  Reduced-order modeling (ROM) commonly refers to the construction, based on a
few solutions (referred to as snapshots) of an expensive discretized partial
differential equation (PDE), and the subsequent application of low-dimensional
discretizations of partial differential equations (PDEs) that can be used to
more efficiently treat problems in control and optimization, uncertainty
quantification, and other settings that require multiple approximate PDE
solutions. In this work, a ROM is developed and tested for the treatment of
nonlinear PDEs whose solutions bifurcate as input parameter values change. In
such cases, the parameter domain can be subdivided into subregions, each of
which corresponds to a different branch of solutions. Popular ROM approaches
such as proper orthogonal decomposition (POD), results in a global
low-dimensional basis that does no respect not take advantage of the often
large differences in the PDE solutions corresponding to different subregions.
Instead, in the new method, the k-means algorithm is used to cluster snapshots
so that within cluster snapshots are similar to each other and are dissimilar
to those in other clusters. This is followed by the construction of local POD
bases, one for each cluster. The method also can detect which cluster a new
parameter point belongs to, after which the local basis corresponding to that
cluster is used to determine a ROM approximation. Numerical experiments show
the effectiveness of the method both for problems for which bifurcation cause
continuous and discontinuous changes in the solution of the PDE.
