Greedy Gauss-Newton algorithm for finding sparse solutions to nonlinear   underdetermined systems of equations
  We consider the problem of finding sparse solutions to a system of underdetermined nonlinear system of equations. The methods are based on a Gauss-Newton approach with line search where the search direction is found by solving a linearized problem using only a subset of the columns in the Jacobian. The choice of columns in the Jacobian is made through a greedy approach looking at either maximum descent or an approach corresponding to orthogonal matching for linear problems. The methods are shown to be convergent and efficient and outperform the $\ell_1$ approach on the test problems presented. 
M{\aa}rten Gulliksson and Anna Oleynik
math.NA