human
3. Affine decomposition
In order to obtain an affine decomposition, we need to recast the problem on a fixed, parameter independent, reference domain, as follows:

Choose $\Omega = \Omega_o((\mu_0, \mu_1) \equiv (1, 1))$ as reference domain, which we generate through the generate_mesh notebook provided in the data folder.
Define a map $\boldsymbol{T}(\boldsymbol{\cdot}.
\boldsymbol{\mu}): \Omega \to \Omega_o(\boldsymbol{\mu})$ to carry out a pull back of the problem from the parametrized domain $\Omega_o(\boldsymbol{\mu})$ to the reference one $\Omega$. Since we aim at obtaining an affine decomposition, the map $\boldsymbol{T}(\boldsymbol{\cdot}.
\boldsymbol{\mu})$ should be affine in its first argument. This is possible by taking a partition of the reference domain in several triangular subdomains, such as the following one

<img src="data/hole_2.png" width="35%"/><br>
3. Define a reference function space $$\mathbb{V} = H^1(\Omega),$$ to which the pulled back solution
$$u(\boldsymbol{\cdot}.
\boldsymbol{\mu}) = u_o(\boldsymbol{T}(\boldsymbol{\cdot}.
\boldsymbol{\mu}).
\boldsymbol{\mu})$$ belongs
4. Pull back the bilinear form $a_o(\cdot, \cdot.
\boldsymbol{\mu})$ and linear form $f_o(\cdot.
\boldsymbol{\mu})$ onto the reference domain $\Omega$ by change of variables. Note that, due to the definition of the map $\boldsymbol{T}(\boldsymbol{\cdot}.
\boldsymbol{\mu})$, the pull back will be different from one subdomain to the other. Call
$$a(\cdot, \cdot.
\boldsymbol{\mu}): \mathbb{V} \times \mathbb{V} \to \mathbb{R}$$
$$f(\cdot.
\boldsymbol{\mu}): \mathbb{V} \to \mathbb{R}$$
the resulting forms (and note that their arguments are now defined on $\Omega$).
5. Solve the equivalent problem
<center>for a given parameter $\boldsymbol{\mu}\in\mathbb{P}$, find $u(\boldsymbol{\mu})\in\mathbb{V}$ such that</center>
$$a\left(u(\boldsymbol{\mu}),v;\boldsymbol{\mu}\right)=f(v;\boldsymbol{\mu})\quad \forall v\in\mathbb{V}$$

Give me a Python program to find $u(\boldsymbol{\mu})\in\mathbb{V}$ such that</center>?
assistant
#Here is Python program to find $u(\boldsymbol{\mu})\in\mathbb{V}$ such that</center>
.
@PullBackFormsToReferenceDomain()
@AffineShapeParametrization("data/hole_vertices_mapping.vmp")
class Hole(EllipticCoerciveProblem):

    # Default initialization of members
    def __init__(self, V, **kwargs):
        # Call the standard initialization
        EllipticCoerciveProblem.__init__(self, V, **kwargs)
        # ... and also store FEniCS data structures for assembly
        assert "subdomains" in kwargs
        assert "boundaries" in kwargs
        self.subdomains, self.boundaries = kwargs["subdomains"], kwargs["boundaries"]
        self.u = TrialFunction(V)
        self.v = TestFunction(V)
        self.dx = Measure("dx")(subdomain_data=subdomains)
        self.ds = Measure("ds")(subdomain_data=boundaries)
        self.subdomains = subdomains
        self.boundaries = boundaries

    # Return custom problem name
    def name(self):
        return "Hole"

    # Return theta multiplicative terms of the affine expansion of the problem.
    def compute_theta(self, term):
        mu = self.mu
        if term == "a":
            theta_a0 = 1.0
            theta_a1 = mu[2]
            return (theta_a0, theta_a1)
        elif term == "f":
            theta_f0 = 1.0
            return (theta_f0, )
        else:
            raise ValueError("Invalid term for compute_theta().")

    # Return forms resulting from the discretization of the affine expansion of the problem operators.
    def assemble_operator(self, term):
        u = self.u
        v = self.v
        dx = self.dx
        ds = self.ds
        if term == "a":
            a0 = inner(grad(u), grad(v)) * dx
            a1 = inner(u, v) * ds(5) + inner(u, v) * ds(6) + inner(u, v) * ds(7) + inner(u, v) * ds(8)
            return (a0, a1)
        elif term == "f":
            f0 = v * ds(1) + v * ds(2) + v * ds(3) + v * ds(4)
            return (f0, )
        elif term == "inner_product":
            x0 = u * v * dx + inner(grad(u), grad(v)) * dx
            return (x0,)
        else:
            raise ValueError("Invalid term for assemble_operator().")