A challenge in high-dimensional inverse problems is developing iterative solvers to find the accurate solution of regularized optimization problems with low computational cost. An important example is computed tomography (CT) where both image and data sizes are large and therefore the forward model is costly to evaluate. Since several years algorithms from stochastic optimization are used for tomographic image reconstruction with great success by subsampling the data. Here we propose a novel way how stochastic optimization can be used to speed up image reconstruction by means of image domain sketching such that at each iteration an image of different resolution is being used. Hence, we coin this algorithm ImaSk. By considering an associated saddle-point problem, we can formulate ImaSk as a gradient-based algorithm where the gradient is approximated in the same spirit as the stochastic average gradient am\'elior\'e (SAGA) and uses at each iteration one of these multiresolution operators at random. We prove that ImaSk is linearly converging for linear forward models with strongly convex regularization functions. Numerical simulations on CT show that ImaSk is effective and increasing the number of multiresolution operators reduces the computational time to reach the modeled solution.

翻译：高维逆问题中的一个挑战在于开发迭代求解器，以较低的计算成本找到正则化优化问题的精确解。计算机断层扫描（CT）是一个重要例子，其图像和数据尺寸均较大，因此前向模型的评估成本高昂。近年来，随机优化算法通过数据子采样在断层图像重建中取得了巨大成功。本文提出一种利用随机优化加速图像重建的新方法：通过图像域草图技术，在每次迭代中使用不同分辨率的图像。因此，我们将该算法命名为ImaSk。通过构建对应的鞍点问题，可将ImaSk表述为基于梯度的算法——其梯度近似思想与随机平均梯度改进法（SAGA）一脉相承，并在每次迭代中随机选用多分辨率算子之一。我们证明对于具有强凸正则化函数的线性前向模型，ImaSk具有线性收敛性。CT数值模拟表明ImaSk算法有效，且增加多分辨率算子的数量能显著缩短达到建模解所需的计算时间。