Reduced-Order Models of Large-Scale Computational SystemsJune 10, 2005
Flow in a representative large indoor structure determined by a finite element code with thousands of unknowns (top) and with a reduced-order model with only eight unknowns (bottom). (This work was partially funded by the Department of Homeland Security, Science and Technology: Advanced Scientific Computing Program.) Images from Max Gunzburger and John Shadid.
By Max Gunzburger and Karen Willcox
Model reduction is a powerful tool that allows the systematic generation of cost-efficient representations of large-scale computational systems, such as those resulting from discretization of partial differential equations. Considerable improvements in model-reduction methodologies in recent years have led to a growing interest in the field across the computational science and engineering community. In the multipart minisymposium "Model Reduction for Large-Scale Systems," 19 speakers covered a broad range of topics.
A number of speakers presented advances in model-reduction methods, including quadrature schemes for more accurate determination of proper orthogonal decomposition (POD) modes, methods for exploiting symmetry properties when determining a reduced basis, centroidal Voronoi tessellation sampling approaches, and structure-preserving reduction methods for second-order systems. Others described new methods, among them a fixed-order transfer function approach that leads to an optimal L-infinity approximation for linear systems, and a POD-based missing-point estimation method that provides a rigorous framework for selective spatial representation and thus yields efficient models for nonlinear systems.
Several speakers described recent developments in control applications. POD-based reduced models, for example, have been successfully used for optimal control of the Navier–Stokes equations. Among the approaches presented are "reduce-then-design," in which a reduced-order representation of the governing equations is used in designing a controller, and "design-then-reduce," in which the full equations are used in the design, after which the coupled system is reduced.
Error estimation and error control continue to be very important in the context of model reduction. Recent work on rigorous a posteriori error estimators for reduced-basis approximations of the incompressible Navier–Stokes equations was presented for multiparameter natural convection bifurcation problems. Speakers also described error estimation methods for Krylov-based reduction approaches, using a theory based on subspace angles, and methods for POD, using small-sample statistical condition estimation combined with an adjoint method.
New and exciting application areas discussed include the use of reduced models in real-time applications. Reduced-order models, for example, provide a viable option for solving complex flow and transport models sufficiently rapidly that inverse problems can be solved in real time. One speaker presented preliminary results for the application of model reduction to the real-time identification of a chemical release source inside an airport building.
A number of challenging and exciting future research directions were discussed throughout the course of the symposium. Several speakers identified the reduction of nonlinear systems as an open question, citing the need for methods that both resolve complex, nonlinear interactions and yield efficient reduced models. A second important area of ongoing research is reduction of systems with large numbers of parametric inputs. In particular, efficient methods for sampling high-dimensional input spaces remain a challenge, along with methods for estimating errors. Finally, the application of model reduction to realistic applications, particularly those with a real-time implementation component, is an area of growing interest and importance.
Max Gunzburger, a professor of mathematics at Florida State University, and Karen Willcox, an assistant professor of aeronautics and astronautics at MIT, organized the five-part minisymposium on model reduction.