"""
=============================================
3.4.2 Divergence Operator
=============================================

This tutorial introduces the divergence operator in CodPy, based on the
transpose of the gradient operator. The divergence is a key concept in vector
calculus and is fundamental in the formulation of many differential equations.

Overview
--------

Let $\\nabla_k$ denote the kernel-based gradient operator defined as:

$$
\\nabla_k f_k(\\cdot) =  \\nabla_k(\\cdot) f(X), \quad \\nabla_k(\\cdot) = (\\nabla K)(\\cdot,X)( K(X,X) + \\epsilon R(X,X) )^{-1}
$$

where:
    - $X \\in \mathbb{R}^{N_x \\times D}$ is the training set,
    - $Y \\in \mathbb{R}^{N_y \\times D}$ is usually set equal to $X$,
    - $Z \\in \mathbb{R}^{N_z \\times D}$ is the evaluation set,
    - $K(X, Y)^{-1}$ is the pseudo-inverse of the kernel Gram matrix of size $\\mathbb{R}^{N_x \\times N_y}$,
    - $\\nabla_k \\in \\mathbb{R}^{D \\times N_z \\times N_y}$ is the kernel gradient with respect to $Z$.


We define the **transpose operator** $\\nabla_k^T$ as the adjoint of the
gradient operator under the standard $L^2$ inner product. This transpose
is consistent with the divergence operator and satisfies:

$$
\\langle \\nabla_k f, g \\rangle = \\langle f, \\nabla_k^T g \\rangle,
$$

for any test functions $f(X) \\in \\mathbb{R}^{N_x \\times D_f}$ and
$g(Z) \\in \\mathbb{R}^{D \\times N_z \\times D_f}$.

Divergence Operator Definition
------------------------------

To compute the transpose of the operator, we consider the structure of
the gradient operator $\\nabla_k$. Then its transpose is given by:

$$
\\nabla_k f_k(\\cdot)^T = ( K(X,X) + \\epsilon R(X,X) )^{-T} (\\nabla K)(\\cdot,X)^T,
$$

where:
    - $K(X, Y)^{-T}$ is the transpose of the inverse kernel matrix,
    - $\\nabla_k^T \\in \\mathbb{R}^{N_y \\times (N_z D)}$ is the transpose of the kernel gradient matrix.

This operator allows one to compute divergence-like quantities in RKHS, and can be used for problems involving conservation laws, variational formulations, etc.
"""

# Importing necessary modules
import os
import sys

from matplotlib import pyplot as plt

curr_f = os.path.join(os.getcwd(), "codpy-book", "utils")
sys.path.insert(0, curr_f)


import numpy as np

# from codpy.plotting import plot1D
# Lets import multi_plot function from codpy utils
from codpy.plot_utils import multi_plot


# Define the sinusoidal function
def periodic_fun(x):
    """
    A sinusoidal function that generates a sum of sines based on the input ``x``.
    """
    from math import pi

    sinss = np.cos(2 * x * pi)
    if x.ndim == 1:
        sinss = np.prod(sinss, axis=0)
        ress = np.sum(x, axis=0)
    else:
        sinss = np.prod(sinss, axis=1)
        ress = np.sum(x, axis=1)
    return ress + sinss


def nabla_my_fun(x):
    from math import pi

    import numpy as np

    sinss = np.cos(2 * x * pi)
    if x.ndim == 1:
        sinss = np.prod(sinss, axis=0)
        D = len(x)
        out = np.ones((D))

        def helper(d):
            out[d] += 2.0 * sinss * pi * np.sin(2 * x[d] * pi) / np.cos(2 * x[d] * pi)

        [helper(d) for d in range(0, D)]
    else:
        sinss = np.prod(sinss, axis=1)
        N = x.shape[0]
        D = x.shape[1]
        out = np.ones((N, D))

        def helper(d):
            out[:, d] += (
                2.0 * sinss * pi * np.sin(2 * x[:, d] * pi) / np.cos(2 * x[:, d] * pi)
            )

        [helper(d) for d in range(0, D)]
    return out


# Function to generate periodic data
def generate_periodic_data_cartesian(size_x, size_z, fun=None, nabla_fun=None):
    """
    Generates 2D structured Cartesian grid data for x and z domains,
    and evaluates a given function and optionally its gradient.

    Parameters:
    - size_x: number of points per axis for x (grid will be size_x^2)
    - size_z: number of points per axis for z (grid will be size_z^2)
    - fun: function to evaluate at each point
    - nabla_fun: optional gradient function to evaluate

    Returns:
    - x, z: 2D Cartesian grids of shape (N, 2)
    - fx, fz: function values at x and z
    - nabla_fx, nabla_fz (if nabla_fun is provided)
    """

    def cartesian_grid(size, box):
        lin = [np.linspace(box[0, d], box[1, d], size) for d in range(2)]
        X, Y = np.meshgrid(*lin)
        return np.stack([X.ravel(), Y.ravel()], axis=1)

    # Define domain boxes
    X_box = np.array([[-1, -1], [1, 1]])
    Z_box = np.array([[-1.5, -1.5], [1.5, 1.5]])

    # Generate Cartesian grids
    x = cartesian_grid(size_x, X_box)
    z = cartesian_grid(size_z, Z_box)

    # Function evaluations
    fx = fun(x).reshape(-1, 1) if fun else None
    fz = fun(z).reshape(-1, 1) if fun else None

    if nabla_fun:
        nabla_fx = nabla_fun(x)
        nabla_fz = nabla_fun(z)
        return x, fx, z, fz, nabla_fx, nabla_fz

    return x, fx, z, fz


# Lets define helper function to plot 3D projection of the function
def plot_trisurf(xfx, ax, legend="", elev=90, azim=-100, **kwargs):
    from matplotlib import cm

    """
    Helper function to plot a 3D surface using a trisurf plot.

    Parameters:
    - xfx: A tuple containing the x-coordinates (2D points) and their 
      corresponding function values.
    - ax: The matplotlib axis object for plotting.
    - legend: The legend/title for the plot.
    - elev, azim: Elevation and azimuth angles for the 3D view.
    - kwargs: Additional keyword arguments for further customization.
    """

    xp, fxp = xfx[0], xfx[1]
    x, fx = xp, fxp

    X, Y = x[:, 0], x[:, 1]
    Z = fx.flatten()
    ax.plot_trisurf(X, Y, Z, antialiased=False, cmap=cm.jet)
    ax.view_init(azim=azim, elev=elev)
    ax.title.set_text(legend)


# import CodPy's core module and Kernel class
from codpy import core
from codpy.kernel import Kernel

#########################################################################
# Divergence operator
#
#########################################################################


def fun_nablaTnabla(size_x=50, size_y=50):
    """
    Runs the experiment applying CodPy and SciPy models on the data and plots the results.

    Parameters:
    - data_x: List of generated x arrays.
    - data_fx: List of function values corresponding to each x.
    - data_z: List of generated z arrays.
    """
    # Apply CodPy and SciPy models for each (x, fx, z) pair

    x, fx, z, fz, _, nabla_fz = generate_periodic_data_cartesian(
        size_x, size_y, periodic_fun, nabla_fun=nabla_my_fun
    )

    nabla_fz = nabla_fz.reshape(-1, 2, 1)

    kernel_ptr = Kernel(
        x=x, fx=fx, set_kernel=core.kernel_setter("gaussianper", None,0, 1e-8)
    ).get_kernel()

    nabla_f_x = core.DiffOps.nabla(
        x=x,
        z=x,
        fx=fx,
        kernel_ptr=kernel_ptr,
        order=2,
        regularization=1e-8,
    )

    nabla_t_f_x = core.DiffOps.nabla_t(
        x=x,
        y=x,
        z=x,
        fz=nabla_f_x,
        kernel_ptr=kernel_ptr,
        order=2,
        regularization=1e-8,
    )

    nablaTnabla_f_x = core.DiffOps.nabla_t_nabla(
        x=x,
        y=x,
        fx=fx,
        kernel_ptr=kernel_ptr,
        order=2,
        regularization=1e-8,
    )
    multi_plot(
        [
            (x, nabla_t_f_x),
            (x, nablaTnabla_f_x),
        ],
        plot_trisurf,
        projection="3d",
        mp_max_items=4,
        mp_ncols=4,
        mp_nrows=1,
        mp_figsize=(12, 3),
        mp_title=[
            "Comparison of the outer product of the gradient to Laplace operator"
        ],
    )
    plt.show()


fun_nablaTnabla()
pass