Geodesic
A numerical method for solving inverse thermoacoustic problem
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 16 (2013) no. 1, pp. 39-44

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In this paper, we consider the inverse problem of determining the initial condition of the initial boundary value problem for the wave equation with additional information about solving the direct initial boundary value problem that is measured at the boundary of the domain. The main objective of our research is to construct a numerical algorithm for solving the inverse problem based on the method of simple iteration (MSI) and to study the resolution of the inverse problem and its dependence on the number and location of measurement points. We consider three two-dimensional inverse problems. The results of numerical calculations are presented. We show that the MSI for each iteration step reduces the value of the object functional. However, due to the ill-posedness of an inverse problem the difference between the exact and the approximate solutions of the inverse problem decreases up to some fixed number kmin and then monotonically increases. This reflects the regularizing properties of the MSI, in which the iteration number is a regularization parameter. 
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     author = {S. I. Kabanikhin and O. I. Krivorotko and M. A. Shishlenin},
     title = {A numerical method for solving inverse thermoacoustic problem},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
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     number = {1},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2013_16_1_a3/}
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S. I. Kabanikhin; O. I. Krivorotko; M. A. Shishlenin. A numerical method for solving inverse thermoacoustic problem. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 16 (2013) no. 1, pp. 39-44. http://geodesic.mathdoc.fr/item/SJVM_2013_16_1_a3/