This work presents a novel methodology for solving inverse problems under uncertainty using stochastic reduced order models (SROMs). Given statistical information about an observed state variable in a system, unknown parameters are estimated probabilistically through the solution of a model-constrained, stochastic optimization problem. The point of departure and crux of the proposed framework is the representation of a random quantity using a SROM - a low dimensional, discrete approximation to a continuous random element that permits e cient and non-intrusive stochastic computations. Characterizing the uncertainties with SROMs transforms the stochastic optimization problem into a deterministic one. The non-intrusive nature of SROMs facilitates e cient gradient computations for random vector unknowns and relies entirely on calls to existing deterministic solvers. Furthermore, the method is naturally extended to handle multiple sources of uncertainty in cases where state variable data, system parameters, and boundary conditions are all considered random. The new and widely-applicable SROM framework is formulated for a general stochastic optimization problem in terms of an abstract objective function and constraining model. For demonstration purposes, however, we study its performance in the specific case of inverse identification of random material parameters in elastodynamics. We demonstrate the ability to efficiently recover random shear moduli given material displacement statistics as input data. We also show that the approach remains effective for the case where the loading in the problem is random as well.

中文翻译：

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不确定性下逆问题的随机降阶模型

这项工作提出了一种使用随机降阶模型 (SROM) 在不确定性下解决逆问题的新方法。给定关于系统中观察到的状态变量的统计信息，未知参数通过模型约束的随机优化问题的解决方案进行概率估计。所提议框架的出发点和关键是使用 SROM 表示随机量 - 一种低维、离散的连续随机元素的近似值，允许进行高效且非侵入性的随机计算。用 SROM 表征不确定性将随机优化问题转化为确定性问题。SROM 的非侵入性有助于对随机向量未知数进行有效的梯度计算，并且完全依赖于对现有确定性求解器的调用。此外，在状态变量数据、系统参数和边界条件都被认为是随机的情况下，该方法自然会扩展到处理多种不确定性来源。新的和广泛适用的 SROM 框架是根据抽象目标函数和约束模型为一般随机优化问题制定的。然而，出于演示目的，我们研究了它在弹性动力学中随机材料参数逆识别的特定情况下的性能。我们展示了在给定材料位移统计数据作为输入数据的情况下有效恢复随机剪切模量的能力。