Nonnegatively constrained least squares and parameter choice by the residual periodogram for the inversion of electrochemical impedance spectroscopy
Abstract
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 nonnegativity is required. Simulations with artificial, but realistic, data demonstrate that the use of nonnegatively constrained least squares with a smoothing norm provides higher quality solutions than those obtained without the nonnegative constraint. Using higherorder smoothing norms also reduces the error in the solutions. The Lcurve 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 nonnegatively 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2013
 DOI:
 10.48550/arXiv.1309.4498
 arXiv:
 arXiv:1309.4498
 Bibcode:
 2013arXiv1309.4498H
 Keywords:

 Mathematics  Numerical Analysis;
 65F10;
 45B05;
 65R32
 EPrint:
 Supplementary materials contain extra figures and results