The authors study Tikhonov regularization of linear ill-posed problems with a general convex penalty defined on a Banach space. It is well known that the error analysis requires smoothness assumptions. Here such assum...
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The authors study Tikhonov regularization of linear ill-posed problems with a general convex penalty defined on a Banach space. It is well known that the error analysis requires smoothness assumptions. Here such assumptions are given in form of inequalities involving only the family of noise-free minimizers along the regularization parameter and the (unknown) penalty-minimizing solution. These inequalities control, respectively, the defect of the penalty, or likewise, the defect of the whole Tikhonov functional. The main results provide error bounds for a Bregman distance, which split into two summands: the first smoothness-dependent term does not depend on the noise level, whereas the second term includes the noise level. This resembles the situation of standard quadratic Tikhonov regularization in Hilbert spaces. It is shown that variational inequalities, as these were studied recently, imply the validity of the assumptions made here. Several examples highlight the results in specific applications.
We deal with finite dimensional differentiable manifolds. All items are concerned with are differentiable as well. The class of differentiability is C-infinity. A metric structure in a vector bundle E is a constant ra...
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ISBN:
(纸本)9783319684451;9783319684444
We deal with finite dimensional differentiable manifolds. All items are concerned with are differentiable as well. The class of differentiability is C-infinity. A metric structure in a vector bundle E is a constant rank symmetric bilinear vector bundle homomorphism of E x E in the trivial bundle line bundle. We address the question whether a given gauge structure in E is metric. That is the main concerns. We use generalized Amari functors of the information geometry for introducing two index functions defined in the moduli space of gauge structures in E. Beside we introduce a differential equation whose analysis allows to link the new index functions just mentioned with the main concerns. We sketch applications in the differential geometry theory of statistics.
The widespread use of composite indices has often been motivated by their practicality to quantify qualitative data in an easy and intuitive way. At the same time, this approach has been challenged due to the subjecti...
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Data on (economic) institutions are often available only as observations on ordinal, inherently incomparable properties, which are then typically aggregated to a composite index in the empirical social science literat...
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Data on (economic) institutions are often available only as observations on ordinal, inherently incomparable properties, which are then typically aggregated to a composite index in the empirical social science literature. From a methodological perspective, the present paper advocates the application of partially ordered set (POSET) theory as an alternative approach. Its main virtue is that it takes the ordinal nature of the data seriously and dispenses with the unavoidably subjective assignment of weights to incomparable properties, maintains a high standard of objectivity, and can be applied in various fields of economics. As an application, the POSET approach is then used to calculate new indices on the stringency of fiscal rules for 81 countries over the period 1985-2012 based on recent data by IMF (2012). The derived measures of fiscal rules are used to test their significance for public finances in a fiscal reaction function and compare the POSET with the composite index approach. (C) 2014 Elsevier Inc. All rights reserved.
The article is concerned with ill-posed operator equations Ax = y where A:X →Y is an injective bounded linear operator with non-closed range (Formula presented.) and X and Y are Hilbert spaces. The solution x=x † is ...
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