## Abstract

In this work we study a Lipschitz stability result in the reconstruction of a compactly supported initial temperature for the heat equation in R^{n}, from measurements along a positive time interval and over an open set containing its support. We employ a nonconstructive method which ensures the existence of the stability constant, but it is not explicit in terms of the parameters of the problem. The main ingredients in our method are the compactness of support of the initial condition and the explicit dependency of solutions to the heat equation with respect to it. By means of Carleman estimates we obtain an analogous result for the case when the observation is made along an exterior region ω × (τ, T), such that the unobserved part R^{n}\ω is bounded. In the latter setting, the method of Carleman estimates gives a general conditional logarithmic stability result when initial temperatures belong to a certain admissible set, without the assumption of compactness of support and allowing an explicit stability constant. Furthermore, we apply these results to deduce similar stability inequalities for the heat equation in R and with measurements available on a curve contained in R ×[0, ∞), leading to the derivation of stability estimates for an inverse problem arising in 2D fluorescence microscopy. In order to further understand this Lipschitz stability, in particular, the magnitude of its stability constant with respect to the parameters of the problem, a numerical reconstruction is presented based on the construction of a linear system for the inverse problem in fluorescence microscopy. We investigate the stability constant by analyzing the condition number of the corresponding matrix.

Original language | English |
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Pages (from-to) | 5948-5978 |

Number of pages | 31 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 53 |

Issue number | 5 |

Early online date | 21 Oct 2021 |

DOIs | |

Publication status | Published - 31 Dec 2021 |

Externally published | Yes |

### Bibliographical note

Funding Information:∗Received by the editors October 27, 2020; accepted for publication (in revised form) July 22, 2021; published electronically October 21, 2021. https://doi.org/10.1137/20M1374183 Funding: The work of the first, second, and fourth authors was partially supported by the Anid-FONDECYT grant 1191903. The work of the first and fourth authors was partially supported by the Basal Program CMM-AFB 170001. The work of the fourth author was also partially supported by the Anid-FONDECYT grant 1201311, FONDAP 15110009, CYAN-CLI2020008, Math-Amsud MATH190008, and ANID Millennium Science Initiative Program NCN17 129. The work of the second, third, and fourth authors was partially funded by Anid–Millennium Science Initiative Program–NCN19 161.

## Keywords

- backward heat equation
- fluorescence microscopy
- inverse problem
- Lipschitz stability

## ASJC Scopus subject areas

- Analysis
- Computational Mathematics
- Applied Mathematics