Efficient nonlinear model reduction for improved predictive uncertainty quantification and optimal design of monitoring networks in coupled groundwater-surface water systems

在复杂的耦合地下水-地表水系统中，相互作用的水流和输运过程的可靠模拟和预测通常需要基于偏微分方程的数值模型，其参数在空间上分布，可能表现出不同程度的非线性。它们是数据饥饿，计算要求高，通常只有很少的是已知的先验这些模型的真实参数值。全局搜索参数估计方案可能需要大量的重复模型运行。预测性不确定性分析或基于模型的优化任务-特别是监测网络的优化-面临更大的问题。这对于复杂的基于物理的过程模型是不可行的，并且模型简化通常是使这些模型可用于指定目的的唯一选择。已经开发了几种有前途的模型简化方法，以减少基于物理模型的计算工作量。这包括基于反演的尺度放大和正式的数学模型简化。然而，许多挑战必须满足之前，这些方法可以稳健地应用于地下水-地表水背景下的真实的情况。最重要的是，模型简化引入了额外的误差，根据现有数据，这些误差可能使简化模型无效。模型简化的总体目标是获得计算工作量的大幅减少，但只有很小的误差。例如，预测概率分布必须仍然包含真值（即，小的误差偏差），并且不应该不必要地宽以补偿该偏差（即，小误差方差）。拟议项目力求在地下水-地表水相互作用建模和优化相应监测网络的背景下应对这些挑战。使用两个真实世界的案例研究，我们将首先进行基准测试，以分析现有方法的相对性能（即，特征模型、适当的正交分解方法和基于反演的放大）在校准性能、预测性能和优化任务中的性能方面的改进。其次，将开发或扩展有效的模型简化方法，以应用于非线性地下水-地表水问题，这是本提案的主要目标。在这里，我们还将调整先验参数分布的Karhunen-Loève变换，以获得简化的参数分布，从而减少模拟时间。最后，我们将应用新开发的非线性模型简化方案优化监测网络，以减少地下水-地表水预测模拟的不确定性，并将结果与复杂模型的等效解决方案进行比较。

The reliable simulation and prediction of interacting flow and transport processes in complex coupled groundwater-surface water systems often requires PDE-based numerical models with spatially distributed parameters, which may exhibit a varying degree of non-linearity. They are data hungry, computationally demanding, and typically only little is known a priori about the true parameter values of these models. Global-search parameter-estimation schemes can require a large number of repeated model runs. Predictive uncertainty analysis or model-based optimization tasks - particularly the optimisation of monitoring networks - suffer from even larger problems. This becomes infeasible for complex physically-based process models, and model simplification is often the only choice to make such models accessible for the named purposes. Several promising model simplification methods have been developed to reduce the computational effort of physically-based models. This includes inversion-based upscaling and formal mathematical model reduction. However, many challenges must be met before these methods can be robustly applied to real cases in groundwater-surface water contexts. Most of all, there is additional error introduced by the model simplification which can invalidate the simplified models in the light of available data. The overall goals of model simplification are to obtain substantial reduction in computational effort, yet with only small errors. For example, the predictive probability distribution must still contain the true values (i.e., small error bias), and should not be unnecessarily wide to compensate for that bias (i.e., small error variance). The proposed project seeks to address these challenges in the context of modelling groundwater-surface water interaction and for the optimization of corresponding monitoring networks. Using two real-world case studies, we will first conduct a benchmark exercise to analyse the relative performance of existing methods (i.e., eigenmodels, proper orthogonal decomposition methods, and inversion-based upscaling) in terms of calibration performance, prediction performance, and the performance in optimization tasks. Secondly, efficient model reduction methods will be developed or extended for application to non-linear groundwater-surface water problems, which is the main goal of this proposal. Here, we will also adapt the Karhunen-Loève transformation of the prior parameter distributions to obtain simplified parameter distributions and thus reduced simulation times. Finally, we will apply the newly developed non-linear model reduction scheme for optimizing monitoring networks to reduce the uncertainty of groundwater - surface water predictive simulations and compare the results to equivalent solutions as achieved with a complex model.