Ali Reza Azmoude Ardalan; Abdorreza Safari; Yahya Tavakkoli
Abstract
The methods applied to regularization of the ill-posed problems can be classified under “direct” and “indirect” methods. Practice has shown that the effects of different regularization techniques on an ill-posed problem are not the same, and as such each ill-posed problem requires its own investigation ...
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The methods applied to regularization of the ill-posed problems can be classified under “direct” and “indirect” methods. Practice has shown that the effects of different regularization techniques on an ill-posed problem are not the same, and as such each ill-posed problem requires its own investigation in order to identify its most suitable regularization method. In the geoid computations without applying Stokes formula, the downward continuation based on Abel-Poisson integral is an inverse problem, which requires regularization. Since so far the regularization of this ill-posed problem has been thoroughly studied, in this paper the regularization of the downward continuation problem based on Abel-Poisson integral, is investigated and various techniques falling into the aforementioned classes of regularizations are applied and their efficiency is compared. From the first class Truncated Singular Value Decomposition (TSVD) and Truncated Generalized Singular Value Decomposition (TGSVD) methods and from the second class Generalized Tikhonov (GT) with the norms and semi-norms in Sobolev subspaces , are applied and their capabilities for the regularization of the problem is compared. Our numerical results derived from simulated studies reveal that the GT method with discretized norm of Sobolev subspace gives the best results among the studied methods for the regularization of the downward continuation problem based on the Abel-Poisson integral. On the contrary, the TGSVD method with the discretized second order derivatives has less consistency with the ill-posed problem and yields less accuracy. Finally, the GT method with discretized norm of Sobolev subspace is applied to the downward continuation of real gravity data of the type modulus of gravity acceleration within the geographical region of Iran to derive a geoid model for this region.
Fereydoun Sabet Ghadam; Vahid Esfahanian; Mohammad Eftekhari Yazdi
Abstract
Capability of the Proper Orthogonal Decomposition (POD) method in extraction of the coherent structures from a spatio-temporal chaotic field is assessed in this paper. As the chaotic field, an ensemble of 40 snapshots, obtained from Direct Numerical Simulation (DNS) of the Kuramoto-Sivashinsky (KS) equation, ...
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Capability of the Proper Orthogonal Decomposition (POD) method in extraction of the coherent structures from a spatio-temporal chaotic field is assessed in this paper. As the chaotic field, an ensemble of 40 snapshots, obtained from Direct Numerical Simulation (DNS) of the Kuramoto-Sivashinsky (KS) equation, has been used. Contrary to the usual methods, where the ergodicity of the field is needed, the ensemble members are generated directly by disturbing the initial conditions in a random fashion. The POD eigenvalues and eigenmodes are extracted, using the POD/SVD technique and an ensemble averaging process is performed on the resulting modes, which are used in reconstruction of the field. The resulted mean field is compared with the Fourier mean field as well as the original (instantaneous) field. The results strictly have shown presence of non-coherent parts in the POD reconstructed field, which can be interpreted as the POD weakness in the filtering of the random part of the field. On the other hand, the ensemble averaged POD modes had obvious superiority, in comparison with the mean Fourier modes, in tracing of the mean behavior of the field and the mean temporal gradients. As a consequence, use of the ensemble averaging in the POD modes can be suggested, at least in some spatio-temporal the fields with dominant coherent structures.
