{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "b4a8c377",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "054abb0c",
   "metadata": {
    "load": "myst_code_init.py",
    "tags": [
     "remove-cell"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Loading pyMOR defaults from file /builds/pymor/pymor/docs/source/pymor_defaults.py\n"
     ]
    }
   ],
   "source": [
    "import warnings\n",
    "\n",
    "import matplotlib as mpl\n",
    "from IPython import get_ipython\n",
    "\n",
    "import pymor.tools.random\n",
    "\n",
    "ip = get_ipython()\n",
    "if ip is not None:\n",
    "    ip.run_line_magic('matplotlib', 'inline')\n",
    "\n",
    "warnings.filterwarnings('ignore', category=UserWarning, module='torch')\n",
    "\n",
    "pymor.tools.random._default_random_state = None\n",
    "\n",
    "mpl.rcParams['figure.facecolor'] = (1.0, 1.0, 1.0, 0.0)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "48b79f00",
   "metadata": {},
   "source": [
    "# Tutorial: Binding an external PDE solver to pyMOR\n",
    "\n",
    "One of pyMOR's main features is easy integration of external solvers that implement the full-order model. In this tutorial\n",
    "we will do this step-by-step for a custom toy solver written in C++.\n",
    "If you use the [FEniCS](<https://fenicsproject.org>) or [NGSolve](<https://ngsolve.org>) PDE solver libraries,\n",
    "you can find ready-to-use pyMOR bindings in the `bindings` package. pyMOR support for\n",
    "[deal.II](<https://dealii.org>) can be found in a [separate repository](<https://github.com/pymor/pymor-deal.II>).\n",
    "\n",
    "## Defining the PDE solver\n",
    "\n",
    "Our solver discretizes the one-dimensional Laplace equation {math}`u''(x)=0` on the interval {math}`[\\text{left},\\text{right}]`\n",
    "using a central differences scheme with {math}`h=\\frac{|\\text{right}-\\text{left}|}{n}`.\n",
    "First, we need a class to store our data in and with some basic linear algebra operations\n",
    "declared on it.\n",
    "\n",
    "```{literalinclude} minimal_cpp_demo/model.hh\n",
    ":language: cpp\n",
    ":lines: 4-19\n",
    "\n",
    "\n",
    "```\n",
    "\n",
    "Next, we need the operator that discretizes our PDE.\n",
    "\n",
    "```{literalinclude} minimal_cpp_demo/model.hh\n",
    ":language: cpp\n",
    ":lines: 22-32\n",
    "\n",
    "\n",
    "```\n",
    "\n",
    "Together with some header guards, these two snippets make up our {download}`model.hh <minimal_cpp_demo/model.hh>`.\n",
    "\n",
    "The definitions for the `Vector` class are pretty straightforward:\n",
    "\n",
    "```{literalinclude} minimal_cpp_demo/model.cc\n",
    ":language: cpp\n",
    ":lines: 7-35\n",
    "\n",
    "```\n",
    "\n",
    "Just like the diffusion operator that computes a central differences stencil:\n",
    "\n",
    "```{literalinclude} minimal_cpp_demo/model.cc\n",
    ":language: cpp\n",
    ":lines: 39-49\n",
    "\n",
    "```\n",
    "\n",
    "This completes all the C++ code needed for the toy solver itself. Next, we will make this code usable from Python.\n",
    "We utilize the [pybind11](<https://github.com/pybind/pybind11>) library to create a Python\n",
    "[extension module](<https://docs.python.org/3/extending/extending.html>) named `model`, that allows us to manipulate\n",
    "instances of the C++ `Vector` and `DiffusionOperator` classes.\n",
    "\n",
    "Compiling the PDE solver as a shared library and creating Python bindings for it using\n",
    "[pybind11](<https://github.com/pybind/pybind11>), [Cython](<https://cython.org>) or\n",
    "[ctypes](<https://docs.python.org/3/library/ctypes.html>) is the preferred way of integrating\n",
    "external solvers, as it offers maximal flexibility and performance. For instance, in this\n",
    "example we will actually completely implement the `Model` in Python using a\n",
    "`time stepper` from pyMOR to\n",
    "`solve` the `Model`.\n",
    "\n",
    "When this is not an option,\n",
    "[RPC](<https://en.wikipedia.org/wiki/Remote_procedure_call>)-based approaches are\n",
    "possible as well. For small to medium-sized linear problems, another option it to import\n",
    "system matrices and snapshot data into pyMOR via file exchange and to use NumPy-based\n",
    "`Operators` and `VectorArrays`\n",
    "to represent the full-order model.\n",
    "\n",
    "## Binding the solver to Python\n",
    "\n",
    "All of the C++ code related to the extension module is defined inside a scope started with\n",
    "\n",
    "```{literalinclude} minimal_cpp_demo/model.cc\n",
    ":language: cpp\n",
    ":lines: 56-57\n",
    "\n",
    "```\n",
    "\n",
    "This tells pybind11 to make the contained symbols accessible in the module instance `m` that will be importable by the\n",
    "name `model`. Now we create a new pybind11 `class\\_` object that wraps the `DiffusionOperator`. Note that the module\n",
    "instance is passed to the constructor alongside a name for the Python class and a docstring. The second\n",
    "line shows how to define an init function for the Python object by using the special `py:init` object to\n",
    "forward arguments to the C++ constructor.\n",
    "\n",
    "```{literalinclude} minimal_cpp_demo/model.cc\n",
    ":lines: 60-61\n",
    ":language: cpp\n",
    "\n",
    "```\n",
    "\n",
    "Next, we define read-only properties on the Python side named after and delegated to the members of the C++ class.\n",
    "\n",
    "```{literalinclude} minimal_cpp_demo/model.cc\n",
    ":lines: 62-63\n",
    ":language: cpp\n",
    "\n",
    "```\n",
    "\n",
    "The last `DiffusionOperator`-related line exposes the function call to apply in the same way:\n",
    "\n",
    "```{literalinclude} minimal_cpp_demo/model.cc\n",
    ":lines: 64\n",
    ":language: cpp\n",
    "\n",
    "```\n",
    "\n",
    "This is everything that is needed to expose the operator to Python. We will now do the same for the `Vector`,\n",
    "with a few more advanced techniques added.\n",
    "\n",
    "```{literalinclude} minimal_cpp_demo/model.cc\n",
    ":lines: 66-68\n",
    ":language: cpp\n",
    "\n",
    "```\n",
    "\n",
    "Again we define a `py:class\\_` with appropriate name and docstring, but now we also indicate to pybind11\n",
    "that this class will implement the [buffer protocol](<https://docs.python.org/3/c-api/buffer.html>), which basically\n",
    "exposes direct access to the chunk of memory associated with a `Vector` instance to Python.\n",
    "We also see how we can dispatch multiple init functions by using `py:init` objects with C++ lambda functions.\n",
    "Note that direct memory access to the vector data from Python is not required to integrate a solver with pyMOR.\n",
    "It is, however, useful for debugging and quickly modifying or extending the solver from within Python. For instance,\n",
    "in our toy example we will use the direct memory access to quickly define a visualization of the solutions and to\n",
    "construct the right-hand side vector for our problem.\n",
    "\n",
    "```{literalinclude} minimal_cpp_demo/model.cc\n",
    ":lines: 70-74\n",
    ":language: cpp\n",
    "\n",
    "```\n",
    "\n",
    "```{literalinclude} minimal_cpp_demo/model.cc\n",
    ":lines: 76-80\n",
    ":language: cpp\n",
    "\n",
    "```\n",
    "\n",
    "This completes the {download}`model.cc <minimal_cpp_demo/model.cc>`.\n",
    "\n",
    "This extension module needs to be compiled to a shared object that the Python interpreter can import.\n",
    "We use a minimal [CMake](<https://cmake.org/>) project that generates makefiles for us to achieve this.\n",
    "\n",
    "First we make sure pybind11 can be used:\n",
    "\n",
    "```{literalinclude} minimal_cpp_demo/CMakeLists.txt\n",
    ":lines: 11-13\n",
    ":language: cmake\n",
    "\n",
    "```\n",
    "\n",
    "Next, we define a new library with our `model.cc` as the single source file and let pybind11 set the proper compile\n",
    "flags.\n",
    "\n",
    "```{literalinclude} minimal_cpp_demo/CMakeLists.txt\n",
    ":lines: 16-19\n",
    ":language: cmake\n",
    "\n",
    "```\n",
    "\n",
    "That is all that is needed for {download}`CMakeLists.txt <minimal_cpp_demo/CMakeLists.txt>`.\n",
    "In the next step, we will switch to a bash terminal and actually compile this module.\n",
    "\n",
    "After creating a build directory for the module, we let cmake initialize the build and call make to execute the\n",
    "compilation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "cd9a82d2",
   "metadata": {
    "tags": [
     "raises-exception"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-- The C compiler identification is GNU 10.2.1\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-- The CXX compiler identification is GNU 10.2.1\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-- Detecting C compiler ABI info\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-- Detecting C compiler ABI info - done\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-- Check for working C compiler: /usr/bin/cc - skipped\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-- Detecting C compile features\n",
      "-- Detecting C compile features - done\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-- Detecting CXX compiler ABI info\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-- Detecting CXX compiler ABI info - done\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-- Check for working CXX compiler: /usr/bin/c++ - skipped\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-- Detecting CXX compile features\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-- Detecting CXX compile features - done\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-- Found Python3: /usr/local/bin/python3 (found version \"3.13.6\") found components: Interpreter Deve"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "lopment.Module \n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-- Performing Test HAS_FLTO_AUTO\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-- Performing Test HAS_FLTO_AUTO - Success\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-- Found pybind11: /usr/local/lib/python3.13/site-packages/pybind11/include (found version \"3.0.1\")\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-- Found pybind11 v3.0.1: /usr/local/lib/python3.13/site-packages/pybind11/include;/usr/local/includ"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "e/python3.13\n",
      "-- Configuring done\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-- Generating done\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-- Build files have been written to: /builds/pymor/pymor/docs/source/minimal_cpp_demo/build\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "make[2]: Entering directory '/builds/pymor/pymor/docs/source/minimal_cpp_demo/build'\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "make[3]: Entering directory '/builds/pymor/pymor/docs/source/minimal_cpp_demo/build'\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "make[4]: Entering directory '/builds/pymor/pymor/docs/source/minimal_cpp_demo/build'\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[35m\u001b[1mScanning dependencies of target model\u001b[0m\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "make[4]: Leaving directory '/builds/pymor/pymor/docs/source/minimal_cpp_demo/build'\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "make[4]: Entering directory '/builds/pymor/pymor/docs/source/minimal_cpp_demo/build'\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 50%] \u001b[32mBuilding CXX object CMakeFiles/model.dir/model.cc.o\u001b[0m\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[100%] \u001b[32m\u001b[1mLinking CXX shared module model.cpython-313-x86_64-linux-gnu.so\u001b[0m\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "make[4]: Leaving directory '/builds/pymor/pymor/docs/source/minimal_cpp_demo/build'\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[100%] Built target model\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "make[3]: Leaving directory '/builds/pymor/pymor/docs/source/minimal_cpp_demo/build'\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "make[2]: Leaving directory '/builds/pymor/pymor/docs/source/minimal_cpp_demo/build'\n"
     ]
    }
   ],
   "source": [
    "%%bash\n",
    "mkdir -p minimal_cpp_demo/build\n",
    "cmake -B minimal_cpp_demo/build -S minimal_cpp_demo\n",
    "make -C minimal_cpp_demo/build"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ffa61957",
   "metadata": {},
   "source": [
    "To be able to use this extension module we need to insert the build directory into the path where the Python\n",
    "interpreter looks for things to import. Afterwards we can import the module and create and use the exported classes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "f77aa549",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['DiffusionOperator',\n",
       " 'Vector',\n",
       " '__doc__',\n",
       " '__file__',\n",
       " '__loader__',\n",
       " '__name__',\n",
       " '__package__',\n",
       " '__spec__']"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sys\n",
    "sys.path.insert(0, 'minimal_cpp_demo/build')\n",
    "\n",
    "import model\n",
    "mymodel = model.DiffusionOperator(10, 0, 1)\n",
    "myvector = model.Vector(10, 0)\n",
    "mymodel.apply(myvector, myvector)\n",
    "dir(model)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "57cc766f",
   "metadata": {},
   "source": [
    "## Using the exported Python classes with pyMOR\n",
    "\n",
    "All of pyMOR's algorithms operate on `VectorArray` and `Operator` objects that all\n",
    "share the same programming interface. To be able to use\n",
    "our Python `model.Vector` and `model.DiffusionOperator` in pyMOR, we have to provide implementations of\n",
    "`VectorArray`, `VectorSpace` and `Operator` that wrap the classes defined in the extension module\n",
    "and translate calls to the interface methods into operations on `model.Vector` and `model.DiffusionOperator`.\n",
    "\n",
    "Instead of writing a full implementation of a `VectorArray` that manages multiple `model.Vector`\n",
    "instances, we can instead implement a wrapper `WrappedVector` for a single `model.Vector` instance based on\n",
    "`CopyOnWriteVector` which will be used to create\n",
    "`ListVectorArrays` via a `ListVectorSpace`-based `WrappedVectorSpace`.\n",
    "\n",
    "The `CopyOnWriteVector` base class manages a reference count for\n",
    "us and automatically copies data when necessary in methods `scal`\n",
    "and `axpy`. To use this, we need to implement\n",
    "{meth}`!_scal`\n",
    "and {meth}`!_axpy` in addition to all the abstract\n",
    "methods from  `CopyOnWriteVector`. We can get away\n",
    "with using just a stub that raises an {class}`~NotImplementedError` in some methods that are not actually called in our example."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "4551d881",
   "metadata": {},
   "outputs": [],
   "source": [
    "from pymor.operators.interface import Operator\n",
    "from pymor.vectorarrays.list import CopyOnWriteVector, ListVectorSpace\n",
    "\n",
    "import numpy as np\n",
    "import math\n",
    "from model import Vector, DiffusionOperator\n",
    "\n",
    "\n",
    "class WrappedVector(CopyOnWriteVector):\n",
    "\n",
    "    def __init__(self, vector):\n",
    "        assert isinstance(vector, Vector)\n",
    "        self._impl = vector\n",
    "\n",
    "    @classmethod\n",
    "    def from_instance(cls, instance):\n",
    "        return cls(instance._impl)\n",
    "\n",
    "    def to_numpy(self, ensure_copy=False):\n",
    "        # Note how this uses the buffer protocol setup to allow efficient\n",
    "        # data access as a Numpy Vector\n",
    "        result = np.frombuffer(self._impl, dtype=np.float64)\n",
    "        if ensure_copy:\n",
    "            result = result.copy()\n",
    "        return result\n",
    "\n",
    "    def _copy_data(self):\n",
    "        # This uses the second exposed 'init' signature to delegate to the C++ copy constructor\n",
    "        self._impl = Vector(self._impl)\n",
    "\n",
    "    def _scal(self, alpha):\n",
    "        self._impl.scal(alpha)\n",
    "\n",
    "    def _axpy(self, alpha, x):\n",
    "        self._impl.axpy(alpha, x._impl)\n",
    "\n",
    "    def inner(self, other):\n",
    "        return self._impl.inner(other._impl)\n",
    "\n",
    "    def norm(self):\n",
    "        return math.sqrt(self.inner(self))\n",
    "\n",
    "    def norm2(self):\n",
    "        return self.inner(self)\n",
    "\n",
    "    def sup_norm(self):\n",
    "        raise NotImplementedError\n",
    "\n",
    "    def dofs(self, dof_indices):\n",
    "        raise NotImplementedError\n",
    "\n",
    "    def amax(self):\n",
    "        raise NotImplementedError"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "43b0acdf",
   "metadata": {},
   "source": [
    "The implementation of the `WrappedVectorSpace` is very short as most of the necessary methods\n",
    "of `VectorSpace` are implemented in `ListVectorSpace`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "e4161b73",
   "metadata": {},
   "outputs": [],
   "source": [
    "class WrappedVectorSpace(ListVectorSpace):\n",
    "\n",
    "    def __init__(self, dim):\n",
    "        self.dim = dim\n",
    "\n",
    "    def zero_vector(self):\n",
    "        return WrappedVector(Vector(self.dim, 0))\n",
    "\n",
    "    def make_vector(self, obj):\n",
    "        # obj is a `model.Vector` instance\n",
    "        return WrappedVector(obj)\n",
    "\n",
    "    def __eq__(self, other):\n",
    "        return type(other) is WrappedVectorSpace and self.dim == other.dim"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e42d2f05",
   "metadata": {},
   "source": [
    "Wrapping the `model.DiffusionOperator` is straightforward as well. We just need to attach\n",
    "suitable `VectorSpaces` to the class and implement the application of the operator on a `VectorArray`\n",
    "as a sequence of applications on single vectors."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "b0b755a8",
   "metadata": {},
   "outputs": [],
   "source": [
    "class WrappedDiffusionOperator(Operator):\n",
    "    def __init__(self, op):\n",
    "        assert isinstance(op, DiffusionOperator)\n",
    "        self.op = op\n",
    "        self.source = WrappedVectorSpace(op.dim_source)\n",
    "        self.range = WrappedVectorSpace(op.dim_range)\n",
    "        self.linear = True\n",
    "\n",
    "    @classmethod\n",
    "    def create(cls, n, left, right):\n",
    "        return cls(DiffusionOperator(n, left, right))\n",
    "\n",
    "    def apply(self, U, mu=None):\n",
    "        assert U in self.source\n",
    "\n",
    "        def apply_one_vector(u):\n",
    "            v = Vector(self.range.dim, 0)\n",
    "            self.op.apply(u._impl, v)\n",
    "            return v\n",
    "\n",
    "        return self.range.make_array([apply_one_vector(u) for u in U.vectors])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "841458ea",
   "metadata": {},
   "source": [
    "## Putting it all together\n",
    "\n",
    "As a demonstration, we will use our toy Laplace solver to compute an approximation for\n",
    "the transient diffusion equation\n",
    "\n",
    "```{math}\n",
    "\\frac{\\partial u}{\\partial t} = {\\alpha_\\mu} \\frac{\\partial^2 u}{\\partial x^2},\n",
    "```\n",
    "\n",
    "with explicit timestepping provided by pyMOR, with a parameterized, block-wise defined,  diffusion\n",
    "coefficient  {math}`\\alpha_\\mu`.\n",
    "\n",
    "First up, we implement a `discretize` function that uses the `WrappedDiffusionOperator` and `WrappedVectorSpace`\n",
    "to assemble an `InstationaryModel`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "5476f2c6",
   "metadata": {},
   "outputs": [],
   "source": [
    "from pymor.algorithms.pod import pod\n",
    "from pymor.algorithms.timestepping import ExplicitEulerTimeStepper\n",
    "from pymor.discretizers.builtin.gui.visualizers import OnedVisualizer\n",
    "from pymor.models.basic import InstationaryModel\n",
    "from pymor.discretizers.builtin import OnedGrid\n",
    "from pymor.operators.constructions import VectorOperator, LincombOperator\n",
    "from pymor.parameters.functionals import ProjectionParameterFunctional\n",
    "from pymor.reductors.basic import InstationaryRBReductor\n",
    "\n",
    "\n",
    "def discretize(n, nt, blocks):\n",
    "    h = 1. / blocks\n",
    "    ops = [WrappedDiffusionOperator.create(n, h * i, h * (i + 1)) for i in range(blocks)]\n",
    "    pfs = [ProjectionParameterFunctional('diffusion_coefficients', blocks, i) for i in range(blocks)]\n",
    "    operator = LincombOperator(ops, pfs)\n",
    "\n",
    "    initial_data = operator.source.zeros()\n",
    "\n",
    "    rhs_vec = operator.range.zeros()\n",
    "    rhs_data = rhs_vec.vectors[0].to_numpy()\n",
    "    rhs_data[:] = np.ones(len(rhs_data))\n",
    "    rhs_data[0] = 0\n",
    "    rhs_data[len(rhs_data) - 1] = 0\n",
    "    rhs = VectorOperator(rhs_vec)\n",
    "\n",
    "    # we can re-use pyMOR's builtin grid and visualizer for our demonstration\n",
    "    grid = OnedGrid(domain=(0, 1), num_intervals=n)\n",
    "    visualizer = OnedVisualizer(grid)\n",
    "\n",
    "    time_stepper = ExplicitEulerTimeStepper(nt)\n",
    "\n",
    "    fom = InstationaryModel(T=1, operator=operator, rhs=rhs, initial_data=initial_data,\n",
    "                            time_stepper=time_stepper, num_values=20,\n",
    "                            visualizer=visualizer, name='C++-Model')\n",
    "    return fom"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d8227309",
   "metadata": {},
   "source": [
    "Now we can build a reduced basis for our model. Note that this code is not specific to our wrapped classes.\n",
    "Those wrapped classes are only directly used in the `discretize` call."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "621dcb34",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "9feb4d2b720e48418bbd677ed1d86bcb",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Accordion(children=(HTML(value='', layout=Layout(height='16em', width='100%')),), titles=('Log Output',))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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       "\n",
       "<link rel=\"stylesheet\"\n",
       "href=\"https://maxcdn.bootstrapcdn.com/font-awesome/4.4.0/css/font-awesome.min.css\">\n",
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       "  function isInternetExplorer() {\n",
       "    ua = navigator.userAgent;\n",
       "    /* MSIE used to detect old browsers and Trident used to newer ones*/\n",
       "    return ua.indexOf(\"MSIE \") > -1 || ua.indexOf(\"Trident/\") > -1;\n",
       "  }\n",
       "\n",
       "  /* Define the Animation class */\n",
       "  function Animation(frames, img_id, slider_id, interval, loop_select_id){\n",
       "    this.img_id = img_id;\n",
       "    this.slider_id = slider_id;\n",
       "    this.loop_select_id = loop_select_id;\n",
       "    this.interval = interval;\n",
       "    this.current_frame = 0;\n",
       "    this.direction = 0;\n",
       "    this.timer = null;\n",
       "    this.frames = new Array(frames.length);\n",
       "\n",
       "    for (var i=0; i<frames.length; i++)\n",
       "    {\n",
       "     this.frames[i] = new Image();\n",
       "     this.frames[i].src = frames[i];\n",
       "    }\n",
       "    var slider = document.getElementById(this.slider_id);\n",
       "    slider.max = this.frames.length - 1;\n",
       "    if (isInternetExplorer()) {\n",
       "        // switch from oninput to onchange because IE <= 11 does not conform\n",
       "        // with W3C specification. It ignores oninput and onchange behaves\n",
       "        // like oninput. In contrast, Microsoft Edge behaves correctly.\n",
       "        slider.setAttribute('onchange', slider.getAttribute('oninput'));\n",
       "        slider.setAttribute('oninput', null);\n",
       "    }\n",
       "    this.set_frame(this.current_frame);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.get_loop_state = function(){\n",
       "    var button_group = document[this.loop_select_id].state;\n",
       "    for (var i = 0; i < button_group.length; i++) {\n",
       "        var button = button_group[i];\n",
       "        if (button.checked) {\n",
       "            return button.value;\n",
       "        }\n",
       "    }\n",
       "    return undefined;\n",
       "  }\n",
       "\n",
       "  Animation.prototype.set_frame = function(frame){\n",
       "    this.current_frame = frame;\n",
       "    document.getElementById(this.img_id).src =\n",
       "            this.frames[this.current_frame].src;\n",
       "    document.getElementById(this.slider_id).value = this.current_frame;\n",
       "  }\n",
       "\n",
       "  Animation.prototype.next_frame = function()\n",
       "  {\n",
       "    this.set_frame(Math.min(this.frames.length - 1, this.current_frame + 1));\n",
       "  }\n",
       "\n",
       "  Animation.prototype.previous_frame = function()\n",
       "  {\n",
       "    this.set_frame(Math.max(0, this.current_frame - 1));\n",
       "  }\n",
       "\n",
       "  Animation.prototype.first_frame = function()\n",
       "  {\n",
       "    this.set_frame(0);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.last_frame = function()\n",
       "  {\n",
       "    this.set_frame(this.frames.length - 1);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.slower = function()\n",
       "  {\n",
       "    this.interval /= 0.7;\n",
       "    if(this.direction > 0){this.play_animation();}\n",
       "    else if(this.direction < 0){this.reverse_animation();}\n",
       "  }\n",
       "\n",
       "  Animation.prototype.faster = function()\n",
       "  {\n",
       "    this.interval *= 0.7;\n",
       "    if(this.direction > 0){this.play_animation();}\n",
       "    else if(this.direction < 0){this.reverse_animation();}\n",
       "  }\n",
       "\n",
       "  Animation.prototype.anim_step_forward = function()\n",
       "  {\n",
       "    this.current_frame += 1;\n",
       "    if(this.current_frame < this.frames.length){\n",
       "      this.set_frame(this.current_frame);\n",
       "    }else{\n",
       "      var loop_state = this.get_loop_state();\n",
       "      if(loop_state == \"loop\"){\n",
       "        this.first_frame();\n",
       "      }else if(loop_state == \"reflect\"){\n",
       "        this.last_frame();\n",
       "        this.reverse_animation();\n",
       "      }else{\n",
       "        this.pause_animation();\n",
       "        this.last_frame();\n",
       "      }\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.anim_step_reverse = function()\n",
       "  {\n",
       "    this.current_frame -= 1;\n",
       "    if(this.current_frame >= 0){\n",
       "      this.set_frame(this.current_frame);\n",
       "    }else{\n",
       "      var loop_state = this.get_loop_state();\n",
       "      if(loop_state == \"loop\"){\n",
       "        this.last_frame();\n",
       "      }else if(loop_state == \"reflect\"){\n",
       "        this.first_frame();\n",
       "        this.play_animation();\n",
       "      }else{\n",
       "        this.pause_animation();\n",
       "        this.first_frame();\n",
       "      }\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.pause_animation = function()\n",
       "  {\n",
       "    this.direction = 0;\n",
       "    if (this.timer){\n",
       "      clearInterval(this.timer);\n",
       "      this.timer = null;\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.play_animation = function()\n",
       "  {\n",
       "    this.pause_animation();\n",
       "    this.direction = 1;\n",
       "    var t = this;\n",
       "    if (!this.timer) this.timer = setInterval(function() {\n",
       "        t.anim_step_forward();\n",
       "    }, this.interval);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.reverse_animation = function()\n",
       "  {\n",
       "    this.pause_animation();\n",
       "    this.direction = -1;\n",
       "    var t = this;\n",
       "    if (!this.timer) this.timer = setInterval(function() {\n",
       "        t.anim_step_reverse();\n",
       "    }, this.interval);\n",
       "  }\n",
       "</script>\n",
       "\n",
       "<style>\n",
       ".animation {\n",
       "    display: inline-block;\n",
       "    text-align: center;\n",
       "}\n",
       "input[type=range].anim-slider {\n",
       "    width: 374px;\n",
       "    margin-left: auto;\n",
       "    margin-right: auto;\n",
       "}\n",
       ".anim-buttons {\n",
       "    margin: 8px 0px;\n",
       "}\n",
       ".anim-buttons button {\n",
       "    padding: 0;\n",
       "    width: 36px;\n",
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       ".anim-state label {\n",
       "    margin-right: 8px;\n",
       "}\n",
       ".anim-state input {\n",
       "    margin: 0;\n",
       "    vertical-align: middle;\n",
       "}\n",
       "</style>\n",
       "\n",
       "<div class=\"animation\">\n",
       "  <img id=\"_anim_img81d530a3c1994dad8c1654d4d48c2086\">\n",
       "  <div class=\"anim-controls\">\n",
       "    <input id=\"_anim_slider81d530a3c1994dad8c1654d4d48c2086\" type=\"range\" class=\"anim-slider\"\n",
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       "      <button title=\"Pause\" aria-label=\"Pause\" onclick=\"anim81d530a3c1994dad8c1654d4d48c2086.pause_animation()\">\n",
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       "          <i class=\"fa fa-plus\"></i></button>\n",
       "    </div>\n",
       "    <form title=\"Repetition mode\" aria-label=\"Repetition mode\" action=\"#n\" name=\"_anim_loop_select81d530a3c1994dad8c1654d4d48c2086\"\n",
       "          class=\"anim-state\">\n",
       "      <input type=\"radio\" name=\"state\" value=\"once\" id=\"_anim_radio1_81d530a3c1994dad8c1654d4d48c2086\"\n",
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       "      <input type=\"radio\" name=\"state\" value=\"loop\" id=\"_anim_radio2_81d530a3c1994dad8c1654d4d48c2086\"\n",
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       "      <input type=\"radio\" name=\"state\" value=\"reflect\" id=\"_anim_radio3_81d530a3c1994dad8c1654d4d48c2086\"\n",
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       "  </div>\n",
       "</div>\n",
       "\n",
       "\n",
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       "\"\n",
       "\n",
       "\n",
       "    /* set a timeout to make sure all the above elements are created before\n",
       "       the object is initialized. */\n",
       "    setTimeout(function() {\n",
       "        anim81d530a3c1994dad8c1654d4d48c2086 = new Animation(frames, img_id, slider_id, 200.0,\n",
       "                                 loop_select_id);\n",
       "    }, 0);\n",
       "  })()\n",
       "</script>\n"
      ],
      "text/plain": [
       "<matplotlib.animation.FuncAnimation at 0x7f2e1f9e5a90>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "# discretize\n",
    "fom = discretize(50, 10000, 4)\n",
    "parameter_space = fom.parameters.space(0.1, 1)\n",
    "\n",
    "# generate solution snapshots\n",
    "snapshots = fom.solution_space.empty()\n",
    "for mu in parameter_space.sample_uniformly(2):\n",
    "    snapshots.append(fom.solve(mu))\n",
    "\n",
    "# apply POD\n",
    "reduced_basis = pod(snapshots, modes=4)[0]\n",
    "\n",
    "# reduce the model\n",
    "reductor = InstationaryRBReductor(fom, reduced_basis, check_orthonormality=True)\n",
    "rom = reductor.reduce()\n",
    "\n",
    "# stochastic error estimation\n",
    "mu_max = None\n",
    "err_max = -1.\n",
    "for mu in parameter_space.sample_randomly(10):\n",
    "    U_RB = reductor.reconstruct(rom.solve(mu))\n",
    "    U = fom.solve(mu)\n",
    "    err = np.max((U_RB-U).norm())\n",
    "    if err > err_max:\n",
    "        err_max = err\n",
    "        mu_max = mu\n",
    "\n",
    "# visualize maximum error solution\n",
    "U_RB = (reductor.reconstruct(rom.solve(mu_max)))\n",
    "U = fom.solve(mu_max)\n",
    "fom.visualize((U_RB, U), title=f'mu = {mu}', legend=('reduced', 'detailed'))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "88658493",
   "metadata": {},
   "source": [
    "As you can see in this comparison, we get a good approximation of the full-order model here and\n",
    "the error plot confirms it:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "e78456ac",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<link rel=\"stylesheet\"\n",
       "href=\"https://maxcdn.bootstrapcdn.com/font-awesome/4.4.0/css/font-awesome.min.css\">\n",
       "<script language=\"javascript\">\n",
       "  function isInternetExplorer() {\n",
       "    ua = navigator.userAgent;\n",
       "    /* MSIE used to detect old browsers and Trident used to newer ones*/\n",
       "    return ua.indexOf(\"MSIE \") > -1 || ua.indexOf(\"Trident/\") > -1;\n",
       "  }\n",
       "\n",
       "  /* Define the Animation class */\n",
       "  function Animation(frames, img_id, slider_id, interval, loop_select_id){\n",
       "    this.img_id = img_id;\n",
       "    this.slider_id = slider_id;\n",
       "    this.loop_select_id = loop_select_id;\n",
       "    this.interval = interval;\n",
       "    this.current_frame = 0;\n",
       "    this.direction = 0;\n",
       "    this.timer = null;\n",
       "    this.frames = new Array(frames.length);\n",
       "\n",
       "    for (var i=0; i<frames.length; i++)\n",
       "    {\n",
       "     this.frames[i] = new Image();\n",
       "     this.frames[i].src = frames[i];\n",
       "    }\n",
       "    var slider = document.getElementById(this.slider_id);\n",
       "    slider.max = this.frames.length - 1;\n",
       "    if (isInternetExplorer()) {\n",
       "        // switch from oninput to onchange because IE <= 11 does not conform\n",
       "        // with W3C specification. It ignores oninput and onchange behaves\n",
       "        // like oninput. In contrast, Microsoft Edge behaves correctly.\n",
       "        slider.setAttribute('onchange', slider.getAttribute('oninput'));\n",
       "        slider.setAttribute('oninput', null);\n",
       "    }\n",
       "    this.set_frame(this.current_frame);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.get_loop_state = function(){\n",
       "    var button_group = document[this.loop_select_id].state;\n",
       "    for (var i = 0; i < button_group.length; i++) {\n",
       "        var button = button_group[i];\n",
       "        if (button.checked) {\n",
       "            return button.value;\n",
       "        }\n",
       "    }\n",
       "    return undefined;\n",
       "  }\n",
       "\n",
       "  Animation.prototype.set_frame = function(frame){\n",
       "    this.current_frame = frame;\n",
       "    document.getElementById(this.img_id).src =\n",
       "            this.frames[this.current_frame].src;\n",
       "    document.getElementById(this.slider_id).value = this.current_frame;\n",
       "  }\n",
       "\n",
       "  Animation.prototype.next_frame = function()\n",
       "  {\n",
       "    this.set_frame(Math.min(this.frames.length - 1, this.current_frame + 1));\n",
       "  }\n",
       "\n",
       "  Animation.prototype.previous_frame = function()\n",
       "  {\n",
       "    this.set_frame(Math.max(0, this.current_frame - 1));\n",
       "  }\n",
       "\n",
       "  Animation.prototype.first_frame = function()\n",
       "  {\n",
       "    this.set_frame(0);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.last_frame = function()\n",
       "  {\n",
       "    this.set_frame(this.frames.length - 1);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.slower = function()\n",
       "  {\n",
       "    this.interval /= 0.7;\n",
       "    if(this.direction > 0){this.play_animation();}\n",
       "    else if(this.direction < 0){this.reverse_animation();}\n",
       "  }\n",
       "\n",
       "  Animation.prototype.faster = function()\n",
       "  {\n",
       "    this.interval *= 0.7;\n",
       "    if(this.direction > 0){this.play_animation();}\n",
       "    else if(this.direction < 0){this.reverse_animation();}\n",
       "  }\n",
       "\n",
       "  Animation.prototype.anim_step_forward = function()\n",
       "  {\n",
       "    this.current_frame += 1;\n",
       "    if(this.current_frame < this.frames.length){\n",
       "      this.set_frame(this.current_frame);\n",
       "    }else{\n",
       "      var loop_state = this.get_loop_state();\n",
       "      if(loop_state == \"loop\"){\n",
       "        this.first_frame();\n",
       "      }else if(loop_state == \"reflect\"){\n",
       "        this.last_frame();\n",
       "        this.reverse_animation();\n",
       "      }else{\n",
       "        this.pause_animation();\n",
       "        this.last_frame();\n",
       "      }\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.anim_step_reverse = function()\n",
       "  {\n",
       "    this.current_frame -= 1;\n",
       "    if(this.current_frame >= 0){\n",
       "      this.set_frame(this.current_frame);\n",
       "    }else{\n",
       "      var loop_state = this.get_loop_state();\n",
       "      if(loop_state == \"loop\"){\n",
       "        this.last_frame();\n",
       "      }else if(loop_state == \"reflect\"){\n",
       "        this.first_frame();\n",
       "        this.play_animation();\n",
       "      }else{\n",
       "        this.pause_animation();\n",
       "        this.first_frame();\n",
       "      }\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.pause_animation = function()\n",
       "  {\n",
       "    this.direction = 0;\n",
       "    if (this.timer){\n",
       "      clearInterval(this.timer);\n",
       "      this.timer = null;\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.play_animation = function()\n",
       "  {\n",
       "    this.pause_animation();\n",
       "    this.direction = 1;\n",
       "    var t = this;\n",
       "    if (!this.timer) this.timer = setInterval(function() {\n",
       "        t.anim_step_forward();\n",
       "    }, this.interval);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.reverse_animation = function()\n",
       "  {\n",
       "    this.pause_animation();\n",
       "    this.direction = -1;\n",
       "    var t = this;\n",
       "    if (!this.timer) this.timer = setInterval(function() {\n",
       "        t.anim_step_reverse();\n",
       "    }, this.interval);\n",
       "  }\n",
       "</script>\n",
       "\n",
       "<style>\n",
       ".animation {\n",
       "    display: inline-block;\n",
       "    text-align: center;\n",
       "}\n",
       "input[type=range].anim-slider {\n",
       "    width: 374px;\n",
       "    margin-left: auto;\n",
       "    margin-right: auto;\n",
       "}\n",
       ".anim-buttons {\n",
       "    margin: 8px 0px;\n",
       "}\n",
       ".anim-buttons button {\n",
       "    padding: 0;\n",
       "    width: 36px;\n",
       "}\n",
       ".anim-state label {\n",
       "    margin-right: 8px;\n",
       "}\n",
       ".anim-state input {\n",
       "    margin: 0;\n",
       "    vertical-align: middle;\n",
       "}\n",
       "</style>\n",
       "\n",
       "<div class=\"animation\">\n",
       "  <img id=\"_anim_img20ae348e34134b55be35337edd1b3908\">\n",
       "  <div class=\"anim-controls\">\n",
       "    <input id=\"_anim_slider20ae348e34134b55be35337edd1b3908\" type=\"range\" class=\"anim-slider\"\n",
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       "           oninput=\"anim20ae348e34134b55be35337edd1b3908.set_frame(parseInt(this.value));\">\n",
       "    <div class=\"anim-buttons\">\n",
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       "          <i class=\"fa fa-minus\"></i></button>\n",
       "      <button title=\"First frame\" aria-label=\"First frame\" onclick=\"anim20ae348e34134b55be35337edd1b3908.first_frame()\">\n",
       "        <i class=\"fa fa-fast-backward\"></i></button>\n",
       "      <button title=\"Previous frame\" aria-label=\"Previous frame\" onclick=\"anim20ae348e34134b55be35337edd1b3908.previous_frame()\">\n",
       "          <i class=\"fa fa-step-backward\"></i></button>\n",
       "      <button title=\"Play backwards\" aria-label=\"Play backwards\" onclick=\"anim20ae348e34134b55be35337edd1b3908.reverse_animation()\">\n",
       "          <i class=\"fa fa-play fa-flip-horizontal\"></i></button>\n",
       "      <button title=\"Pause\" aria-label=\"Pause\" onclick=\"anim20ae348e34134b55be35337edd1b3908.pause_animation()\">\n",
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       "          <i class=\"fa fa-play\"></i></button>\n",
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       "          <i class=\"fa fa-step-forward\"></i></button>\n",
       "      <button title=\"Last frame\" aria-label=\"Last frame\" onclick=\"anim20ae348e34134b55be35337edd1b3908.last_frame()\">\n",
       "          <i class=\"fa fa-fast-forward\"></i></button>\n",
       "      <button title=\"Increase speed\" aria-label=\"Increase speed\" onclick=\"anim20ae348e34134b55be35337edd1b3908.faster()\">\n",
       "          <i class=\"fa fa-plus\"></i></button>\n",
       "    </div>\n",
       "    <form title=\"Repetition mode\" aria-label=\"Repetition mode\" action=\"#n\" name=\"_anim_loop_select20ae348e34134b55be35337edd1b3908\"\n",
       "          class=\"anim-state\">\n",
       "      <input type=\"radio\" name=\"state\" value=\"once\" id=\"_anim_radio1_20ae348e34134b55be35337edd1b3908\"\n",
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       "      <label for=\"_anim_radio1_20ae348e34134b55be35337edd1b3908\">Once</label>\n",
       "      <input type=\"radio\" name=\"state\" value=\"loop\" id=\"_anim_radio2_20ae348e34134b55be35337edd1b3908\"\n",
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       "      <label for=\"_anim_radio2_20ae348e34134b55be35337edd1b3908\">Loop</label>\n",
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       "      <label for=\"_anim_radio3_20ae348e34134b55be35337edd1b3908\">Reflect</label>\n",
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       "  </div>\n",
       "</div>\n",
       "\n",
       "\n",
       "<script language=\"javascript\">\n",
       "  /* Instantiate the Animation class. */\n",
       "  /* The IDs given should match those used in the template above. */\n",
       "  (function() {\n",
       "    var img_id = \"_anim_img20ae348e34134b55be35337edd1b3908\";\n",
       "    var slider_id = \"_anim_slider20ae348e34134b55be35337edd1b3908\";\n",
       "    var loop_select_id = \"_anim_loop_select20ae348e34134b55be35337edd1b3908\";\n",
       "    var frames = new Array(20);\n",
       "    \n",
       "  frames[0] = \"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAoAAAAHgCAYAAAA10dzkAAAAOnRFWHRTb2Z0d2FyZQBNYXRwbG90\\\n",
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       "\"\n",
       "\n",
       "\n",
       "    /* set a timeout to make sure all the above elements are created before\n",
       "       the object is initialized. */\n",
       "    setTimeout(function() {\n",
       "        anim20ae348e34134b55be35337edd1b3908 = new Animation(frames, img_id, slider_id, 200.0,\n",
       "                                 loop_select_id);\n",
       "    }, 0);\n",
       "  })()\n",
       "</script>\n"
      ],
      "text/plain": [
       "<matplotlib.animation.FuncAnimation at 0x7f2e1f8f3d90>"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fom.visualize((U-U_RB), title=f'mu = {mu}', legend=('error'))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1bfd6f2f",
   "metadata": {},
   "source": [
    "You can download this demonstration plus the wrapper definitions as a\n",
    "notebook {nb-download}`tutorial_external_solver.ipynb` or\n",
    "as Markdown text {download}`tutorial_external_solver.md`."
   ]
  }
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