac{f(x + h e_i) - f(x)}{h} \]

with a step width \(h>0\).

Parameters:

Name	Type	Description	Default
func	Callable	Callable to use for FD computation	required
eval_point	npt.NDArray	Parameter value at which to approximate the derivative	required
step_width	Real	Step width of the finite difference	required
index	int	Vector component for which to compute partial derivative	required

Returns:

Type	Description
jtReal[npt.NDArray, N]	npt.NDArray: Partial derivative approximation

eikonax.finitediff.finite_diff_1_backward

finite_diff_1_backward(
    func: Callable[[jtReal[npt.NDArray, M]], jtReal[npt.NDArray, M]],
    eval_point: jtReal[npt.NDArray, M],
    step_width: float,
    index: int,
) -> jtReal[npt.NDArray, N]

Backward finite difference approximation of a first order derivative.

This method expects vector-valued functions \(f: \mathbb{R}^M \to \mathbb{R}^N\), and approximates the first derivative of \(f\) at a given point \(x \in \mathbb{R}^M\) with respect to the \(i\)-th component of \(x\) as

\[ \frac{\partial f(x)}{\partial x_i} \approx \frac{f(x)- f(x - h e_i)}{h} \]

with a step width \(h>0\).

Parameters:

Name	Type	Description	Default
func	Callable	Callable to use for FD computation	required
eval_point	npt.NDArray	Parameter value at which to approximate the derivative	required
step_width	Real	Step width of the finite difference	required
index	int	Vector component for which to compute partial derivative	required

Returns:

Type	Description
jtReal[npt.NDArray, N]	npt.NDArray: Partial derivative approximation

eikonax.finitediff.finite_diff_1_central

finite_diff_1_central(
    func: Callable[[jtReal[npt.NDArray, M]], jtReal[npt.NDArray, M]],
    eval_point: jtReal[npt.NDArray, M],
    step_width: float,
    index: int,
) -> jtReal[npt.NDArray, N]

Central finite difference approximation of a first order derivative.

This method expects vector-valued functions \(f: \mathbb{R}^M \to \mathbb{R}^N\), and approximates the first derivative of \(f\) at a given point \(x \in \mathbb{R}^M\) with respect to the \(i\)-th component of \(x\) as

\[ \frac{\partial f(x)}{\partial x_i} \approx \frac{f(x + h e_i)- f(x - h e_i)}{2h} \]

with a step width \(h>0\).

Parameters:

Name	Type	Description	Default
func	Callable	Callable to use for FD computation	required
eval_point	npt.NDArray	Parameter value at which to approximate the derivative	required
step_width	Real	Step width of the finite difference	required
index	int	Vector component for which to compute partial derivative	required

Returns:

Type	Description
jtReal[npt.NDArray, N]	npt.NDArray: Partial derivative approximation

eikonax.finitediff.finite_diff_2

finite_diff_2(
    func: Callable[[jtReal[npt.NDArray, M]], jtReal[npt.NDArray, M]],
    eval_point: jtReal[npt.NDArray, M],
    step_width: float,
    index_1: int,
    index_2: int,
) -> None

Implement second order finite differences.

Not implemented yet

eikonax.finitediff.run_eikonax_with_tensorfield

run_eikonax_with_tensorfield(
    parameter_vector: jtReal[npt.NDArray, M],
    eikonax_solver: solver.Solver,
    tensor_field: tensorfield.TensorField,
) -> jtReal[npt.NDArray, N]

Wrapper function for Eikonax runs.

Parameters:

Name	Type	Description	Default
parameter_vector	npt.NDArray	Parameter vector at which to compute eikonal solution	required
eikonax_solver	solver.Solver	Initialized solver object	required
tensor_field	tensorfield.TensorField	Initialized tensor field object	required

Returns:

Type	Description
jtReal[npt.NDArray, N]	npt.NDArray: Solution of the Eikonal equation

eikonax.finitediff.compute_fd_jacobian

compute_fd_jacobian(
    eikonax_solver: solver.Solver,
    tensor_field: tensorfield.TensorField,
    stencil: Callable,
    eval_point: jtReal[npt.NDArray | jax.Array, M],
    step_width: float,
) -> jtReal[npt.NDArray, "N M"]

Finite Difference Jacobian.

Compute the Jacobian of the discrete Eikonal equation solution \(\mathbf{u}\in\mathbb{R}^N\) w.r.t. to a parameter vector \(\mathbf{m}\in\mathbb{R}^M\) with finite differences.

Warning

This method should only be used for small problems.

Parameters:

Name	Type	Description	Default
eikonax_solver	solver.Solver	Initialized solver object	required
tensor_field	tensorfield.TensorField	Initialized tensor field object	required
stencil	Callable	Finite difference stencil to use for computation	required
eval_point	npt.NDArray	Parameter vector at which to approximate derivative	required
step_width	float	Step with in FD stencil	required

Returns:

Type	Description
jtReal[npt.NDArray, 'N M']	npt.NDArray: (Dense) Jacobian matrix

eikonax.finitediff.compute_fd_hessian

compute_fd_hessian(
    func: Callable,
    stencil: Callable,
    eval_point: jtReal[npt.NDArray | jax.Array, M],
    step_width: float,
) -> None

Implement finite difference Hessian computation.

Not implemented yet