Regularization of an Inverse Problem by Controlling the Stiffness of the Graphs of Approximate Solutions ⋆ ⋆⋆ Micha l Cia lkowski 1 , Nikolai Botkin 2[0000-0003-2724-4817] , Jan Ko lodziej 3 , Andrzej Fra¸ ckowiak 1 , and Wiktor Hoffmann 1[0000-0001-7446-1369] 1 Institute of Thermal Engineering, Poznan University of Technology, Poznan, Poland {Michal.Cialkowski,andrzej.frackowiak}@put.poznan.pl, j.hoffmann@doctorate.put.poznan.pl 2 Mathematical Faculty, Technical University of Munich, Garching bei M¨ unchen, Germany botkin@ma.tum.de 3 Institute of Applied Mechanics, Poznan University of Technology, Poznan, Poland jan.kolodziej@put.poznan.pl Abstract. In this paper, an one-dimensional heat conductivity equa- tion is considered. Such an equation describes, for example, a wall which exhibits temperature changes across the thickness, whereas the temper- ature remains constant along the in-plane directions. The problem of recovering the unknown temperature at the left end point of the domain is studied. It is assumed that the temperature and the heat flux are mea- sured at the right end point of the domain. Using the Laplace transform, the problem is reduced to an integral equation defining the unknown temperature at the left end point as function of time. An approximation of the integral equation yields a linear system defining the values of the unknown function. Additionally, the graph of the unknown function is considered as a sequence of segments or overlapping quadratic or cubic parabolas, and the condition of common tangents at common points of neighboring parabolas is imposed. The resulting overdetermined system is solved using the least square method whose fitting function consists of two parts: a residual responsible for satisfying the integral equation and a term responsible for the matching of the segments/parabolas. The last term is multiplied by a regularization parameter that defines the stiffness of the solution graph. Appropriate values of the regularization parameter are being chosen as local minimizers of a discrepancy. Numerical experi- ments show that one of such values provides the best choice. Numerical simulations exhibit a very exact reconstruction of solutions even in the case of large measurement errors (up to 10%). Keywords: Heat transfer · Inverse problems · Cauchy problems · Laplace transform · Regularization. ⋆ The paper was carried out in the framework of the grant N0. 4917/B/T02/2010/39 funded by the Ministry of Highschool Education (Poland). ⋆⋆ Copyright c 2020 for this paper by its author. Use permitted under Creative Com- mons License Attribution 4.0 International (CC BY 4.0).