On weakly nonlinear inverse problems

In this paper the class of weakly nonlinear inverse problems is introduced. These problems are characterized by the property that the second derivative of the parameter-to-observation mapping can be bounded by the square of the first derivative of that mapping. Using geometric techniques it is shown that weakly nonlinear inverse problems behave similarly to linear inverse problems. In particular, their Tikhonov regularization leads to a family of quadratically well-posed problems. Examples involving the determination of source terms in semilinear reaction diffusion equations are given.