{ "cells": [ { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "%matplotlib inline" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n# Quadratic approximations\n\nIn this example, we illustrate the use of the :class:`.QuadApprox` plot\non the Sobieski's SSBJ problem.\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "from gemseo.api import configure_logger\nfrom gemseo.api import create_discipline\nfrom gemseo.api import create_scenario\nfrom gemseo.problems.sobieski.core.problem import SobieskiProblem\nfrom matplotlib import pyplot as plt" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Import\nThe first step is to import some functions from the API\nand a method to get the design space.\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "configure_logger()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Description\n\nThe :class:`~gemseo.post.quad_approx.QuadApprox` post-processing\nperforms a quadratic approximation of a given function\nfrom an optimization history\nand plot the results as cuts of the approximation.\n\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Create disciplines\nThen, we instantiate the disciplines of the Sobieski's SSBJ problem:\nPropulsion, Aerodynamics, Structure and Mission\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "disciplines = create_discipline(\n [\n \"SobieskiPropulsion\",\n \"SobieskiAerodynamics\",\n \"SobieskiStructure\",\n \"SobieskiMission\",\n ]\n)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Create design space\nWe also read the design space from the :class:`.SobieskiProblem`.\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "design_space = SobieskiProblem().design_space" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Create and execute scenario\nThe next step is to build an MDO scenario in order to maximize the range,\nencoded 'y_4', with respect to the design parameters, while satisfying the\ninequality constraints 'g_1', 'g_2' and 'g_3'. We can use the MDF formulation,\nthe SLSQP optimization algorithm\nand a maximum number of iterations equal to 100.\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "scenario = create_scenario(\n disciplines,\n formulation=\"MDF\",\n objective_name=\"y_4\",\n maximize_objective=True,\n design_space=design_space,\n)\nscenario.set_differentiation_method(\"user\")\nfor constraint in [\"g_1\", \"g_2\", \"g_3\"]:\n scenario.add_constraint(constraint, \"ineq\")\nscenario.execute({\"algo\": \"SLSQP\", \"max_iter\": 10})" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Post-process scenario\nLastly, we post-process the scenario by means of the :class:`.QuadApprox`\nplot which performs a quadratic approximation of a given function\nfrom an optimization history and plot the results as cuts of the\napproximation.\n\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ ".. tip::\n\n Each post-processing method requires different inputs and offers a variety\n of customization options. Use the API function\n :meth:`~gemseo.api.get_post_processing_options_schema` to print a table with\n the options for any post-processing algorithm.\n Or refer to our dedicated page:\n `gen_post_algos`.\n\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The first plot shows an approximation of the Hessian matrix\n$\\frac{\\partial^2 f}{\\partial x_i \\partial x_j}$ based on the\n*Symmetric Rank 1* method (SR1) :cite:`Nocedal2006`. The\ncolor map uses a symmetric logarithmic (symlog) scale.\nThis plots the cross influence of the design variables on the objective function\nor constraints. For instance, on the last figure, the maximal second-order\nsensitivity is $\\frac{\\partial^2 -y_4}{\\partial^2 x_0} = 2.10^5$,\nwhich means that the $x_0$ is the most influential variable. Then,\nthe cross derivative\n$\\frac{\\partial^2 -y_4}{\\partial x_0 \\partial x_2} = 5.10^4$\nis positive and relatively high compared to the previous one but the combined\neffects of $x_0$ and $x_2$ are non-negligible in comparison.\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "scenario.post_process(\"QuadApprox\", function=\"-y_4\", save=False, show=False)\n# Workaround for HTML rendering, instead of ``show=True``\nplt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The second plot represents the quadratic approximation of the objective around the\noptimal solution : $a_{i}(t)=0.5 (t-x^*_i)^2\n\\frac{\\partial^2 f}{\\partial x_i^2} + (t-x^*_i) \\frac{\\partial\nf}{\\partial x_i} + f(x^*)$, where $x^*$ is the optimal solution.\nThis approximation highlights the sensitivity of the :term:`objective function`\nwith respect to the :term:`design variables`: we notice that the design\nvariables $x\\_1, x\\_5, x\\_6$ have little influence , whereas\n$x\\_0, x\\_2, x\\_9$ have a huge influence on the objective. This\ntrend is also noted in the diagonal terms of the :term:`Hessian` matrix\n$\\frac{\\partial^2 f}{\\partial x_i^2}$.\n\n" ] } ], "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": "3.9.13" } }, "nbformat": 4, "nbformat_minor": 0 }