Novel Multi-level Projected Iteration to Solve Inverse Problems with Nearly Optimal Accuracy

• Published in: Journal of optimization theory and applications Vol. 194; no. 2; pp. 643 - 680
• Main Authors: Mittal, Gaurav, Giri, Ankik Kumar
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.08.2022; Springer Nature B.V
• Subjects: Accuracy; Algorithms; Applications of Mathematics; Approximation; Banach spaces; Calculus of Variations and Optimal Control; Optimization; Convex analysis; Engineering; Estimates; Ill posed problems; Inverse problems; Iterative methods; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Regularization methods; Stability; Stability constants; Theory of Computation; Inverse problems; Nonlinear ill-posed operator equations; 35R30; 65J22; Multi-level approach; 47J25; Regularization; Iterative regularization methods
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-022-02044-9

In this paper, we introduce a novel nonlinear projected iterative method to solve the ill-posed inverse problems in Banach spaces. This method is motivated by the well-known iteratively regularized Landweber iteration method. We analyze the convergence of our novel method by assuming the conditional stability of the inverse problem on a convex and compact set. Further, we consider a nested family of convex and compact sets on which stability holds, and based on this family, we develop a multi-level algorithm with nearly optimal accuracy. To enhance the accuracy between neighboring levels, we couple the increase in accuracy with the growth of stability constants. This ensures that the algorithm terminates within a finite number of iterations after achieving a certain discrepancy criterion. Moreover, we discuss example of an ill-posed problem on which our both the methods are applicable and deduce various constants appearing in our work.