10,056 research outputs found
Notes on Feynman Integrals and Renormalization
I review various aspects of Feynman integrals, regularization and
renormalization. Following Bloch, I focus on a linear algebraic approach to the
Feynman rules, and I try to bring together several renormalization methods
found in the literature from a unifying point of view, using resolutions of
singularities. In the second part of the paper, I briefly sketch the work of
Belkale, Brosnan resp. Bloch, Esnault and Kreimer on the motivic nature of
Feynman integrals.Comment: 39
Unified theory of bound and scattering molecular Rydberg states as quantum maps
Using a representation of multichannel quantum defect theory in terms of a
quantum Poincar\'e map for bound Rydberg molecules, we apply Jung's scattering
map to derive a generalized quantum map, that includes the continuum. We show,
that this representation not only simplifies the understanding of the method,
but moreover produces considerable numerical advantages. Finally we show under
what circumstances the usual semi-classical approximations yield satisfactory
results. In particular we see that singularities that cause problems in
semi-classics are irrelevant to the quantum map
Permutation and Grouping Methods for Sharpening Gaussian Process Approximations
Vecchia's approximate likelihood for Gaussian process parameters depends on
how the observations are ordered, which can be viewed as a deficiency because
the exact likelihood is permutation-invariant. This article takes the
alternative standpoint that the ordering of the observations can be tuned to
sharpen the approximations. Advantageously chosen orderings can drastically
improve the approximations, and in fact, completely random orderings often
produce far more accurate approximations than default coordinate-based
orderings do. In addition to the permutation results, automatic methods for
grouping calculations of components of the approximation are introduced, having
the result of simultaneously improving the quality of the approximation and
reducing its computational burden. In common settings, reordering combined with
grouping reduces Kullback-Leibler divergence from the target model by a factor
of 80 and computation time by a factor of 2 compared to ungrouped
approximations with default ordering. The claims are supported by theory and
numerical results with comparisons to other approximations, including tapered
covariances and stochastic partial differential equation approximations.
Computational details are provided, including efficiently finding the orderings
and ordered nearest neighbors, and profiling out linear mean parameters and
using the approximations for prediction and conditional simulation. An
application to space-time satellite data is presented
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