Title: Toward real-time analysis of large-scale nonlinear systems:
a system-approximated Gauss-Newton model reduction method

Speaker: Kevin Carlberg, Stanford University

Date/Time:  Tuesday, February 1, 2011, 1:00 pm in NM & 12:00 pm in CA              

Location: CSRI Building/Room 279 (Sandia NM) 915/S101 in SNL/CA

Brief Abstract: Despite the development of efficient parallel solution algorithms and high-performance supercomputers, the computational cost of analyzing large-scale, high-fidelity mathematical models remains a significant barrier in many engineering applications. For example, uncertainty quantification and ``in the field'' analysis require simulations to be completed in mere seconds or minutes; in situ parameter estimation and embedded control applications require high-fidelity models to be optimized in real time.  For systems with general nonlinearities, typical surrogate modeling methods (e.g. response surfaces, mesh coarsening, POD-Galerkin model reduction) often fail to meet these time constraints without introducing unacceptable errors.
 
In this talk, I will present a recently developed model reduction method that has generated accurate, near-real-time solutions for a broad range of nonlinear systems.  The method first executes high-fidelity simulations for several system configurations during an expensive ``offline'' stage; then, it computes a proper orthogonal decomposition (POD) subspace that optimally represents the state evolution observed during these simulations. To compute fast, accurate solutions at new system configurations ``online'', the method introduces two approximations. First, the method decreases the system dimension by searching for solutions in the low-dimensional POD subspace. The method handles the resulting overdetermined system by formulating a least-squares problem and solving it by the Gauss-Newton method. Second, the method decreases the cost of solving this least-squares problem by approximating the residual and Jacobian using a ``Gappy POD'' approach; this requires computing only a few rows of the approximated quantities online. To demonstrate the ability of the method to deliver fast, accurate solutions, I will present results for nonlinear problems in fluid mechanics, circuits, and structural dynamics.
 
The talk will conclude with a discussion of this work's relevance to Sandia's ongoing projects on uncertainty quantification of the power grid and design optimization of nanoporous structures. Also, I will discuss future research directions, which include augmenting the method with experimental data, and decreasing the number of time steps at which the model is evaluated.

CSRI POC: Scott Collis, 505-284-1123