.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_ch3\ch3_4_4.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_4.py: =================================================== 3.4.4 Integral operator - inverse gradient operator =================================================== Given a kernel gradient operator ∇ₖ, CodPy defines an integral-type inverse operator denoted by: $$ \nabla_k^{-1} = \Delta_k^{-1} \nabla_k^T \in \mathbb{R}^{N_x \times D N_z} $$ **Matrix Interpretation** To compute this operator, the gradient tensor $$ \nabla_k(\cdot, X) \in \mathbb{R}^{D \times N_z \times N_x} $$ is reshaped into a matrix of shape $\mathbb{R}^{D N_z \times N_x}$. Then, its transpose is multiplied by the inverse Laplacian $\Delta_k^{-1}$ to obtain $\nabla_k^{-1}$. This operator acts on a vector field $v_z \in \mathbb{R}^{D \times N_z \times D_{v_z}}$ and returns: $$ \nabla_k^{-1}(\cdot, X) \cdot v_z \in \mathbb{R}^{N_x \times D_{v_z}}. $$ ** Least-Squares Formulation ** Conceptually, this operation solves the following minimization problem: $$ \bar{h} = \arg \min_{h \in \mathbb{R}^{N_x \times D_{v_z}}} \| \nabla_k h - v_z \|_{\ell^2}^2 $$ That is, it finds the best function $h$ whose kernel gradient approximates a given vector field $v_z$. **Example: 2D Case** In 2D, we can verify the behavior of the inverse by checking whether the composition: $$ \nabla_k^{-1}(\cdot, X) \cdot \nabla_k(\cdot, X) f(X) $$ recovers the original function $f(X)$. This test confirms whether $\nabla_k^{-1} \nabla_k$ approximates the identity. **Extrapolation** We can also evaluate generalization by applying: $$ \nabla_k^{-1}(\cdot, Z) \cdot \nabla_k(\cdot, X) f(X) $$ This measures how well the inverse-gradient operator extrapolates from $X$ to unseen points $Z$. .. GENERATED FROM PYTHON SOURCE LINES 61-199 .. 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 200-203 Integral operator - inverse gradient operator ######################################################################## .. GENERATED FROM PYTHON SOURCE LINES 203-271 .. code-block:: Python def fun_NablainvNabla1(size_x=50, size_y=50): 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("tensornorm", "scale_to_unitcube") ).get_kernel() fz_inv = core.DiffOps.nabla_inv( x=x, y=x, z=x, kernel_ptr=kernel_ptr, fz=core.DiffOps.nabla(x=x, y=x, z=x, fx=fx, kernel_ptr=kernel_ptr), ) multi_plot( [(x, fx), (x, fz_inv)], plot_trisurf, projection="3d", mp_nrows=1, mp_figsize=(12, 3), mp_titles=[ "Comparison between original function to the product of the gradient operator and its inverse" ], ) plt.show() fun_NablainvNabla1() def fun_NablainvNabla2(size_x=50, size_y=50): 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("tensornorm", "scale_to_unitcube") ).get_kernel() fz_inv = core.DiffOps.nabla_inv( x=x, y=x, z=z, kernel_ptr=kernel_ptr, fz=core.DiffOps.nabla(x=x, y=x, z=z, fx=fx, kernel_ptr=kernel_ptr), ) multi_plot( [(x, fx), (x, fz_inv)], plot_trisurf, projection="3d", mp_nrows=1, mp_figsize=(12, 3), mp_titles=[ "Comparison between original function to the product of the inverse of the gradient operator and the gradient operator" ], ) plt.show() fun_NablainvNabla2() .. rst-class:: sphx-glr-horizontal * .. image-sg:: /auto_ch3/images/sphx_glr_ch3_4_4_001.png :alt: ch3 4 4 :srcset: /auto_ch3/images/sphx_glr_ch3_4_4_001.png :class: sphx-glr-multi-img * .. image-sg:: /auto_ch3/images/sphx_glr_ch3_4_4_002.png :alt: ch3 4 4 :srcset: /auto_ch3/images/sphx_glr_ch3_4_4_002.png :class: sphx-glr-multi-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 2.915 seconds) .. _sphx_glr_download_auto_ch3_ch3_4_4.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_4.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: ch3_4_4.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: ch3_4_4.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_