A localized reduced-order modeling approach for PDEs with bifurcating solutions

Reduced-order modeling has become an indispensable tool for the simulation and control of systems governed by complex, nonlinear partial differential equations (PDEs), as well as for the quantification of uncertainties in outputs of interest that depend on the solutions of those systems. The works cited in this section attest to the broad success such modeling has had in many settings. Here, reduced-order models generally refer to inexpensive surrogates for expensive models that are built based on a relatively few solutions of the latter model and for which the expense incurred in the construction process is then amortized over many solutions of the surrogate. The focus here is on the construction of efficient reduced-order models (ROMs) for PDEs whose solutions bifurcate as the values of parameters appearing in the PDE change. In such cases, using a single ROM results in surrogates that themselves are expensive enough so that their repeated use can incur high costs. Here, we instead build multiple local ROMs, each of which can serve to inexpensively determine a set of approximations of PDE solutions that do not span discontinuous or other large changes due to parameter changes. In our methodology, the construction of the multiple local ROMs is automated as is the determination of which local basis one should use for values of the parameters not used in the construction process. A more complete general discussion of ROMs and of our localized ROM approach is given in the rest of this section and details are about the latter are provided in subsequent sections.