Model Reduction of Nonlinear Fire Dynamics Models

Due to the complexity, multi-scale, and multi-physics nature of the mathematical models for fires, current numerical models require too much computational effort to be useful in design and real-time decision making, especially when dealing with fires over large domains. To reduce the computational time while retaining the complexity of the domain and physics, our research has focused on several reduced-order modeling techniques. Our contributions are improving wildland fire reduced-order models (ROMs), creating new ROM techniques for nonlinear systems, and preserving optimality when discretizing a continuous-time ROM. Currently, proper orthogonal decomposition (POD) is being used to reduce wildland fire-spread models with limited success. We use a technique known as the discrete empirical interpolation method (DEIM) to address the slowness due to the nonlinearity. We create new methods to reduce nonlinear models, such as the Burgers' equation, that perform better than POD over a wider range of input conditions. Further, these ROMs can often be constructed without needing to capture full-order solutions a priori. This significantly reduces the off-line costs associated with creating the ROM. Finally, we investigate methods of time-discretization that preserve the optimality conditions in a certain norm associated with the input to output mapping of a dynamical system. In particular, we are able to show that the Crank-Nicholson method preserves the optimality conditions, but other single-step methods do not. We further clarify the need for these discrete-time ROMs to match at infinity in order to ensure local optimality.