发表机构
Dartmouth College（达特茅斯学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对双曲守恒律变分数据同化中高斯平滑破坏间断的问题，提出基于稀疏促进残差变换的正则化项，结合广义稀疏贝叶斯学习，在三维变分框架下解决非凸目标，实验表明比全变分更准确解析间断。
AI 中文摘要
双曲守恒律为变分数据同化带来了具有挑战性的环境，因为高斯假设施加了平滑性，从而模糊了跳跃间断。稀疏促进正则化可以通过将结构先验纳入变分目标来帮助缓解这一问题。在选择该先验时必须小心，因为它需要同时抵消结构上错误指定的背景误差协方差（该协方差在平滑区域引入虚假振荡）以及空间稀疏的观测（这些观测留下的数据太少，无法约束状态）。此外，双曲守恒律的状态变量通常不具有分段常数结构，而这是使用标准稀疏促进算子（如全变分）时所做的假设。实际上，双曲守恒律解的潜在变异性既不是预先已知的，也不是固定的。因此，使用高阶全变分也不合适。在这里，我们引入了一种新的数据同化正则化项，基于稀疏促进残差变换，它既不承诺固定的平滑阶数，也不要求预先知道潜在的变异性。我们在三维变分框架内使用广义稀疏贝叶斯学习来解决由此产生的非凸目标。该方法用残差变换上的分层条件高斯先验取代了全局ℓ1惩罚，其中学习到的超先验为每个空间点提供特定位置的权重。跨标量、浅水和欧拉测试问题的数值实验表明，该正则化项比全变分更准确地解析间断，并且其优势随着背景误差协方差变得错误指定和观测变得更稀疏而增长。
英文摘要
Hyperbolic conservation laws pose a challenging setting for variational data assimilation, since the Gaussian assumption imposes a smoothness that smears jump discontinuities. A sparsity promoting regularization can help to mitigate this problem by incorporating a structural prior into the variational objective. Care must be taken when choosing this prior because it serves to counterbalance both a structurally misspecified background-error covariance, which introduces spurious oscillations over smooth regions, as well as spatially sparse observations, which leave too little data to constrain the state. Moreover, state variables of hyperbolic conservation laws typically do not have piecewise constant structure, which is an assumption made when using standard sparsity-promoting operators, such as total variation. Indeed the underlying {\em variability} of hyperbolic conservation law solutions are neither known in advance nor fixed. Hence using higher order total variation is also not suitable. Here we introduce a new regularization term for data assimilation, built on the sparsity promoting residual transform that neither commits to a fixed smoothness order nor requires the underlying variability to be known a priori. We solve the resulting nonconvex objective within a three-dimensional variational framework using generalized sparse Bayesian learning. This approach replaces a global \(\ell _{1}\) penalty with a hierarchical conditional Gaussian prior on the residual transform, where a learned hyper-prior yields location-specific weights attached to each spatial point. Numerical experiments across scalar, shallow-water, and Euler test problems show that this regularization term resolves discontinuities more accurately than total variation, and that its advantage grows as the background-error covariance becomes misspecified and the observations become sparser.
