参数化降阶模型反演的随机化方法
Randomization for the Efficient Computation of Parametric Reduced Order Models for Inversion
摘要：
非线性参数反问题出现在许多应用中。在这里，我们专注于扩散光学断层扫描（DOT）在医学成像中恢复未知的图像的兴趣，如癌组织在给定的介质中，使用数学（前向）模型。DOT中的前向模型是光子通量的扩散-吸收模型。这些问题的主要瓶颈是大规模正演模型的重复评估。对于DOT，这对应于在每个优化步骤处针对每个源和频率求解大型线性系。此外，牛顿型方法，通常是选择的方法，需要额外的线性求解伴随计算导数信息。新兴技术允许大量的源和检测器，使得这些问题过于昂贵。降阶模型（ROM）已被用来大大减少系统的大小在每个优化步骤，同时精确地解决逆问题。然而，对于大量的源和检测器，仅仅是ROM投影空间的候选基的构造就产生了相当大的成本，因为在插值模型简化中匹配全参数梯度矩阵需要对所有源和频率以及每个参数插值点的所有检测器和频率进行大的线性求解。由于该候选基在数值上具有低秩，因此该构造之后是秩揭示因子分解，其通常大幅减少候选基中的向量的数量。我们建议使用随机化来近似这个基础，大幅减少大型线性求解的数量。我们还提供了一个详细的分析，为我们感兴趣的问题的候选基础的低秩结构。虽然我们专注于DOT问题，提出的想法是相关的许多其他大规模的反问题和最优化问题。
英文摘要：
Nonlinear parametric inverse problems appear in many applications. Here, we focus on diffuse optical tomography (DOT) in medical imaging to recover unknown images of interest, such as cancerous tissue in a given medium, using a mathematical (forward) model. The forward model in DOT is a diffusion-absorption model for the photon flux. The main bottleneck in these problems is the repeated evaluation of the large-scale forward model. For DOT, this corresponds to solving large linear systems for each source and frequency at each optimization step. Moreover, Newton-type methods, often the method of choice, require additional linear solves with the adjoint to compute derivative information. Emerging technology allows for large numbers of sources and detectors, making these problems prohibitively expensive. Reduced order models (ROM) have been used to drastically reduce the system size in each optimization step, while solving the inverse problem accurately. However, for large numbers of sources and detectors, just the construction of the candidate basis for the ROM projection space incurs a substantial cost, as matching the full parameter gradient matrix in interpolatory model reduction requires large linear solves for all sources and frequencies and all detectors and frequencies for each parameter interpolation point. As this candidate basis numerically has low rank, this construction is followed by a rank-revealing factorization that typically reduces the number of vectors in the candidate basis substantially. We propose to use randomization to approximate this basis with a drastically reduced number of large linear solves. We also provide a detailed analysis for the low-rank structure of the candidate basis for our problem of interest. Even though we focus on the DOT problem, the ideas presented are relevant to many other large scale inverse problems and optimization problems.
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