4,489 research outputs found
The Complexity of Linear Tensor Product Problems in (Anti-) Symmetric Hilbert Spaces
We study linear problems defined on tensor products of Hilbert spaces with an
additional (anti-) symmetry property. We construct a linear algorithm that uses
finitely many continuous linear functionals and show an explicit formula for
its worst case error in terms of the singular values of the univariate problem.
Moreover, we show that this algorithm is optimal with respect to a wide class
of algorithms and investigate its complexity. We clarify the influence of
different (anti-) symmetry conditions on the complexity, compared to the
classical unrestricted problem. In particular, for symmetric problems we give
characterizations for polynomial tractability and strong polynomial
tractability in terms of the amount of the assumed symmetry. Finally, we apply
our results to the approximation problem of solutions of the electronic
Schr\"odinger equation.Comment: Extended version (53 pages); corrected typos, added journal referenc
Segre maps and entanglement for multipartite systems of indistinguishable particles
We elaborate the concept of entanglement for multipartite system with bosonic
and fermionic constituents and its generalization to systems with arbitrary
parastatistics. The entanglement is characterized in terms of generalized Segre
maps, supplementing thus an algebraic approach to the problem by a more
geometric point of view.Comment: 16 pages, the version to appear in J. Phys. A. arXiv admin note: text
overlap with arXiv:1012.075
Several Approaches to Break the Curse of Dimensionality
In modern science the efficient numerical treatment of high-dimensional
problems becomes more and more important. A fundamental insight of the theory
of information-based complexity (IBC for short) is that the computational
hardness of a problem can not be described properly only by the rate of
convergence. There exist problems for which an exponential number of
information operations is needed in order to reduce the initial error, although
there are algorithms which provide an arbitrary large rate of convergence.
Problems that yield this exponential dependence are said to suffer from the
curse of dimensionality. While analyzing numerical problems it turns out that
we can often vanquish this curse by exploiting additional structural
properties. The aim of this thesis is to present several approaches of this
type. Moreover, a detailed introduction to the field of IBC is given.Comment: 133 pages, my Ph.D. thesis for becoming Dr. rer. nat. at
Friedrich-Schiller-University Jen
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