### Projection-Based Model Reduction

**
Geoffrey Oxberry, Massachusetts Institute of Technology
**

Most practical problems in combustion, such as combustion in an engine, occur under inhomogeneous, transient conditions. In order to simulate faithfully the physics involved, the chemistry in these applications is typically modeled using a large, detailed chemical mechanism (tens to hundreds of species, hundreds to thousands of reactions). Due to the large number of species involved, the range of active time scales in these problems can span up to ten orders of magnitude, so ODEs and PDEs solved in these applications are very stiff. Consequently, simulations of combustion in these settings typically have prohibitive computational costs. For this reason, model reduction techniques are used to reduce the computational requirements of these simulations; here, a "model" is the right-hand side of an ODE or PDE. Many techniques reduce models using a projection operator, but at first glance, the mathematical form of the reduced model can look very different, depending on the model reduction technique. In this work, we define a class of model reduction techniques called projection-based model reduction techniques, and show that they can be represented mathematically three different ways. Graphical sketch proofs will be used to illustrate how to interconvert among the three forms and the sources of approximation error in each form. Based on these proofs, it can be shown that six different techniques in the literature are projection-based, demonstrating the similarities between existing techniques. The existence of three interconvertible representations of projection-based model reduction suggests the possibility that computational costs may be reduced by selecting a representation with structure that can be exploited by algorithms for solving ODEs and DAEs. Such a finding would enable additional reduction in computational costs using existing model reduction techniques, further facilitating the use of detailed reaction mechanisms in simulations.

**Abstract Author(s): **Geoffrey M. Oxberry, William H. Green, Paul I. Barton