SIAM Journal on Scientific Computing, Ahead of Print.
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 nonlinear objective, which is hard to solve. An example of 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 in medical imaging. In an attempt to relieve the nonlinearity of FWI, several approaches suggested expanding the optimization search space and relaxing 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：

---

使用扩展和同时源的全波形反演

SIAM 科学计算杂志，提前印刷。
PDE 约束优化问题通常使用减少 PDE 约束的简化公式来处理。众所周知，这种方法在大规模计算上比其他替代方案更具计算可行性。然而，约束的消除迫使优化过程始终满足约束。在某些问题中，这可能会导致高度非线性的目标，这很难解决。我们在这项工作中关注的此类问题的一个例子是全波形反演 (FWI)，它出现在油气藏的地震勘探和医学成像中。为了减轻 FWI 的非线性，几种方法建议扩大优化搜索空间并放宽 PDE 约束。然而，这会带来严重的内存和计算成本，我们的目标是减少这些成本。在这项工作中，我们采用扩展搜索空间方法并建议使用扩展源函数的 FWI 新公式。为了使源扩展问题在内存和计算中更加可行，我们以低秩矩阵的形式耦合源扩展。通过这种方式，我们有一个大但可管理的附加参数空间，它具有相当低的内存占用，并且比全秩附加空间更适合解决问题的大规模实例。此外，我们展示了我们的源扩展方法如何与流行的同时源技术一起应用——一种显着减少 FWI 反演所需计算的随机优化技术。我们展示了我们仅使用带有高频数据的 2D 和 3D 模型解决 FWI 问题的方法。