SIAM/ASA Journal on Uncertainty Quantification, Volume 9, Issue 2, Page 922-952, January 2021.
The primary objective of this work is to study the inverse problem of identifying a stochastic parameter in partial differential equations with random data. In the framework of stochastic Sobolev spaces, we prove the Lipschitz continuity and the differentiability of the parameter-to-solution map and provide a new derivative characterization. We introduce a new energy-norm based modified output least-squares (OLS) objective functional and prove its smoothness and convexity. For stable inversion, we develop a regularization framework and prove an existence result for the regularized stochastic optimization problem. We also consider the OLS based stochastic optimization problem and provide an adjoint approach to compute the derivative of the OLS-functional. In the finite-dimensional noise setting, we give a parameterization of the inverse problem. We develop a computational framework by using the stochastic Galerkin discretization scheme and derive explicit discrete formulas for the considered objective functionals and their gradient. We provide detailed computational results to illustrate the feasibility and efficacy of the developed inversion framework. Encouraging numerical results demonstrate some of the advantages of the new framework over the existing approaches.

中文翻译：

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识别随机偏微分方程中随机参数的逆问题的凸优化框架

SIAM/ASA 不确定性量化杂志，第 9 卷，第 2 期，第 922-952 页，2021 年 1 月。
这项工作的主要目的是研究在具有随机数据的偏微分方程中识别随机参数的逆问题。在随机 Sobolev 空间的框架中，我们证明了参数到解映射的 Lipschitz 连续性和可微性，并提供了一种新的导数表征。我们引入了一种新的基于能量范数的修正输出最小二乘 (OLS) 目标函数，并证明了它的平滑性和凸性。对于稳定反演，我们开发了一个正则化框架并证明了正则化随机优化问题的存在性结果。我们还考虑了基于 OLS 的随机优化问题，并提供了一种计算 OLS 泛函导数的伴随方法。在有限维噪声设置中，我们给出了逆问题的参数化。我们通过使用随机伽辽金离散化方案开发了一个计算框架，并为所考虑的目标函数及其梯度推导出显式离散公式。我们提供了详细的计算结果来说明所开发的反演框架的可行性和有效性。令人鼓舞的数值结果证明了新框架相对于现有方法的一些优势。