EIM
The Empirical Interpolation Method (EIM) is used with the POD-Galerkin to obtain an affine decomposition on the parameter.
Indeed, as explained in the Galerkin-POD section (here), we need an affine decomposition with respects to the parameter, then the parameter-independent terms are computed offline, making the online computation faster.
The EIM build a linear combination of fully determined solutions from basis functions $(q_i)_{i=1,...,M}$ depending on some interpolating points (called “magic points”) and some values $(\alpha_i)_{i=1,...,M}$ relying on certain instances of the parameter $\nu$, selected within the algorithm. Let us introduce the method with the following example:
Consider a function
$$ g(x,\nu)= \frac1{\sqrt{(x_1-\nu_1)^2+(y_1-\nu_2)}},$$with $x=(x_1,x_2)$ and $\nu=(\nu_1,\nu_2)$.
“OFFLINE”
The first chosen parameter $\nu^1$ is the one that maximizes $g$ on norm $\mathcal{L}^{\infty}$ (we want to retrieve most information on $g$ with few points) and the associated magic point $x^1$ is the point that gives the most information on $g(\cdot,\nu^1)$ i.e. which maximizes its modulus.
Then the first basis function is $q_1(\cdot) = \frac{g(\cdot,\nu^1)}{g(x^1, \nu^1)}$.
We can then find $\alpha_1$ as the coefficient corresponding to this basis function:
We compute for each training parameters $\nu \in \mathcal{G}$ the real $G$ such that $G=g(x^1,\nu)$ and then we solve the problem $Q \alpha^1(\nu)=G$ where $Q$ at this initialization step is just $q_1(x^1)$.
Then, we find recursively the $M$ basis functions with the following interpolation problem
$$ \forall 1 \leq i \leq M-1,\ \mathcal{I}_{M-1}[g](x^i)= g(x^i),$$$$\mathcal{I}_{M-1}[g]=\sum_{j=1}^{M-1} \alpha_j^{M-1} q_j.$$" ONLINE "
Now we are interested by a new parameter $\nu^{target}$. To obtain the linear decomposition, we solve a reduced problem from the previously computed basis functions and with the magic points (see details in the Python notebook).
$$ g^M(x,\nu^{target})=\sum_{i=1}^M \alpha(\nu^{target}) q(x). $$There exists a generalized form of this method (GEIM) and a discrete version named DEIM. The GEIM replaces the $M$ pointwise evaluations used by the EIM by general measures. In the presence of measurement noise, a stabilization of the method can be employed.