Abstract
In this work, we consider the problem of identifying the time independent source for full parabolic equations in Rn from noisy data. This is an ill-posed problem in the sense of Hadamard. To compensate the factor that causes the instability, a family of parametric regularization operators is introduced, where the rule to select the value of the regularization parameter is included. This rule, known as regularization parameter choice rule, depends on the data noise level and the degree of smoothness that it is assumed for the source. The proof for the stability and convergence of the regularization criteria is presented and a Hölder type bound is obtained for the estimation error. Numerical examples are included to illustrate the effectiveness of this regularization approach.
Highlights
The problem of source identification has been studied and analyzed in different areas of applied mathematics for the last decades
This work focus on the problem of the inverse source for full parabolic equations in Rn
A family of regularization operators is defined in order to deal with the ill-posedness the problem
Summary
The problem of source identification has been studied and analyzed in different areas of applied mathematics for the last decades. This work aims to the determination, from noisy measurements taken at an arbitrary fixed time, of the real-valued function of n real variables, independent of time, in an evolutionary equation of transport in an unbounded domain This is an ill-posed problem because the high frequency components of arbitrarily small data errors can lead to arbitrarily large errors in the solution [9, 19]. The regularization operator family proposed here turns out to be an n-dimensional generalization of the modified regularization method considered in [23, 30, 31, 46, 50] In these articles the authors estimate the source of the one-dimensional equation of heat from data measured in a fixed moment of time (t = 1) by adding a penalizing term and the parameter choice rule depends on the norm of the unknown function. In order to illustrate the regularization performance, some numerical examples for the 1D, 2D and 3D cases are included
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