Residual Minimizing Model Interpolation for Parameterized Nonlinear Dynamical Systems

We present a method for approximating the solution of a parameterized, nonlinear dynamical system using an affine combination of solutions computed at other points in the input parameter space. The coefficients of the affine combination are computed with a nonlinear least squares procedure that minimizes the residual of the governing equations. The approximation properties of this residual minimizing scheme are comparable to existing reduced basis and POD-Galerkin model reduction methods, but its implementation requires only independent evaluations of the nonlinear forcing function. It is particularly appropriate when one wishes to approximate the states at a few points in time without time marching from the initial conditions. We prove some interesting characteristics of the scheme including an interpolatory property, and we present heuristics for mitigating the effects of the ill-conditioning and reducing the overall cost of the method. We apply the method to representative numerical examples from kinetics - a three state system with one parameter controlling the stiffness - and conductive heat transfer - a nonlinear parabolic PDE with a random field model for the thermal conductivity.Comment: 28 pages, 8 figures, 2 table

The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows

The Gauss--Newton with approximated tensors (GNAT) method is a nonlinear model reduction method that operates on fully discretized computational models. It achieves dimension reduction by a Petrov--Galerkin projection associated with residual minimization; it delivers computational efficency by a hyper-reduction procedure based on the `gappy POD' technique. Originally presented in Ref. [1], where it was applied to implicit nonlinear structural-dynamics models, this method is further developed here and applied to the solution of a benchmark turbulent viscous flow problem. To begin, this paper develops global state-space error bounds that justify the method's design and highlight its advantages in terms of minimizing components of these error bounds. Next, the paper introduces a `sample mesh' concept that enables a distributed, computationally efficient implementation of the GNAT method in finite-volume-based computational-fluid-dynamics (CFD) codes. The suitability of GNAT for parameterized problems is highlighted with the solution of an academic problem featuring moving discontinuities. Finally, the capability of this method to reduce by orders of magnitude the core-hours required for large-scale CFD computations, while preserving accuracy, is demonstrated with the simulation of turbulent flow over the Ahmed body. For an instance of this benchmark problem with over 17 million degrees of freedom, GNAT outperforms several other nonlinear model-reduction methods, reduces the required computational resources by more than two orders of magnitude, and delivers a solution that differs by less than 1% from its high-dimensional counterpart