Tradeoffs between Convergence Speed and Reconstruction Accuracy in Inverse Problems
School of Electrical Engineering, Tel Aviv University（特拉维夫大学电子工程学院）；Electrical Engineering Department, Technion - IIT（技术学院-理工学院电子工程系）；Computer Science Department, Technion - IIT（技术学院-理工学院计算机科学系）；Electrical and Computer Engineering Department, Duke University（杜克大学电气与计算机工程系）
Raja Giryes, Yonina C. Eldar, Alex M. Bronstein, Guillermo Sapiro
备注：
To appear in IEEE Transactions on Signal Processing
Solving inverse problems with iterative algorithms is popular, especially for large data. Due to time constraints, the number of possible iterations is usually limited, potentially affecting the achievable accuracy. Given an error one is willing to tolerate, an important question is whether it is possible to modify the original iterations to obtain faster convergence to a minimizer achieving the allowed error without increasing the computational cost of each iteration considerably. Relying on recent recovery techniques developed for settings in which the desired signal belongs to some low-dimensional set, we show that using a coarse estimate of this set may lead to faster convergence at the cost of an additional reconstruction error related to the accuracy of the set approximation. Our theory ties to recent advances in sparse recovery, compressed sensing, and deep learning. Particularly, it may provide a possible explanation to the successful approximation of the l1-minimization solution by neural networks with layers representing iterations, as practiced in the learned iterative shrinkage-thresholding algorithm (LISTA).
A Unified DeepONet Framework for Logarithmically Stable Infinite-Dimensional Inverse Problems
School of Mathematics and Shing-Tung Yau Center, Southeast University（东南大学数学系及施一公中心）；Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）
Inverse boundary value problem for the Helmholtz equation: Multi-level approach and iterative reconstruction
Politecnico di Milano（米兰理工大学）；Universita’ di Roma ”La Sapienza”（罗马第一大学）；Purdue University（普渡大学）；University of Minnesota（明尼苏达大学）；University of Vienna（维也纳大学）
Domain derivative and shape reconstruction for an inverse backscattering problem for the wave equation
Institut für Angewandte und Numerische Mathematik, Karlsruher Institut für Technologie（应用与数值数学研究所，卡尔斯鲁厄理工学院）；Department of Mathematics and Statistics, University of Helsinki（赫尔辛基大学数学与统计系）
A direct sampling method for inverse time-dependent electromagnetic source problems: reconstruction of the radiating time and spatial support
School of Mathematical Sciences, Tianjin Normal University（天津师范大学数学科学学院）；Institute of Mathematics and Interdisciplinary Sciences, Tianjin Normal University（天津师范大学数学与交叉科学研究院）
