عنوان مقاله [English]
The ill-posed problems could be seen anywhere in our daily lives. An ill-posed problem is a problem that there are no uniqueness solutions (there is no solution or two or more solutions for the problem) or the solutions are unstable; i.e., an arbitrarily small error in the observation may lead to extremely large errors in the solutions. The main difficulty in solving ill-posed problems is instability of their solutions with respect to small variations of input data. A regularized estimation of an ill-posed problem is always biased; thus, it's worth obtaining the solution from different methods for reliable evaluation of the uncertainty in our
estimation. The regularization methods, such as Tikhonov's method, are used to obtain stable solutions for solving ill-posed problems. In Tikhonov's regularization method, a scalar
quantity is used as the stabilization parameter to solve ill-posed problems; whereas, In the Optimally Scaled Vector Regularization Method (OSVRM), a vector is used as the
stabilization parameter. In this paper, a comparison has been made between the results of Tikhonov, TSVD, and OSVRM methods in terms of accuracy of the results for estimation of the earth gravity field from GOCE satellite data. The RMSE of the results of Tikhonov, EGM96 model, and the OSVRM method – that use a vector instead of scalar as
regularization parameter – in the order of 10-5, 10-9, and 10-13, respectively. It is seen that the results obtained from the OSVRM method are much better compared to the Tikhonov method and EGM96 model for solving linear ill-posed problems. On the other hand, a
significant improvement has been achieved in the stability and accuracy of numerical results for linear problems solution.
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