  Implicit numerical integration of nonlinear ODEs requires solving a system of
nonlinear algebraic equations at each time step. Each of these systems is often
solved by a Newton-like method, which incurs a sequence of linear-system
solves. Most model-reduction techniques for nonlinear ODEs exploit knowledge of
system's spatial behavior to reduce the computational complexity of each
linear-system solve. However, the number of linear-system solves for the
reduced-order simulation often remains roughly the same as that for the
full-order simulation. We propose exploiting knowledge of the model's temporal
behavior to 1) forecast the unknown variable of the reduced-order system of
nonlinear equations at future time steps, and 2) use this forecast as an
initial guess for the Newton-like solver during the reduced-order-model
simulation. To compute the forecast, we propose using the Gappy POD technique.
The goal is to generate an a ccurate initial guess so that the Newton solver
requires many fewer iterations to converge, thereby decreasing the number of
lin ear-system solves in the reduced-order-model simulation.
