(196c) Model Reduction-Based Constrained Optimisation for Large-Scale Steady State Systems Using Black-Box Simulators

The design of all industrial systems involves the concept of optimization. Many, if not most, of those systems are complex and are typically described accurately by a set of partial differential equations (PDEs). The latter are discretized over a mesh for the numerical simulation of the system at hand, which leads to a large-scale system. Steady state simulators employ iterative methods for the solution of those systems. Optimization of those processes can be based on deterministic, or stochastic/meta-heuristic methods. Stochastic methods perform a large number of function evaluations, invoking the system simulator. Thus they are more appropriate for moderate-sized problems [1]. On the other hand, the application of deterministic optimization methods to large-scale systems with constraints is often problematic or even unrealistic, having increased requirements regarding computing power and memory size. The task is deemed even more tedious in the case of black-box solvers (commercial simulators or legacy codes), since the system equations are not available to the user.

In recent years we have developed a model reduction-based framework for gradient-based steady-state [2] and dynamic [3] optimisation that employs input/output dynamic simulators. Here we extend this work by presenting a novel framework for steady-state optimization that uses black-box steady-state simulators based on solvers using iterative linear algebra. The proposed algorithm relies solely on the computation of low-dimensional Jacobian and reduced Hessian matrices [4], which correspond to the dominant modes of the system at hand. A basis for the dominant subspace of the system is computed using subspace iterations and is exploited for the calculation of the reduced Jacobian matrices through a small number of numerical directional perturbations. The reduced Hessian matrices are calculated from a 2-step projection scheme, firstly onto the dominant subspace of the system and secondly onto the subspace of the decision variables.

We have demonstrated the performance of the algorithm and its efficiency in handling large-scale input/output simulators, through an illustrative example: The optimisation of a Counter-Flow Jet Reactor [5] which is simulated via the state-of-the-art massively parallel finite element code MPSALSA developed at SANDIA National Laboratories [6]. The convergence of the optimization algorithm and its relation with the convergence of the iterative solver is analyzed and its efficiency for handling both equality and inequality constraints is discussed.

References

1. Blum, C. and A. Roli, Metaheuristics in combinatorial optimization: Overview and conceptual comparison. Acm Computing Surveys, 2003. 35(3): p. 268-308.

2. Luna-Ortiz, E. and C. Theodoropoulos. Multiscale Modeling & Simulation, 2005. 4(2): p. 691-708.

3. Theodoropoulos, C. and E. Luna-Ortiz, in Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena, A. Gorban, et al. eds 2006, pp. 535-560.

4. Biegler, L.T., J. Nocedal, and C. Schmid. Siam Journal on Optimization, 1995. 5: 314-347.

5. Safvi, S.A. and T.J. Mountziaris, AIChE Journal, 1994. 40: 1535-1548.

6. Shadid J, Hutchinson S, Hennigan G, Moffat H, Devine K, Salinger AG, Parallel Computing 1997. 23: 1307-1325.