Abstract
We study the complexity of Banach space valued integration. The input data are assumed to be r-smooth. We consider both definite and indefinite integration and analyse the deterministic and the randomized setting. We develop algorithms, estimate their error, and prove lower bounds. In the randomized setting the optimal convergence rate turns out to be related to the geometry of the underlying Banach space. Then we study the corresponding problems for parameter dependent scalar integration. For this purpose we use the Banach space results and develop a multilevel scheme which connects Banach space and parametric case.
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References
Defant, A., Floret, K.: Tensor Norms and Operator Ideals. North Holland, Amsterdam (1993)
Heinrich, S.: Monte Carlo complexity of global solution of integral equations. J. Complexity 14, 151–175 (1998)
Heinrich, S.: Monte Carlo approximation of weakly singular integral operators. J. Complexity 22, 192–219 (2006)
Heinrich, S.: The randomized information complexity of elliptic PDE. J. Complexity 22, 220–249 (2006)
Heinrich, S., Milla, B.: The randomized complexity of indefinite integration. J. Complexity 27, 352–382 (2011)
Heinrich, S., Sindambiwe, E.: Monte Carlo complexity of parametric integration. J. Complexity 15, 317–341 (1999)
Ledoux, M., Talagrand, M.: Probability in Banach Spaces. Springer, Berlin (1991)
Light, W.A., Cheney, W.: Approximation Theory in Tensor Product Spaces. Lecture Notes in Mathematics 1169. Springer, Berlin (1985)
Maurey, B., Pisier, G.: Series de variables aléatoires vectorielles independantes et propriétés geométriques des espaces de Banach. Studia Mathematica 58, 45–90 (1976)
Novak, E.: Deterministic and Stochastic Error Bounds in Numerical Analysis. Lecture Notes in Mathematics 1349. Springer, Berlin (1988)
Novak, E., Woźniakowski, H.: Tractability of Multivariate Problems, Volume 2, Standard Information for Functionals. European Mathematical Society, Zürich (2010)
Traub, J.F., Wasilkowski, G.W., Woźniakowski, H.: Information-Based Complexity. Academic, New York (1988)
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Daun, T., Heinrich, S. (2013). Complexity of Banach Space Valued and Parametric Integration. In: Dick, J., Kuo, F., Peters, G., Sloan, I. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2012. Springer Proceedings in Mathematics & Statistics, vol 65. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41095-6_12
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DOI: https://doi.org/10.1007/978-3-642-41095-6_12
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