Reduced order modeling of complex systems

Solutions of (nonlinear) complex systems are expensive with respect to both storage and CPU costs. As a result, it is difficult if not impossible to deal with a number of situations such as: continuation or homotopy methods for computing state solutions; parametric studies of state solutions; optimization and control problems (multiple state solutions); and feedback control settings (real-time state solutions). Not surprisingly, a lot of attention has been paid to reducing the costs of the nonlinear state solutions by using reduced-order models for the state; these are low-dimensional approximations to the state. Reduced-order modeling has been and remains a very active research direction in many seemingly disparate fields. We will focus on three approaches to reduced-order modeling: reduced basis methods; proper orthogonal decomposition (POD); andcentroidal Voronoi tessellations (CVT). Before describing the three approaches, we first discuss what we exactly mean by reduced-ordering modeling and make some general comments that apply to all reduced-order models. For a state simulation, a reduced-order method would proceed as follows. One first chooses a reduced basis ui, i = 1, . . . , n, where n is hopefully very small compared to the usual number of functions used in a finite element approximation or the number of grid points used in a finite difference approximation. Next, one seeks an approximation ũ to the state of the form ũ = ∑n i=1 ciui ∈ V ≡ span{u1, . . . ,un}. Then, one determines the coefficients ci, i = 1, . . . , n, by solving the state equations in the set V , e.g., one could find a Galerkin solution of the state equations in a standard way, using V for the space of approximations. The cost of such a computation would be very small if n is small (ignoring the cost of the off-line determination of the reduced basis {u1, . . . ,un}). In control or optimization settings, one is faced with multiple state solves