  The inverse problem associated with electrochemical impedance spectroscopy
requiring the solution of a Fredholm integral equation of the first kind is
considered. If the underlying physical model is not clearly determined, the
inverse problem needs to be solved using a regularized linear least squares
problem that is obtained from the discretization of the integral equation. For
this system, it is shown that the model error can be made negligible by a
change of variables and by extending the effective range of quadrature. This
change of variables serves as a right preconditioner that significantly
improves the condition of the system. Still, to obtain feasible solutions the
additional constraint of non-negativity is required. Simulations with
artificial, but realistic, data demonstrate that the use of non-negatively
constrained least squares with a smoothing norm provides higher quality
solutions than those obtained without the non-negative constraint. Using
higher-order smoothing norms also reduces the error in the solutions. The
L-curve and residual periodogram parameter choice criteria, which are used for
parameter choice with regularized linear least squares, are successfully
adapted to be used for the non-negatively constrained Tikhonov least squares
problem. Although these results have been verified within the context of the
analysis of electrochemical impedance spectroscopy, there is no reason to
suppose that they would not be relevant within the broader framework of solving
Fredholm integral equations for other applications.
