Collaborative Research: CMG: Uncertainty Quantification in Geophysical State Estimation

0530858/0530867海洋学家、数值分析员和计算机科学家之间的这种合作是针对使用超大型、复杂的模型和数据集的问题。这样的问题出现在海洋学中，也出现在气象学、经济学、工程学和所有进行复杂模拟的领域。自动微分(AD)工具在确定模型对初始和边界条件以及内部参数扰动的敏感性方面已被证明是极其强大的。在最近使这些模型与现代大规模全球数据集(气象学中的状态估计或“数据同化”)保持一致的努力中，AD是至关重要的。然而，仅凭敏感性来描述模型行为的价值是有限的。拟议中的不确定性量化研究代表了我们对模型理解的质的进步，并最终将指导模型的改进；这需要针对本征求解器、海森计算和非光滑优化的新的数学方法来处理计算复杂的模型。该项目寻求对地球物理模型和地球物理系统状态估计中固有的不确定性的深入的新见解。如果没有它们，就不可能将这些模型中遇到的问题归因于模型数值的非光滑表述，或者归因于基本光滑动力系统的理论极限。然而，用于不确定性量化的算法需要取得重大进展，这就需要计算以地球物理模型为特征的大而密集的黑森的本征解。在计算海森向量积的自动微分算法方面取得了实质性进展，再加上基于拟牛顿更新和尺度探测的新型预条件，有望实现对数值不确定性的有效表征。此外，通过使用非光滑优化算法实现的高度不连续系统的状态估计的进展和相应的微分算法的进展，将提供通过使用非光滑参数化方案引入的模型不确定性的洞察。更广泛的影响数值模型被用于广泛的科学和工程问题，包括地球物理；经济学；物理学；机械、核、航空和化学工程；以及医学。通常在模拟模式中使用的这些模型的复杂性随着时间的推移而增加，直到没有一个人理解代码如何以及为什么响应外部或内部参数的变化。所提出的数学算法和AD方法将产生即使在非常大的规模的问题中也在计算上可行的灵敏度测试。通常，模型状态模拟与各种观测相结合，以产生对真实状态的最佳估计。事实证明，在使用基于梯度的优化算法解决由此产生的极大优化问题方面，AD工具非常有用。这项拟议的工作同时解决了大规模模型中的不连续性和非线性，提供了洞察力，将使不确定性量化之前难以解决的一大类问题受益。该项目将提供数学和计算工具，以便研究人员能够以近乎自动的方式评估模型的不确定性。为了最大限度地提高算法的可用性和影响力，所有算法都将作为开源软件实现。

0530858/0530867This collaboration between oceanographers, numerical analysts, and computer scientists is directed at the problems of using extremely large, complex models and data sets. Such problems appear in oceanography, but also in meteorology, economics, engineering, and all fields in which complex simulations are carried out. Automatic differentiation (AD) tools have proved extremely powerful in determining model sensitivities to perturbations in initial and boundary conditions, as well as in internal parameters. AD is critical in recent efforts to bring these models into consistency with modern, massive global data sets (state estimation or "data assimilation" in meteorology). However, sensitivities alone are of limited value for characterizing model behavior. The proposed research on uncertainty quantification represents a qualitative advance in our understanding of the models and will ultimately guide model improvements; this requires new mathematical approaches for eigen-solvers, Hessian computations, and non-smooth optimization to handle the computationally complex models.Intellectual Merit This project seeks to gain deep new insight into geophysical models and the uncertainty inherent in the state estimation of geophysical systems. Without them it is impossible to attribute problems encountered in such models to either non-smooth formulation of the model numerics, or to theoretical limits of the underlying smooth dynamical system. However, significant advances are needed in the algorithms used for uncertainty quantification, which necessitate computing the eigen-solutions of the large, dense Hessians that characterize geophysical models. Substantial advances in automatic differentiation algorithms for computing Hessian-vector products, coupled with novel pre-conditioners based on quasi-Newton updates and scale probing are expected to enable an efficient characterization of the numerical uncertainties. Furthermore, advances in state estimation for highly discontinuous systems, achieved via the use of non-smooth optimization algorithms and corresponding advances in differentiation algorithms, will provide insight into the model uncertainties introduced through the use of non-smooth parameterization schemes. Broader Impacts Numerical models are used in a wide range of scientific and engineering problems, including geophysics; economics; physics; mechanical, nuclear, aeronautical and chemical engineering; and medicine. The complexity of these models, which are typically used in simulation mode, increases over time until no single individual understands how and why the code responds to changes in external or internal parameters. The proposed mathematical algorithms and AD methods will produce sensitivity tests that are computationally feasible even in very large scale problems. Often, the model state simulations are combined with a wide variety of observations so as to produce best estimates of the true state. AD tools have proven extremely useful in enabling the resulting extremely large optimization problem to be solved by using gradient-based optimization algorithms. The proposed work addresses both discontinuities and nonlinearities in large scale models, offering insights that will benefit a wide class of problems where uncertainty quantification has previously been intractable. This project will provide the mathematical and computational tools so that researchers can evaluate model uncertainties in near-automatic fashion. To maximize their availability and impact all algorithms will be implemented as open source software.