PDE-constrained optimization problems are often treated using the reduced formulation where the PDE constraints are eliminated. This approach is known to be more computationally feasible than other alternatives at large scales. However, the elimination of the constraints forces the optimization process to fulfill the constraints at all times. In some problems this may lead to a highly non-linear objective, which is hard to solve. An example to such a problem, which we focus on in this work, is Full Waveform Inversion (FWI), which appears in seismic exploration of oil and gas reservoirs, and medical imaging. In an attempt to relieve the non-linearity of FWI, several approaches suggested to expand the optimization search space and relax the PDE constraints. This comes, however, with severe memory and computational costs, which we aim to reduce. In this work we adopt the expanded search space approach, and suggest a new formulation of FWI using extended source functions. To make the source-extended problem more feasible in memory and computations, we couple the source extensions in the form of a low-rank matrix. This way, we have a large-but-manageable additional parameter space, which has a rather low memory footprint, and is much more suitable for solving large scale instances of the problem than the full rank additional space. In addition, we show how our source-extended approach is applied together with the popular simultaneous sources technique -- a stochastic optimization technique that significantly reduces the computations needed for FWI inversions. We demonstrate our approaches for solving FWI problems using 2D and 3D models with high frequency data only.

中文翻译：

---

使用扩展和同步源进行全波形反转

经常使用减少了PDE约束的简化公式来处理PDE约束的优化问题。已知这种方法在规模上比其他替代方法在计算上更可行。但是，消除约束会迫使优化过程始终满足约束条件。在某些问题中，这可能导致高度非线性的目标，很难解决。我们在本工作中重点解决的一个问题是全波形反演（FWI），它出现在油气藏的地震勘探和医学成像中。为了缓解FWI的非线性，建议采用几种方法来扩展优化搜索空间并放宽PDE约束。但是，这会带来大量的内存和计算成本，我们希望减少这些成本。在这项工作中，我们采用扩展的搜索空间方法，并建议使用扩展源功能的FWI的新公式。为了使源扩展问题在内存和计算中更可行，我们以低秩矩阵的形式耦合源扩展。这样，我们拥有一个大但可管理的附加参数空间，该空间具有相当低的内存占用，并且比全等级附加空间更适合于解决问题的大规模实例。此外，我们展示了如何将我们的源扩展方法与流行的同时源技术（一种随机优化技术，可以显着减少FWI反演所需的计算量）一起应用。我们演示了仅使用具有高频数据的2D和3D模型来解决FWI问题的方法。