37,716 research outputs found
Probabilistic Numerics and Uncertainty in Computations
We deliver a call to arms for probabilistic numerical methods: algorithms for
numerical tasks, including linear algebra, integration, optimization and
solving differential equations, that return uncertainties in their
calculations. Such uncertainties, arising from the loss of precision induced by
numerical calculation with limited time or hardware, are important for much
contemporary science and industry. Within applications such as climate science
and astrophysics, the need to make decisions on the basis of computations with
large and complex data has led to a renewed focus on the management of
numerical uncertainty. We describe how several seminal classic numerical
methods can be interpreted naturally as probabilistic inference. We then show
that the probabilistic view suggests new algorithms that can flexibly be
adapted to suit application specifics, while delivering improved empirical
performance. We provide concrete illustrations of the benefits of probabilistic
numeric algorithms on real scientific problems from astrometry and astronomical
imaging, while highlighting open problems with these new algorithms. Finally,
we describe how probabilistic numerical methods provide a coherent framework
for identifying the uncertainty in calculations performed with a combination of
numerical algorithms (e.g. both numerical optimisers and differential equation
solvers), potentially allowing the diagnosis (and control) of error sources in
computations.Comment: Author Generated Postprint. 17 pages, 4 Figures, 1 Tabl
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
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