206 research outputs found
Quantum Approximation II. Sobolev Embeddings
A basic problem of approximation theory, the approximation of functions from
the Sobolev space W_p^r([0,1]^d) in the norm of L_q([0,1]^d), is considered
from the point of view of quantum computation. We determine the quantum query
complexity of this problem (up to logarithmic factors). It turns out that in
certain regions of the domain of parameters p,q,r,d quantum computation can
reach a speedup of roughly squaring the rate of convergence of classical
deterministic or randomized approximation methods. There are other regions were
the best possible rates coincide for all three settings.Comment: 23 pages, paper submitted to the Journal of Complexit
Discontinuous information in the worst case and randomized settings
We believe that discontinuous linear information is never more powerful than
continuous linear information for approximating continuous operators. We prove
such a result in the worst case setting. In the randomized setting we consider
compact linear operators defined between Hilbert spaces. In this case, the use
of discontinuous linear information in the randomized setting cannot be much
more powerful than continuous linear information in the worst case setting.
These results can be applied when function evaluations are used even if
function values are defined only almost everywhere
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
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