We present an uncertainty quantification (UQ) algorithm using the intrusive generalized polynomial chaos (gPC) expansion in combination with dimension reduction techniques and compare the UQ accuracy and computational efficiency of the intrusive gPC-based UQ algorithm to other sampling-based nonintrusive methods. The successful application of intrusive gPC-based UQ is associated with the stochastic Galerkin (SG) projection, which yields a family of models described by several coupled equations of gPC coefficients. Using these coefficients, the evolution of uncertainty in a dynamic system can be quickly determined when there is probabilistic uncertainty in the system. While elegant, when dealing with models that involve complex functions (e.g., nonpolynomial terms) and larger numbers of uncertainties, SG projection becomes computationally intractable and cannot be applied directly to solve gPC coefficients in real-time. To address this issue, the generalized dimension reduction method (gDRM) is used to convert a high-dimensional integral involved in the SG projection into several lower-dimensional integrals that can be easily solved. To show the accuracy of UQ, the algorithm in this work is compared to sampling-based approaches such as the nonintrusive stochastic collocation (SC) and Monte Carlo (MC) simulations using three cases: a nonlinear algebraic benchmark, a penicillin manufacturing process, and autocrine signalling networks of living cells.

我们提出了一种使用侵入式广义多项式混沌（gPC）扩展与降维技术相结合的不确定性量化（UQ）算法，并将基于侵入式gPC的UQ算法与其他基于采样的非侵入式方法的UQ准确性和计算效率进行了比较。侵入式基于gPC的UQ的成功应用与随机Galerkin（SG）投影相关，该投影产生了一系列由gPC系数的几个耦合方程式描述的模型。使用这些系数，当系统中存在概率不确定性时，可以快速确定动态系统中不确定性的演变。虽然优雅，但是在处理涉及复杂功能（例如，非多项式项）和大量不确定性的模型时，SG投影在计算上变得棘手，无法直接应用于实时求解gPC系数。为了解决这个问题，使用广义降维方法（gDRM）将SG投影中涉及的高维积分转换为可以轻松解决的几个低维积分。为了显示UQ的准确性，将本文中的算法与基于采样的方法进行了比较，例如使用三种情况的非侵入式随机搭配（SC）和蒙特卡洛（MC）模拟：非线性代数基准，青霉素制造过程和活细胞的自分泌信号网络。广义降维方法（gDRM）用于将SG投影中涉及的高维积分转换为易于解决的几个低维积分。为了显示UQ的准确性，将本文中的算法与基于采样的方法进行了比较，例如使用三种情况的非侵入式随机搭配（SC）和蒙特卡洛（MC）模拟：非线性代数基准，青霉素制造过程和活细胞的自分泌信号网络。广义降维方法（gDRM）用于将SG投影中涉及的高维积分转换为易于解决的几个低维积分。为了显示UQ的准确性，将本文中的算法与基于采样的方法进行了比较，例如使用三种情况的非侵入式随机搭配（SC）和蒙特卡洛（MC）模拟：非线性代数基准，青霉素制造过程和活细胞的自分泌信号网络。