{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Solving adjoint equations using `firedrake-adjoint`" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This notebook demonstrates how to use `firedrake-adjoint` to solve an adjoint equation and extract the solution thereof.\n", "\n", "Suppose we have a PDE in 'residual form'\n", "\n", "$$F(u)=0,\\quad u\\in V,$$\n", "\n", "for some function space $V$.\n", "Given an objective functional $J:V\\rightarrow\\mathbb R$, the adjoint equation is given by\n", "\n", "$$\\frac{\\partial F}{\\partial u}^T\\lambda=\\frac{\\partial J}{\\partial u}^T,\\quad\\lambda\\in V.$$\n", "\n", "We seek to compute the adjoint solution, $\\lambda$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Suppose the PDE involves a parameter $\\nu$ which is defined as a Firedrake `Constant` or `Function` (rather than a float).\n", "Then `firedrake-adjoint` enables us to compute the gradient $\\mathrm dJ/\\mathrm d\\nu$.\n", "Whilst we might not actually need this for our application, performing the computation endows the `SolveVarFormBlock`s with adjoint solutions.\n", "Why is that?\n", "\n", "Taking the transpose of the adjoint equation gives\n", "\n", "$$\\frac{\\partial J}{\\partial u}=\\lambda^T\\frac{\\partial F}{\\partial u}.$$\n", "\n", "Now, expanding the gradient $\\mathrm dJ/\\mathrm d\\nu$ using the chain rule gives\n", "\n", "$$\n", "\\frac{\\mathrm dJ}{\\mathrm d\\nu}\n", "=\\frac{\\partial J}{\\partial u}\\frac{\\mathrm du}{\\mathrm d\\nu}+\\frac{\\partial J}{\\partial\\nu}\n", "=\\lambda^T\\frac{\\partial F}{\\partial u}\\frac{\\mathrm du}{\\mathrm d\\nu}+\\frac{\\partial J}{\\partial\\nu}\n", "=\\lambda^T\\frac{\\partial F}{\\partial\\nu}+\\frac{\\partial J}{\\partial\\nu}\n", "$$\n", "\n", "So in order to compute the gradient $\\mathrm dJ/\\mathrm d\\nu$, it is sufficient to evaluate:\n", "1. the adjoint solution, $\\lambda$;\n", "2. the _partial_ derivative of the PDE residual w.r.t. the parameter $\\nu$;\n", "3. the _partial_ derivative of the objective functional w.r.t. $\\nu$.\n", "\n", "In order to compute the gradient, `firedrake-adjoint` does all three of these things." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## The forward problem" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Import Firedrake with adjoint mode activated." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "%matplotlib notebook\n", "import matplotlib.pyplot as plt\n", "from firedrake import *\n", "from firedrake_adjoint import *" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We employ the same finite element discretisation of Burgers equation in $\\mathbb P2$ space as in the [Firedrake demo](https://firedrakeproject.org/demos/burgers.py.html). Here $\\nu$ is the viscosity." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "# Define a simple mesh\n", "n = 32\n", "mesh = UnitSquareMesh(n, n)\n", "\n", "# Define P2 function space and corresponding test function\n", "V = VectorFunctionSpace(mesh, \"CG\", 2)\n", "v = TestFunction(V)\n", "\n", "# Create Functions for the solution and time-lagged solution\n", "u = Function(V, name=\"Velocity\")\n", "u_ = Function(V)\n", "\n", "# Assign initial condition\n", "x, y = SpatialCoordinate(mesh)\n", "u_ = interpolate(as_vector([sin(pi*x), 0]), V)\n", "u.assign(u_)\n", "\n", "# Set diffusivity constant\n", "nu = Constant(0.0001)\n", "\n", "# Define nonlinear form\n", "dt = 1.0/n\n", "F = (inner((u - u_)/dt, v) + inner(dot(u, nabla_grad(u)), v) + nu*inner(grad(u), grad(v)))*dx" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Having set up the residual,`F`, we are able to compute weak solutions of the PDE." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "# Timestepping details\n", "end_time = 0.5\n", "timesteps_per_export = 4\n", "num_timesteps = int(end_time/dt)\n", "num_exports = num_timesteps//timesteps_per_export\n", "\n", "# Store forward solution at exports so we can plot again later\n", "forward_solutions = [u.copy(deepcopy=True), ]\n", "\n", "# Time integrate\n", "i = 0\n", "t = 0.0\n", "while (t < end_time):\n", " solve(F == 0, u)\n", " u_.assign(u)\n", " t += dt\n", " i += 1\n", " if i % timesteps_per_export == 0:\n", " forward_solutions.append(u.copy(deepcopy=True))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Plot solution at each export timestep." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support. 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