An efficient projection method for nonlinear inverse problems with sparsity constraints

In this paper, we propose a modification of the accelerated projective steepest descent method for solving nonlinear inverse problems with an l₁ constraint on the variable, which was recently proposed by Teschke and Borries (2010 Inverse Problems 26 025007). In their method, there are some parameters need to be estimated, which is a difficult task for many applications. We overcome this difficulty by introducing a self-adaptive strategy in choosing the parameters. Theoretically, the convergence of their algorithm was guaranteed under the assumption that the underlying mapping F is twice Fréchet differentiable together with some other conditions, while we can prove weak and strong convergence of the proposed algorithm under the condition that F is Fréchet differentiable, which is a relatively weak condition. We also report some preliminary computational results and compare our algorithm with that of Teschke and Borries, which indicate that our method is efficient.