### Abstract

Determining parameters which describe the performance of a microbial fuel cell requires the solution of an inverse problem. Two formulations have been presented in the literature: a convolutional approach or a direct quadrature approach. A complete study and analysis of the direct quadrature method, which leads to two systems for the unknown signal given measured complex data, known as the distribution function of relaxation times, is presented. A theoretical analysis justifies the minimal range of integration that is appropriate for the quadrature and suggests that the systems should be combined giving an overdetermined system that is not well posed but not as ill-posed as either system considered separately. All measures of ill-posedness support using the combined data when the level of error in both components of the complex measurements is equivalent. Tikhonov regularization for the filtered singular value and truncated singular value decomposition are used to find solutions of the underlying model system. Given such solutions the application requires the determination of the model parameters that define the signal, among which are the location and peaks of the individual processes of the cell. A nonlinear data fitting approach is presented which consistently estimates these parameters. Simulations support the use of the combined systems for finding the underlying distribution function of relaxation times and the subsequent nonlinear data fitting to these curves. The approach is also illustrated for measured practical data, demonstrating that without the theoretical analysis incorrect conclusions on the underlying physical system would arise. This work justifies the use of Tikhonov regularization combined with nonlinear data fitting for finding reliable solutions for the specific model, when the signal is comprised of a mixture of signals from a small number of processes.

Original language | English (US) |
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Article number | 045006 |

Journal | Inverse Problems |

Volume | 29 |

Issue number | 4 |

DOIs | |

State | Published - Apr 1 2013 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Signal Processing
- Mathematical Physics
- Computer Science Applications
- Applied Mathematics

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## Cite this

*Inverse Problems*,

*29*(4), [045006]. https://doi.org/10.1088/0266-5611/29/4/045006