An important theme in modern inverse problems is the reconstruction of time-dependent data from only finitely many measurements. To obtain satisfactory reconstruction results in this setting it is essential to strongly exploit temporal consistency between the different measurement times. The strongest consistency can be achieved by reconstructing data directly in phase space, the space of positions and velocities. However, this space is usually too high-dimensional for feasible computations. We introduce a novel dimension reduction technique, based on projections of phase space onto lower-dimensional subspaces, which provably circumvents this curse of dimensionality: Indeed, in the exemplary framework of superresolution we prove that known exact reconstruction results stay true after dimension reduction, and we additionally prove new error estimates of reconstructions from noisy data in optimal transport metrics which are of the same quality as one would obtain in the non-dimension-reduced case.

翻译：现代反向问题的一个重要主题是从有限的许多测量中重建基于时间的数据。在这一背景下,为了取得令人满意的重建结果,必须大力利用不同测量时间之间的时间一致性。通过直接在阶段空间、位置空间和速度空间中重建数据,可以实现最强烈的一致性。然而,这一空间通常太高,无法进行可行的计算。我们根据对阶段空间的预测,引入了一个新的减少维度技术,可以避免这种维度的诅咒:事实上,在超分辨率的模范框架内,我们证明已知的准确重建结果在降低维度后仍然真实存在,我们还证明了从最优运输指标中的噪音数据进行重建的新的错误估计,这些数据的质量与在非分散减少的情况下获得的数据相同。