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The inverse problem associated with electrochemical impedance spectroscopyrequiring the solution of a Fredholm integral equation of the first kind isconsidered. If the underlying physical model is not clearly determined, theinverse problem needs to be solved using a regularized linear least squaresproblem that is obtained from the discretization of the integral equation. Forthis system, it is shown that the model error can be made negligible by achange of variables and by extending the effective range of quadrature. Thischange of variables serves as a right preconditioner that significantlyimproves the condition of the system. Still, to obtain feasible solutions theadditional constraint of non-negativity is required. Simulations withartificial, but realistic, data demonstrate that the use of non-negativelyconstrained least squares with a smoothing norm provides higher qualitysolutions than those obtained without the non-negative constraint. Usinghigher-order smoothing norms also reduces the error in the solutions. TheL-curve and residual periodogram parameter choice criteria, which are used forparameter choice with regularized linear least squares, are successfullyadapted to be used for the non-negatively constrained Tikhonov least squaresproblem. Although these results have been verified within the context of theanalysis of electrochemical impedance spectroscopy, there is no reason tosuppose that they would not be relevant within the broader framework of solvingFredholm integral equations for other applications.

journal of computational and applied mathematics

Non-negatively constrained least squares and parameter choice by the  residual periodogram for the inversion of electrochemical impedance  spectroscopy