  The usual approach to model reduction for parametric partial differential
equations (PDEs) is to construct a linear space $V_n$ which approximates well
the solution manifold $\mathcal{M}$ consisting of all solutions $u(y)$ with $y$
the vector of parameters. This linear reduced model $V_n$ is then used for
various tasks such as building an online forward solver for the PDE or
estimating parameters from data observations. It is well understood in other
problems of numerical computation that nonlinear methods such as adaptive
approximation, $n$-term approximation, and certain tree-based methods may
provide improved numerical efficiency. For model reduction, a nonlinear method
would replace the linear space $V_n$ by a nonlinear space $\Sigma_n$. This idea
has already been suggested in recent papers on model reduction where the
parameter domain is decomposed into a finite number of cells and a linear space
of low dimension is assigned to each cell.
  Up to this point, little is known in terms of performance guarantees for such
a nonlinear strategy. Moreover, most numerical experiments for nonlinear model
reduction use a parameter dimension of only one or two. In this work, a step is
made towards a more cohesive theory for nonlinear model reduction. Framing
these methods in the general setting of library approximation allows us to give
a first comparison of their performance with those of standard linear
approximation for any general compact set. We then turn to the study these
methods for solution manifolds of parametrized elliptic PDEs. We study a very
specific example of library approximation where the parameter domain is split
into a finite number $N$ of rectangular cells and where different reduced
affine spaces of dimension $m$ are assigned to each cell. The performance of
this nonlinear procedure is analyzed from the viewpoint of accuracy of
approximation versus $m$ and $N$.
