8,081 research outputs found
Quadrature Points via Heat Kernel Repulsion
We discuss the classical problem of how to pick weighted points on a
dimensional manifold so as to obtain a reasonable quadrature rule
This problem, naturally, has a long history; the purpose of our paper is to
propose selecting points and weights so as to minimize the energy functional
\sum_{i,j =1}^{N}{ a_i a_j \exp\left(-\frac{d(x_i,x_j)^2}{4t}\right) }
\rightarrow \min, \quad \mbox{where}~t \sim N^{-2/d}, is the
geodesic distance and is the dimension of the manifold. This yields point
sets that are theoretically guaranteed, via spectral theoretic properties of
the Laplacian , to have good properties. One nice aspect is that the
energy functional is universal and independent of the underlying manifold; we
show several numerical examples
Bayesian Quadrature for Multiple Related Integrals
Bayesian probabilistic numerical methods are a set of tools providing
posterior distributions on the output of numerical methods. The use of these
methods is usually motivated by the fact that they can represent our
uncertainty due to incomplete/finite information about the continuous
mathematical problem being approximated. In this paper, we demonstrate that
this paradigm can provide additional advantages, such as the possibility of
transferring information between several numerical methods. This allows users
to represent uncertainty in a more faithful manner and, as a by-product,
provide increased numerical efficiency. We propose the first such numerical
method by extending the well-known Bayesian quadrature algorithm to the case
where we are interested in computing the integral of several related functions.
We then prove convergence rates for the method in the well-specified and
misspecified cases, and demonstrate its efficiency in the context of
multi-fidelity models for complex engineering systems and a problem of global
illumination in computer graphics.Comment: Proceedings of the 35th International Conference on Machine Learning
(ICML), PMLR 80:5369-5378, 201
The SLH framework for modeling quantum input-output networks
Many emerging quantum technologies demand precise engineering and control
over networks consisting of quantum mechanical degrees of freedom connected by
propagating electromagnetic fields, or quantum input-output networks. Here we
review recent progress in theory and experiment related to such quantum
input-output networks, with a focus on the SLH framework, a powerful modeling
framework for networked quantum systems that is naturally endowed with
properties such as modularity and hierarchy. We begin by explaining the
physical approximations required to represent any individual node of a network,
eg. atoms in cavity or a mechanical oscillator, and its coupling to quantum
fields by an operator triple . Then we explain how these nodes can be
composed into a network with arbitrary connectivity, including coherent
feedback channels, using algebraic rules, and how to derive the dynamics of
network components and output fields. The second part of the review discusses
several extensions to the basic SLH framework that expand its modeling
capabilities, and the prospects for modeling integrated implementations of
quantum input-output networks. In addition to summarizing major results and
recent literature, we discuss the potential applications and limitations of the
SLH framework and quantum input-output networks, with the intention of
providing context to a reader unfamiliar with the field.Comment: 60 pages, 14 figures. We are still interested in receiving
correction
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