(46d) A Nonlinear Interior Point Framework for Stochastic PDEs Over Networks

We present a nonlinear interior point framework for stochastic programs with network constraints in which each link is described by a set of partial differential equations (PDE). This problem class arises in the design and operation of national-scale infrastructure systems such as natural gas and water. Deterministic operational models have been previously proposed in [1,2]. We present a stochastic operational model to determine optimal inventory levels in the face of spatio-temporal uncertainty of demands. We discuss computational complexity and limitations of the state-of-the-art and propose alternatives to enable scalability in high-performance computing systems. Our framework uses  inexact linear algebra and multi-level preconditioners for nested block-bordered and PDE systems.  The framework does not require inertia information which is key because this is difficult to obtain in the problems of interest. This strategy has been recently proposed by Curtis, Schenk, and Waechter [3]. We present numerical results to demonstrate the developments. 

[1] Steinbach, Marc C. "On PDE solution in transient optimization of gas networks." Journal of computational and applied mathematics 203.2 (2007): 345-361.

[2] Burgschweiger, Jens, Bernd Gnädig, and Marc C. Steinbach. "Optimization models for operative planning in drinking water networks." Optimization and Engineering 10.1 (2009): 43-73.

[3] Curtis, Frank E., Olaf Schenk, and Andreas Wächter. "An interior-point algorithm for large-scale nonlinear optimization with inexact step computations." SIAM Journal on Scientific Computing 32.6 (2010): 3447-3475.