[Paper Review] The network uncertainty quantification method for propagating uncertainties in component-based systems

Kevin Carlberg, Sofia Guzzetti+2 more|arXiv (Cornell University)|Aug 29, 2019

Probabilistic and Robust Engineering Design5 citations

TL;DR

This paper introduces the Network Uncertainty Quantification (NetUQ) method, a domain-decomposition approach for propagating uncertainties in component-based systems using relaxation methods (Jacobi/Gauss–Seidel with Anderson acceleration). It enables component independence, supports arbitrary network topologies, and propagates functional representations of random variables, allowing efficient joint distribution estimation and strong scalability.

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, endogenous-input random variables, output random variables, and a local uncertainty-propagation operator 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 black-box evaluations of component uncertainty-propagation operators. Compared with other available methods, this approach is applicable to any network topology, promotes component independence by enabling components to employ tailored uncertainty-propagation 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.

Motivation & Objective

• Address the challenge of uncertainty propagation in large-scale, interconnected systems composed of heterogeneous components with different physics and modeling codes.
• Overcome limitations of existing methods that require full-system model integration, restrict network topology (e.g., feed-forward only), or rely on specific stochastic representations like polynomial chaos.
• Enable each component to use its own tailored uncertainty-propagation operator without requiring coupled deterministic solves or offline preprocessing.
• Support general functional representations of random variables to allow flexible post-processing, including joint distribution estimation.
• Develop a scalable, black-box-compatible method that requires only component-level uncertainty-propagation evaluations and avoids mesh compatibility or coupling constraints.

Proposed method

• Formalize each component as a black-box operator mapping exogenous and endogenous inputs to output random variables, with local uncertainty propagation via stochastic collocation or similar methods.
• Construct the full-system network by connecting components through direct association of endogenous-input variables with output variables from other components, without requiring compatibility conditions.
• Apply classical relaxation methods—Jacobi and Gauss–Seidel iteration—augmented with Anderson acceleration to solve the coupled system of component equations.
• Use a permutation-based graph coloring strategy to minimize sequential processing steps in Gauss–Seidel updates by analyzing the sparsity pattern of component dependencies.
• Maintain functional representations of random variables throughout the network, enabling a posteriori estimation of joint distributions and variance-based decomposition.
• Ensure black-box compatibility by requiring only evaluations of component uncertainty-propagation operators, avoiding internal model access or preprocessing.

Experimental results

Research questions

• RQ1 Can a scalable, component-independent uncertainty propagation method be developed that supports arbitrary network topologies without requiring full-system model integration?
• RQ2 How can functional representations of random variables be preserved and propagated through complex, coupled component networks to enable flexible post-processing?
• RQ3 To what extent do relaxation methods with Anderson acceleration converge and scale in uncertainty propagation for large-scale component-based systems?
• RQ4 Can the method maintain high performance and convergence without requiring offline preprocessing or compatibility constraints between components?
• RQ5 How does the choice of component permutation affect convergence rate and the number of sequential processing steps in Gauss–Seidel iterations?

Key findings

• The NetUQ method successfully propagates uncertainties across component networks with arbitrary topologies, including those with two-way coupling, without requiring feed-forward or two-component restrictions.
• The method enables component independence by allowing each component to use its own tailored uncertainty-propagation operator, such as stochastic collocation or polynomial chaos, without coupling constraints.
• Functional representations of random variables are preserved throughout the network, enabling straightforward estimation of joint distributions and variance-based decomposition after the fact.
• Numerical experiments demonstrate weak- and strong-scaling performance, indicating the method scales efficiently with increasing system size and number of processors.
• Anderson acceleration significantly improves convergence speed of relaxation iterations, and permutation-based graph coloring reduces the number of sequential steps in Gauss–Seidel updates.
• The method requires no offline preprocessing or mesh compatibility checks, and operates purely via black-box evaluations of component uncertainty operators.