Smoothing by cubic spline modified applied to solve inverse thermal problem
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Smoothing by cubic spline modified applied to solve inverse thermal problem Leticia Hiromi Kubo1 · Juliana de Oliveira1
Received: 22 September 2016 / Accepted: 29 September 2016 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2016
Abstract This paper presents an alternative to cubic spline regularization and its weighted form applied in solving inverse thermal problems. The inverse heat transfer problems are classified as ill-posed, that is, the solution may become unstable, mainly because they are sensitive to random errors deriving from the input data, necessitating a regularization method to soften these effects. The smoothing technique proposed by cubic spline regularization ensures that the global data tend to be more stable, with fewer data oscillations and dependent on a single arbitrary parameter input. It also shows that the weighted cubic spline is able to enhance filter action. The methods have been implemented in order for the search engine to optimize the choice of parameters and weight and, thus, the smoothing gains more flexibility and accuracy. The simulated and experimental tests confirm that the techniques are effective in reducing the amplified noise by inverse thermal problem presented. Keywords Cubic spline · Regularization · Ponderation · Inverse problem Mathematics Subject Classification 65F22 · 65D07 · 65D10 · 80A23
1 Introduction Recently, the application and theory of the Inverse Heat Transfer Problems (IHTP) have grown increasingly to be found in almost all branches of science and engineering. Chemical, mechanical, nuclear and aerospace engineers, statisticians, physicists, mathematicians, among others,
Communicated by Cristina Turner.
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Juliana de Oliveira [email protected] Leticia Hiromi Kubo [email protected]
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Department of Biological Sciences, Faculty of Sciences and Letters of Assis FCLA, University of São Paulo State, UNESP, Av. Dom Antonio 2100, Assis, SP 19806-900, Brazil
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L. H. Kubo, J. de Oliveira
are focused on the subject, even with their different needs and applications (Özisik and Orlande 2000; Kim et al. 2006; Kabala 1997; Park et al. 1999; Kaipio and Fox 2011). The IHTP are, however, classified as ill-posed. While the notion of a well-posed or “well-designed” mathematical problem came in 1923 with discussions on the Hadamard’s work (Hadamard 1923) in which the determining conditions are generally the existence properties, uniqueness and (by implication) stability of solutions (Beck et al. 1985), which the inverse problems do not. This occurs because the mathematical formulation of physical processes in thermal sciences has an unknown amount of constants, such as temperature measurements, heat flow, radiation intensity, among others (Orlande 2011). Difficulties associated with IHTP occur by bringing solutions that can become unstable as the results originated from measurements with inherent errors in analyses (Özisik and Orlande 2000), mainly because they have input data sensitive to random errors (Orlande 2011), as inserted
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