  We propose a calibrated filtered reduced order model (CF-ROM) framework for
the numerical simulation of general nonlinear PDEs that are amenable to reduced
order modeling. The novel CF-ROM framework consists of two steps: (i) In the
first step, we use explicit ROM spatial filtering of the nonlinear PDE to
construct a filtered ROM. This filtered ROM is low-dimensional, but is not
closed (because of the nonlinearity in the given PDE). (ii) In the second step,
we use a calibration procedure to close the filtered ROM, i.e., to model the
interaction between the resolved and unresolved modes. To this end, we use a
linear or quadratic ansatz to model this interaction and close the filtered
ROM. To find the new coefficients in the closed filtered ROM, we solve an
optimization problem that minimizes the difference between the full order model
data and our ansatz. Although we use a fluid dynamics setting to illustrate how
to construct and use the CF-ROM framework, we emphasize that it is built on
general ideas of spatial filtering and optimization and is independent of
(restrictive) phenomenological arguments. Thus, the CF-ROM framework can be
applied to a wide variety of PDEs.
