An Accelerated Landweber Iteration Based on Barzilai–Borwein Step‐Size for Solving Nonlinear Inverse Problems

ABSTRACT Iterative regularization methods are vital for solving mathematical physics inverse problems. Landweber iteration is one of the most well‐known iterative regularization methods, whose most attractive property is its stability with respect to noise. However, its convergence speed is usually very slow, which greatly limits its widespread applications in practical problems. In this study, we propose an accelerated Landweber iteration based on Barzilai–Borwein (BB) step‐size for solving nonlinear inverse problems. The convergence analysis of our proposed method is presented under standard assumptions. For validating the effectiveness of our method from the numerical computation point of view, the auto‐convolution inverse problem based on integral equation and 1D, 2D parameter identification inverse problems based on elliptic equation are illustrated. Numerical results show that BB step‐size can greatly accelerate Landweber iteration.