.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_ch3\ch3_4_2.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_ch3_ch3_4_2.py: ============================================= 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. .. GENERATED FROM PYTHON SOURCE LINES 54-192 .. code-block:: Python # 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 .. GENERATED FROM PYTHON SOURCE LINES 193-196 Divergence operator ######################################################################## .. GENERATED FROM PYTHON SOURCE LINES 196-266 .. code-block:: Python 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 .. image-sg:: /auto_ch3/images/sphx_glr_ch3_4_2_001.png :alt: ['Comparison of the outer product of the gradient to Laplace operator'] :srcset: /auto_ch3/images/sphx_glr_ch3_4_2_001.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 3.609 seconds) .. _sphx_glr_download_auto_ch3_ch3_4_2.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: ch3_4_2.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: ch3_4_2.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: ch3_4_2.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_