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General Gradient Norm Stepper¤

exponax.stepper.generic.GeneralGradientNormStepper ¤

Bases: BaseStepper

Source code in exponax/stepper/generic/_gradient_norm.py 
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class GeneralGradientNormStepper(BaseStepper):
    linear_coefficients: tuple[float, ...]
    gradient_norm_scale: float
    dealiasing_fraction: float

    def __init__(
        self,
        num_spatial_dims: int,
        domain_extent: float,
        num_points: int,
        dt: float,
        *,
        linear_coefficients: tuple[float, ...] = (0.0, 0.0, -1.0, 0.0, -1.0),
        gradient_norm_scale: float = 1.0,
        order=2,
        dealiasing_fraction: float = 2 / 3,
        num_circle_points: int = 16,
        circle_radius: float = 1.0,
    ):
        """
        Timestepper for d-dimensional (`d ∈ {1, 2, 3}`) semi-linear PDEs
        consisting of a gradient norm nonlinearity and an arbitrary combination
        of (isotropic) linear operators.

        In 1d, the equation is given by

        ```
            uₜ + b₂ 1/2 (uₓ)² = sum_j a_j uₓˢ
        ```

        with `b₂` the gradient norm coefficient and `a_j` the coefficients of
        the linear operators. `uₓˢ` denotes the s-th derivative of `u` with
        respect to `x`. Oftentimes `b₂ = 1`.

        The number of channels is always one, no matter the number of spatial
        dimensions. The higher dimensional equation reads

        ```
            uₜ + b₂ 1/2 ‖ ∇u ‖₂² = sum_j a_j (1⋅∇ʲ)u
        ```

        The default configuration coincides with a Kuramoto-Sivashinsky equation
        in combustion format (see `exponax.stepper.KuramotoSivashinsky`). Note
        that this requires negative values (because the KS usually defines their
        linear operators on the left hand side of the equation)

        **Arguments:**

        - `num_spatial_dims`: The number of spatial dimensions `d`.
        - `domain_extent`: The size of the domain `L`; in higher dimensions
            the domain is assumed to be a scaled hypercube `Ω = (0, L)ᵈ`.
        - `num_points`: The number of points `N` used to discretize the
            domain. This **includes** the left boundary point and **excludes**
            the right boundary point. In higher dimensions; the number of points
            in each dimension is the same. Hence, the total number of degrees of
            freedom is `Nᵈ`.
        - `dt`: The timestep size `Δt` between two consecutive states.
        - `linear_coefficients` (keyword-only): The list of coefficients `a_j`
            corresponding to the derivatives. The length of this tuple
            represents the highest occuring derivative. The default value `(0.0,
            0.0, -1.0, 0.0, -1.0)` corresponds to the Kuramoto- Sivashinsky
            equation in combustion format.
        - `gradient_norm_scale` (keyword-only): The scale of the gradient
            norm term `b₂`. Default: 1.0.
        - `order`: The order of the Exponential Time Differencing Runge
            Kutta method. Must be one of {0, 1, 2, 3, 4}. The option `0` only
            solves the linear part of the equation. Use higher values for higher
            accuracy and stability. The default choice of `2` is a good
            compromise for single precision floats.
        - `dealiasing_fraction`: The fraction of the wavenumbers to keep
            before evaluating the nonlinearity. The default 2/3 corresponds to
            Orszag's 2/3 rule. To fully eliminate aliasing, use 1/2. Default:
            2/3.
        - `num_circle_points`: How many points to use in the complex contour
            integral method to compute the coefficients of the exponential time
            differencing Runge Kutta method. Default: 16.
        - `circle_radius`: The radius of the contour used to compute the
            coefficients of the exponential time differencing Runge Kutta
            method. Default: 1.0.
        """
        self.linear_coefficients = linear_coefficients
        self.gradient_norm_scale = gradient_norm_scale
        self.dealiasing_fraction = dealiasing_fraction
        super().__init__(
            num_spatial_dims=num_spatial_dims,
            domain_extent=domain_extent,
            num_points=num_points,
            dt=dt,
            num_channels=1,
            order=order,
            num_circle_points=num_circle_points,
            circle_radius=circle_radius,
        )

    def _build_linear_operator(
        self,
        derivative_operator: Complex[Array, "D ... (N//2)+1"],
    ) -> Complex[Array, "1 ... (N//2)+1"]:
        linear_operator = sum(
            jnp.sum(
                c * (derivative_operator) ** i,
                axis=0,
                keepdims=True,
            )
            for i, c in enumerate(self.linear_coefficients)
        )
        return linear_operator

    def _build_nonlinear_fun(
        self,
        derivative_operator: Complex[Array, "D ... (N//2)+1"],
    ) -> GradientNormNonlinearFun:
        return GradientNormNonlinearFun(
            self.num_spatial_dims,
            self.num_points,
            derivative_operator=derivative_operator,
            dealiasing_fraction=self.dealiasing_fraction,
            scale=self.gradient_norm_scale,
            zero_mode_fix=True,  # Todo: check this
        )
__init__ ¤
__init__(
    num_spatial_dims: int,
    domain_extent: float,
    num_points: int,
    dt: float,
    *,
    linear_coefficients: tuple[float, ...] = (
        0.0,
        0.0,
        -1.0,
        0.0,
        -1.0,
    ),
    gradient_norm_scale: float = 1.0,
    order=2,
    dealiasing_fraction: float = 2 / 3,
    num_circle_points: int = 16,
    circle_radius: float = 1.0
)

Timestepper for d-dimensional (d ∈ {1, 2, 3}) semi-linear PDEs consisting of a gradient norm nonlinearity and an arbitrary combination of (isotropic) linear operators.

In 1d, the equation is given by

    uₜ + b₂ 1/2 (uₓ)² = sum_j a_j uₓˢ

with b₂ the gradient norm coefficient and a_j the coefficients of the linear operators. uₓˢ denotes the s-th derivative of u with respect to x. Oftentimes b₂ = 1.

The number of channels is always one, no matter the number of spatial dimensions. The higher dimensional equation reads

    uₜ + b₂ 1/2 ‖ ∇u ‖₂² = sum_j a_j (1⋅∇ʲ)u

The default configuration coincides with a Kuramoto-Sivashinsky equation in combustion format (see exponax.stepper.KuramotoSivashinsky). Note that this requires negative values (because the KS usually defines their linear operators on the left hand side of the equation)

Arguments:

  • num_spatial_dims: The number of spatial dimensions d.
  • domain_extent: The size of the domain L; in higher dimensions the domain is assumed to be a scaled hypercube Ω = (0, L)ᵈ.
  • num_points: The number of points N used to discretize the domain. This includes the left boundary point and excludes the right boundary point. In higher dimensions; the number of points in each dimension is the same. Hence, the total number of degrees of freedom is Nᵈ.
  • dt: The timestep size Δt between two consecutive states.
  • linear_coefficients (keyword-only): The list of coefficients a_j corresponding to the derivatives. The length of this tuple represents the highest occuring derivative. The default value (0.0, 0.0, -1.0, 0.0, -1.0) corresponds to the Kuramoto- Sivashinsky equation in combustion format.
  • gradient_norm_scale (keyword-only): The scale of the gradient norm term b₂. Default: 1.0.
  • order: The order of the Exponential Time Differencing Runge Kutta method. Must be one of {0, 1, 2, 3, 4}. The option 0 only solves the linear part of the equation. Use higher values for higher accuracy and stability. The default choice of 2 is a good compromise for single precision floats.
  • dealiasing_fraction: The fraction of the wavenumbers to keep before evaluating the nonlinearity. The default 2/3 corresponds to Orszag's 2/3 rule. To fully eliminate aliasing, use 1/2. Default: 2/3.
  • num_circle_points: How many points to use in the complex contour integral method to compute the coefficients of the exponential time differencing Runge Kutta method. Default: 16.
  • circle_radius: The radius of the contour used to compute the coefficients of the exponential time differencing Runge Kutta method. Default: 1.0.
Source code in exponax/stepper/generic/_gradient_norm.py 
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def __init__(
    self,
    num_spatial_dims: int,
    domain_extent: float,
    num_points: int,
    dt: float,
    *,
    linear_coefficients: tuple[float, ...] = (0.0, 0.0, -1.0, 0.0, -1.0),
    gradient_norm_scale: float = 1.0,
    order=2,
    dealiasing_fraction: float = 2 / 3,
    num_circle_points: int = 16,
    circle_radius: float = 1.0,
):
    """
    Timestepper for d-dimensional (`d ∈ {1, 2, 3}`) semi-linear PDEs
    consisting of a gradient norm nonlinearity and an arbitrary combination
    of (isotropic) linear operators.

    In 1d, the equation is given by

    ```
        uₜ + b₂ 1/2 (uₓ)² = sum_j a_j uₓˢ
    ```

    with `b₂` the gradient norm coefficient and `a_j` the coefficients of
    the linear operators. `uₓˢ` denotes the s-th derivative of `u` with
    respect to `x`. Oftentimes `b₂ = 1`.

    The number of channels is always one, no matter the number of spatial
    dimensions. The higher dimensional equation reads

    ```
        uₜ + b₂ 1/2 ‖ ∇u ‖₂² = sum_j a_j (1⋅∇ʲ)u
    ```

    The default configuration coincides with a Kuramoto-Sivashinsky equation
    in combustion format (see `exponax.stepper.KuramotoSivashinsky`). Note
    that this requires negative values (because the KS usually defines their
    linear operators on the left hand side of the equation)

    **Arguments:**

    - `num_spatial_dims`: The number of spatial dimensions `d`.
    - `domain_extent`: The size of the domain `L`; in higher dimensions
        the domain is assumed to be a scaled hypercube `Ω = (0, L)ᵈ`.
    - `num_points`: The number of points `N` used to discretize the
        domain. This **includes** the left boundary point and **excludes**
        the right boundary point. In higher dimensions; the number of points
        in each dimension is the same. Hence, the total number of degrees of
        freedom is `Nᵈ`.
    - `dt`: The timestep size `Δt` between two consecutive states.
    - `linear_coefficients` (keyword-only): The list of coefficients `a_j`
        corresponding to the derivatives. The length of this tuple
        represents the highest occuring derivative. The default value `(0.0,
        0.0, -1.0, 0.0, -1.0)` corresponds to the Kuramoto- Sivashinsky
        equation in combustion format.
    - `gradient_norm_scale` (keyword-only): The scale of the gradient
        norm term `b₂`. Default: 1.0.
    - `order`: The order of the Exponential Time Differencing Runge
        Kutta method. Must be one of {0, 1, 2, 3, 4}. The option `0` only
        solves the linear part of the equation. Use higher values for higher
        accuracy and stability. The default choice of `2` is a good
        compromise for single precision floats.
    - `dealiasing_fraction`: The fraction of the wavenumbers to keep
        before evaluating the nonlinearity. The default 2/3 corresponds to
        Orszag's 2/3 rule. To fully eliminate aliasing, use 1/2. Default:
        2/3.
    - `num_circle_points`: How many points to use in the complex contour
        integral method to compute the coefficients of the exponential time
        differencing Runge Kutta method. Default: 16.
    - `circle_radius`: The radius of the contour used to compute the
        coefficients of the exponential time differencing Runge Kutta
        method. Default: 1.0.
    """
    self.linear_coefficients = linear_coefficients
    self.gradient_norm_scale = gradient_norm_scale
    self.dealiasing_fraction = dealiasing_fraction
    super().__init__(
        num_spatial_dims=num_spatial_dims,
        domain_extent=domain_extent,
        num_points=num_points,
        dt=dt,
        num_channels=1,
        order=order,
        num_circle_points=num_circle_points,
        circle_radius=circle_radius,
    )
__call__ ¤
__call__(
    u: Float[Array, "C ... N"]
) -> Float[Array, "C ... N"]

Perform one step of the time integration for a single state.

Arguments:

  • u: The state vector, shape (C, ..., N,).

Returns:

  • u_next: The state vector after one step, shape (C, ..., N,).

Tip

Use this call method together with exponax.rollout to efficiently produce temporal trajectories.

Info

For batched operation, use jax.vmap on this function.

Source code in exponax/_base_stepper.py 
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def __call__(
    self,
    u: Float[Array, "C ... N"],
) -> Float[Array, "C ... N"]:
    """
    Perform one step of the time integration for a single state.

    **Arguments:**

    - `u`: The state vector, shape `(C, ..., N,)`.

    **Returns:**

    - `u_next`: The state vector after one step, shape `(C, ..., N,)`.

    !!! tip
        Use this call method together with `exponax.rollout` to efficiently
        produce temporal trajectories.

    !!! info
        For batched operation, use `jax.vmap` on this function.
    """
    expected_shape = (self.num_channels,) + spatial_shape(
        self.num_spatial_dims, self.num_points
    )
    if u.shape != expected_shape:
        raise ValueError(
            f"Expected shape {expected_shape}, got {u.shape}. For batched operation use `jax.vmap` on this function."
        )
    return self.step(u)