Mean anisotropy of homogeneous Gaussian random fields and anisotropic norms of linear translation-invariant operators on multidimensional integer lattices

  • Sensitivity of output of a linear operator to its input can be quantified in various ways. In Control Theory, the input is usually interpreted as disturbance and the output is to be minimized in some sense. In stochastic worst-case design settings, the disturbance is considered random with imprecisely known probability distribution. The prior set of probability measures can be chosen so as to quantify how far the disturbance deviates from the white-noise hypothesis of Linear Quadratic Gaussian control. Such deviation can be measured by the minimal Kullback-Leibler informational divergence from the Gaussian distributions with zero mean and scalar covariance matrices. The resulting anisotropy functional is defined for finite power random vectors. Originally, anisotropy was introduced for directionally generic random vectors as the relative entropy of the normalized vector with respect to the uniform distribution on the unit sphere. The associated a-anisotropic norm of a matrix is then its maximum root mean square or average energy gain with respect to finite power or directionally generic inputs whose anisotropy is bounded above by a >= 0. We give a systematic comparison of the anisotropy functionals and the associated norms. These are considered for unboundedly growing fragments of homogeneous Gaussian random fields on multidimensional integer lattice to yield mean anisotropy. Correspondingly, the anisotropic norms of finite matrices are extended to bounded linear translation invariant operators over such fields.

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Metadaten
Author:Phil Diamond, Peter E. Kloeden, Igor Vladimirov
URN:urn:nbn:de:hebis:30:3-236446
DOI:https://doi.org/10.1155/S1048953303000169
Parent Title (English):Journal of Applied Mathematics and Stochastic Analysis
Publisher:Hindawi Publ.
Place of publication:New York, NY
Document Type:Article
Language:English
Date of Publication (online):2012/06/08
Year of first Publication:2003
Publishing Institution:Universitätsbibliothek Johann Christian Senckenberg
Release Date:2012/06/08
Tag:Anisotropic Norm; Gaussian Random Field; Kullback-Leibler Informational Divergence; Mean Anisotropy
Volume:16
Issue:3
Page Number:23
First Page:209
Last Page:231
HeBIS-PPN:302907238
Institutes:Informatik und Mathematik / Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Sammlungen:Universitätspublikationen
Licence (German):License LogoCreative Commons - Namensnennung 3.0