A Modern Retrospective on Probabilistic Numerics

arXiv.org Machine Learning

The field of probabilistic numerics (PN), loosely speaking, attempts to provide a statistical treatment of the errors and/or approximations that are made en route to the output of a deterministic numerical method, e.g. the approximation of an integral by quadrature, or the discretised solution of an ordinary or partial differential equation. This decade has seen a surge of activity in this field. In comparison with historical developments that can be traced back over more than a hundred years, the most recent developments are particularly interesting because they have been characterised by simultaneous input from multiple scientific disciplines: mathematics, statistics, machine learning, and computer science. The field has, therefore, advanced on a broad front, with contributions ranging from the building of overarching generaltheory to practical implementations in specific problems of interest. Over the same period of time, and because of increased interaction among researchers coming from different communities, the extent to which these developments were -- or were not -- presaged by twentieth-century researchers has also come to be better appreciated. Thus, the time appears to be ripe for an update of the 2014 Tübingen Manifesto on probabilistic numerics[Hennig, 2014, Osborne, 2014d,c,b,a] and the position paper[Hennig et al., 2015] to take account of the developments between 2014 and 2019, an improved awareness of the history of this field, and a clearer sense of its future directions. In this article, we aim to summarise some of the history of probabilistic perspectives on numerics (Section 2), to place more recent developments into context (Section 3), and to articulate a vision for future research in, and use of, probabilistic numerics (Section 4).

Merkel Calls Russia a Partner, Urges Global Cooperation

U.S. News

"We are proud of our cars and so we should be," Merkel said, adding, however, that many were built in the United States and exported to China. "If that is viewed as a security threat to the United States, then we are shocked," she told the Munich Security Conference to applause from the audience.

D Lookup: Fast Protpin Structure Database Searches at 90% ReHabi!iW

AAAI Conferences

Litsa Holm and Chris Sander European Molecular Biology Laboratory D-69012 Heidelberg, Germany Holm @EMBL-HeidelbergAe Abstract The problem of structure comparison is much There are far fewer classes of threedimensional protein folds than sequence comparison because a 3-D match requires co-more complicated than sequence string families but dieproblem of detecting threedimensional similarities is NPcomplete. Structure alignment operative similarity in the relative disposition of present a novel heuristic for identifying 3-D similarities between a query structure and the is an optimization problem that requires the database of known protein structures. Here, the top-down variables, the search landscape contains very approach is to start with the global many local optima due to the recurrence of comparison and select a rough secondary structure elements (helices and superimposition using a fast 3-D lookup of strands) and small tertiary structural motifs, i.e., secondary..structure motifs. An all-againstall compaaSson of 385 representative proteins necessary to locate the absolute optimum of the object function in each pair comparison. This is (150,000 pair comparisons) took 1 day because one is usually only interested in those computer time on a single R8000 processor. The method is rated at 90 % fast if not complete.