{ "cells": [ { "cell_type": "markdown", "id": "b7a59aec", "metadata": {}, "source": [ "# 5 · Solving the Minimization: Newton vs. Gradient Descent\n", "\n", "Time to open the solver box. We pin a beam's left edge, grab its **whole right\n", "edge** as a handle, fix a target down-and-to-the-right, and drive the beam to\n", "equilibrium from rest with two methods:\n", "\n", "* **Gradient descent** — step along $-\\nabla E$.\n", "* **Newton** — step along $-H^{-1}\\nabla E$, using curvature.\n", "\n", "Per iteration we track the energy, the gradient norm $\\lVert\\nabla E\\rVert$,\n", "and the **Newton decrement** $\\lambda = \\sqrt{\\nabla E^\\top H^{-1}\\nabla E}$." ] }, { "cell_type": "code", "execution_count": 1, "id": "7cd33993", "metadata": { "execution": { "iopub.execute_input": "2026-06-09T05:22:29.598441Z", "iopub.status.busy": "2026-06-09T05:22:29.598300Z", "iopub.status.idle": "2026-06-09T05:22:30.100818Z", "shell.execute_reply": "2026-06-09T05:22:30.100523Z" } }, "outputs": [], "source": [ "%matplotlib inline\n", "import numpy as np\n", "import scipy as sp\n", "import matplotlib.pyplot as plt\n", "import simkit\n", "import utils" ] }, { "cell_type": "markdown", "id": "b8241de3", "metadata": {}, "source": [ "## The objective\n", "\n", "Elastic + pinned-left-edge + right-edge handle, all stable Neo-Hookean. We keep\n", "the **handle spring soft** (similar magnitude to the elastic forces) so the\n", "problem is not horribly ill-conditioned — otherwise gradient descent just\n", "yanks the handle and leaves the rest of the beam untouched.\n", "\n", "`BeamObjective` is built like every other simulator class: `precompute` runs\n", "once for a **soft beam** (`ym=0.5, pr=0.45`), and it composes the Neo-Hookean\n", "material, a gentle **pin** spring (`K=1e2`), and a soft **handle** spring\n", "(`K=25`). The `energy / gradient / hessian` methods read like the math, summing\n", "their terms **one per line** — exactly what the solver loop wants." ] }, { "cell_type": "code", "execution_count": 2, "id": "8d157bb0", "metadata": { "execution": { "iopub.execute_input": "2026-06-09T05:22:30.102067Z", "iopub.status.busy": "2026-06-09T05:22:30.101980Z", "iopub.status.idle": "2026-06-09T05:22:30.104252Z", "shell.execute_reply": "2026-06-09T05:22:30.104034Z" } }, "outputs": [], "source": [ "class BeamObjective:\n", " \"\"\"elastic(x) + pinned-left-edge spring + right-edge handle spring.\"\"\"\n", "\n", " def __init__(self, X, T):\n", " self.p = utils.precompute(X, T, ym=0.5, pr=0.45) # soft beam\n", " self.psi = utils.make_material(\"Neo-Hookean\") # elastic term\n", " self.pin = utils.PenaltySpring(self.p.n, self.p.dim, K=1e2) # gentle pin\n", " self.handle = utils.PenaltySpring(self.p.n, self.p.dim, K=25.0) # soft handle\n", "\n", " def set_pin(self, idx, targets): self.pin.set(idx, targets); return self\n", " def set_handle(self, idx, targets): self.handle.set(idx, targets); return self\n", "\n", " def energy(self, x):\n", " E_elastic = self.psi.energy(x, self.p)\n", " E_pin = self.pin.energy(x)\n", " E_handle = self.handle.energy(x)\n", " return E_elastic + E_pin + E_handle\n", "\n", " def gradient(self, x):\n", " g_elastic = self.psi.gradient(x, self.p)\n", " g_pin = self.pin.gradient(x)\n", " g_handle = self.handle.gradient(x)\n", " return g_elastic + g_pin + g_handle\n", "\n", " def hessian(self, x):\n", " H_elastic = self.psi.hessian(x, self.p)\n", " H_pin = self.pin.hessian(x)\n", " H_handle = self.handle.hessian(x)\n", " return H_elastic + H_pin + H_handle" ] }, { "cell_type": "markdown", "id": "e114406a", "metadata": {}, "source": [ "Build the objective **once**: pin the left edge, and aim the\n", "right-edge handle down-and-to-the-right. Both solvers minimize this *same*\n", "objective from the *same* rest state — the only thing that changes is the\n", "search direction." ] }, { "cell_type": "code", "execution_count": 3, "id": "b8025de0", "metadata": { "execution": { "iopub.execute_input": "2026-06-09T05:22:30.105189Z", "iopub.status.busy": "2026-06-09T05:22:30.105135Z", "iopub.status.idle": "2026-06-09T05:22:30.108974Z", "shell.execute_reply": "2026-06-09T05:22:30.108782Z" } }, "outputs": [], "source": [ "X, T = utils.triangulated_grid(nx=12, ny=4, width=2.0, height=0.4)\n", "n, dim = X.shape\n", "\n", "pin_idx = np.where(X[:, 0] <= X[:, 0].min() + 1e-6)[0]\n", "right_idx = np.where(X[:, 0] >= X[:, 0].max() - 1e-6)[0]\n", "targets = X[right_idx] + np.array([0.5, -0.7]) # pull down-and-right\n", "\n", "beam = BeamObjective(X, T).set_pin(pin_idx, X[pin_idx]).set_handle(right_idx, targets)" ] }, { "cell_type": "markdown", "id": "90052bad", "metadata": {}, "source": [ "## The two solver loops\n", "\n", "They differ only in the **search direction**. `run_solver` is a thin\n", "solver-iteration loop: given a direction rule it records the energy, gradient\n", "norm, and Newton decrement per iteration and returns the deformed states. Both\n", "use a backtracking line search (tutorial 6)." ] }, { "cell_type": "code", "execution_count": 4, "id": "1516ab18", "metadata": { "execution": { "iopub.execute_input": "2026-06-09T05:22:30.109863Z", "iopub.status.busy": "2026-06-09T05:22:30.109815Z", "iopub.status.idle": "2026-06-09T05:22:30.149748Z", "shell.execute_reply": "2026-06-09T05:22:30.149557Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Newton: 11 iters, final |grad| = 2.57e-11\n", "Gradient Descent: 60 iters, final |grad| = 2.01e-01\n" ] } ], "source": [ "def run_solver(direction, n_iters=60):\n", " \"\"\"direction(g, H) -> step. Returns (states, metrics).\"\"\"\n", " x = X.flatten().reshape(-1, 1).copy()\n", " states, E, G, DEC = [x.reshape(n, dim).copy()], [], [], []\n", " for _ in range(n_iters):\n", " g, H = beam.gradient(x), beam.hessian(x)\n", " dx = direction(g, H)\n", " dec = float(np.sqrt(max(-(g.T @ dx)[0, 0], 0.0))) # sqrt(g^T H^-1 g) for Newton\n", " alpha, _, _ = simkit.backtracking_line_search(beam.energy, x, g, dx)\n", " x = x + alpha * dx\n", " E.append(beam.energy(x)); G.append(float(np.linalg.norm(g))); DEC.append(dec)\n", " states.append(x.reshape(n, dim).copy())\n", " if np.linalg.norm(alpha * dx) < 1e-9:\n", " break\n", " return states, {\"energy\": E, \"grad\": G, \"decrement\": DEC}\n", "\n", "newton_dir = lambda g, H: sp.sparse.linalg.spsolve(H.tocsc(), -g).reshape(-1, 1)\n", "gd_dir = lambda g, H: -g.reshape(-1, 1)\n", "\n", "newton_states, newton_metrics = run_solver(newton_dir)\n", "gd_states, gd_metrics = run_solver(gd_dir)\n", "print(f\"Newton: {len(newton_metrics['energy'])} iters, final |grad| = {newton_metrics['grad'][-1]:.2e}\")\n", "print(f\"Gradient Descent: {len(gd_metrics['energy'])} iters, final |grad| = {gd_metrics['grad'][-1]:.2e}\")" ] }, { "cell_type": "markdown", "id": "bd637ba4", "metadata": {}, "source": [ "## Convergence curves\n", "\n", "Newton's quadratic convergence crushes the gradient norm in a handful of steps;\n", "gradient descent crawls on the stiff, ill-conditioned elastic Hessian. (GD never\n", "forms $H$, so it has no Newton decrement.)" ] }, { "cell_type": "code", "execution_count": 5, "id": "18791a41", "metadata": { "execution": { "iopub.execute_input": "2026-06-09T05:22:30.150702Z", "iopub.status.busy": "2026-06-09T05:22:30.150651Z", "iopub.status.idle": "2026-06-09T05:22:30.292262Z", "shell.execute_reply": "2026-06-09T05:22:30.292043Z" } }, "outputs": [ { "data": { "image/png": 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig, _ = utils.convergence_plot(\n", " {\"Newton\": newton_metrics, \"Gradient Descent\": gd_metrics},\n", " title=\"Newton vs. Gradient Descent on the same problem\")\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "096b3226", "metadata": {}, "source": [ "## Watching the beam converge (one frame per iteration)" ] }, { "cell_type": "code", "execution_count": 6, "id": "7cbe8341", "metadata": { "execution": { "iopub.execute_input": "2026-06-09T05:22:30.293270Z", "iopub.status.busy": "2026-06-09T05:22:30.293208Z", "iopub.status.idle": "2026-06-09T05:22:30.475799Z", "shell.execute_reply": "2026-06-09T05:22:30.475560Z" } }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "lims = ((-1.2, 1.8), (-1.3, 0.6))\n", "fig, anim = utils.animate_mesh(newton_states, T, lims=lims, title=\"Newton's method\",\n", " pin_pts=X[pin_idx], handle_traj=[s[right_idx] for s in newton_states],\n", " target_pts=[targets] * len(newton_states), fps=6)\n", "utils.show_video(fig, anim, \"media/05_newton.mp4\", fps=6)" ] }, { "cell_type": "code", "execution_count": 7, "id": "269b687e", "metadata": { "execution": { "iopub.execute_input": "2026-06-09T05:22:30.476891Z", "iopub.status.busy": "2026-06-09T05:22:30.476826Z", "iopub.status.idle": "2026-06-09T05:22:31.275673Z", "shell.execute_reply": "2026-06-09T05:22:31.275387Z" } }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "fig, anim = utils.animate_mesh(gd_states, T, lims=lims, title=\"Gradient descent\",\n", " pin_pts=X[pin_idx], handle_traj=[s[right_idx] for s in gd_states],\n", " target_pts=[targets] * len(gd_states), fps=10)\n", "utils.show_video(fig, anim, \"media/05_gradient_descent.mp4\", fps=10)" ] }, { "cell_type": "markdown", "id": "09bf7693", "metadata": {}, "source": [ "### Takeaways\n", "* **Newton** uses curvature and converges *quadratically* — ideal for stiff\n", " elasticity.\n", "* **Gradient descent** ignores curvature and stalls.\n", "* The **Newton decrement** is a clean, scale-aware stopping measure.\n", "\n", "But the Newton *step length* needs care — that's tutorial 6." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.13.9" } }, "nbformat": 4, "nbformat_minor": 5 }