### 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 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.

Original language | English (US) |
---|---|

Pages (from-to) | 52-74 |

Number of pages | 23 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 278 |

DOIs | |

State | Published - Apr 15 2015 |

### Fingerprint

### Keywords

- Ill-posed
- Inverse problem
- Non-negative least squares
- Regularization
- Residual periodogram

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics

### Cite this

*Journal of Computational and Applied Mathematics*,

*278*, 52-74. https://doi.org/10.1016/j.cam.2014.09.017

**Non-negatively constrained least squares and parameter choice by the residual periodogram for the inversion of electrochemical impedance spectroscopy data.** / Hansen, Jakob K.; Hogue, Jarom D.; Sander, Grant K.; Renaut, Rosemary; Popat, Sudeep C.

Research output: Contribution to journal › Article

*Journal of Computational and Applied Mathematics*, vol. 278, pp. 52-74. https://doi.org/10.1016/j.cam.2014.09.017

}

TY - JOUR

T1 - Non-negatively constrained least squares and parameter choice by the residual periodogram for the inversion of electrochemical impedance spectroscopy data

AU - Hansen, Jakob K.

AU - Hogue, Jarom D.

AU - Sander, Grant K.

AU - Renaut, Rosemary

AU - Popat, Sudeep C.

PY - 2015/4/15

Y1 - 2015/4/15

N2 - 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.

AB - 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.

KW - Ill-posed

KW - Inverse problem

KW - Non-negative least squares

KW - Regularization

KW - Residual periodogram

UR - http://www.scopus.com/inward/record.url?scp=84908405885&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84908405885&partnerID=8YFLogxK

U2 - 10.1016/j.cam.2014.09.017

DO - 10.1016/j.cam.2014.09.017

M3 - Article

AN - SCOPUS:84908405885

VL - 278

SP - 52

EP - 74

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

ER -