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The power of random information for numerical approximation and integration

  • This thesis investigates the quality of randomly collected data by employing a framework built on information-based complexity, a field related to the numerical analysis of abstract problems. The quality or power of gathered information is measured by its radius which is the uniform error obtainable by the best possible algorithm using it. The main aim is to present progress towards understanding the power of random information for approximation and integration problems. In the first problem considered, information given by linear functionals is used to recover vectors, in particular from generalized ellipsoids. This is related to the approximation of diagonal operators which are important objects of study in the theory of function spaces. We obtain upper bounds on the radius of random information both in a convex and a quasi-normed setting, which extend and, in some cases, improve existing results. We conjecture and partially establish that the power of random information is subject to a dichotomy determined by the decay of theThis thesis investigates the quality of randomly collected data by employing a framework built on information-based complexity, a field related to the numerical analysis of abstract problems. The quality or power of gathered information is measured by its radius which is the uniform error obtainable by the best possible algorithm using it. The main aim is to present progress towards understanding the power of random information for approximation and integration problems. In the first problem considered, information given by linear functionals is used to recover vectors, in particular from generalized ellipsoids. This is related to the approximation of diagonal operators which are important objects of study in the theory of function spaces. We obtain upper bounds on the radius of random information both in a convex and a quasi-normed setting, which extend and, in some cases, improve existing results. We conjecture and partially establish that the power of random information is subject to a dichotomy determined by the decay of the length of the semiaxes of the generalized ellipsoid. Second, we study multivariate approximation and integration using information given by function values at sampling point sets. We obtain an asymptotic characterization of the radius of information in terms of a geometric measure of equidistribution, the distortion, which is well known in the theory of quantization of measures. This holds for isotropic Sobolev as well as Hölder and Triebel-Lizorkin spaces on bounded convex domains. We obtain that for these spaces, depending on the parameters involved, typical point sets are either asymptotically optimal or worse by a logarithmic factor, again extending and improving existing results. Further, we study isotropic discrepancy which is related to numerical integration using linear algorithms with equal weights. In particular, we analyze the quality of lattice point sets with respect to this criterion and obtain that they are suboptimal compared to uniform random points. This is in contrast to the approximation of Sobolev functions and resolves an open question raised in the context of a possible low discrepancy construction on the two-dimensional sphere.show moreshow less

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Metadaten
Author:Mathias SonnleitnerORCiD
URN:urn:nbn:de:bvb:739-opus4-11305
Advisor:Joscha Prochno, Aicke Hinrichs, Alexander Litvak
Document Type:Doctoral Thesis
Language:English
Year of Completion:2022
Date of Publication (online):2022/09/13
Date of first Publication:2022/09/13
Publishing Institution:Universität Passau
Granting Institution:Universität Passau, Fakultät für Informatik und Mathematik
Date of final exam:2022/05/18
Release Date:2022/09/13
Tag:Monte Carlo integration; cubature; discrepancy; information-based complexity; numerical approximation
GND Keyword:Komplexität / Algorithmus; Numerische IntegrationGND; Monte-Carlo-IntegrationGND; Gewichteter FunktionenraumGND; Scattered-Data-InterpolationGND
Page Number:x, 165 Seiten
Institutes:Fakultät für Informatik und Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
open_access (DINI-Set):open_access
Licence (German):License LogoCreative Commons - CC BY - Namensnennung 4.0 International