├── .github └── workflows │ └── build.yml ├── .gitignore ├── .hgignore ├── CITATION.cff ├── CONTRIBUTING.md ├── README.md ├── docs ├── Makefile ├── conf.py ├── images │ ├── waves_drift_ratios.png │ └── waves_spectrum.png ├── index.rst ├── notebooks │ ├── Mixed Layer Instability.ipynb │ └── Orr Somerfeld pseudospectra.ipynb ├── pages │ ├── getting_started.rst │ ├── installation.rst │ └── upgrading.rst └── requirements.rtd.txt ├── eigentools ├── __init__.py ├── criticalfinder.py ├── eigenproblem.py └── tools.py ├── examples ├── mri.py ├── orr_sommerfeld.py └── rayleigh_benard_2d.py ├── license.txt ├── setup.py └── tests ├── non_constant_test.py └── test_rbc_growth.py /.github/workflows/build.yml: -------------------------------------------------------------------------------- 1 | name: build 2 | 3 | on: 4 | pull_request: 5 | branches: [ master ] 6 | 7 | jobs: 8 | build: 9 | runs-on: ubuntu-latest 10 | container: dedalusproject/dedalus-conda 11 | steps: 12 | - uses: actions/checkout@v2 13 | with: 14 | ref: 'v2.0-refactor' 15 | - name: build 16 | run: /opt/conda/envs/dedalus/bin/pip install -e . 17 | 18 | - name: test 19 | run: /opt/conda/envs/dedalus/bin/pytest 20 | 21 | 22 | -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | # Byte-compiled / optimized / DLL files 2 | __pycache__/ 3 | *.py[cod] 4 | *$py.class 5 | 6 | # C extensions 7 | *.so 8 | 9 | # Distribution / packaging 10 | .Python 11 | build/ 12 | develop-eggs/ 13 | dist/ 14 | downloads/ 15 | eggs/ 16 | .eggs/ 17 | lib/ 18 | lib64/ 19 | parts/ 20 | sdist/ 21 | var/ 22 | wheels/ 23 | share/python-wheels/ 24 | *.egg-info/ 25 | .installed.cfg 26 | *.egg 27 | MANIFEST 28 | 29 | # ipython 30 | .ipynb_checkpoints/ 31 | 32 | -------------------------------------------------------------------------------- /.hgignore: -------------------------------------------------------------------------------- 1 | # use glob syntax. 2 | syntax: glob 3 | *~ 4 | *__pycache__/* 5 | *.ipynb_checkpoints/* -------------------------------------------------------------------------------- /CITATION.cff: -------------------------------------------------------------------------------- 1 | cff-version: 1.2.0 2 | preferred-citation: 3 | type: article 4 | authors: 5 | - family-names: "Oishi" 6 | given-names: "Jeffrey S" 7 | orcid: "https://orcid.org/0000-0001-8531-6570" 8 | - family-names: "Burns" 9 | given-names: "Keaton J" 10 | orcid: "https://orcid.org/0000-0003-4761-4766" 11 | - family-names: "Clark" 12 | given-names: "S E" 13 | orcid: "https://orcid.org/0000-0002-7633-3376" 14 | - family-names: "Anders" 15 | given-names: "Evan H" 16 | orcid: "https://orcid.org/0000-0002-3433-4733" 17 | - family-names: "Brown" 18 | given-names: "Benjamin P" 19 | orcid: "https://orcid.org/0000-0001-8935-219X" 20 | - family-names: "Vasil" 21 | given-names: "Geoffrey M" 22 | orcid: "https://orcid.org/0000-0002-8902-5030" 23 | - family-names: "Lecoanet" 24 | given-names: "Daniel" 25 | orcid: "https://orcid.org/0000-0002-7635-9728" 26 | title: "eigentools: A Python package for studying differential eigenvalue problems with an emphasis on robustness" 27 | journal: "Journal of Open Source Software" 28 | doi: "10.21105/joss.03079" 29 | volume: 6 30 | issue: 62 31 | start: 3079 32 | month: 6 33 | year: 2021 34 | -------------------------------------------------------------------------------- /CONTRIBUTING.md: -------------------------------------------------------------------------------- 1 | # Contributing to Eigentools # 2 | 3 | We welcome contributions, including issue reports, bug fixes, and feature implementations. 4 | Contributions are reviewed on Github via pull request; to get started, fork the repository, make changes, and issue a pull request. 5 | You can also contribute by submitting an issue. 6 | 7 | ## Reporting issues ## 8 | 9 | If you find a bug or unexpected behavior, please file an issue report on [github](https://github.com/DedalusProject/eigentools/issues). 10 | Please provide as much detail as possible, including version of both eigentools and dedalus, platform (Mac/Linux), and a stand alone `.py` file that demonstrates the problem in as simple a manner as possible. 11 | 12 | ## Proposing features ## 13 | 14 | You can propose new features on the issue tracker using the "enhancement" tag. 15 | 16 | ## Contributing code ## 17 | 18 | Code contributions ranging from fixing typos to implementing additional features are most welcome! We use [pull requests](https://github.com/DedalusProject/eigentools/pulls) to integrate contributions into the main codebase. If you would like to contribute and are looking for a place to start, please don't hesitate to contact the authors (emails are in [readme.md](readme.md)). 19 | 20 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Eigentools 2 | 3 | Eigentools is a set of tools for studying linear eigenvalue problems. The underlying eigenproblems are solved using [Dedalus](http://dedalus-project.org), which provides a domain-specific language for partial differential equations. Eigentools extends Dedalus's `EigenvalueProblem` object and provides 4 | 5 | * automatic rejection of unresolved eigenvalues 6 | * simple plotting of specified eigenmodes 7 | * simple plotting of spectra 8 | * computation of pseudospectra for any Differential-Algebraic Equations with **user-specifiable norms** 9 | * tools to find critical parameters for linear stability analysis 10 | * ability to project eigenmode onto 2- or 3-D domain for visualization 11 | * ability to output projected eigenmodes as Dedalus-formatted HDF5 file to be used as initial conditions for Initial Value Problems 12 | * simple plotting of drift ratios (both ordinal and nearest) to evaluate tolerance for eigenvalue rejection 13 | 14 | ## Installation 15 | 16 | Eigentools can be `pip` installed, though it requires [Dedalus](http://dedalus-project.org/), which has non-`pip` installable dependencies. See the [installation instructions](https://eigentools.readthedocs.io/en/latest/pages/installation.html) for details. 17 | 18 | ## Documentation 19 | 20 | Documentation (including detailed API documentation) can be found at [Read the Docs](https://eigentools.readthedocs.io/). 21 | 22 | If you are upgrading from version 1 to version 2, you can find a guide to API changes [here](https://eigentools.readthedocs.io/en/latest/pages/upgrading.html) 23 | 24 | ## Contributing 25 | 26 | Eigentools welcomes community contributions from issue reports to code contributions. For details, please see [our contribution policy](CONTRIBUTING.md). 27 | 28 | ## Developers 29 | The core development team consists of 30 | 31 | * Jeff Oishi () 32 | * Keaton Burns () 33 | * Susan Clark () 34 | * Evan Anders () 35 | * Ben Brown () 36 | * Geoff Vasil () 37 | * Daniel Lecoanet () 38 | 39 | ## Support 40 | Eigentools was developed with support from the Research Corporation under award Scialog Collaborative Award (TDA) ID# 24231. 41 | 42 | 43 | 45 | 47 | 49 | -------------------------------------------------------------------------------- /docs/Makefile: -------------------------------------------------------------------------------- 1 | # Minimal makefile for Sphinx documentation 2 | # 3 | 4 | # You can set these variables from the command line. 5 | SPHINXOPTS = 6 | SPHINXBUILD = sphinx-build 7 | SPHINXPROJ = Eigentools 8 | SOURCEDIR = . 9 | BUILDDIR = _build 10 | 11 | # Put it first so that "make" without argument is like "make help". 12 | help: 13 | @$(SPHINXBUILD) -M help "$(SOURCEDIR)" "$(BUILDDIR)" $(SPHINXOPTS) $(O) 14 | 15 | .PHONY: help Makefile 16 | 17 | # Catch-all target: route all unknown targets to Sphinx using the new 18 | # "make mode" option. $(O) is meant as a shortcut for $(SPHINXOPTS). 19 | %: Makefile 20 | @$(SPHINXBUILD) -M $@ "$(SOURCEDIR)" "$(BUILDDIR)" $(SPHINXOPTS) $(O) 21 | -------------------------------------------------------------------------------- /docs/conf.py: -------------------------------------------------------------------------------- 1 | # -*- coding: utf-8 -*- 2 | # 3 | # Configuration file for the Sphinx documentation builder. 4 | # 5 | # This file does only contain a selection of the most common options. For a 6 | # full list see the documentation: 7 | # http://www.sphinx-doc.org/en/master/config 8 | 9 | # -- Path setup -------------------------------------------------------------- 10 | 11 | # If extensions (or modules to document with autodoc) are in another directory, 12 | # add these directories to sys.path here. If the directory is relative to the 13 | # documentation root, use os.path.abspath to make it absolute, like shown here. 14 | # 15 | # import os 16 | # import sys 17 | # sys.path.insert(0, os.path.abspath('.')) 18 | 19 | 20 | # -- Project information ----------------------------------------------------- 21 | 22 | project = 'Eigentools' 23 | copyright = '2020 Dedalus Collaboration' 24 | author = 'Dedalus Collaboration' 25 | 26 | # The short X.Y version 27 | version = '' 28 | # The full version, including alpha/beta/rc tags 29 | release = '' 30 | 31 | 32 | # -- General configuration --------------------------------------------------- 33 | 34 | # If your documentation needs a minimal Sphinx version, state it here. 35 | # 36 | # needs_sphinx = '1.0' 37 | 38 | # Add any Sphinx extension module names here, as strings. They can be 39 | # extensions coming with Sphinx (named 'sphinx.ext.*') or your custom 40 | # ones. 41 | extensions = [] 42 | extensions += ['sphinx.ext.mathjax'] 43 | extensions += ['autoapi.extension'] 44 | extensions += ['sphinx.ext.viewcode'] 45 | extensions += ['sphinx.ext.napoleon'] 46 | extensions += ['nbsphinx'] 47 | 48 | add_module_names = False 49 | autoapi_type = 'python' 50 | autoapi_dirs = ['../eigentools'] 51 | autoapi_file_patterns = ['*.py'] 52 | autoapi_options = ['members', 'undoc-members'] 53 | autoapi_python_class_content = 'both' 54 | autoapi_add_toctree_entry = False 55 | 56 | napoleon_use_param = False 57 | napoleon_use_keyword = False 58 | napoleon_use_ivar = True 59 | 60 | # Add any paths that contain templates here, relative to this directory. 61 | templates_path = ['_templates'] 62 | 63 | # The suffix(es) of source filenames. 64 | # You can specify multiple suffix as a list of string: 65 | # 66 | # source_suffix = ['.rst', '.md'] 67 | source_suffix = '.rst' 68 | 69 | # The master toctree document. 70 | master_doc = 'index' 71 | 72 | # The language for content autogenerated by Sphinx. Refer to documentation 73 | # for a list of supported languages. 74 | # 75 | # This is also used if you do content translation via gettext catalogs. 76 | # Usually you set "language" from the command line for these cases. 77 | language = None 78 | 79 | # List of patterns, relative to source directory, that match files and 80 | # directories to ignore when looking for source files. 81 | # This pattern also affects html_static_path and html_extra_path . 82 | exclude_patterns = ['_build', 'Thumbs.db', '.DS_Store'] 83 | 84 | # The name of the Pygments (syntax highlighting) style to use. 85 | pygments_style = 'sphinx' 86 | 87 | 88 | # -- Options for HTML output ------------------------------------------------- 89 | 90 | # The theme to use for HTML and HTML Help pages. See the documentation for 91 | # a list of builtin themes. 92 | # 93 | html_theme = 'sphinx_rtd_theme' 94 | html_logo = 'epic12_4_exp_2_1.25.png' 95 | 96 | # Theme options are theme-specific and customize the look and feel of a theme 97 | # further. For a list of options available for each theme, see the 98 | # documentation. 99 | # 100 | # html_theme_options = {} 101 | 102 | # Add any paths that contain custom static files (such as style sheets) here, 103 | # relative to this directory. They are copied after the builtin static files, 104 | # so a file named "default.css" will overwrite the builtin "default.css". 105 | html_static_path = ['_static'] 106 | 107 | # Custom sidebar templates, must be a dictionary that maps document names 108 | # to template names. 109 | # 110 | # The default sidebars (for documents that don't match any pattern) are 111 | # defined by theme itself. Builtin themes are using these templates by 112 | # default: ``['localtoc.html', 'relations.html', 'sourcelink.html', 113 | # 'searchbox.html']``. 114 | # 115 | # html_sidebars = {} 116 | 117 | 118 | # -- Options for HTMLHelp output --------------------------------------------- 119 | 120 | # Output file base name for HTML help builder. 121 | htmlhelp_basename = 'Eigentoolsdoc' 122 | 123 | 124 | # -- Options for LaTeX output ------------------------------------------------ 125 | 126 | latex_elements = { 127 | # The paper size ('letterpaper' or 'a4paper'). 128 | # 129 | # 'papersize': 'letterpaper', 130 | 131 | # The font size ('10pt', '11pt' or '12pt'). 132 | # 133 | # 'pointsize': '10pt', 134 | 135 | # Additional stuff for the LaTeX preamble. 136 | # 137 | # 'preamble': '', 138 | 139 | # Latex figure (float) alignment 140 | # 141 | # 'figure_align': 'htbp', 142 | } 143 | 144 | # Grouping the document tree into LaTeX files. List of tuples 145 | # (source start file, target name, title, 146 | # author, documentclass [howto, manual, or own class]). 147 | latex_documents = [ 148 | (master_doc, 'Eigentools.tex', 'Eigentools Documentation', 149 | 'Dedalus Collaboration', 'manual'), 150 | ] 151 | 152 | 153 | # -- Options for manual page output ------------------------------------------ 154 | 155 | # One entry per manual page. List of tuples 156 | # (source start file, name, description, authors, manual section). 157 | man_pages = [ 158 | (master_doc, 'dedalusproject', 'Eigentools Documentation', 159 | [author], 1) 160 | ] 161 | 162 | 163 | # -- Options for Texinfo output ---------------------------------------------- 164 | 165 | # Grouping the document tree into Texinfo files. List of tuples 166 | # (source start file, target name, title, author, 167 | # dir menu entry, description, category) 168 | texinfo_documents = [ 169 | (master_doc, 'Eigentools', 'Eigentools Documentation', 170 | author, 'DedalusProject', 'One line description of project.', 171 | 'Miscellaneous'), 172 | ] 173 | -------------------------------------------------------------------------------- /docs/images/waves_drift_ratios.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DedalusProject/eigentools/3fc8556dcbe3712043d93527cb943f3f7beef732/docs/images/waves_drift_ratios.png -------------------------------------------------------------------------------- /docs/images/waves_spectrum.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DedalusProject/eigentools/3fc8556dcbe3712043d93527cb943f3f7beef732/docs/images/waves_spectrum.png -------------------------------------------------------------------------------- /docs/index.rst: -------------------------------------------------------------------------------- 1 | Eigentools 2 | ********** 3 | 4 | Eigentools is a set of tools for studying linear eigenvalue problems. The underlying eigenproblems are solved using `Dedalus `_, which provides a domain-specific language for partial differential equations. Each entry in the following list of features links to a Jupyter notebook giving an example of its use. 5 | 6 | * :ref:`automatic rejection of unresolved eigenvalues ` 7 | * :ref:`simple plotting of drift ratios (both ordinal and nearest) to evaluate tolerance for eigenvalue rejection ` 8 | 9 | * :ref:`simple plotting of specified eigenmodes ` 10 | * :ref:`simple plotting of spectra ` 11 | * :ref:`computation of pseudospectra for any Differential-Algebraic Equations ` with :ref:`user-specifiable norms ` 12 | * :ref:`tools to find critical parameters for linear stability analysis ` with :ref:`user-specifiable definitions of growth and stability ` 13 | * :ref:`ability to project eigenmode onto 2- or 3-D domain for visualization ` 14 | * :ref:`ability to output projected eigenmodes as Dedalus-formatted HDF5 file to be used as initial conditions for Initial Value Problems ` 15 | 16 | Contents 17 | ======== 18 | 19 | .. toctree:: 20 | :maxdepth: 2 21 | 22 | pages/installation 23 | pages/getting_started 24 | pages/upgrading 25 | 26 | Example notebooks 27 | ----------------- 28 | 29 | .. toctree:: 30 | :maxdepth: 1 31 | 32 | Example 1: Orr-Somerfield pseudospectra 33 | Example 2: Mixed Layer instability 34 | 35 | API reference 36 | ------------- 37 | 38 | .. toctree:: 39 | :maxdepth: 2 40 | 41 | Eigentools API reference 42 | 43 | Developers 44 | ========== 45 | The core development team consists of 46 | 47 | * Jeff Oishi () 48 | * Keaton Burns () 49 | * Susan Clark () 50 | * Evan Anders () 51 | * Ben Brown () 52 | * Geoff Vasil () 53 | * Daniel Lecoanet () 54 | 55 | Support 56 | ======= 57 | Eigentools was developed with support from the Research Corporation under award Scialog Collaborative Award (TDA) ID# 24231. 58 | 59 | -------------------------------------------------------------------------------- /docs/pages/getting_started.rst: -------------------------------------------------------------------------------- 1 | Getting Started 2 | *************** 3 | 4 | eigentools comes with several `examples `_ to get you started, but let's outline some basics in a very simple problem, the 1-D wave equation with :math:`u = 0` at both ends. 5 | This is not quite as trivial a problem as it might seem, because we are expanding the solution in Chebyshev polynomials, but the eigenmodes are sines and cosines. 6 | 7 | 8 | .. code-block:: python 9 | 10 | from eigentools import Eigenproblem 11 | import dedalus.public as de 12 | 13 | Nx = 128 14 | x = de.Chebyshev('x',Nx, interval=(-1, 1)) 15 | d = de.Domain([x]) 16 | 17 | string = de.EVP(d, ['u','u_x'], eigenvalue='omega') 18 | string.add_equation("omega*u + dx(u_x) = 0") 19 | string.add_equation("u_x - dx(u) = 0") 20 | string.add_bc("left(u) = 0") 21 | string.add_bc("right(u) = 0") 22 | 23 | EP = Eigenproblem(string) 24 | EP.solve(sparse=False) 25 | ax = EP.plot_spectrum() 26 | print("there are {} good eigenvalues.".format(len(EP.evalues))) 27 | ax.set_ylim(-1,1) 28 | ax.figure.savefig('waves_spectrum.png') 29 | 30 | ax = EP.plot_drift_ratios() 31 | ax.figure.savefig('waves_drift_ratios.png') 32 | 33 | That code takes about 10 seconds to run on a 2020 Core-i7 laptop, produces about 68 "good" eigenvalues, and produces the following output: 34 | 35 | .. image:: ../images/waves_spectrum.png 36 | :width: 400 37 | :alt: A spectrum for waves on a string 38 | 39 | eigentools has taken a Dedalus eigenvalue problem, automatically run it at 1.5 times the specified resolution, rejected any eigenvalues that do not agree to a default precision of one part in :math:`10^{-6}` and plotted a spectrum in six extra lines of code! 40 | 41 | Most of the plotting functions in eigentools return a `matplotlib` `axes` object, making it easy to modify the plot defaults. 42 | Here, we set the y-limits manually, because the eigenvalues of a string are real. 43 | Try removing the `ax.set_ylim(-1,1)` line and see what happens. 44 | 45 | Mode Rejection 46 | -------------- 47 | One of the most important tasks eigentools performs is spurious mode rejection. It does so by computing the "drift ratio" [Boyd2000]_ between the eigenvalues at the given resolution and a higher resolution problem that eigentools automatically assembles. By default, the "high" resolution case is 1.5 times the given resolution, though this is user configurable via the `factor` keyword option to `Eigenproblem()`. 48 | 49 | The drift ratio :math:`\delta` is calculated using either the **ordinal** (e.g. first mode of low resolution to first mode of high resolution) or **nearest** (mode with smallest difference between a given high mode and all low modes). In order to visualize this, `EP.plot_drift_ratios()` in the above code returns an `axes` object making a plot of the *inverse drift ratio* (:math:`1/\delta`), 50 | 51 | .. image:: ../images/waves_drift_ratios.png 52 | :width: 400 53 | :alt: Plot of inverse drift ratios vs. mode number for waves on a string. 54 | 55 | Good modes are those *above* the horizontal line at :math:`10^{6}`; bad modes are also grayed out. In this case, the **nearest** and **ordinal** methods produce identical results. If the problem contains more than one wave *family*, **nearest** typically fails. For an example, see the `MRI example script `_. Note that **nearest** is the default criterion used by eigentools. 56 | 57 | 58 | .. [Boyd2000] Boyd, J (2000). "Chebyshev and Fourier Spectral Methods." Dover. ``_ 59 | -------------------------------------------------------------------------------- /docs/pages/installation.rst: -------------------------------------------------------------------------------- 1 | Installing eigentools 2 | ********************* 3 | 4 | eigentools requires Dedalus, which you can install via any of the methods found in `the Dedalus installation instructions `_. 5 | 6 | Once Dedalus is installed, eigentools is `pip` installable:: 7 | 8 | pip install eigentools 9 | 10 | If you would like the development version, you can clone the repository and install locally:: 11 | 12 | git clone https://github.com/DedalusProject/eigentools.git 13 | pip install -e eigentools 14 | 15 | 16 | 17 | -------------------------------------------------------------------------------- /docs/pages/upgrading.rst: -------------------------------------------------------------------------------- 1 | Upgrading eigentools scripts 2 | **************************** 3 | 4 | Version 2 of eigentools has made significant changes to the API and will necessitate some changes (for the better, we hope) to the user experience. The guiding principle behind the new API is that one should no longer need to touch the Dedalus :code:`EVP` object that defines the eigenvalue problem at hand. 5 | 6 | **Most importantly**, no changes need to be made to the underlying Dedalus :code:`EVP` object. 7 | 8 | Basic :code:`eigenproblem` usage 9 | -------------------------------- 10 | 11 | Choosing a sparse or dense solve is no longer done when instantiating :code:`Eigenproblem` objects. Instead, this is a choice at *solve* time: 12 | 13 | .. code-block:: python 14 | 15 | EP = Eigenproblem(string, reject=True) 16 | EP.solve(sparse=False) 17 | 18 | Also, notice that rejection of spurious modes is now done automatically with :code:`EP.solve` if :code:`reject=True` is selected at instantiation time. Note that although in the above code, we explicitly set :code:`reject=True`, this is **unnecessary**, as it is the default. The :code:`EP.reject_spurious()` function has been removed 19 | 20 | In addition, solving again with different parameters has been greatly simplified from the previous version. You now simply *pass a dictionary* with the parameters you wish to change to solve itself. Let's revisit the simple waves-on-a-string problem from :ref:`the getting started page `, but add a parameter, :code:`c2`, the wave speed squared. 21 | 22 | Here, we solve twice, once with :code:`c1 = 1` and once with :code:`c2 = 2`. Given the dispersion relation for this problem is :math:`\omega^2 = c^2 k` and our eigenvalue :code:`omega` is really :math:`\omega^2`, we expect the eigenvalues for the second solve to be twice those for the first. 23 | 24 | .. code-block:: python 25 | 26 | import numpy as np 27 | from eigentools import Eigenproblem 28 | import dedalus.public as de 29 | 30 | Nx = 128 31 | x = de.Chebyshev('x',Nx, interval=(-1, 1)) 32 | d = de.Domain([x]) 33 | 34 | string = de.EVP(d, ['u','u_x'], eigenvalue='omega') 35 | string.parameters['c2'] = 1 36 | string.add_equation("omega*u + c2*dx(u_x) = 0") 37 | string.add_equation("u_x - dx(u) = 0") 38 | string.add_bc("left(u) = 0") 39 | string.add_bc("right(u) = 0") 40 | EP = Eigenproblem(string) 41 | EP.solve(sparse=False) 42 | evals_c1 = EP.evalues 43 | EP.solve(sparse=False, parameters={'c2':2}) 44 | evals_c2 = EP.evalues 45 | 46 | print(np.allclose(evals_c2, 2*evals_c1)) 47 | 48 | Getting eigenmodes 49 | ================== 50 | 51 | Getting eigenmodes has also been simplified and significantly extended. Previously, getting an eigenmode corresponding to an eigenvalue required using the :code:`set_state()` method on the underlying :code:`EVP` object. In keeping with the principle of not needing to manipulate the :code:`EVP`, we provide a new :code:`.eigenmode(index)`, where :code:`index` is the mode number corresponding to the eigenvalue index in :code:`EP.evalues`. By default, with mode rejection on, these are the "good" eigenmodes. 52 | `.eigenmode(index)` returns a Dedalus :code:`FieldSystem` object, with a Dedalus :code:`Field` for each field in the eigenmode: 53 | 54 | .. code-block:: python 55 | 56 | emode = EP.eigenmode(0) 57 | print([f.name for f in emode.fields]) 58 | u = emode.fields[0] 59 | u_x = emode.fields[1] 60 | 61 | 62 | Finding critical parameters 63 | --------------------------- 64 | 65 | This has been considerably cleaned up. The two major things to note are that 66 | 67 | 1. one no longer needs to create a shim function to translate between an x-y grid and the parameter names within the :code:`EVP`. 68 | 2. The parameter grid is no longer defined inside :code:`CriticalFinder`, but is instead created by the user and passed in 69 | 70 | For example, here are the relevant changes necessary for the `MRI test problem `_. 71 | 72 | First, replace 73 | 74 | .. code-block:: python 75 | 76 | EP = Eigenproblem(mri, sparse=True) 77 | 78 | # create a shim function to translate (x, y) to the parameters for the eigenvalue problem: 79 | def shim(x,y): 80 | iRm = 1/x 81 | iRe = (iRm*Pm) 82 | print("Rm = {}; Re = {}; Pm = {}".format(1/iRm, 1/iRe, Pm)) 83 | gr, indx, freq = EP.growth_rate({"Q":y,"iRm":iRm,"iR":iRe}) 84 | ret = gr+1j*freq 85 | return ret 86 | 87 | cf = CriticalFinder(shim, comm) 88 | 89 | with 90 | 91 | .. code-block:: python 92 | 93 | EP = Eigenproblem(mri) 94 | 95 | cf = CriticalFinder(EP, ("Q", "Rm"), comm, find_freq=False) 96 | 97 | **Important:** note that :code:`find_freq` is specified at instantiation rather than when calling :code:`cf.crit_finder` later. 98 | 99 | Once this is done, the grid generation changes from 100 | 101 | .. code-block:: python 102 | 103 | mins = np.array((4.6, 0.5)) 104 | maxs = np.array((5.5, 1.5)) 105 | ns = np.array((10,10)) 106 | logs = np.array((False, False)) 107 | 108 | cf.grid_generator(mins, maxs, ns, logs=logs) 109 | 110 | to 111 | 112 | .. code-block:: python 113 | 114 | nx = 20 115 | ny = 20 116 | xpoints = np.linspace(0.5, 1.5, nx) 117 | ypoints = np.linspace(4.6, 5.5, ny) 118 | 119 | cf.grid_generator((xpoints, ypoints), sparse=True) 120 | 121 | 122 | -------------------------------------------------------------------------------- /docs/requirements.rtd.txt: -------------------------------------------------------------------------------- 1 | setuptools >= 18.0 2 | sphinx-autoapi 3 | nbsphinx 4 | pygments>=2.4.1 5 | -------------------------------------------------------------------------------- /eigentools/__init__.py: -------------------------------------------------------------------------------- 1 | # Copyright (c) 2016, Jeffrey S. Oishi & Susan E. Clark 2 | 3 | # This file is part of Dedalus, which is free software distributed 4 | # under the terms of the GPLv3 license. A copy of the license should 5 | # have been included in the file 'LICENSE.txt', and is also available 6 | # online at . 7 | 8 | from .eigenproblem import Eigenproblem 9 | from .criticalfinder import CriticalFinder 10 | -------------------------------------------------------------------------------- /eigentools/criticalfinder.py: -------------------------------------------------------------------------------- 1 | import logging 2 | import numpy as np 3 | from mpi4py import MPI 4 | import h5py 5 | from scipy import interpolate, optimize 6 | import matplotlib.pyplot as plt 7 | from matplotlib import transforms 8 | 9 | from dedalus.tools.cache import CachedAttribute 10 | 11 | logger = logging.getLogger(__name__.split('.')[-1]) 12 | 13 | class CriticalFinder: 14 | """finds critical parameters for eigenvalue problems. 15 | 16 | This class provides simple tools for finding the critical parameters 17 | for the linear (in)stability of a given flow. The parameter space must 18 | be 2D; typically this will be (k, Re), where k is a wavenumber and Re 19 | is some control parameter (e. g. Reynolds or Rayleigh). The parameters 20 | are defined by the underlying Eigenproblem object. 21 | 22 | Parameters 23 | ---------- 24 | eigenproblem: Eigenproblem 25 | An eigentools eigenproblem object over which to find critical 26 | parameters 27 | param_names : tuple of str 28 | The names of parameters to search over 29 | comm : mpi4py.MPI.Intracomm, optional 30 | The MPI comm group to share jobs across (default: MPI.COMM_WORLD) 31 | find_freq : bool, optional 32 | If True, also find frequency at critical point 33 | 34 | Attributes 35 | ---------- 36 | parameter_grids: 37 | NumPy mesh grids containing the parameter values for the EVP 38 | evalue_grid: 39 | NumPy array of complex values, containing the maximum growth rates 40 | of the EVP for the corresponding input values. 41 | roots : ndarray 42 | Array of roots along axis 1 of parameter_grid 43 | """ 44 | 45 | def __init__(self, eigenproblem, param_names, comm=MPI.COMM_WORLD, find_freq=False): 46 | self.eigenproblem = eigenproblem 47 | self.param_names = param_names 48 | self.comm = comm 49 | self.size = self.comm.size 50 | self.rank = self.comm.rank 51 | self.find_freq = find_freq 52 | 53 | self.roots = None 54 | 55 | def grid_generator(self, points, sparse=False): 56 | """Generates a grid of eigenvalues over the specified parameter 57 | space of an eigenvalue problem. 58 | 59 | Parameters 60 | ---------- 61 | points : tuple of ndarray 62 | The parameter values over which to find the critical value 63 | """ 64 | self.parameter_grids = np.meshgrid(*points) 65 | self.evalue_grid = np.zeros(self.parameter_grids[0].shape, dtype=np.complex128) 66 | dims = self.evalue_grid.shape 67 | # Split parameter load across processes 68 | index = np.arange(np.prod(dims)) 69 | load_indices = np.array_split(index,self.size) 70 | my_indices = load_indices[self.rank] 71 | 72 | # Calculate growth values for local process grid 73 | local_grid = np.empty(my_indices.size,dtype=np.complex128) 74 | for n, index in enumerate(my_indices): 75 | logger.info("Solving Local EVP {}/{}".format(n+1, len(my_indices))) 76 | unraveled_index = np.unravel_index(index, dims) 77 | values = [self.parameter_grids[i][unraveled_index] for i,v in enumerate(self.parameter_grids)] 78 | 79 | gr, indx, freq = self._growth_rate(values, sparse=sparse) 80 | local_grid[n] = gr + 1j*freq 81 | 82 | # Communicate growth modes to root 83 | data = np.empty(dims, dtype=np.complex128) 84 | rec_counts = np.array([s.size for s in load_indices]) 85 | displacements = np.cumsum(rec_counts) - rec_counts 86 | self.comm.Gatherv(local_grid,[data,rec_counts,displacements, MPI.F_DOUBLE_COMPLEX]) 87 | self.evalue_grid = data 88 | 89 | def _growth_rate(self, values, **kwargs): 90 | """Compute growth rate at values 91 | 92 | Parameters 93 | ---------- 94 | values : dict 95 | Dictionary of parameter names and values 96 | """ 97 | var_dict = {self.param_names[i]: v for i,v in enumerate(values)} 98 | return self.eigenproblem.growth_rate(var_dict, **kwargs) #solve 99 | 100 | @CachedAttribute 101 | def _interpolator(self): 102 | """Creates and then uses a 2D grid interpolator for growth rate 103 | 104 | NB: this transposes x and y for the root finding step, because that 105 | requires the function to be interpolated along the FIRST axis 106 | """ 107 | xx = self.parameter_grids[0] 108 | yy = self.parameter_grids[1] 109 | return interpolate.interp2d(yy.T, xx.T, self.evalue_grid.real.T) 110 | 111 | @CachedAttribute 112 | def _freq_interpolator(self): 113 | """Creates and then uses a 2D grid interpolator for growth rate 114 | """ 115 | xx = self.parameter_grids[0] 116 | yy = self.parameter_grids[1] 117 | return interpolate.interp2d(xx, yy, self.evalue_grid.imag) 118 | 119 | def load_grid(self, filename): 120 | """ 121 | Load a grid file, in the format as created in save_grid. 122 | 123 | Parameters 124 | ---------- 125 | filename : str 126 | The name of the .h5 file containing the grid data 127 | """ 128 | with h5py.File(filename,'r') as infile: 129 | self.parameter_grids = [k[()] for k in infile.values() if 'xyz' in k.name] 130 | self.N = len(self.parameter_grids) 131 | logger.info("Read an {}-dimensional grid".format(self.N)) 132 | self.evalue_grid = infile['/grid'][:] 133 | 134 | def save_grid(self, filename): 135 | """ 136 | Saves the grids of all input parameters as well as the growth rate 137 | grid that has been solved for. 138 | 139 | Parameters 140 | ---------- 141 | filename : str 142 | A file stem, which DOES NOT include the file type extension. The 143 | grid will be saved to a file called filen.h5 144 | """ 145 | if self.comm.rank == 0: 146 | with h5py.File(filename+'.h5','w') as outfile: 147 | outfile.create_dataset('grid',data=self.evalue_grid) 148 | for i, grid in enumerate(self.parameter_grids): 149 | outfile.create_dataset('xyz_{}'.format(i),data=grid) 150 | 151 | def _root_finder(self): 152 | """Find rooots from interpolated values at each point along zero axis of parameter_grid 153 | 154 | """ 155 | yy = self.parameter_grids[1] 156 | xx = self.parameter_grids[0] 157 | self.roots = np.zeros_like(xx[0,:]) 158 | for j,x in enumerate(xx[0,:]): 159 | try: 160 | self.roots[j] = optimize.brentq(self._interpolator,yy[0,0],yy[-1,0],args=(x)) 161 | except ValueError: 162 | self.roots[j] = np.nan 163 | 164 | def crit_finder(self, polish_roots=False, polish_sparse=True, tol=1e-3, method='Powell', maxiter=200, **kwargs): 165 | """returns parameters at which critical eigenvalue occurs and optionally frequency at that value. 166 | 167 | The critical parameter is defined as the absolute minimum of the 168 | growth rate, defined in the Eigenproblem via its grow_func. If 169 | frequency is to be found also, returns the frequnecy defined in the 170 | Eigenproblem via its freq_func. 171 | 172 | If find_freq is True, returns (critical parameter 1, critical 173 | parameter 2, frequency); otherwise returns (critical parameter 1, 174 | critical parameter 2) 175 | 176 | Parameters 177 | ---------- 178 | polish_roots : bool, optional 179 | If true, use optimization routines to polish critical value (default: False) 180 | polish_sparse : bool, optional 181 | If true, use the sparse solver when polishing roots (default: True) 182 | tol : float, optional 183 | Tolerance for polishing routine (default: 1e-3) 184 | method : str, optional 185 | Method for scipy.optimize used for polishing (default: Powell) 186 | maxiter : int, optional 187 | Maximum number of optimization iterations used for polishing (default: 200) 188 | 189 | Returns 190 | ------ 191 | tuple 192 | """ 193 | if self.rank != 0: 194 | return 195 | self._root_finder() 196 | mask = np.isfinite(self.roots) 197 | xx_root = self.parameter_grids[0][0,mask] 198 | rroot = self.roots[mask] 199 | 200 | self.root_fn = interpolate.interp1d(xx_root,rroot,kind='cubic') 201 | 202 | mid = xx_root.shape[0]//2 203 | 204 | bracket = [xx_root[0],xx_root[mid],xx_root[-1]] 205 | 206 | self.opt = optimize.minimize_scalar(self.root_fn,bracket=bracket) 207 | 208 | x_crit = self.opt['x'] 209 | y_crit = self.opt['fun'].item() 210 | if self.find_freq: 211 | crit_freq = self._freq_interpolator(x_crit, y_crit)[0] 212 | crits = (x_crit, y_crit, crit_freq) 213 | if polish_roots: 214 | crits = self.critical_polisher(crits, sparse=polish_sparse, 215 | tol=tol, method=method, maxiter=maxiter, **kwargs) 216 | 217 | return crits 218 | 219 | crits = (x_crit, y_crit) 220 | if polish_roots: 221 | crits = self.critical_polisher(crits, sparse=polish_sparse, 222 | tol=tol, method=method, maxiter=maxiter, **kwargs) 223 | 224 | return crits 225 | 226 | def critical_polisher(self, guess, sparse=True, tol=1e-3, method='Powell', maxiter=200, **kwargs): 227 | """ 228 | Polishes a guess for the critical value using scipy's optimization 229 | routines to find a more precise location of the critical value. 230 | 231 | Parameters 232 | ---------- 233 | guess : complex 234 | Initial guess for optimization routines 235 | sparse : bool, optional 236 | If true, use the sparse solver when polishing roots (default: True) 237 | tol : float, optional 238 | Tolerance for polishing routine (default: 1e-3) 239 | method : str, optional 240 | Method for scipy.optimize used for polishing (default: Powell) 241 | maxiter : int, optional 242 | Maximum number of optimization iterations used for polishing 243 | (default: 200) 244 | """ 245 | 246 | # minimize absolute value of growth rate 247 | function = lambda args: np.abs(self._growth_rate(args, sparse=sparse)[0]) 248 | if self.find_freq: 249 | x0 = guess[:-1] 250 | else: 251 | x0 = guess 252 | search_result = optimize.minimize(function, x0, 253 | tol=tol, options={'maxiter': maxiter}, method=method) 254 | 255 | logger.debug("Optimize results: {}".format(search_result)) 256 | 257 | if self.find_freq: 258 | freq = self._freq_interpolator(search_result.x[0],search_result.x[1]) 259 | 260 | if search_result.success: 261 | logger.info('Minimum growth rate of {} found'.format(search_result.fun)) 262 | results = list(search_result.x) 263 | if self.find_freq: 264 | results += list(freq) 265 | return results 266 | else: 267 | logger.warning('Optimize results not fully converged, returning crit_finder results.') 268 | return guess 269 | 270 | def plot_crit(self, axes=None, transpose=False, xlabel = None, ylabel = None, zlabel="growth rate", cmap="viridis"): 271 | """Create a 2D colormap of the grid of growth rates. 272 | 273 | If available, the root values that have been found will be plotted 274 | over the colormap. 275 | 276 | Parameters 277 | ---------- 278 | transpose : bool, optional 279 | If True, plot dim 0 on the y axis and dim 1 on the x axis. 280 | xlabel : str, optional 281 | If not None, the x-label of the plot. Otherwise, use parameter name from EVP 282 | ylabel : str, optional 283 | If not None, the y-label of the plot. Otherwise, use parameter name from EVP 284 | zlabel : str, optional 285 | Label for the colorbar. (default: growth rate) 286 | cmp : str, optional 287 | matplotlib colormap name (default: viridis) 288 | """ 289 | if self.rank != 0: 290 | return 291 | 292 | if axes is None: 293 | fig = plt.figure(figsize=[8,8]) 294 | ax = fig.add_subplot(111) 295 | else: 296 | ax = axes 297 | fig = axes.figure 298 | 299 | # Grab out grid data for colormap 300 | if transpose: 301 | xx = self.parameter_grids[1].T 302 | yy = self.parameter_grids[0].T 303 | grid = self.evalue_grid.real.T 304 | else: 305 | xx = self.parameter_grids[0] 306 | yy = self.parameter_grids[1] 307 | grid = self.evalue_grid.real 308 | # Plot colormap, only plot 2 stdevs off zero 309 | biggest_val = 2*np.abs(grid).std() 310 | 311 | # Setup axes 312 | # Bounds (left, bottom, width, height) relative-to-axes 313 | pbbox = transforms.Bbox.from_bounds(0.03, 0, 0.94, 0.94) 314 | cbbox = transforms.Bbox.from_bounds(0.03, 0.95, 0.94, 0.05) 315 | # Convert to relative-to-figure 316 | to_axes_bbox = transforms.BboxTransformTo(ax.get_position()) 317 | pbbox = pbbox.transformed(to_axes_bbox) 318 | cbbox = cbbox.transformed(to_axes_bbox) 319 | # Create new axes and suppress base axes 320 | pax = ax.figure.add_axes(pbbox) 321 | cax = ax.figure.add_axes(cbbox) 322 | 323 | plot = pax.pcolormesh(xx,yy,grid,cmap=cmap,vmin=-biggest_val,vmax=biggest_val) 324 | ax.axis('off') 325 | cbar = plt.colorbar(plot, cax=cax, label=zlabel, orientation='horizontal') 326 | cbar.outline.set_visible(False) 327 | cax.xaxis.set_ticks_position('top') 328 | cax.xaxis.set_label_position('top') 329 | # Plot root data if they're available 330 | if self.roots is not None: 331 | if transpose: 332 | x = self.roots[:] 333 | y = self.parameter_grids[0][0,:] 334 | else: 335 | x = self.parameter_grids[0][0,:] 336 | y = self.roots[:] 337 | 338 | if transpose: 339 | y, x = y[np.isfinite(x)], x[np.isfinite(x)] 340 | else: 341 | y, x = y[np.isfinite(y)], x[np.isfinite(y)] 342 | pax.scatter(x,y, color='k') 343 | 344 | # Pretty up the plot, save. 345 | pax.set_ylim(yy.min(),yy.max()) 346 | pax.set_xlim(xx.min(),xx.max()) 347 | if xlabel is None: 348 | xlabel = self.param_names[0] 349 | if ylabel is None: 350 | ylabel = self.param_names[1] 351 | pax.set_xlabel(xlabel) 352 | pax.set_ylabel(ylabel) 353 | 354 | return pax,cax 355 | -------------------------------------------------------------------------------- /eigentools/eigenproblem.py: -------------------------------------------------------------------------------- 1 | from dedalus.tools.cache import CachedAttribute 2 | import logging 3 | from dedalus.core.field import Field 4 | from dedalus.core.evaluator import Evaluator 5 | from dedalus.core.system import FieldSystem 6 | from dedalus.tools.post import merge_process_files 7 | import dedalus.public as de 8 | import matplotlib.pyplot as plt 9 | import numpy as np 10 | from scipy.interpolate import interp1d 11 | import scipy.sparse.linalg 12 | from . import tools 13 | 14 | logger = logging.getLogger(__name__.split('.')[-1]) 15 | 16 | class Eigenproblem(): 17 | def __init__(self, EVP, reject=True, factor=1.5, scales=1, drift_threshold=1e6, use_ordinal=False, grow_func=lambda x: x.real, freq_func=lambda x: x.imag): 18 | """An object for feature-rich eigenvalue analysis. 19 | 20 | Eigenproblem provides support for common tasks in eigenvalue 21 | analysis. Dedalus EVP objects compute raw eigenvalues and 22 | eigenvectors for a given problem; Eigenproblem provides support for 23 | numerous common tasks required for scientific use of those 24 | solutions. This includes rejection of inaccurate eigenvalues and 25 | analysis of those rejection criteria, plotting of eigenmodes and 26 | spectra, and projection of 1-D eigenvectors onto 2- or 3-D domains 27 | for use as initial conditions in subsequent initial value problems. 28 | 29 | Additionally, Eigenproblems can compute epsilon-pseudospectra for 30 | arbitrary Dedalus differential-algebraic equations. 31 | 32 | 33 | Parameters 34 | ---------- 35 | EVP : dedalus.core.problems.EigenvalueProblem 36 | The Dedalus EVP object containing the equations to be solved 37 | reject : bool, optional 38 | whether or not to reject spurious eigenvalues (default: True) 39 | factor : float, optional 40 | The factor by which to multiply the resolution. 41 | NB: this must be a rational number such that factor times the 42 | resolution of EVP is an integer. (default: 1.5) 43 | scales : float, optional 44 | A multiple for setting the grid resolution. (default: 1) 45 | drift_threshold : float, optional 46 | Inverse drift ratio threshold for keeping eigenvalues during 47 | rejection (default: 1e6) 48 | use_ordinal : bool, optional 49 | If true, use ordinal method from Boyd (1989); otherwise use 50 | nearest (default: False) 51 | grow_func : func 52 | A function that takes a complex input and returns the growth 53 | rate as defined by the EVP (default: uses real part) 54 | freq_func : func 55 | A function that takes a complex input and returns the frequency 56 | as defined by the EVP (default: uses imaginary part) 57 | 58 | Attributes 59 | ---------- 60 | evalues : ndarray 61 | Lists "good" eigenvalues 62 | evalues_low : ndarray 63 | Lists eigenvalues from low resolution solver (i.e. the 64 | resolution of the specified EVP) 65 | evalues_high : ndarray 66 | Lists eigenvalues from high resolution solver (i.e. factor 67 | times specified EVP resolution) 68 | pseudospectrum : ndarray 69 | epsilon-pseudospectrum computed at specified points in the 70 | complex plane 71 | ps_real : ndarray 72 | real coordinates for epsilon-pseudospectrum 73 | ps_imag : ndarray 74 | imaginary coordinates for epsilon-pseudospectrum 75 | 76 | Notes 77 | ----- 78 | See references for algorithms in individual method docstrings. 79 | 80 | """ 81 | self.reject = reject 82 | self.factor = factor 83 | self.EVP = EVP 84 | self.solver = EVP.build_solver() 85 | if self.reject: 86 | self._build_hires() 87 | 88 | self.grid_name = self.EVP.domain.bases[0].name 89 | self.evalues = None 90 | self.evalues_low = None 91 | self.evalues_high = None 92 | self.pseudospectrum = None 93 | self.ps_real = None 94 | self.ps_imag = None 95 | 96 | self.drift_threshold = drift_threshold 97 | self.use_ordinal = use_ordinal 98 | self.scales = scales 99 | self.grow_func = grow_func 100 | self.freq_func = freq_func 101 | 102 | def _set_parameters(self, parameters): 103 | """set the parameters in the underlying EVP object 104 | 105 | Parameters 106 | ---------- 107 | parameters : dict 108 | Dict of parameter names and values (keys and values 109 | respectively) to set in EVP 110 | 111 | 112 | """ 113 | for k,v in parameters.items(): 114 | tools.update_EVP_params(self.EVP, k, v) 115 | if self.reject: 116 | tools.update_EVP_params(self.EVP_hires, k, v) 117 | 118 | def grid(self): 119 | """get grid points for eigenvectors. 120 | 121 | """ 122 | return self.EVP.domain.grids(scales=self.scales)[0] 123 | 124 | def solve(self, sparse=False, parameters=None, pencil=0, N=15, target=0, **kwargs): 125 | """solve underlying eigenvalue problem. 126 | 127 | Parameters 128 | ---------- 129 | sparse : bool, optional 130 | If true, use sparse solver, otherwise use dense solver 131 | (default: False) 132 | parameters : dict, optional 133 | A dict giving parameter names and values to the EVP. If None, 134 | use values specified at EVP construction time. (default: None) 135 | pencil : int, optional 136 | The EVP pencil to be solved. (default: 0) 137 | N : int, optional 138 | The number of eigenvalues to find if using a sparse solver 139 | (default: 15) 140 | target : complex, optional 141 | The target value to search for when using sparse solver 142 | (default: 0+0j) 143 | 144 | 145 | """ 146 | if parameters: 147 | self._set_parameters(parameters) 148 | self.pencil = pencil 149 | self.N = N 150 | self.target = target 151 | self.solver_kwargs = kwargs 152 | 153 | self._run_solver(self.solver, sparse) 154 | self.evalues_low = self.solver.eigenvalues 155 | 156 | if self.reject: 157 | self._run_solver(self.hires_solver, sparse) 158 | self.evalues_high = self.hires_solver.eigenvalues 159 | self._reject_spurious() 160 | else: 161 | self.evalues = self.evalues_low 162 | self.evalues_index = np.arange(len(self.evalues),dtype=int) 163 | 164 | def _run_solver(self, solver, sparse): 165 | """wrapper method to run solver. 166 | 167 | Parameters 168 | ---------- 169 | solver : dedalus.core.problems.EigenvalueProblem 170 | The Dedalus EVP object containing the equations to be solved 171 | sparse : bool 172 | If True, use sparse solver; otherwise use dense. 173 | """ 174 | if sparse: 175 | solver.solve_sparse(solver.pencils[self.pencil], N=self.N, target=self.target, rebuild_coeffs=True, **self.solver_kwargs) 176 | else: 177 | solver.solve_dense(solver.pencils[self.pencil], rebuild_coeffs=True, **self.solver_kwargs) 178 | 179 | def _set_eigenmode(self, index, all_modes=False): 180 | """use EVP solver's set_state to access eigenmode in grid or coefficient space 181 | 182 | The index parameter is either the index of the ordered good 183 | eigenvalues or the direct index of the low-resolution EVP depending 184 | on the all_modes option. 185 | 186 | Parameters 187 | ---------- 188 | index : int 189 | index of eigenvalue corresponding to desired eigenvector 190 | all_modes : bool, optional 191 | If True, index specifies the unsorted index of the 192 | low-resolution EVP; otherwise it is the index corresponding to 193 | the self.evalues order (default: False) 194 | """ 195 | if all_modes: 196 | good_index = index 197 | else: 198 | good_index = self.evalues_index[index] 199 | self.solver.set_state(good_index) 200 | 201 | def eigenmode(self, index, scales=None, all_modes=False): 202 | """Returns Dedalus FieldSystem object containing the eigenmode 203 | given by index. 204 | 205 | 206 | Parameters 207 | ---------- 208 | index : int 209 | index of eigenvalue corresponding to desired eigenvector 210 | scales : float 211 | A multiple for setting the grid resolution. If not None, will 212 | overwrite self.scales. (default: None) 213 | all_modes : bool, optional 214 | If True, index specifies the unsorted index of the 215 | low-resolution EVP; otherwise it is the index corresponding to 216 | the self.evalues order (default: False) 217 | """ 218 | self._set_eigenmode(index, all_modes=all_modes) 219 | if scales is not None: 220 | self.scales = scales 221 | for f in self.solver.state.fields: 222 | f.set_scales(self.scales,keep_data=True) 223 | 224 | return self.solver.state 225 | 226 | def growth_rate(self, parameters=None, **kwargs): 227 | """returns the maximum growth rate, defined by self.grow_func(), 228 | the index of the maximal mode, and the frequency of that mode. If 229 | there is no growing mode, returns the slowest decay rate. 230 | 231 | also returns the index of the fastest growing mode. If there are 232 | no good eigenvalues, returns np.nan for all three quantities. 233 | 234 | Returns 235 | ------- 236 | growth_rate, index, freqency : tuple of ints 237 | 238 | """ 239 | try: 240 | self.solve(parameters=parameters, **kwargs) 241 | gr_rate = np.max(self.grow_func(self.evalues)) 242 | gr_indx = np.where(self.grow_func(self.evalues) == gr_rate)[0] 243 | freq = self.freq_func(self.evalues[gr_indx[0]]) 244 | 245 | return gr_rate, gr_indx[0], freq 246 | 247 | except np.linalg.linalg.LinAlgError: 248 | logger.warning("Dense eigenvalue solver failed for parameters {}".format(params)) 249 | return np.nan, np.nan, np.nan 250 | except (scipy.sparse.linalg.eigen.arpack.ArpackNoConvergence, scipy.sparse.linalg.eigen.arpack.ArpackError): 251 | logger.warning("Sparse eigenvalue solver failed to converge for parameters {}".format(params)) 252 | return np.nan, np.nan, np.nan 253 | 254 | def plot_mode(self, index, fig_height=8, norm_var=None, scales=None, all_modes=False): 255 | """plots eigenvector corresponding to specified index. 256 | 257 | By default, the plot will show the real and complex parts of the 258 | unnormalized components of the eigenmode. If a norm_var is 259 | specified, all components will be scaled such that variable chosen 260 | is purely real and has unit amplitude. 261 | 262 | Parameters 263 | ---------- 264 | index : int 265 | index of eigenvalue corresponding to desired eigenvector 266 | fig_height : float, optional 267 | Height of constructed figure (default: 8) 268 | norm_var : str 269 | If not None, selects the field in the eigenmode with which to 270 | normalize. Otherwise, plots the unnormalized 271 | eigenmode. (default: None) 272 | scales : float 273 | A multiple for setting the grid resolution. If not None, will 274 | overwrite self.scales. (default: None) 275 | all_modes : bool, optional 276 | If True, index specifies the unsorted index of the 277 | low-resolution EVP; otherwise it is the index corresponding to 278 | the self.evalues order (default: False) 279 | 280 | Returns 281 | ------- 282 | matplotlib.figure.Figure 283 | 284 | """ 285 | state = self.eigenmode(index, scales=scales, all_modes=all_modes) 286 | 287 | z = self.grid() 288 | nrow = 2 289 | nvars = len(self.EVP.variables) 290 | ncol = int(np.ceil(nvars/nrow)) 291 | 292 | if norm_var: 293 | rotation = self.solver.state[norm_var]['g'].conj() 294 | else: 295 | rotation = 1. 296 | 297 | fig = plt.figure(figsize=[fig_height*ncol/nrow,fig_height]) 298 | for i,v in enumerate(self.EVP.variables): 299 | ax = fig.add_subplot(nrow,ncol,i+1) 300 | ax.plot(z, (rotation*state[v]['g']).real, label='real') 301 | ax.plot(z, (rotation*state[v]['g']).imag, label='imag') 302 | ax.set_xlabel(self.grid_name) 303 | ax.set_ylabel(v) 304 | if i == 0: 305 | ax.legend() 306 | 307 | fig.tight_layout() 308 | 309 | return fig 310 | 311 | def project_mode(self, index, domain, transverse_modes, all_modes=False): 312 | """projects a mode specified by index onto a domain of higher 313 | dimension. 314 | 315 | Parameters 316 | ---------- 317 | index : 318 | an integer giving the eigenmode to project 319 | domain : 320 | a domain to project onto 321 | transverse_modes : 322 | a tuple of mode numbers for the transverse directions 323 | 324 | Returns 325 | ------- 326 | dedalus.core.system.FieldSystem 327 | """ 328 | 329 | if len(transverse_modes) != (len(domain.bases) - 1): 330 | raise ValueError("Must specify {} transverse modes for a domain with {} bases; {} specified".format(len(domain.bases)-1, len(domain.bases), len(transverse_modes))) 331 | 332 | field_slice = tuple(transverse_modes) + (slice(None),) 333 | 334 | self._set_eigenmode(index, all_modes=all_modes) 335 | 336 | fields = [] 337 | 338 | for v in self.EVP.variables: 339 | fields.append(domain.new_field(name=v)) 340 | fields[-1]['c'][field_slice] = self.solver.state[v]['c'] 341 | field_system = FieldSystem(fields) 342 | 343 | return field_system 344 | 345 | def write_global_domain(self, field_system, base_name="IVP_output"): 346 | """Given a field system, writes a Dedalus HDF5 file. 347 | 348 | Typically, one would use this to write a field system constructed by project_mode. 349 | 350 | Parameters 351 | ---------- 352 | field_system : dedalus.core.system.FieldSystem 353 | A field system containing the data to be written 354 | base_name : str, optional 355 | The base filename of the resulting HDF5 file. (default: IVP_output) 356 | 357 | """ 358 | output_evaluator = Evaluator(field_system.domain, self.EVP.namespace) 359 | output_handler = output_evaluator.add_file_handler(base_name) 360 | output_handler.add_system(field_system) 361 | 362 | output_evaluator.evaluate_handlers(output_evaluator.handlers, timestep=0,sim_time=0, world_time=0, wall_time=0, iteration=0) 363 | 364 | merge_process_files(base_name, cleanup=True, comm=output_evaluator.domain.distributor.comm) 365 | 366 | def calc_ps(self, k, zgrid, mu=0., pencil=0, inner_product=None, norm=-2, maxiter=10, rtol=1e-3, parameters=None, **kw): 367 | """computes epsilon-pseudospectrum for the eigenproblem. 368 | 369 | Uses the algorithm described in section 5 of 370 | 371 | Embree & Keeler (2017). SIAM J. Matrix Anal. Appl. 38, 3: 372 | 1028-1054. 373 | 374 | to enable the approximation of epsilon-pseudospectra for arbitrary 375 | differential-algebraic equation systems. 376 | 377 | 378 | Parameters: 379 | ----------- 380 | k : int 381 | number of eigenmodes in invariant subspace 382 | zgrid : tuple 383 | (real, imag) points 384 | mu : complex 385 | center point for pseudospectrum. 386 | pencil : int 387 | pencil holding EVP 388 | inner_product : function 389 | a function that takes two field systems and computes their 390 | inner product 391 | parameters : dict, optional 392 | A dict giving parameter names and values to the EVP. If None, 393 | use values specified at EVP construction time. (default: None) 394 | """ 395 | 396 | self.solve(sparse=True, N=k, pencil=pencil, parameters=parameters, **kw) # O(N k)? 397 | pre_right = self.solver.pencils[pencil].pre_right 398 | pre_right_LU = scipy.sparse.linalg.splu(pre_right.tocsc()) # O(N) 399 | V = pre_right_LU.solve(self.solver.eigenvectors) # O(N k) 400 | 401 | # Orthogonalize invariant subspace 402 | Q, R = np.linalg.qr(V) # O(N k^2) 403 | 404 | # Compute approximate Schur factor 405 | E = -(self.solver.pencils[pencil].M_exp) 406 | A = (self.solver.pencils[pencil].L_exp) 407 | A_mu_E = A - mu*E 408 | A_mu_E_LU = scipy.sparse.linalg.splu(A_mu_E.tocsc()) # O(N) 409 | Ghat = Q.conj().T @ A_mu_E_LU.solve(E @ Q) # O(N k^2) 410 | 411 | # Invert-shift Schur factor 412 | I = np.identity(k) 413 | if inner_product is not None: 414 | M = self.compute_mass_matrix(pre_right@Q, inner_product) 415 | Z, S = np.linalg.qr(scipy.linalg.cholesky(M)) 416 | Gmu = S@np.linalg.inv(S@Ghat) + mu*I 417 | else: 418 | logger.warning("No inner product given. Using 2-norm of state vector coefficients. This is probably not physically meaningful, especially if you are using Chebyshev polynomials.") 419 | Gmu = np.linalg.inv(Ghat) + mu*I # O(k^3) 420 | 421 | self.pseudospectrum = self._pseudo(Gmu, zgrid, maxiter=maxiter, rtol=rtol) 422 | self.ps_real = zgrid[0] 423 | self.ps_imag = zgrid[1] 424 | 425 | def compute_mass_matrix(self, Q, inner_product): 426 | """Compute the mass matrix M using a given inner product 427 | 428 | M must be hermitian, so we compute only half the inner products. 429 | 430 | Parameters 431 | ---------- 432 | Q : ndarray 433 | Matrix of eigenvectors 434 | inner_product : function 435 | a function that takes two field systems and computes their 436 | inner product 437 | 438 | Returns 439 | ------- 440 | ndarray 441 | 442 | """ 443 | k = Q.shape[1] 444 | M = np.zeros((k,k), dtype=np.complex128) 445 | Xj = self._copy_system(self.solver.state) 446 | Xi = self._copy_system(self.solver.state) 447 | 448 | for j in range(k): 449 | self.set_state(Xj, Q[:,j]) 450 | for i in range(j,k): # M must be hermitian 451 | self.set_state(Xi, Q[:,i]) 452 | M[j,i] = inner_product(Xj, Xi) 453 | M[i,j] = M[j,i].conj() 454 | 455 | return M 456 | 457 | def set_state(self, system, evector): 458 | """ 459 | Set system to given evector 460 | 461 | Parameters 462 | ---------- 463 | system : FieldSystem 464 | system to fill in 465 | evector : ndarray 466 | eigenvector 467 | """ 468 | system.data[:] = 0 469 | system.set_pencil(self.solver.eigenvalue_pencil, evector) 470 | system.scatter() 471 | 472 | def _copy_system(self, state): 473 | """copies a field system. 474 | 475 | Parameters 476 | ---------- 477 | state : dedalus.core.system.FieldSystem 478 | 479 | Returns 480 | ------- 481 | dedalus.core.system.FieldSystem 482 | """ 483 | fields = [] 484 | for f in state.fields: 485 | field = f.copy() 486 | field.name = f.name 487 | fields.append(field) 488 | 489 | return FieldSystem(fields) 490 | 491 | def _pseudo(self, L, zgrid, maxiter=10, rtol=1e-3): 492 | """computes epsilon-pseudospectrum for a regular eigenvalue 493 | problem. 494 | 495 | If maxiter is zero, uses a direct algorithm: at point z in the 496 | complex plane, the resolvant R is calculated 497 | 498 | R = ||z*I - L||_{-2} 499 | 500 | finding the maximum singular value. 501 | 502 | If maxiter is not zero, uses the iterative algorithm from figure 503 | 39.3 (p.375) of 504 | 505 | Trefethen & Embree, "Spectra and Pseudospectra: The Behavior of 506 | Nonnormal Matrices and Operators" (2005, Princeton University 507 | Press) 508 | 509 | Parameters 510 | ---------- 511 | L : square 2D ndarray 512 | the matrix to be analyzed 513 | zgrid : tuple 514 | (real, imag) points 515 | 516 | Returns 517 | ------- 518 | ndarray 519 | """ 520 | xx = zgrid[0] 521 | yy = zgrid[1] 522 | R = np.zeros((len(xx), len(yy))) 523 | matsize = L.shape[0] 524 | T, Z = scipy.linalg.schur(L, output='complex') 525 | if maxiter == 0: 526 | logger.debug("Using direct solver for calculating pseudospectrum") 527 | else: 528 | logger.debug("Using iterative solver for calculating pseudospectrum") 529 | for j, y in enumerate(yy): 530 | for i, x in enumerate(xx): 531 | z = (x + 1j*y) 532 | # if _maxiter is set to zero 533 | if maxiter == 0: 534 | R[j,i] = np.linalg.norm((z*np.eye(matsize) - L), ord=-2) 535 | else: 536 | T1 = z*np.eye(matsize) - T 537 | T2 = T1.conj().T 538 | sigold = 0 539 | qold = np.zeros(matsize,dtype=np.complex128) 540 | beta = 0 541 | 542 | q = np.random.randn(matsize)+1j*np.random.randn(matsize) 543 | q /= np.linalg.norm(q) 544 | H = np.zeros((maxiter+1, maxiter+1), dtype=np.complex128) 545 | for p in range(maxiter): 546 | v = scipy.linalg.solve_triangular(T1, scipy.linalg.solve_triangular(T2,q,lower=True)) - beta*qold 547 | alpha = np.dot(q.conj(), v) 548 | v -= alpha*q 549 | beta = np.linalg.norm(v) 550 | qold = q 551 | q = v/beta 552 | H[p+1,p] = beta 553 | H[p,p+1] = beta 554 | H[p,p] = alpha 555 | sig = np.max(np.linalg.eigvalsh(H[:p+1,:p+1])) 556 | if np.abs(sigold/sig - 1) < rtol: 557 | break 558 | sigold = sig 559 | if p == (maxiter - 1): 560 | logger.warning("Iterative solver did not converge for (x, y) = ({},{})".format(x,y)) 561 | R[j, i] = 1/np.sqrt(sig) 562 | return R 563 | 564 | def plot_spectrum(self, axes=None, spectype='good', xlog=True, ylog=True, real_label="real", imag_label="imag"): 565 | """Plots the spectrum. 566 | 567 | The spectrum plots real parts on the x axis and imaginary parts on 568 | the y axis. 569 | 570 | Parameters 571 | ---------- 572 | spectype : {'good', 'low', 'high'}, optional 573 | specifies whether to use good, low, or high eigenvalues 574 | xlog : bool, optional 575 | Use symlog on x axis 576 | ylog : bool, optional 577 | Use symlog on y axis 578 | real_label : str, optional 579 | Label to be applied to the real axis 580 | imag_label : str, optional 581 | Label to be applied to the imaginary axis 582 | """ 583 | if spectype == 'low': 584 | ev = self.evalues_low 585 | elif spectype == 'high': 586 | ev = self.evalues_high 587 | elif spectype == 'good': 588 | ev = self.evalues_good 589 | else: 590 | raise ValueError("Spectrum type is not one of {low, high, good}") 591 | 592 | if axes is None: 593 | fig = plt.figure() 594 | ax = fig.add_subplot(111) 595 | else: 596 | ax = axes 597 | fig = axes.figure 598 | 599 | ax.scatter(ev.real, ev.imag) 600 | 601 | if xlog: 602 | ax.set_xscale('symlog') 603 | if ylog: 604 | ax.set_yscale('symlog') 605 | ax.set_xlabel(real_label) 606 | ax.set_ylabel(imag_label) 607 | if axes is None: 608 | fig.tight_layout() 609 | 610 | return ax 611 | 612 | def _reject_spurious(self): 613 | """perform eigenvalue rejection 614 | 615 | """ 616 | evg, indx = self._discard_spurious_eigenvalues() 617 | self.evalues_good = evg 618 | self.evalues_index = indx 619 | self.evalues = self.evalues_good 620 | 621 | def _build_hires(self): 622 | """builds a high-resolution EVP from the EVP passed in at 623 | construction 624 | 625 | """ 626 | old_evp = self.EVP 627 | old_x = old_evp.domain.bases[0] 628 | 629 | x = tools.basis_from_basis(old_x, self.factor) 630 | d = de.Domain([x],comm=old_evp.domain.dist.comm) 631 | self.EVP_hires = de.EVP(d,old_evp.variables,old_evp.eigenvalue, ncc_cutoff=old_evp.ncc_kw['cutoff'], max_ncc_terms=old_evp.ncc_kw['max_terms'], tolerance=self.EVP.tol) 632 | 633 | for k,v in old_evp.substitutions.items(): 634 | self.EVP_hires.substitutions[k] = v 635 | 636 | for k,v in old_evp.parameters.items(): 637 | if type(v) == Field: #NCCs 638 | new_field = d.new_field() 639 | v.set_scales(self.factor, keep_data=True) 640 | new_field['g'] = v['g'] 641 | self.EVP_hires.parameters[k] = new_field 642 | else: #scalars 643 | self.EVP_hires.parameters[k] = v 644 | 645 | for e in old_evp.equations: 646 | self.EVP_hires.add_equation(e['raw_equation']) 647 | 648 | try: 649 | for b in old_evp.boundary_conditions: 650 | self.EVP_hires.add_bc(b['raw_equation']) 651 | except AttributeError: 652 | # after version befc23584fea, Dedalus no longer 653 | # distingishes BCs from other equations 654 | pass 655 | 656 | self.hires_solver = self.EVP_hires.build_solver() 657 | 658 | def _discard_spurious_eigenvalues(self): 659 | """ Solves the linear eigenvalue problem for two different 660 | resolutions. Returns trustworthy eigenvalues using nearest delta, 661 | from Boyd chapter 7. 662 | """ 663 | eval_low = self.evalues_low 664 | eval_hi = self.evalues_high 665 | 666 | # Reverse engineer correct indices to make unsorted list from sorted 667 | reverse_eval_low_indx = np.arange(len(eval_low)) 668 | reverse_eval_hi_indx = np.arange(len(eval_hi)) 669 | 670 | eval_low_and_indx = np.asarray(list(zip(eval_low, reverse_eval_low_indx))) 671 | eval_hi_and_indx = np.asarray(list(zip(eval_hi, reverse_eval_hi_indx))) 672 | 673 | # remove nans 674 | eval_low_and_indx = eval_low_and_indx[np.isfinite(eval_low)] 675 | eval_hi_and_indx = eval_hi_and_indx[np.isfinite(eval_hi)] 676 | 677 | # Sort eval_low and eval_hi by real parts 678 | eval_low_and_indx = eval_low_and_indx[np.argsort(eval_low_and_indx[:, 0].real)] 679 | eval_hi_and_indx = eval_hi_and_indx[np.argsort(eval_hi_and_indx[:, 0].real)] 680 | 681 | eval_low_sorted = eval_low_and_indx[:, 0] 682 | eval_hi_sorted = eval_hi_and_indx[:, 0] 683 | 684 | # Compute sigmas from lower resolution run (gridnum = N1) 685 | sigmas = np.zeros(len(eval_low_sorted)) 686 | sigmas[0] = np.abs(eval_low_sorted[0] - eval_low_sorted[1]) 687 | sigmas[1:-1] = [0.5*(np.abs(eval_low_sorted[j] - eval_low_sorted[j - 1]) + np.abs(eval_low_sorted[j + 1] - eval_low_sorted[j])) for j in range(1, len(eval_low_sorted) - 1)] 688 | sigmas[-1] = np.abs(eval_low_sorted[-2] - eval_low_sorted[-1]) 689 | 690 | if not (np.isfinite(sigmas)).all(): 691 | logger.warning("At least one eigenvalue spacings (sigmas) is non-finite (np.inf or np.nan)!") 692 | 693 | # Ordinal delta 694 | self.delta_ordinal = np.array([np.abs(eval_low_sorted[j] - eval_hi_sorted[j])/sigmas[j] for j in range(len(eval_low_sorted))]) 695 | 696 | # Nearest delta 697 | self.delta_near = np.array([np.nanmin(np.abs(eval_low_sorted[j] - eval_hi_sorted)/sigmas[j]) for j in range(len(eval_low_sorted))]) 698 | 699 | # Discard eigenvalues with 1/delta_near < drift_threshold 700 | if self.use_ordinal: 701 | inverse_drift = 1/self.delta_ordinal 702 | else: 703 | inverse_drift = 1/self.delta_near 704 | eval_low_and_indx = eval_low_and_indx[np.where(inverse_drift > self.drift_threshold)] 705 | 706 | eval_low = eval_low_and_indx[:, 0] 707 | indx = eval_low_and_indx[:, 1].real.astype(int) 708 | 709 | return eval_low, indx 710 | 711 | def plot_drift_ratios(self, axes=None): 712 | """Plot drift ratios (both ordinal and nearest) vs. mode number. 713 | 714 | The drift ratios give a measure of how good a given eigenmode is; 715 | this can help set thresholds. 716 | 717 | Returns 718 | ------- 719 | matplotlib.figure.Figure 720 | 721 | """ 722 | if self.reject is False: 723 | raise NotImplementedError("Can't plot drift ratios unless eigenvalue rejection is True.") 724 | 725 | if axes is None: 726 | fig = plt.figure() 727 | ax = fig.add_subplot(111) 728 | else: 729 | ax = axes 730 | fig = axes.figure 731 | 732 | mode_numbers = np.arange(len(self.delta_near)) 733 | ax.semilogy(mode_numbers,1/self.delta_near,'o',alpha=0.4) 734 | ax.semilogy(mode_numbers,1/self.delta_ordinal,'x',alpha=0.4) 735 | 736 | ax.set_prop_cycle(None) 737 | good_near = 1/self.delta_near > self.drift_threshold 738 | good_ordinal = 1/self.delta_ordinal > self.drift_threshold 739 | ax.semilogy(mode_numbers[good_near],1/self.delta_near[good_near],'o', label='nearest') 740 | ax.semilogy(mode_numbers[good_ordinal],1/self.delta_ordinal[good_ordinal],'x',label='ordinal') 741 | ax.axhline(self.drift_threshold,alpha=0.4, color='black') 742 | ax.set_xlabel("mode number") 743 | ax.set_ylabel(r"$1/\delta$") 744 | ax.legend() 745 | 746 | return ax 747 | -------------------------------------------------------------------------------- /eigentools/tools.py: -------------------------------------------------------------------------------- 1 | import dedalus.public as de 2 | 3 | # these are the currently supported dedalus eigenvalue bases 4 | bases_register = {"Chebyshev": de.Chebyshev, "Fourier": de.Fourier, "Legendre": de.Legendre} 5 | 6 | def update_EVP_params(EVP, key, value): 7 | # Dedalus workaround: must change values in two places 8 | vv = EVP.namespace[key] 9 | vv.value = value 10 | EVP.parameters[key] = value 11 | 12 | def basis_from_basis(basis, factor): 13 | """duplicates input basis with number of modes multiplied by input factor. 14 | 15 | the new number of modes will be cast to an integer 16 | 17 | inputs 18 | ------ 19 | basis : a dedalus basis 20 | factor : a float that will multiply the grid size by basis 21 | 22 | """ 23 | basis_type = basis.__class__.__name__ 24 | n_hi = int(basis.base_grid_size*factor) 25 | 26 | if type(basis) == de.Compound: 27 | sub_bases = [] 28 | for sub_basis in basis.subbases: 29 | sub_basis_type = sub_basis.__class__.__name__ 30 | try: 31 | nb = bases_register[sub_basis_type](basis.name, n_hi, interval=sub_basis.interval) 32 | except KeyError: 33 | raise KeyError("Don't know how to make a basis of type {}".format(basis_type)) 34 | sub_bases.append(nb) 35 | new_basis = de.Compound(basis.name, tuple(sub_bases)) 36 | else: 37 | try: 38 | new_basis = bases_register[basis_type](basis.name, n_hi, interval=basis.interval) 39 | except KeyError: 40 | raise KeyError("Don't know how to make a basis of type {}".format(basis_type)) 41 | 42 | return new_basis 43 | -------------------------------------------------------------------------------- /examples/mri.py: -------------------------------------------------------------------------------- 1 | """ 2 | finds the critical magnetic Renoylds number and wave number for the magnetorotational instability (MRI). 3 | 4 | This script can be run in parallel by using 5 | 6 | $ mpirun -np 4 python3 mri.py 7 | 8 | It will parallelize over the grid generation portion and save that 9 | 10 | """ 11 | import sys 12 | from mpi4py import MPI 13 | from eigentools import Eigenproblem, CriticalFinder 14 | import time 15 | import dedalus.public as de 16 | import numpy as np 17 | import matplotlib.pylab as plt 18 | 19 | import logging 20 | 21 | logger = logging.getLogger(__name__.split('.')[-1]) 22 | 23 | comm = MPI.COMM_WORLD 24 | 25 | 26 | # Define the MRI problem in Dedalus: 27 | 28 | x = de.Chebyshev('x',64) 29 | d = de.Domain([x],comm=MPI.COMM_SELF) 30 | 31 | mri = de.EVP(d,['psi','u', 'A', 'B', 'psix', 'psixx', 'psixxx', 'ux', 'Ax', 'Bx'],'sigma') 32 | 33 | 34 | Rm = 4.879 35 | Pm = 0.001 36 | mri.parameters['q'] = 1.5 37 | mri.parameters['beta'] = 25.0 38 | mri.parameters['iR'] = Pm/Rm 39 | mri.parameters['Rm'] = Rm 40 | mri.parameters['Q'] = 0.748 41 | mri.substitutions['iRm'] = '1/Rm' 42 | 43 | mri.add_equation("sigma*psixx - Q**2*sigma*psi - iR*dx(psixxx) + 2*iR*Q**2*psixx - iR*Q**4*psi - 2*1j*Q*u - (2/beta)*1j*Q*dx(Ax) + (2/beta)*Q**3*1j*A = 0") 44 | mri.add_equation("sigma*u - iR*dx(ux) + iR*Q**2*u - (q - 2)*1j*Q*psi - (2/beta)*1j*Q*B = 0") 45 | mri.add_equation("sigma*A - iRm*dx(Ax) + iRm*Q**2*A - 1j*Q*psi = 0") 46 | mri.add_equation("sigma*B - iRm*dx(Bx) + iRm*Q**2*B - 1j*Q*u + q*1j*Q*A = 0") 47 | 48 | mri.add_equation("dx(psi) - psix = 0") 49 | mri.add_equation("dx(psix) - psixx = 0") 50 | mri.add_equation("dx(psixx) - psixxx = 0") 51 | mri.add_equation("dx(u) - ux = 0") 52 | mri.add_equation("dx(A) - Ax = 0") 53 | mri.add_equation("dx(B) - Bx = 0") 54 | 55 | mri.add_bc("left(u) = 0") 56 | mri.add_bc("right(u) = 0") 57 | mri.add_bc("left(psi) = 0") 58 | mri.add_bc("right(psi) = 0") 59 | mri.add_bc("left(A) = 0") 60 | mri.add_bc("right(A) = 0") 61 | mri.add_bc("left(psix) = 0") 62 | mri.add_bc("right(psix) = 0") 63 | mri.add_bc("left(Bx) = 0") 64 | mri.add_bc("right(Bx) = 0") 65 | 66 | # create an Eigenproblem object 67 | EP = Eigenproblem(mri) 68 | 69 | cf = CriticalFinder(EP, ("Q", "Rm"), comm, find_freq=False) 70 | 71 | # generating the grid is the longest part 72 | nx = 20 73 | ny = 20 74 | xpoints = np.linspace(0.5, 1.5, nx) 75 | ypoints = np.linspace(4.6, 5.5, ny) 76 | 77 | file_name = 'mri_growth_rate' 78 | try: 79 | cf.load_grid('{}.h5'.format(file_name)) 80 | except: 81 | start = time.time() 82 | cf.grid_generator((xpoints, ypoints), sparse=True) 83 | end = time.time() 84 | 85 | if comm.rank == 0: 86 | cf.save_grid(file_name) 87 | logger.info("grid generation time: {:10.5f} sec".format(end-start)) 88 | 89 | crit = cf.crit_finder(polish_roots=False) 90 | 91 | if comm.rank == 0: 92 | logger.info("critical Rm = {:10.5f}, Q = {:10.5f}".format(crit[1], crit[0])) 93 | # create plot of critical parameter space 94 | pax,cax = cf.plot_crit() 95 | fig = pax.figure 96 | # add an interpolated critical line 97 | x_lim = cf.parameter_grids[0][0,np.isfinite(cf.roots)] 98 | x_hires = np.linspace(x_lim[0], x_lim[-1], 100) 99 | pax.plot(x_hires, cf.root_fn(x_hires), color='k') 100 | fig.savefig('{}.png'.format(file_name), dpi=300) 101 | 102 | # plot the spectrum for the critical mode 103 | logger.info("solving dense eigenvalue problem for critical parameters") 104 | EP.solve(parameters = {"Q": crit[0], "Rm": crit[1]}, sparse=False) 105 | ax = EP.plot_spectrum() 106 | 107 | # mark critical mode 108 | eps = 1e-2 109 | mask = np.abs(EP.evalues.real) < eps 110 | ax.scatter(EP.evalues[mask].real, EP.evalues[mask].imag, c='red') 111 | ax.figure.savefig('mri_critical_spectrum.png', dpi=300) 112 | 113 | # plot drift ratio for critical mode 114 | ax = EP.plot_drift_ratios() 115 | ax.figure.savefig('mri_critical_drift_ratios.png', dpi=300) 116 | -------------------------------------------------------------------------------- /examples/orr_sommerfeld.py: -------------------------------------------------------------------------------- 1 | """finds the critical Renoylds number, wave number, and frequency for the 2 | Orr-Somerfeld eigenvalue equation. 3 | 4 | NB: This formulation uses a slightly different scaling of the eigenvalue than Orszag (1971). In order to convert, use 5 | 6 | sigma = -1j*alpha*Re*lambda, 7 | 8 | where sigma is our eigenvalue and Lambda is Orszag's. 9 | 10 | """ 11 | import matplotlib 12 | matplotlib.use('Agg') 13 | from mpi4py import MPI 14 | from eigentools import Eigenproblem, CriticalFinder 15 | import time 16 | import dedalus.public as de 17 | import numpy as np 18 | import matplotlib.pylab as plt 19 | import sys 20 | import logging 21 | logger = logging.getLogger(__name__.split('.')[-1]) 22 | 23 | file_name = sys.argv[0].strip('.py') 24 | comm = MPI.COMM_WORLD 25 | 26 | 27 | # Define the Orr-Somerfeld problem in Dedalus: 28 | 29 | z = de.Chebyshev('z',50) 30 | d = de.Domain([z],comm=MPI.COMM_SELF) 31 | 32 | orr_somerfeld = de.EVP(d,['w','wz','wzz','wzzz'],'sigma') 33 | orr_somerfeld.parameters['alpha'] = 1. 34 | orr_somerfeld.parameters['Re'] = 10000. 35 | 36 | orr_somerfeld.add_equation('dz(wzzz) - 2*alpha**2*wzz + alpha**4*w - sigma*(wzz-alpha**2*w)-1j*alpha*(Re*(1-z**2)*(wzz-alpha**2*w) + 2*Re*w) = 0 ') 37 | orr_somerfeld.add_equation('dz(w)-wz = 0') 38 | orr_somerfeld.add_equation('dz(wz)-wzz = 0') 39 | orr_somerfeld.add_equation('dz(wzz)-wzzz = 0') 40 | 41 | orr_somerfeld.add_bc('left(w) = 0') 42 | orr_somerfeld.add_bc('right(w) = 0') 43 | orr_somerfeld.add_bc('left(wz) = 0') 44 | orr_somerfeld.add_bc('right(wz) = 0') 45 | 46 | # create an Eigenproblem object 47 | EP = Eigenproblem(orr_somerfeld) 48 | 49 | # create a shim function to translate (x, y) to the parameters for the eigenvalue problem: 50 | 51 | cf = CriticalFinder(EP,("alpha", "Re"), comm, find_freq=True) 52 | 53 | # generating the grid is the longest part 54 | start = time.time() 55 | nx = 20 56 | ny = 20 57 | xpoints = np.linspace(1.0, 1.1, nx) 58 | ypoints = np.linspace(5500, 6000, ny) 59 | try: 60 | cf.load_grid('{}.h5'.format(file_name)) 61 | except: 62 | cf.grid_generator((xpoints, ypoints), sparse=True) 63 | if comm.rank == 0: 64 | cf.save_grid(file_name) 65 | end = time.time() 66 | if comm.rank == 0: 67 | logger.info("grid generation time: {:10.5f} sec".format(end-start)) 68 | 69 | crit = cf.crit_finder(polish_roots=True, tol=1e-5, method='Nelder-Mead') 70 | 71 | Re_orszag = 5772.22 72 | alpha_orszag = 1.02056 73 | omega_orszag = -1555.2070 74 | 75 | if comm.rank == 0: 76 | alpha = crit[0] 77 | Re = crit[1] 78 | omega = crit[2] 79 | 80 | Re_err = (Re-Re_orszag)/Re_orszag 81 | alpha_err = (alpha-alpha_orszag)/alpha_orszag 82 | L2 = np.sqrt((Re-Re_orszag)**2 + (alpha-alpha_orszag)**2) 83 | logger.info("critical wavenumber alpha = {:10.5f}".format(alpha)) 84 | logger.info("critical Re = {:10.5f}".format(Re)) 85 | logger.info("critical omega = {:10.5f}".format(omega)) 86 | logger.info("critical Re error = {:10.5e}".format(Re_err)) 87 | logger.info("critical alpha error = {:10.5}".format(alpha_err)) 88 | logger.info("L2 norm from Orszag 71 solution = {:10.5e}".format(L2)) 89 | 90 | cf.save_grid('orr_sommerfeld_growth_rates') 91 | cf.plot_crit() 92 | 93 | -------------------------------------------------------------------------------- /examples/rayleigh_benard_2d.py: -------------------------------------------------------------------------------- 1 | """ 2 | Finds the critical Rayleigh number and wavenumber for the 2-dimensional, 3 | incompressible, Boussinesq Navier-Stokes equations in order to determine 4 | the onset of convection in such a system. 5 | """ 6 | import matplotlib 7 | matplotlib.use('Agg') 8 | from mpi4py import MPI 9 | from eigentools import Eigenproblem, CriticalFinder 10 | import time 11 | import dedalus.public as de 12 | import numpy as np 13 | import sys 14 | import logging 15 | 16 | logger = logging.getLogger(__name__.split('.')[-1]) 17 | 18 | 19 | comm = MPI.COMM_WORLD 20 | 21 | 22 | no_slip = False 23 | stress_free = True 24 | file_name = sys.argv[0].strip('.py') 25 | if no_slip: 26 | file_name += '_no_slip' 27 | elif stress_free: 28 | file_name += '_stress_free' 29 | 30 | Nz = 16 31 | z = de.Chebyshev('z',Nz, interval=(0, 1)) 32 | d = de.Domain([z],comm=MPI.COMM_SELF) 33 | 34 | rayleigh_benard = de.EVP(d,['p', 'b', 'u', 'w', 'bz', 'uz', 'wz'], eigenvalue='omega') 35 | rayleigh_benard.parameters['k'] = 3.117 #horizontal wavenumber 36 | rayleigh_benard.parameters['Ra'] = 1708. #Rayleigh number, rigid-rigid 37 | rayleigh_benard.parameters['Pr'] = 1 #Prandtl number 38 | rayleigh_benard.parameters['dzT0'] = 1 39 | rayleigh_benard.substitutions['dt(A)'] = 'omega*A' 40 | rayleigh_benard.substitutions['dx(A)'] = '1j*k*A' 41 | 42 | #Boussinesq eqns -- nondimensionalized on thermal diffusion timescale 43 | #Incompressibility 44 | rayleigh_benard.add_equation("dx(u) + wz = 0") 45 | #Momentum eqns 46 | rayleigh_benard.add_equation("dt(u) - Pr*(dx(dx(u)) + dz(uz)) + dx(p) = -u*dx(u) - w*uz") 47 | rayleigh_benard.add_equation("dt(w) - Pr*(dx(dx(w)) + dz(wz)) + dz(p) - Ra*Pr*b = -u*dx(w) - w*wz") 48 | #Temp eqn 49 | rayleigh_benard.add_equation("dt(b) - w*dzT0 - (dx(dx(b)) + dz(bz)) = -u*dx(b) - w*bz") 50 | #Derivative defns 51 | rayleigh_benard.add_equation("dz(u) - uz = 0") 52 | rayleigh_benard.add_equation("dz(w) - wz = 0") 53 | rayleigh_benard.add_equation("dz(b) - bz = 0") 54 | 55 | 56 | 57 | #fixed temperature 58 | rayleigh_benard.add_bc('left(b) = 0') 59 | rayleigh_benard.add_bc('right(b) = 0') 60 | #Impenetrable 61 | rayleigh_benard.add_bc('left(w) = 0') 62 | rayleigh_benard.add_bc('right(w) = 0') 63 | 64 | 65 | if no_slip: 66 | rayleigh_benard.add_bc('left(u) = 0') 67 | rayleigh_benard.add_bc('right(u) = 0') 68 | elif stress_free: 69 | rayleigh_benard.add_bc('left(uz) = 0') 70 | rayleigh_benard.add_bc('right(uz) = 0') 71 | 72 | # create an Eigenproblem object 73 | EP = Eigenproblem(rayleigh_benard) 74 | 75 | cf = CriticalFinder(EP, ("k","Ra"), comm, find_freq = True) 76 | 77 | # generating the grid is the longest part 78 | start = time.time() 79 | if no_slip: 80 | nx = 20 81 | ny = 20 82 | xpoints = np.linspace(2, 4, ny) 83 | ypoints = np.linspace(1000, 3000, nx) 84 | elif stress_free: 85 | #657.5, 2.221 86 | nx = 10 87 | ny = 10 88 | xpoints = np.linspace(2, 2.4, ny) 89 | ypoints = np.linspace(550, 700, nx) 90 | 91 | try: 92 | cf.load_grid('{}.h5'.format(file_name)) 93 | except: 94 | cf.grid_generator((xpoints, ypoints), sparse=True) 95 | cf.save_grid(file_name) 96 | 97 | end = time.time() 98 | if comm.rank == 0: 99 | logger.info("grid generation time: {:10.5f} sec".format(end-start)) 100 | 101 | logger.info("Beginning critical finding with root polishing...") 102 | begin = time.time() 103 | crit = cf.crit_finder(polish_roots=True, tol=1e-5) 104 | end = time.time() 105 | logger.info("critical finding/root polishing time: {:10.5f} sec".format(end-start)) 106 | 107 | if comm.rank == 0: 108 | print("crit = {}".format(crit)) 109 | print("critical wavenumber k = {:10.5f}".format(crit[0])) 110 | print("critical Ra = {:10.5f}".format(crit[1])) 111 | print("critical freq = {:10.5f}".format(crit[2])) 112 | 113 | pax, cax = cf.plot_crit(xlabel=r'$k_x$', ylabel=r'$\mathrm{Ra}$') 114 | pax.figure.savefig("rayleigh_benard_2d_growth_rates.png",dpi=300) 115 | -------------------------------------------------------------------------------- /license.txt: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU General Public License is a free, copyleft license for 11 | software and other kinds of works. 12 | 13 | The licenses for most software and other practical works are designed 14 | to take away your freedom to share and change the works. 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But first, please read 674 | . 675 | -------------------------------------------------------------------------------- /setup.py: -------------------------------------------------------------------------------- 1 | import setuptools 2 | 3 | with open("README.md", "r") as fh: 4 | long_description = fh.read() 5 | 6 | setuptools.setup( 7 | name="eigentools", 8 | version="2.2112", 9 | author="J. S. Oishi", 10 | author_email="jsoishi@gmail.com", 11 | description="A toolkit for solving eigenvalue problems with Dedalus", 12 | long_description=long_description, 13 | long_description_content_type="text/markdown", 14 | url="https://github.com/dedalusproject/eigentools", 15 | packages=setuptools.find_packages(), 16 | classifiers=[ 17 | "Programming Language :: Python :: 3", 18 | "License :: OSI Approved :: GNU General Public License v3 (GPLv3)" 19 | ], 20 | python_requires='>=3.5', 21 | ) 22 | -------------------------------------------------------------------------------- /tests/non_constant_test.py: -------------------------------------------------------------------------------- 1 | """test problem for eigentools for non-constant coefficients: 2 | 3 | problem from equation 26 of 4 | 5 | Huang, Chen, and Luo, Applied Mathematics Letters (2013) 6 | https://www.sciencedirect.com/science/article/pii/S0893965913000748 7 | 8 | y''''(x) - 0.02 x^2 y'' - 0.04 x y' + (0.0001 x^4 - 0.02) y = lambda y 9 | 10 | with boundary conditions 11 | 12 | y(0) = y(5) = y'(0) = y'(5) = 0 13 | 14 | this is their Case 1 15 | 16 | NB: I corrected a typo 17 | 18 | Table 3 from that paper gives 19 | 0.86690250239956 20 | 6.35768644786998 21 | 23.99274694653769 22 | 64.97869559403952 23 | 144.2841396045761 24 | 25 | NB: THIS IS NOT A HIGH PRECISION TEST! It's unclear from the reference what the "true" values actually are. We agree much more closely with their reference 14, but I'm not sure if that is a more trustworthy calculation anyway. 26 | 27 | """ 28 | import pytest 29 | import numpy as np 30 | import dedalus.public as de 31 | from eigentools import Eigenproblem, CriticalFinder 32 | 33 | @pytest.mark.parametrize('Nx', [60]) 34 | @pytest.mark.parametrize('sparse', [False]) 35 | @pytest.mark.parametrize('ordinal', [False, True]) 36 | def test_non_constant(Nx, sparse, ordinal): 37 | x = de.Chebyshev('x',Nx,interval=(0,5)) 38 | d = de.Domain([x,]) 39 | 40 | prob = de.EVP(d,['y','yx','yxx','yxxx'],'sigma') 41 | 42 | prob.add_equation("dx(yxxx) -0.02*x*x*yxx -0.04*x*yx + (0.0001*x*x*x*x - 0.02)*y - sigma*y = 0") 43 | prob.add_equation("dx(yxx) - yxxx = 0") 44 | prob.add_equation("dx(yx) - yxx = 0") 45 | prob.add_equation("dx(y) - yx = 0") 46 | 47 | prob.add_bc("left(y) = 0") 48 | prob.add_bc("right(y) = 0") 49 | prob.add_bc("left(yx) = 0") 50 | prob.add_bc("right(yx) = 0") 51 | 52 | EP = Eigenproblem(prob, use_ordinal=ordinal) 53 | 54 | EP.solve(sparse=sparse) 55 | indx = EP.evalues_good.argsort() 56 | 57 | five_evals = EP.evalues_good[indx][0:5] 58 | print("First five good eigenvalues are: ") 59 | print(five_evals) 60 | print(five_evals[-1]) 61 | 62 | reference = np.array([0.86690250239956+0j, 6.35768644786998+0j, 23.99274694653769+0j, 64.97869559403952+0j, 144.2841396045761+0j]) 63 | 64 | assert np.allclose(reference, five_evals,rtol=1e-4) 65 | -------------------------------------------------------------------------------- /tests/test_rbc_growth.py: -------------------------------------------------------------------------------- 1 | import pytest 2 | import dedalus.public as de 3 | import eigentools as eig 4 | import numpy as np 5 | from mpi4py import MPI 6 | 7 | def rbc_problem(problem_type, domain, stress_free=False): 8 | problems = {'EVP': de.EVP, 'IVP': de.IVP} 9 | 10 | try: 11 | args = [domain,['p', 'b', 'u', 'w', 'bz', 'uz', 'wz']] 12 | if problem_type == 'EVP': 13 | args.append('omega') 14 | rayleigh_benard = problems[problem_type](*args) 15 | except KeyError: 16 | raise ValueError("problem_type must be one of 'EVP' or 'IVP', not {}".format(problem)) 17 | 18 | rayleigh_benard.parameters['k'] = 3.117 #horizontal wavenumber 19 | rayleigh_benard.parameters['Ra'] = 1708. #Rayleigh number, rigid-rigid 20 | rayleigh_benard.parameters['Pr'] = 1 #Prandtl number 21 | rayleigh_benard.parameters['dzT0'] = 1 22 | if problem_type == 'EVP': 23 | rayleigh_benard.substitutions['dt(A)'] = 'omega*A' 24 | rayleigh_benard.substitutions['dx(A)'] = '1j*k*A' 25 | 26 | rayleigh_benard.add_equation("dx(u) + wz = 0") 27 | rayleigh_benard.add_equation("dt(u) - Pr*(dx(dx(u)) + dz(uz)) + dx(p) = -u*dx(u) - w*uz") 28 | rayleigh_benard.add_equation("dt(w) - Pr*(dx(dx(w)) + dz(wz)) + dz(p) - Ra*Pr*b = -u*dx(w) - w*wz") 29 | rayleigh_benard.add_equation("dt(b) - w*dzT0 - (dx(dx(b)) + dz(bz)) = -u*dx(b) - w*bz") 30 | rayleigh_benard.add_equation("dz(u) - uz = 0") 31 | rayleigh_benard.add_equation("dz(w) - wz = 0") 32 | rayleigh_benard.add_equation("dz(b) - bz = 0") 33 | rayleigh_benard.add_bc('left(b) = 0') 34 | rayleigh_benard.add_bc('right(b) = 0') 35 | rayleigh_benard.add_bc('left(w) = 0') 36 | rayleigh_benard.add_bc('right(w) = 0') 37 | if stress_free: 38 | rayleigh_benard.add_bc('left(uz) = 0') 39 | rayleigh_benard.add_bc('right(uz) = 0') 40 | else: 41 | rayleigh_benard.add_bc('left(u) = 0') 42 | rayleigh_benard.add_bc('right(u) = 0') 43 | 44 | return rayleigh_benard 45 | 46 | @pytest.mark.parametrize('z', [de.Chebyshev('z',16, interval=(0, 1)), de.Compound('z',(de.Chebyshev('z',10, interval=(0, 0.5)),de.Chebyshev('z',10, interval=(0.5, 1))))]) 47 | @pytest.mark.parametrize('sparse', [True, False]) 48 | def test_rbc_growth(z, sparse): 49 | d = de.Domain([z]) 50 | 51 | rayleigh_benard = rbc_problem('EVP',d) 52 | 53 | EP = eig.Eigenproblem(rayleigh_benard) 54 | 55 | growth, index, freq = EP.growth_rate(sparse=sparse) 56 | assert np.allclose((growth, freq), (0.0018125573647729994,0.)) 57 | 58 | @pytest.mark.parametrize('z', [de.Chebyshev('z',16, interval=(0, 1))]) 59 | def test_rbc_output(z): 60 | d = de.Domain([z]) 61 | rb_evp = rbc_problem('EVP',d) 62 | EP = eig.Eigenproblem(rb_evp) 63 | 64 | growth, index, freq = EP.growth_rate(sparse=False) 65 | 66 | x = de.Fourier('x', 32) 67 | ivp_domain = de.Domain([x,z],grid_dtype=np.float64) 68 | 69 | fields = EP.project_mode(index, ivp_domain, [1,]) 70 | EP.write_global_domain(fields) 71 | 72 | rb_IVP = rbc_problem('IVP', ivp_domain) 73 | solver = rb_IVP.build_solver(de.timesteppers.RK222) 74 | solver.load_state("IVP_output/IVP_output_s1.h5",-1) 75 | 76 | @pytest.mark.parametrize('z', [de.Chebyshev('z',16, interval=(0, 1))]) 77 | def test_rbc_crit_find(z): 78 | d = de.Domain([z], comm=MPI.COMM_SELF) 79 | rb_evp = rbc_problem('EVP', d, stress_free=True) 80 | EP = eig.Eigenproblem(rb_evp) 81 | comm = MPI.COMM_WORLD 82 | cf = eig.CriticalFinder(EP, ("k", "Ra"), comm, find_freq=True) 83 | 84 | nx = 10 85 | ny = 10 86 | xpoints = np.linspace(2, 2.4, nx) 87 | ypoints = np.linspace(550, 700, ny) 88 | 89 | cf.grid_generator((xpoints, ypoints),sparse=True) 90 | crit = cf.crit_finder(polish_roots=True, tol=1e-6, method='Powell') 91 | 92 | Rac = 27*np.pi**4/4. 93 | kc = 2*np.pi/2**1.5 94 | 95 | assert np.allclose(crit, [kc, Rac, 0.], rtol=1e-5) 96 | --------------------------------------------------------------------------------