Journal
INVERSE PROBLEMS IN SCIENCE AND ENGINEERING
Volume 25, Issue 8, Pages 1138-1167Publisher
TAYLOR & FRANCIS LTD
DOI: 10.1080/17415977.2016.1229778
Keywords
Parabolic equations; determining the initial condition; integral observations; ill-posed problems; singular values; regularization; conjugate gradient method; splitting method
Funding
- Vietnam National Foundation for Science and Technology Development (NAFOSTED) [101.02-2014.54]
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We study the problem of determining the initial condition in parabolic equations with time-dependent coefficients from integral observations which can be regarded as generalizations of point-wise interior observations. Our approach is new in the sense that for determining the initial condition we do not assume that the data available in the whole space domain at the final moment or in a subset of the space domain during a certain time interval, but some integral observations during a time interval. We propose a variational method in combination with Tikhonov regularization for solving the problem and then discretize it by finite difference splitting methods. The discretized minimization problem is solved by the conjugate gradient method and tested on computer to show its efficiency. Also as a by-product of the variational method, we propose a numerical scheme for estimating the degree of ill-posedness of the problem.
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