64 research outputs found
On Weak Tractability of the Clenshaw-Curtis Smolyak Algorithm
We consider the problem of integration of d-variate analytic functions
defined on the unit cube with directional derivatives of all orders bounded by
1. We prove that the Clenshaw Curtis Smolyak algorithm leads to weak
tractability of the problem. This seems to be the first positive tractability
result for the Smolyak algorithm for a normalized and unweighted problem. The
space of integrands is not a tensor product space and therefore we have to
develop a different proof technique. We use the polynomial exactness of the
algorithm as well as an explicit bound on the operator norm of the algorithm.Comment: 18 page
Some Results on the Complexity of Numerical Integration
This is a survey (21 pages, 124 references) written for the MCQMC 2014
conference in Leuven, April 2014. We start with the seminal paper of Bakhvalov
(1959) and end with new results on the curse of dimension and on the complexity
of oscillatory integrals. Some small errors of earlier versions are corrected
Smolyak's algorithm: A powerful black box for the acceleration of scientific computations
We provide a general discussion of Smolyak's algorithm for the acceleration
of scientific computations. The algorithm first appeared in Smolyak's work on
multidimensional integration and interpolation. Since then, it has been
generalized in multiple directions and has been associated with the keywords:
sparse grids, hyperbolic cross approximation, combination technique, and
multilevel methods. Variants of Smolyak's algorithm have been employed in the
computation of high-dimensional integrals in finance, chemistry, and physics,
in the numerical solution of partial and stochastic differential equations, and
in uncertainty quantification. Motivated by this broad and ever-increasing
range of applications, we describe a general framework that summarizes
fundamental results and assumptions in a concise application-independent
manner
Optimal Algorithms for Numerical Integration: Recent Results and Open Problems
We present recent results on optimal algorithms for numerical integration and
several open problems. The paper has six parts:
1. Introduction
2. Lower Bounds
3. Universality
4. General Domains
5. iid Information
6. Concluding RemarksComment: Survey written for the MCQMC conference in Linz, 26 pages. arXiv
admin note: text overlap with arXiv:2108.0205
09391 Abstracts Collection -- Algorithms and Complexity for Continuous Problems
From 20.09.09 to 25.09.09, the Dagstuhl Seminar 09391
Algorithms and Complexity for Continuous Problems was held in the
International Conference and Research Center (IBFI), Schloss Dagstuhl.
During the seminar, participants presented their current research, and
ongoing work and open problems were discussed. Abstracts of the
presentations given during the seminar are put together in this paper. The
first section describes the seminar topics and goals in general. Links to
extended abstracts or full papers are provided, if available
- …