We study the efficient numerical solution of linear inverse problems with operator valued data which arise, e.g., in seismic exploration, inverse scattering, or tomographic imaging. The high-dimensionality of the data space implies extremely high computational cost already for the evaluation of the forward operator which makes a numerical solution of the inverse problem, e.g., by iterative regularization methods, practically infeasible. To overcome this obstacle, we take advantage of the underlying tensor product structure of the problem and propose a strategy for constructing low-dimensional certified reduced order models of quasi-optimal rank for the forward operator which can be computed much more efficiently than the truncated singular value decomposition. A complete analysis of the proposed model reduction approach is given in a functional analytic setting and the efficient numerical construction of the reduced order models as well as of their application for the numerical solution of the inverse problem is discussed. In summary, the setup of a low-rank approximation can be achieved in an offline stage at essentially the same cost as a single evaluation of the forward operator, while the actual solution of the inverse problem in the online phase can be done with extremely high efficiency. The theoretical results are illustrated by application to a typical model problem in fluorescence optical tomography.

我们研究了线性逆问题的有效数值解，其中包含在地震勘探、逆散射或断层成像中出现的算子值数据。数据空间的高维意味着对于前向算子的评估已经具有极高的计算成本，这使得逆问题的数值解，例如，通过迭代正则化方法，实际上是不可行的。为了克服这个障碍，我们利用问题的潜在张量积结构，并提出了一种策略，用于为前向算子构建准最优秩的低维证明降阶模型，该模型可以比截断奇异值更有效地计算价值分解。在函数分析设置中给出了所提出的模型简化方法的完整分析，并讨论了简化阶模型的有效数值构造及其在逆问题的数值解中的应用。总之，可以在离线阶段以与前向算子的单个评估基本相同的成本实现低秩近似的设置，而在线阶段的逆问题的实际解决可以以极高的成本完成效率。通过应用于荧光光学断层扫描中的典型模型问题来说明理论结果。可以在离线阶段以与前向算子的单个评估基本相同的成本实现低秩近似的设置，而在线阶段可以以极高的效率实际解决逆问题。通过应用于荧光光学断层扫描中的典型模型问题来说明理论结果。可以在离线阶段以与前向算子的单个评估基本相同的成本实现低秩近似的设置，而在线阶段可以以极高的效率实际解决逆问题。通过应用于荧光光学断层扫描中的典型模型问题来说明理论结果。