Randomization for the Efficient Computation of Parametric Reduced Order Models for Inversion

Abstract

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.