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.

现代反问题的一个重要主题是仅根据有限的多次测量来重建与时间相关的数据。为了在这种情况下获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。通过直接在相空间（位置和速度的空间）中重建数据可以实现最强的一致性。然而，这个空间通常维度太高，无法进行可行的计算。我们引入了一种新颖的降维技术，该技术基于相空间到低维子空间的投影，这可以证明避免了这种维数灾难：事实上，在超分辨率的示例框架中，我们证明已知的精确重建结果在降维后仍然正确，并且我们还证明了最佳传输指标中噪声数据重建的新误差估计，其质量与非降维情况下获得的质量相同。