The network uncertainty quantification method for propagating uncertainties in component-based systems

Abstract

This work introduces the network uncertainty quantification (NetUQ) method for performing uncertainty propagation in systems composed of interconnected components. The method assumes the existence of a collection of components, each of which is characterized by exogenous-input random variables (e.g., material properties), endogenous-input random variables (e.g., boundary conditions defined by another component), output random variables (e.g., quantities of interest), and a local uncertainty-propagation operator (e.g., provided by stochastic collocation) that computes output random variables from input random variables. The method assembles the full-system network by connecting components, which is achieved simply by associating endogenous-input random variables for each component with output random variables from other components; no other inter-component compatibility conditions are required. The network uncertainty-propagation problem is: Compute output random variables for all components given all exogenous-input random variables. To solve this problem, the method applies classical relaxation methods (i.e., Jacobi and Gauss-Seidel iteration with Anderson acceleration), which require only blackbox evaluations of component uncertainty-propagation operators. Compared with other available methods, this approach is applicable to any network topology (e.g., no restriction to feed-forward or two-component networks), promotes component independence by enabling components to employ tailored uncertaintypropagation operators, supports general functional representations of random variables, and requires no offline preprocessing stage. Also, because the method propagates functional representations of random variables throughout the network (and not, e.g., probability density functions), the joint distribution of any set of random variables throughout the network can be estimated a posteriori in a straightforward manner. We perform supporting convergence and error analysis and execute numerical experiments that demonstrate the weak-and strong-scaling performance of the method.

Localized model order reduction and domain decomposition methods for coupled heterogeneous systems

“…A detailed discussion on the two approaches can be found in other works. 6 Extending the presented formulation to more than two subdomains is rather straightforward. Denoting by  i the index set of the problems  j that share an interface Γ ij with  i , the coupling conditions of each  i can be written as…”

“…This problem is strictly related to the generation of a directed acyclic graph and graph coloring. 6…”

“…To conclude, we mention that in our simulations we processed the elements using a two-color, checkerboard pattern. A different choice may lead to a different number of iterations and a different number of sequential step required in a parallel implementation, 6 both at the full and the reduced level.…”

“…Additional complexities arise if the components are modeled using different simulation tools, as they can be difficult to integrate. 6 We propose a model reduction technique that relies on local surrogates, in the form of reduced models constructed in a component-wise, decoupled fashion. In the offline phase, the components are considered separately, and the snapshots are collected through a parametrization of the interface data.…”

Reduced basis methods for time-dependent problems

“…An open question then arises as to whether the availability of a collection of reduced models, coupled in a predefined way, is sufficient to characterize the behaviour of the global system. There are some early results in this direction for the harmonic Maxwell's equations in Ganesh, Hesthaven and Stamm (2012) and recently some also for time-dependent problems (Carlberg, Guzzetti, Khalil andSargsyan 2019, Discacciati and. Nonetheless, how to achieve this for more general time-dependent multi-physics coupled systems remains open -in particular the question of how to guarantee that global emergent behaviour, arising as a consequence of the model coupling, can be established.…”