Uncertainty quantification (UQ) tasks, such as sensitivity analysis and parameter estimation, entail a huge computational complexity when dealing with input-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-reduced basis (RB) method, have been extensively developed in the last decades to overcome the computational complexity of high fidelity full order models (FOMs), providing remarkable speedups when addressing UQ tasks related with parameterized differential problems. Nonetheless, constructing a projection-based ROM that can be efficiently queried usually requires extensive modifications to the original code, a task which is non-trivial for nonlinear problems, 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 recently developed to overcome this issue. In this work, we consider ROMs exploiting proper orthogonal decomposition to construct a reduced basis from a set of FOM snapshots, and Gaussian process regression (GPR) to approximate the RB projection coefficients. Two different approaches, namely a global GPR and a tensor-decomposition-based GPR, are explored on a set of 3D time-dependent solid mechanics examples. Finally, the non-intrusive ROM is exploited to perform global sensitivity analysis (relying on both screening and variance-based methods) and parameter estimation (through Markov chain Monte Carlo methods), showing remarkable computational speedups and very good accuracy compared to high-fidelity FOMs.

翻译：不确定性量化（UQ）任务（如灵敏度分析与参数估计）在处理涉及非线性微分问题求解的输入-输出映射时，因需重复调用计算成本高昂的数值求解器而面临巨大计算复杂度。过去数十年间，基于投影的降阶模型（ROMs），如伽辽金约化基（RB）方法，已被广泛开发以克服高保真全阶模型（FOM）的计算复杂度，在处理参数化微分问题相关的不确定性量化任务时实现了显著加速。然而，构建可高效调用的投影型降阶模型通常需要对原始代码进行大量修改——这对非线性问题而言颇具挑战性，甚至在使用专有软件时完全不可行。近年来，非侵入式降阶模型（将全阶模型视为黑箱）被开发以解决此问题。本研究采用基于本征正交分解的降阶模型，通过全阶模型快照构建约化基，并利用高斯过程回归（GPR）近似约化基投影系数。我们探索了两种不同方法——全局高斯过程回归与基于张量分解的高斯过程回归——并将其应用于一组三维时变固体力学算例。最后，利用非侵入式降阶模型执行全局灵敏度分析（结合筛选法与基于方差的方法）与参数估计（通过马尔可夫链蒙特卡洛方法），相较于高保真全阶模型展现出显著计算加速与极佳精度。