Tutorial: Binding an external PDE solver to pyMOR ================================================= One of pyMOR's main features is easy integration of external solvers that implement the full-order model. In this tutorial we will do this step-by-step for a custom toy solver written in C++. If you use the FEniCS _ or NGSovle _ PDE solver libraries, you can find ready-to-use pyMOR bindings in the :mod:~pymor.bindings package. pyMOR support for deal.II _ can be found in a separate repository _. Defining the PDE solver ----------------------- Our solver discretizes the one-dimensional Laplace equation :math:u''(x)=0 on the interval :math:[\text{left},\text{right}] using a central differences scheme with :math:h=\frac{|\text{right}-\text{left}|}{n}. First, we need a class to store our data in and with some basic linear algebra operations declared on it. .. literalinclude:: minimal_cpp_demo/model.hh :lines: 4-19 :language: cpp Next, we need the operator that discretizes our PDE. .. literalinclude:: minimal_cpp_demo/model.hh :lines: 22-32 :language: cpp Together with some header guards, these two snippets make up our :download:model.hh . The definitions for the Vector class are pretty straightforward: .. literalinclude:: minimal_cpp_demo/model.cc :lines: 7-35 :language: cpp Just like the diffusion operator that computes a central differences stencil: .. literalinclude:: minimal_cpp_demo/model.cc :lines: 39-49 :language: cpp This completes all the C++ code needed for the toy solver itself. Next, we will make this code usable from Python. We utilize the pybind11 _ library to create a Python extension module _ named model, that allows us to manipulate instances of the C++ Vector and DiffusionOperator classes. Compiling the PDE solver as a shared library and creating Python bindings for it using pybind11 _, Cython _ or ctypes _ is the preferred way of integrating external solvers, as it offers maximal flexibility and performance. For instance, in this example we will actually completely implement the |Model| in Python using a :mod:time stepper  from pyMOR to :meth:~pymor.models.interface.Model.solve the |Model|. When this is not an option, RPC _-based approaches are possible as well. For small to medium-sized linear problems, another option it to import system matrices and snapshot data into pyMOR via file exchange and to use NumPy-based :mod:Operators  and :mod:VectorArrays  to represent the full-order model. Binding the solver to Python ---------------------------- All of the C++ code related to the extension module is defined inside a scope started with .. literalinclude:: minimal_cpp_demo/model.cc :lines: 56-57 :language: cpp This tells pybind11 to make the contained symbols accessible in the module instance m that will be importable by the name model. Now we create a new pybind11 class\_ object that wraps the DiffusionOperator. Note that the module instance is passed to the constructor alongside a name for the Python class and a docstring. The second line shows how to define an init function for the Python object by using the special py:init object to forward arguments to the C++ constructor. .. literalinclude:: minimal_cpp_demo/model.cc :lines: 60-61 :language: cpp Next, we define read-only properties on the Python side named after and delegated to the members of the C++ class. .. literalinclude:: minimal_cpp_demo/model.cc :lines: 62-63 :language: cpp The last DiffusionOperator-related line exposes the function call to apply in the same way: .. literalinclude:: minimal_cpp_demo/model.cc :lines: 64 :language: cpp This is everything that is needed to expose the operator to Python. We will now do the same for the Vector, with a few more advanced techniques added. .. literalinclude:: minimal_cpp_demo/model.cc :lines: 66-68 :language: cpp Again we define a py:class\_ with appropiate name and docstring, but now we also indicate to pybind11 that this class will implement the buffer protocol _, which basically exposes direct access to the chunk of memory associated with a Vector instance to Python. We also see how we can dispatch multiple init functions by using py:init objects with C++ lambda functions. Note that direct memory access to the vector data from Python is not required to integrate a solver with pyMOR. It is, however, useful for debugging and quickly modifying or extending the solver from within Python. For instance, in our toy example we will use the direct memory access to quickly define a visualization of the solutions and to construct the right-hand side vector for our problem. .. literalinclude:: minimal_cpp_demo/model.cc :lines: 70-74 :language: cpp .. literalinclude:: minimal_cpp_demo/model.cc :lines: 76-80 :language: cpp This completes the :download:model.cc . This extension module needs to be compiled to a shared object that the Python interpreter can import. We use a minimal CMake _ project that generates makefiles for us to achieve this. First we make sure pybind11 can be used: .. literalinclude:: minimal_cpp_demo/CMakeLists.txt :lines: 1-6 :language: cmake Next, we define a new library with our model.cc as the single source file and let pybind11 set the proper compile flags. .. literalinclude:: minimal_cpp_demo/CMakeLists.txt :lines: 9-12 :language: cmake That is all that is needed for :download:CMakeLists.txt . In the next step, we will switch to a bash terminal and actually compile this module. After creating a build directory for the module, we let cmake initialize the build and call make to execute the compilation. .. jupyter-kernel:: bash :id: make .. jupyter-execute:: mkdir -p source/minimal_cpp_demo/build cd source/minimal_cpp_demo/build cmake .. -DPYTHON_EXECUTABLE=\$(which python) -DCMAKE_COLOR_MAKEFILE=OFF # prevent bash control chars in output make You can download this snippet as a notebook file to be used with a bash kernel :jupyter-download:notebook:make. To be able to use this extension module we need to insert the build directory into the path where the Python interpreter looks for things to import. Afterwards we can import the module and create and use the exported classes. .. jupyter-kernel:: .. include:: jupyter_init.txt .. jupyter-execute:: import sys sys.path.insert(0, 'source/minimal_cpp_demo/build') import model mymodel = model.DiffusionOperator(10, 0, 1) myvector = model.Vector(10, 0) mymodel.apply(myvector, myvector) dir(model) Using the exported Python classes with pyMOR -------------------------------------------- All of pyMOR's algorithms operate on |VectorArray| and |Operator| objects that all share the same programming interface. To be able to use our Python model.Vector and model.DiffusionOperator in pyMOR, we have to provide implementations of |VectorArray|, |VectorSpace| and |Operator| that wrap the classes defined in the extension module and translate calls to the interface methods into operations on model.Vector and model.DiffusionOperator. Instead of writing a full implementaion of a |VectorArray| that manages multiple model.Vector instances, we can instead implement a wrapper WrappedVector for a single model.Vector instance based on :class:~pymor.vectorarrays.list.CopyOnWriteVector which will be used to create |ListVectorArrays| via a :class:~pymor.vectorarrays.list.ListVectorSpace-based WrappedVectorSpace. The :class:~pymor.vectorarrays.list.CopyOnWriteVector base class manages a reference count for us and automatically copies data when necessary in methods :meth:~pymor.vectorarrays.list.CopyOnWriteVector.scal and :meth:~pymor.vectorarrays.list.CopyOnWriteVector.axpy. To use this, we need to implement :meth:~pymor.vectorarrays.list.CopyOnWriteVector._scal and :meth:~pymor.vectorarrays.list.CopyOnWriteVector._axpy in addition to all the abstract methods from :class:~pymor.vectorarrays.list.CopyOnWriteVector. We can get away with using just a stub that raises an :class:~NotImplementedError in some methods that are not actually called in our example. .. jupyter-execute:: from pymor.operators.interface import Operator from pymor.vectorarrays.list import CopyOnWriteVector, ListVectorSpace import numpy as np import math from model import Vector, DiffusionOperator class WrappedVector(CopyOnWriteVector): def __init__(self, vector): assert isinstance(vector, Vector) self._impl = vector @classmethod def from_instance(cls, instance): return cls(instance._impl) def to_numpy(self, ensure_copy=False): # Note how this uses the buffer protocol setup to allow efficient # data access as a Numpy Vector result = np.frombuffer(self._impl, dtype=np.float) if ensure_copy: result = result.copy() return result def _copy_data(self): # This uses the second exposed 'init' signature to delegate to the C++ copy constructor self._impl = Vector(self._impl) def _scal(self, alpha): self._impl.scal(alpha) def _axpy(self, alpha, x): self._impl.axpy(alpha, x._impl) def inner(self, other): return self._impl.inner(other._impl) def norm(self): return math.sqrt(self.inner(self)) def norm2(self): return self.inner(self) def sup_norm(self): raise NotImplementedError def dofs(self, dof_indices): raise NotImplementedError def amax(self): raise NotImplementedError The implementation of the WrappedVectorSpace is very short as most of the necessary methods of |VectorSpace| are implemented in :class:~pymor.vectorarrays.list.ListVectorSpace. .. jupyter-execute:: class WrappedVectorSpace(ListVectorSpace): def __init__(self, dim): self.dim = dim def zero_vector(self): return WrappedVector(Vector(self.dim, 0)) def make_vector(self, obj): # obj is a model.Vector instance return WrappedVector(obj) def __eq__(self, other): return type(other) is WrappedVectorSpace and self.dim == other.dim Wrapping the model.DiffusionOperator is straightforward as well. We just need to attach suitable |VectorSpaces| to the class and implement the application of the operator on a |VectorArray| as a sequence of applications on single vectors. .. jupyter-execute:: class WrappedDiffusionOperator(Operator): def __init__(self, op): assert isinstance(op, DiffusionOperator) self.op = op self.source = WrappedVectorSpace(op.dim_source) self.range = WrappedVectorSpace(op.dim_range) self.linear = True @classmethod def create(cls, n, left, right): return cls(DiffusionOperator(n, left, right)) def apply(self, U, mu=None): assert U in self.source def apply_one_vector(u): v = Vector(self.range.dim, 0) self.op.apply(u._impl, v) return v return self.range.make_array([apply_one_vector(u) for u in U._list]) Putting it all together ----------------------- As a demonstration, we will use our toy Laplace solver to compute an approximation for the transient diffusion equation .. math:: \frac{\partial u}{\partial t} = {\alpha_\mu} \frac{\partial^2 u}{\partial x^2}, with explicit timestepping provided by pyMOR, with a parameterized, block-wise defined, diffusion coefficient :math:\alpha_\mu. First up, we implement a discretize function that uses the WrappedDiffusionOperator and WrappedVectorSpace to assemble an |InstationaryModel|. .. jupyter-execute:: from pymor.algorithms.pod import pod from pymor.algorithms.timestepping import ExplicitEulerTimeStepper from pymor.discretizers.builtin.gui.visualizers import OnedVisualizer from pymor.models.basic import InstationaryModel from pymor.discretizers.builtin import OnedGrid from pymor.operators.constructions import VectorOperator, LincombOperator from pymor.parameters.functionals import ProjectionParameterFunctional from pymor.reductors.basic import InstationaryRBReductor def discretize(n, nt, blocks): h = 1. / blocks ops = [WrappedDiffusionOperator.create(n, h * i, h * (i + 1)) for i in range(blocks)] pfs = [ProjectionParameterFunctional('diffusion_coefficients', blocks, i) for i in range(blocks)] operator = LincombOperator(ops, pfs) initial_data = operator.source.zeros() rhs_vec = operator.range.zeros() rhs_data = rhs_vec._data[0] rhs_data[:] = np.ones(len(rhs_data)) rhs_data[0] = 0 rhs_data[len(rhs_data) - 1] = 0 rhs = VectorOperator(rhs_vec) # we can re-use pyMOR's builtin grid and visualizer for our demonstration grid = OnedGrid(domain=(0, 1), num_intervals=n) visualizer = OnedVisualizer(grid) time_stepper = ExplicitEulerTimeStepper(nt) fom = InstationaryModel(T=1, operator=operator, rhs=rhs, initial_data=initial_data, time_stepper=time_stepper, num_values=20, visualizer=visualizer, name='C++-Model') return fom Now we can build a reduced basis for our model. Note that this code is not specific to our wrapped classes. Those wrapped classes are only directly used in the discretize call. .. jupyter-execute:: %matplotlib inline # discretize fom = discretize(50, 10000, 4) parameter_space = fom.parameters.space(0.1, 1) # generate solution snapshots snapshots = fom.solution_space.empty() for mu in parameter_space.sample_uniformly(2): snapshots.append(fom.solve(mu)) # apply POD reduced_basis = pod(snapshots, modes=4)[0] # reduce the model reductor = InstationaryRBReductor(fom, reduced_basis, check_orthonormality=True) rom = reductor.reduce() # stochastic error estimation mu_max = None err_max = -1. for mu in parameter_space.sample_randomly(10): U_RB = reductor.reconstruct(rom.solve(mu)) U = fom.solve(mu) err = np.max((U_RB-U).norm()) if err > err_max: err_max = err mu_max = mu # visualize maximum error solution U_RB = (reductor.reconstruct(rom.solve(mu_max))) U = fom.solve(mu_max) fom.visualize((U_RB, U), title=f'mu = {mu}', legend=('reduced', 'detailed')) As you can see in this comparison, we get a good approximation of the full-order model here and the error plot confirms it: .. jupyter-execute:: fom.visualize((U-U_RB), title=f'mu = {mu}', legend=('error')) You can download this demonstration plus the wrapper definitions as a notebook :jupyter-download:notebook:tutorial_external_solver or as a plain Python script :jupyter-download:script:tutorial_external_solver.