2017/06/17 by Selin Aslan, Aslan, Selin, Eric de Sturler +3
Engineering · Medicine · #Photoacoustic and Ultrasonic Imaging #Optical Imaging and Spectroscopy Techniques #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.1706.05586
In partial differential equations-based (PDE-based) inverse problems with\nmany measurements, many large-scale discretized PDEs must be solved for each\nevaluation of the misfit or objective function. In the nonlinear case,\nevaluating the Jacobian requires solving an additional set of systems. This\nleads to a tremendous computational cost, and this is by far the dominant cost\nfor these problems. Several authors have proposed randomization and stochastic\nprogramming techniques to drastically reduce the number of system solves by\nestimating the objective function using only a few appropriately chosen random\nlinear combinations of the sources. While some have reported good solution\nquality at a greatly reduced cost, for our problem of interest, diffuse optical\ntomography, the approach often does not lead to sufficiently accurate\nsolutions.\n We propose two improvements. First, to efficiently exploit Newton-type\nmethods, we modify the stochastic estimates to include random linear\ncombinations of detectors, drastically reducing the number of adjoint solves.\nSecond, after solving to a modest tolerance, we compute a few simultaneous\nsources and detectors that maximize the Frobenius norm of the sampled Jacobian\nto improve the rate of convergence and obtain more accurate solutions. We\ncomplement these optimized simultaneous sources and detectors by random\nsimultaneous sources and detectors constrained to a complementary subspace. Our\napproach leads to solutions of the same quality as obtained using all sources\nand detectors but at a greatly reduced computational cost, as the number of\nlarge-scale linear systems to be solved is significantly reduced.\n