![\"Open](\"https://colab.research.google.com/assets/colab-badge.svg\" "\\"Open"){kind=icon}
![\"Download\"](\"https://img.shields.io/badge/Github-Download-blue.svg\" "\\"Download"){kind=icon}
"
]
},
{
"cell_type": "markdown",
"metadata": {
"nbpages": {
"level": 1,
"link": "[6.2 A Simple Example of Adjoint Sensitivity Analysis](https://ndcbe.github.io/cbe67701-uncertainty-quantification/06.02-Contributed-Example.html#6.2-A-Simple-Example-of-Adjoint-Sensitivity-Analysis)",
"section": "6.2 A Simple Example of Adjoint Sensitivity Analysis"
}
},
"source": [
"# 6.2 A Simple Example of Adjoint Sensitivity Analysis"
]
},
{
"cell_type": "markdown",
"metadata": {
"nbpages": {
"level": 1,
"link": "[6.2 A Simple Example of Adjoint Sensitivity Analysis](https://ndcbe.github.io/cbe67701-uncertainty-quantification/06.02-Contributed-Example.html#6.2-A-Simple-Example-of-Adjoint-Sensitivity-Analysis)",
"section": "6.2 A Simple Example of Adjoint Sensitivity Analysis"
}
},
"source": [
"Han Gao (hgao1@nd.edu)"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"nbpages": {
"level": 1,
"link": "[6.2 A Simple Example of Adjoint Sensitivity Analysis](https://ndcbe.github.io/cbe67701-uncertainty-quantification/06.02-Contributed-Example.html#6.2-A-Simple-Example-of-Adjoint-Sensitivity-Analysis)",
"section": "6.2 A Simple Example of Adjoint Sensitivity Analysis"
}
},
"outputs": [],
"source": [
"## import all needed Python libraries here\n",
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"from matplotlib.ticker import LinearLocator, FormatStrFormatter\n",
"from matplotlib import cm\n",
"from mpl_toolkits.mplot3d import Axes3D"
]
},
{
"cell_type": "markdown",
"metadata": {
"nbpages": {
"level": 1,
"link": "[6.2 A Simple Example of Adjoint Sensitivity Analysis](https://ndcbe.github.io/cbe67701-uncertainty-quantification/06.02-Contributed-Example.html#6.2-A-Simple-Example-of-Adjoint-Sensitivity-Analysis)",
"section": "6.2 A Simple Example of Adjoint Sensitivity Analysis"
}
},
"source": [
"Here we give a very simple example of adjoint sensitivity analysis (SA) with analytic solution.\n",
"\n",
"Consider a system $$Q=\\int_0^{t=1}u(t)dt$$\n",
"$$s.t.\\;\\dot{u}=bu$$\n",
"$$\\quad\\quad u(0)=a$$\n",
"We want to find the $\\frac{\\partial Q}{\\partial a}$ and $\\frac{\\partial Q}{\\partial b}$ by SA.\n"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"nbpages": {
"level": 1,
"link": "[6.2 A Simple Example of Adjoint Sensitivity Analysis](https://ndcbe.github.io/cbe67701-uncertainty-quantification/06.02-Contributed-Example.html#6.2-A-Simple-Example-of-Adjoint-Sensitivity-Analysis)",
"section": "6.2 A Simple Example of Adjoint Sensitivity Analysis"
}
},
"outputs": [],
"source": [
"\"\"\"define hyperparameters of the problem\"\"\"\n",
"ts=0 # Start Time\n",
"te=1 # End Time\n",
"\"\"\"define the time domain of the problem\"\"\"\n",
"T=np.linspace(ts,te,1000)\n",
"\"\"\"define the QOI function \"\"\"\n",
"qoi=lambda a,b:a/b*(np.exp(b*te)-1)"
]
},
{
"cell_type": "markdown",
"metadata": {
"nbpages": {
"level": 1,
"link": "[6.2 A Simple Example of Adjoint Sensitivity Analysis](https://ndcbe.github.io/cbe67701-uncertainty-quantification/06.02-Contributed-Example.html#6.2-A-Simple-Example-of-Adjoint-Sensitivity-Analysis)",
"section": "6.2 A Simple Example of Adjoint Sensitivity Analysis"
}
},
"source": [
"Itergrate the system we can get\n",
"$$u(t)=a\\cdot e^{bt}$$\n"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"nbpages": {
"level": 1,
"link": "[6.2 A Simple Example of Adjoint Sensitivity Analysis](https://ndcbe.github.io/cbe67701-uncertainty-quantification/06.02-Contributed-Example.html#6.2-A-Simple-Example-of-Adjoint-Sensitivity-Analysis)",
"section": "6.2 A Simple Example of Adjoint Sensitivity Analysis"
}
},
"outputs": [
{
"data": {
"text/plain": [
""
]
}
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"display_name": "Python 3",
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