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Uncertainty quantification (UQ) tasks, such as sensitivity analysis andparameter estimation, entail a huge computational complexity when dealing withinput-output maps involving the solution of nonlinear differential problems,because of the need to query expensive numerical solvers repeatedly.Projection-based reduced order models (ROMs), such as the Galerkin-reducedbasis (RB) method, have been extensively developed in the last decades toovercome the computational complexity of high fidelity full order models(FOMs), providing remarkable speedups when addressing UQ tasks related withparameterized differential problems. Nonetheless, constructing aprojection-based ROM that can be efficiently queried usually requires extensivemodifications to the original code, a task which is non-trivial for nonlinearproblems, or even not possible at all when proprietary software is used.Non-intrusive ROMs - which rely on the FOM as a black box - have been recentlydeveloped to overcome this issue. In this work, we consider ROMs exploitingproper orthogonal decomposition to construct a reduced basis from a set of FOMsnapshots, and Gaussian process regression (GPR) to approximate the RBprojection coefficients. Two different approaches, namely a global GPR and atensor-decomposition-based GPR, are explored on a set of 3D time-dependentsolid mechanics examples. Finally, the non-intrusive ROM is exploited toperform global sensitivity analysis (relying on both screening andvariance-based methods) and parameter estimation (through Markov chain MonteCarlo methods), showing remarkable computational speedups and very goodaccuracy compared to high-fidelity FOMs.

Uncertainty quantification for nonlinear solid mechanics using reduced order models with Gaussian process regression