261 lines
27 KiB
Plaintext
261 lines
27 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "47ceb846",
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"metadata": {},
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"source": [
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"# Exercise sheet\n",
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"\n",
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"Some general remarks about the exercises:\n",
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"* For your convenience functions from the lecture are included below. Feel free to reuse them without copying to the exercise solution box.\n",
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"* For each part of the exercise a solution box has been added, but you may insert additional boxes. Do not hesitate to add Markdown boxes for textual or LaTeX answers (via `Cell > Cell Type > Markdown`). But make sure to replace any part that says `YOUR CODE HERE` or `YOUR ANSWER HERE` and remove the `raise NotImplementedError()`.\n",
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"* Please make your code readable by humans (and not just by the Python interpreter): choose informative function and variable names and use consistent formatting. Feel free to check the [PEP 8 Style Guide for Python](https://www.python.org/dev/peps/pep-0008/) for the widely adopted coding conventions or [this guide for explanation](https://realpython.com/python-pep8/).\n",
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"* Make sure that the full notebook runs without errors before submitting your work. This you can do by selecting `Kernel > Restart & Run All` in the jupyter menu.\n",
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"* For some exercises test cases have been provided in a separate cell in the form of `assert` statements. When run, a successful test will give no output, whereas a failed test will display an error message.\n",
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"* Each sheet has 100 points worth of exercises. Note that only the grades of sheets number 2, 4, 6, 8 count towards the course examination. Submitting sheets 1, 3, 5, 7 & 9 is voluntary and their grades are just for feedback.\n",
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"\n",
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"Please fill in your name here:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "be2b5237",
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"metadata": {},
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"outputs": [],
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"source": [
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"NAME = \"\"\n",
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"NAMES_OF_COLLABORATORS = \"\""
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]
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},
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{
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"cell_type": "markdown",
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"id": "6037899a",
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"metadata": {},
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"source": [
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"---"
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]
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},
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{
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"cell_type": "markdown",
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"id": "3ff5eacf",
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"metadata": {
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"deletable": false,
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"editable": false,
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"nbgrader": {
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"cell_type": "markdown",
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"checksum": "a8ee042e1668d99f0c9d1d4176a2a66f",
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"grade": false,
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"grade_id": "cell-02fe75d787fe0e01",
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"locked": true,
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"schema_version": 3,
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"solution": false,
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"task": false
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}
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},
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"source": [
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"**Exercise sheet 9**"
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]
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},
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{
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"attachments": {
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"image.png": {
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"
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}
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},
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"id": "2e578009",
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"source": [
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"## Running code on the compute cluster: lattice scalar field\n",
|
|
"\n",
|
|
"**Goal**: Learn how to scale up simulations by transitioning from running them in the notebook to stand-alone scripts on the compute cluster.\n",
|
|
"\n",
|
|
"In lecture 7 we implemented the Metropolis-Hastings simulation of a lattice scalar field using the following code:\n",
|
|
"\n",
|
|
"```Python\n",
|
|
"import numpy as np\n",
|
|
"rng = np.random.default_rng() \n",
|
|
"import matplotlib.pylab as plt\n",
|
|
"%matplotlib inline\n",
|
|
"\n",
|
|
"def potential_v(x,lamb):\n",
|
|
" '''Compute the potential function V(x).'''\n",
|
|
" return lamb*(x*x-1)*(x*x-1)+x*x\n",
|
|
"\n",
|
|
"def neighbor_sum(phi,s):\n",
|
|
" '''Compute the sum of the state phi on all 8 neighbors of the site s.'''\n",
|
|
" w = len(phi)\n",
|
|
" return (phi[(s[0]+1)%w,s[1],s[2],s[3]] + phi[(s[0]-1)%w,s[1],s[2],s[3]] +\n",
|
|
" phi[s[0],(s[1]+1)%w,s[2],s[3]] + phi[s[0],(s[1]-1)%w,s[2],s[3]] +\n",
|
|
" phi[s[0],s[1],(s[2]+1)%w,s[3]] + phi[s[0],s[1],(s[2]-1)%w,s[3]] +\n",
|
|
" phi[s[0],s[1],s[2],(s[3]+1)%w] + phi[s[0],s[1],s[2],(s[3]-1)%w] )\n",
|
|
"\n",
|
|
"def scalar_action_diff(phi,site,newphi,lamb,kappa):\n",
|
|
" '''Compute the change in the action when phi[site] is changed to newphi.'''\n",
|
|
" return (2 * kappa * neighbor_sum(phi,site) * (phi[site] - newphi) +\n",
|
|
" potential_v(newphi,lamb) - potential_v(phi[site],lamb) )\n",
|
|
"\n",
|
|
"def scalar_MH_step(phi,lamb,kappa,delta):\n",
|
|
" '''Perform Metropolis-Hastings update on state phi with range delta.'''\n",
|
|
" site = tuple(rng.integers(0,len(phi),4))\n",
|
|
" newphi = phi[site] + rng.uniform(-delta,delta)\n",
|
|
" deltaS = scalar_action_diff(phi,site,newphi,lamb,kappa)\n",
|
|
" if deltaS < 0 or rng.uniform() < np.exp(-deltaS):\n",
|
|
" phi[site] = newphi\n",
|
|
" return True\n",
|
|
" return False\n",
|
|
"\n",
|
|
"def run_scalar_MH(phi,lamb,kappa,delta,n):\n",
|
|
" '''Perform n Metropolis-Hastings updates on state phi and return number of accepted transtions.'''\n",
|
|
" total_accept = 0\n",
|
|
" for _ in range(n):\n",
|
|
" total_accept += scalar_MH_step(phi,lamb,kappa,delta)\n",
|
|
" return total_accept\n",
|
|
"\n",
|
|
"def batch_estimate(data,observable,k):\n",
|
|
" '''Devide data into k batches and apply the function observable to each.\n",
|
|
" Returns the mean and standard error.'''\n",
|
|
" batches = np.reshape(data,(k,-1))\n",
|
|
" values = np.apply_along_axis(observable, 1, batches)\n",
|
|
" return np.mean(values), np.std(values)/np.sqrt(k-1)\n",
|
|
"\n",
|
|
"lamb = 1.5\n",
|
|
"kappas = np.linspace(0.08,0.18,11)\n",
|
|
"width = 3\n",
|
|
"num_sites = width**4\n",
|
|
"delta = 1.5 # chosen to have ~ 50% acceptance\n",
|
|
"equil_sweeps = 1000\n",
|
|
"measure_sweeps = 2\n",
|
|
"measurements = 2000\n",
|
|
"\n",
|
|
"mean_magn = []\n",
|
|
"for kappa in kappas:\n",
|
|
" phi_state = np.zeros((width,width,width,width))\n",
|
|
" run_scalar_MH(phi_state,lamb,kappa,delta,equil_sweeps * num_sites)\n",
|
|
" magnetizations = np.empty(measurements)\n",
|
|
" for i in range(measurements):\n",
|
|
" run_scalar_MH(phi_state,lamb,kappa,delta,measure_sweeps * num_sites)\n",
|
|
" magnetizations[i] = np.mean(phi_state)\n",
|
|
" mean, err = batch_estimate(np.abs(magnetizations),lambda x:np.mean(x),10)\n",
|
|
" mean_magn.append([mean,err])\n",
|
|
" \n",
|
|
"plt.errorbar(kappas,[m[0] for m in mean_magn],yerr=[m[1] for m in mean_magn],fmt='-o')\n",
|
|
"plt.xlabel(r\"$\\kappa$\")\n",
|
|
"plt.ylabel(r\"$|m|$\")\n",
|
|
"plt.title(r\"Absolute field average on $3^4$ lattice, $\\lambda = 1.5$\")\n",
|
|
"plt.show()\n",
|
|
"```\n",
|
|
"The goal will be to reproduce and extend its output.\n",
|
|
"\n",
|
|
""
|
|
]
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},
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{
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"cell_type": "markdown",
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"id": "029e890b",
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"deletable": false,
|
|
"editable": false,
|
|
"nbgrader": {
|
|
"cell_type": "markdown",
|
|
"checksum": "9f8edfc2002ed79ed92e093411435ca4",
|
|
"grade": false,
|
|
"grade_id": "cell-577d0bd92cfc0731",
|
|
"locked": true,
|
|
"points": 40,
|
|
"schema_version": 3,
|
|
"solution": false,
|
|
"task": true
|
|
}
|
|
},
|
|
"source": [
|
|
"**(a)** Turn the simulation into a standalone script `latticescalar.py` (similar to `ising.py`) that takes the relevant parameters e.g.\n",
|
|
"```bash\n",
|
|
"$ python3 latticescalar.py -l 1.5 -k 0.12 -w 3 -n 1000\n",
|
|
"```\n",
|
|
"for $\\lambda=1.5$, $\\kappa=0.12$, $w=3$, and $1000$ measurements, together with optional parameters $\\delta$ and numbers of sweeps, as command line arguments and stores the relevant simulation outcomes in an hdf5-file. **(40 pts)**"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "d4640925",
|
|
"metadata": {
|
|
"deletable": false,
|
|
"editable": false,
|
|
"nbgrader": {
|
|
"cell_type": "markdown",
|
|
"checksum": "261f895518cc39ffdc0eadcef4c3dd65",
|
|
"grade": false,
|
|
"grade_id": "cell-51479ca8a1c07968",
|
|
"locked": true,
|
|
"points": 30,
|
|
"schema_version": 3,
|
|
"solution": false,
|
|
"task": true
|
|
}
|
|
},
|
|
"source": [
|
|
"**(b)** Write a bash script `job_latticescalar.sh` that submits an array job to the cluster for $w=3$ and $2000$ measurements and $\\lambda = 1.0, 1.5, 2.0$ and $\\kappa = 0.08, 0.09, ..., 0.18$ (so 33 simulations in total). Submit the job to the `hefstud` slurm partition (do not run all 33 in parallel). **(30 pts)**"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "711111db",
|
|
"metadata": {
|
|
"deletable": false,
|
|
"editable": false,
|
|
"nbgrader": {
|
|
"cell_type": "markdown",
|
|
"checksum": "0181a075dd61572ba8c2f5027bbb8b74",
|
|
"grade": false,
|
|
"grade_id": "cell-88a18366f80a2168",
|
|
"locked": true,
|
|
"points": 30,
|
|
"schema_version": 3,
|
|
"solution": false,
|
|
"task": true
|
|
}
|
|
},
|
|
"source": [
|
|
"**(c)** Load the stored data into this notebook and reproduce the plot above (with $\\lambda = 1$ and $\\lambda=2$ added). **(30 pts)**"
|
|
]
|
|
}
|
|
],
|
|
"metadata": {
|
|
"kernelspec": {
|
|
"display_name": "Python 3 (ipykernel)",
|
|
"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": "3.9.12"
|
|
}
|
|
},
|
|
"nbformat": 4,
|
|
"nbformat_minor": 5
|
|
}
|