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We ontwikkelen een snel algoritme voor de constructie van goede rang-1-roosterregels voor het benaderen van multivariate integralen. Een gekende methode voor de constructie van deze roosterregels is de component-per-component methode. De zo bekomen roosterregels hebben een optimale orde van convergentie. Hun constructiekost is O(s² n²) voor een regel in s dimensies met n punten. We tonen aan hoe we goede rang-1-roosterregels kunnen construeren in tijd O(s n log(n)) en geheugen O(n) met een nieuw algoritme dat we snelle component-per-component constructie noemen. We tonen dit eerst aan in het geval dat n een priemgetal is en de roosterregel wordt opgesteld voor een gewogen, verschuivingsinvariante en tensorproductvorm Hilbertruimte met reproducerende kern. We tonen dit ook aan voor het geval de gewichten orde-afhankelijk zijn en de ruimte hierdoor geen tensorproductvorm meer heeft. Ook indien n geen priemgetal is, blijkt de snelle constructie mogelijk te zijn. De constructie wordt wel moeilijker indien n uit veel verschillende priemfactoren bestaat. Een interessant geval doet zich voor bij machten van een priemgetal. In dat geval kunnen we op een snelle manier roosterrijen construeren die punt per punt gebruikt kunnen worden. Twee natuurlijke uitbreidingen van het component-per-component algoritme zijn de constructie van veeltermroosterregels en de constructie van stempelregels. We tonen aan dat ook in deze gevallen een snelle constructie mogelijk is. De kwaliteit van de roosterregels wordt gedemonstreerd aan de hand van enkele praktijkvoorbeelden. We develop a fast algorithm for the construction of good rank-1 lattice rules which are a quasi-Monte Carlo method for the approximation of multivariate integrals. A popular method to construct such rules is the component-by-component algorithm which is able to construct good lattice rules that achieve the optimal theoretical rate of convergence. The construction time of this algorithm is O(s² n²), or O(s n²) when using O(n) memory, for an s-dimensional lattice rule with n points. We show how to construct good lattice rules in time O(s n log(n)), using O(n) memory, by means of a new algorithm, called the fast component-by-component algorithm. First this is shown for the base case when n is a prime number and the underlying function space is a weighted, shift-invariant and tensor-product reproducing kernel Hilbert space. Then we show that, by a minor increase in construction cost, also more generally weighted function spaces can be handled by the fast algorithm. In particular we show this for order-dependent weights. When n is not a prime number it turns out that fast construction is also possible, although the construction is more involved for numbers n which have a large number of unique prime factors. An additional advantage is obtained when choosing n to be a prime power, since then the rules are embedded for increasing powers of the prime. Using this embedding, we propose a new fast algorithm to construct lattice sequences which can be used point by point. Two natural extensions of the algorithm are the construction of polynomial lattice rules and so called copy rules. We show that also here the fast component-by-component algorithm can be applied. The quality of the constructed point sets is finally demonstrated on some finance and statistics examples. Goede puntenrijen in duizenden dimensies Meer en meer wil men hoogdimensionale integralen kunnen uitrekenen. In bijvoorbeeld de financiële wiskunde en de fysica komt men gemakkelijk integralen tegen met honderden of duizenden dimensies. Voor het benaderen van zulke integralen maakt men vaak gebruik van de Monte Carlo methode die hiervoor een gemiddelde neemt van een groot aantal functiewaarden. Deze methode werkt echter nogal traag doordat ze veel functie-evaluaties nodig heeft. Een alternatief is de quasi-Monte Carlo methode waarbij men de functie in welgekozen punten evalueert. Hierdoor bekomt men veel sneller een goed resultaat met veel minder functie-evaluaties. Het zoeken van zulke goede puntenverzamelingen vergt heel wat rekentijd. Deze thesis stelt een nieuw algoritme voor om dit zoekproces danig te versnellen. Zo kunnen we met het nieuwe algoritme bijvoorbeeld 100 miljoen punten construeren in 20 dimensies op 10 minuten tijd. Voordien zou dit meer dan 100 jaar gekost hebben. Op basis van dit nieuwe algoritme kunnen we op een snelle manier puntenrijen construeren voor het benaderen van integralen met duizenden dimensies.

519.64 <043> --- Academic collection --- Numerical methods for solution of integral equations. Quadrature formulae--Dissertaties --- Theses

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This book contains a wealth of inequalities used in linear analysis, and explains in detail how they are used. The book begins with Cauchy's inequality and ends with Grothendieck's inequality, in between one finds the Loomis-Whitney inequality, maximal inequalities, inequalities of Hardy and of Hilbert, hypercontractive and logarithmic Sobolev inequalities, Beckner's inequality, and many, many more. The inequalities are used to obtain properties of function spaces, linear operators between them, and of special classes of operators such as absolutely summing operators. This textbook complements and fills out standard treatments, providing many diverse applications: for example, the Lebesgue decomposition theorem and the Lebesgue density theorem, the Hilbert transform and other singular integral operators, the martingale convergence theorem, eigenvalue distributions, Lidskii's trace formula, Mercer's theorem and Littlewood's 4/3 theorem. It will broaden the knowledge of postgraduate and research students, and should also appeal to their teachers, and all who work in linear analysis.

Functional analysis. --- Inequalities (Mathematics). --- Inequalities (Mathematics) --- Processes, Infinite --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations

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At roughly 100 years of age operator theory remains a vibrant and exciting subject area with wide ranging applications. Many of the papers found here expand on lectures given at the 15th International Workshop on Operator Theory and Its Applications, held at the University of Newcastle upon Tyne from the 12th to the 16th of July 2004. Contributions cover the main themes of the workshop: invariant subspaces, Krein space operator theory and its applications, multivariate operator theory and operator model theory, operator theory and function theory, systems theory including inverse scattering, structured matrices, and spectral theory of non-selfadjoint operators, including pseudodifferential and singular integral operators. The book will appeal to both researchers and graduate students in mathematics, physics and engineering.

Operator theory --- Functional analysis --- Operator theory. --- Functional analysis. --- Operator Theory. --- Functional Analysis. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations

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Concentration compactness is an important method in mathematical analysis which has been widely used in mathematical research for two decades. This unique volume fulfills the need for a source book that usefully combines a concise formulation of the method, a range of important applications to variational problems, and background material concerning manifolds, non-compact transformation groups and functional spaces. Highlighting the role in functional analysis of invariance and, in particular, of non-compact transformation groups, the book uses the same building blocks, such as partitions of d

Functional analysis. --- Mathematical analysis. --- 517.1 Mathematical analysis --- Mathematical analysis --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Analyse fonctionnelle

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Inverse problems arise whenever one tries to calculate a required quantity from given measurements of a second quantity that is associated to the first one. Besides medical imaging and non-destructive testing, inverse problems also play an increasing role in other disciplines such as industrial and financial mathematics. Hence, there is a need for stable and efficient solvers. The book is concerned with the method of approximate inverse which is a regularization technique for stably solving inverse problems in various settings such as L2-spaces, Hilbert spaces or spaces of distributions. The performance and functionality of the method is demonstrated on several examples from medical imaging and non-destructive testing such as computerized tomography, Doppler tomography, SONAR, X-ray diffractometry and thermoacoustic computerized tomography. The book addresses graduate students and researchers interested in the numerical analysis of inverse problems and regularization techniques or in efficient solvers for the applications mentioned above.

Inverse problems (Differential equations) --- Tomography --- Mathematical Theory --- Algebra --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Tomography. --- Body section radiography --- Computed tomography --- Computerized tomography --- CT (Computer tomography) --- Laminagraphy --- Laminography --- Radiological stratigraphy --- Stratigraphy, Radiological --- Tomographic imaging --- Zonography --- Mathematics. --- Matrix theory. --- Algebra. --- Integral equations. --- Partial differential equations. --- Numerical analysis. --- Linear and Multilinear Algebras, Matrix Theory. --- Partial Differential Equations. --- Integral Equations. --- Numerical Analysis. --- Mathematical analysis --- Partial differential equations --- Equations, Integral --- Functional equations --- Functional analysis --- Math --- Science --- Cross-sectional imaging --- Radiography, Medical --- Geometric tomography --- Differential equations --- Differential equations, partial. --- Computer tomography --- CT (Computed tomography)