6,237 research outputs found
Quasi-optimal multiplication of linear differential operators
We show that linear differential operators with polynomial coefficients over
a field of characteristic zero can be multiplied in quasi-optimal time. This
answers an open question raised by van der Hoeven.Comment: To appear in the Proceedings of the 53rd Annual IEEE Symposium on
Foundations of Computer Science (FOCS'12
Computing derivative-based global sensitivity measures using polynomial chaos expansions
In the field of computer experiments sensitivity analysis aims at quantifying
the relative importance of each input parameter (or combinations thereof) of a
computational model with respect to the model output uncertainty. Variance
decomposition methods leading to the well-known Sobol' indices are recognized
as accurate techniques, at a rather high computational cost though. The use of
polynomial chaos expansions (PCE) to compute Sobol' indices has allowed to
alleviate the computational burden though. However, when dealing with large
dimensional input vectors, it is good practice to first use screening methods
in order to discard unimportant variables. The {\em derivative-based global
sensitivity measures} (DGSM) have been developed recently in this respect. In
this paper we show how polynomial chaos expansions may be used to compute
analytically DGSMs as a mere post-processing. This requires the analytical
derivation of derivatives of the orthonormal polynomials which enter PC
expansions. The efficiency of the approach is illustrated on two well-known
benchmark problems in sensitivity analysis
Non-commutative Painleve' equations and Hermite-type matrix orthogonal polynomials
We study double integral representations of Christoffel-Darboux kernels
associated with two examples of Hermite-type matrix orthogonal polynomials. We
show that the Fredholm determinants connected with these kernels are related
through the Its-Izergin-Korepin-Slavnov (IIKS) theory with a certain
Riemann-Hilbert problem. Using this Riemann-Hilbert problem we obtain a Lax
pair whose compatibility conditions lead to a non-commutative version of the
Painleve' IV differential equation for each family.Comment: Final version, accepted for publication on CMP: Communications in
Mathematical Physics. 24 pages, 1 figur
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