Uncertainty Management in Networked Dynamical Systems
Monte Carlo methods do not scale well with system size. To overcome this problem a large system can for example be decomposed into subsystems evolving on different time scales using graph decomposition methods . This time scale separation can be exploited to increase computational efficiency when propagating input uncertainty in a subsystem-by-subsystem manner . Another interesting alternative could be decomposition based on state space behavior. Subsets of the state space of a dynamical system where typical trajectories stay longer before entering different regions are called almost invariant sets . Such decomposition could be used in uncertainty analysis based on regions of different dynamics.
Finally, to obtain the global results due to uncertain parameters or initial conditions, the weak coupling between subsystems should be taken into account by using an appropriate iteration scheme. The Waveform Relaxation method has been successfully employed in to address large scale systems, such as integrated circuits.
Even after reduction of the size of the state space of the dynamical systems using the above methods, there still remains the problem of propagating uncertainty through the individual subsystems. Thus, a solution to this sub-problem and the “coherent combination” of the outputs of these sub-problems is absolutely critical to the success of any of these schemes. The exact description of this problem is provided by the well known Fokker Planck Kolmogorov Equation (FPKE), or simply the Fokker Planck Equation (FPE), the solution to which contains complete information about the state PDF. While the FPK equation holds the key for uncertainty propagation through nonlinear dynamical systems, it is a formidable problem to solve because of the issues like positivity, normality, discretization, and most importantly, the dimensionality of the system. For small dimensions (<4), the FPE can be solved numerically after discretization, but for (N > 3), advanced numerical methods are required due to the exponential growth of computational work with increasing N (“the curse of dimensionality”). In particular, there is need for efficient, robust and accurate methods for solving the FPE for the propagation of uncertainties through high dimensional complex nonlinear systems and for assessing their effect on system performance.
The core tool at the heart of our approach to solving the FPE is a generalized FEM method known as partition of unity FEM (PUFEM) and a recently developed adaptive, multi-resolution approximation algorithm known as: Global-Local Orthogonal MAPping (GLO-MAP) . In conjunction with adaptive homotopic methods , the above methodology will focus on the development of a computational tool to obtain the stationary and transient solution to the FPE for general N-dimensional dynamical system. Preliminary results obtained on have shown orders of magnitude improvement in the computational efficiency over those of conventional methods (please see the figure below for the adaptive refinement scheme and the references below for more details). The resulting algorithm can be used in conjunction with conventional methods such as Monte-Carlo, Gaussian closure and statistical linearization for uncertainty propagation through large-scale systems.
Candidate PIs: J. Junkins, Suman Chakravorty, Raktim Bhattacharya, Tamas Kalmar-Nagy, J. Hurtado, S. R. Vadali
• Kalmár-Nagy, T. and Huzmezan, M.: Propagation of Uncertain Inputs Through Networks of Nonlinear Components, in Proceedings of the 43rd CDC, pp. 1799-1802, 2004.
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