*Article* **Data-Informed Decomposition for Localized Uncertainty Quantification of Dynamical Systems**

**Waad Subber \*, Sayan Ghosh, Piyush Pandita, Yiming Zhang and Liping Wang**

Probabilistic Design and Optimization Group, GE Research, 1 Research Circle, Niskayuna, NY 12309, USA; sayan.ghosh1@ge.com (S.G.); piyush.pandita@ge.com (P.P.); yiming.zhang@ge.com (Y.Z.); wangli@ge.com (L.W.) **\*** Correspondence: Waad.Subber@ge.com

**Abstract:** Industrial dynamical systems often exhibit multi-scale responses due to material heterogeneity and complex operation conditions. The smallest length-scale of the systems dynamics controls the numerical resolution required to resolve the embedded physics. In practice however, high numerical resolution is only required in a confined region of the domain where fast dynamics or localized material variability is exhibited, whereas a coarser discretization can be sufficient in the rest majority of the domain. Partitioning the complex dynamical system into smaller easier-to-solve problems based on the localized dynamics and material variability can reduce the overall computational cost. The region of interest can be specified based on the localized features of the solution, user interest, and correlation length of the material properties. For problems where a region of interest is not evident, Bayesian inference can provide a feasible solution. In this work, we employ a Bayesian framework to update the prior knowledge of the localized region of interest using measurements of the system response. Once, the region of interest is identified, the localized uncertainty is propagate forward through the computational domain. We demonstrate our framework using numerical experiments on a three-dimensional elastodynamic problem.


**Citation:** Subber, W.; Ghosh, S.; Pandita, P.; Zhang, Y.; Wang, L. Data-Informed Decomposition for Localized Uncertainty Quantification of Dynamical Systems. *Vibration* **2021**, *4*, 49–63. https://doi.org/10.3390/ vibration4010004

Received: 19 November 2020 Accepted: 16 December 2020 Published: 31 December 2020

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**Keywords:** Bayesian inference; uncertainty quantification; dynamical systems; inverse problem; machine learning; system Identification; Gaussian process; polynomial chaos
