{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "In this post, we'll look at the *obstacle problem*.\n", "We've seen in previous posts examples of variational problems -- minimization of some functional with respect to a field.\n", "The classic example of a variational problem is to find the function $u$ that minimizes the Dirichlet energy\n", "\n", "$$\\mathscr{J}(u) = \\int_\\Omega\\left(\\frac{1}{2}|\\nabla u|^2 - fu\\right)dx$$\n", "\n", "subject to the homogeneous Dirichlet boundary condition $u|_{\\partial\\Omega} = 0$.\n", "The Poisson equation is especially convenient because the objective is convex and quadratic.\n", "The obstacle problem is what you get when you add the additional constraint\n", "\n", "$$u \\ge g$$\n", "\n", "throughout the domain.\n", "More generally, we can look at the problem of minimizing a convex functional $\\mathscr{J}$ subject to the constraint that $u$ has to live in a closed, convex set $K$ of a function space $Q$.\n", "For a totally unconstrained problem, $K$ would just be the whole space $Q$.\n", "\n", "Newton's method with line search is a very effective algorithm for solving unconstrained convex problems, even for infinite-dimensional problems like PDEs.\n", "Things get much harder when you include inequality constraints.\n", "To make matters worse, much of the literature you'll find on this subject is focused on finite-dimensional problems, where techniques like the active-set method work quite well.\n", "It's not so obvious how to generalize these methods to variational problems.\n", "In the following, I'll follow the approach in section 4.1 of [this paper](https://www.tandfonline.com/doi/full/10.1080/10556788.2019.1613655) by Farrell, Croci, and Surowiec, whch was my inspiration for writing this post.\n", "\n", "Minimizing the action functional $\\mathscr{J}$ over the convex set $K$ can be rephrased as an unconstrained problem to minimize the functional\n", "\n", "$$\\mathscr{J}(u) + \\mathscr{I}(u),$$\n", "\n", "where $\\mathscr{I}$ is the *indicator function* of the set $K$:\n", "\n", "$$\\mathscr{I}(u) = \\begin{cases}0 & u \\in K \\\\ \\infty & u \\notin K\\end{cases}.$$\n", "\n", "This functional is still convex, but it can take the value $\\infty$.\n", "The reformulation isn't especially useful by itself, but we can approximate it using the *Moreau envelope*.\n", "The envelope of $\\mathscr{I}$ is defined as\n", "\n", "$$\\mathscr{I}_\\gamma(u) = \\min_v\\left(\\mathscr{I}(v) + \\frac{1}{2\\gamma^2}\\|u - v\\|^2\\right).$$\n", "\n", "In the limit as $\\gamma \\to 0$, $\\mathscr{I}_\\gamma(u) \\to \\mathscr{I}(u)$.\n", "The Moreau envelope is much easier to work with than the original functional because it's differentiable.\n", "In some cases it can be computed analytically; for example, when $\\mathscr{I}$ is an indicator function,\n", "\n", "$$\\mathscr{I}_\\gamma(u) = \\frac{1}{2\\gamma^2}\\text{dist}\\,(u, K)^2$$\n", "\n", "where $\\text{dist}$ is the distance to a convex set.\n", "We can do even better for our specific case, where $K$ is the set of all functions greater than $g$.\n", "For this choice of $K$, the distance to $K$ is\n", "\n", "$$\\text{dist}(u, K)^2 = \\int_\\Omega(u - g)_-^2dx,$$\n", "\n", "where $v_- = \\min(v, 0)$ is the negative part of $v$.\n", "So, our approach to solving the obstacle problem will be to find the minimzers of\n", "\n", "$$\\mathscr{J}_\\gamma(u) = \\int_\\Omega\\left(\\frac{1}{2}|\\nabla u|^2 - fu\\right)dx + \\frac{1}{2\\gamma^2}\\int_\\Omega(u - g)_-^2dx$$\n", "\n", "as $\\gamma$ goes to 0.\n", "I've written things in such a way that $\\gamma$ has units of length.\n", "Rather than take $\\gamma$ to 0 we can instead stop at some fraction of the finite element mesh spacing.\n", "At that point, the errors in the finite element approximation are comparable to the distance of the approximate solution to the constraint set.\n", "\n", "This is a lot like the penalty method for optimization problems with equality constraints.\n", "One of the main practical considerations when applying this regularization method is that the solution $u$ only satisfies the inequality constraints approximately.\n", "For the obstacle problem this deficiency isn't so severe, but for other problems we may need the solution to stay strictly feasible.\n", "In those cases, another approach like the logarithmic barrier method might be more appropriate." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Demonstration\n", "\n", "For our problem, the domain will be the unit square and the obstacle function $g$ will be the upper half of a sphere." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import firedrake\n", "nx, ny = 64, 64\n", "mesh = firedrake.UnitSquareMesh(nx, ny, quadrilateral=True)\n", "Q = firedrake.FunctionSpace(mesh, family='CG', degree=1)" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "from firedrake import max_value, sqrt, inner, as_vector, Constant\n", "\n", "def make_obstacle(mesh):\n", " x = firedrake.SpatialCoordinate(mesh)\n", " y = as_vector((1/2, 1/2))\n", " z = 1/4\n", " return sqrt(max_value(z**2 - inner(x - y, x - y), 0))\n", "\n", "g = firedrake.interpolate(make_obstacle(mesh), Q)" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", 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