Content deleted Content added
m tag "celerite" as proper name |
Wikilink |
||
Line 7:
Recently, this method has been extended ('''Generalized Rybicki Press algorithm''') for inverting matrices whose entries of the form <math>A(i,j) = \sum_{k=1}^p a_k \exp(-\beta_k \vert t_i - t_j \vert)</math>.<ref>{{Cite journal|last=Ambikasaran|first=Sivaram|date=2015-12-01|title=Generalized Rybicki Press algorithm|journal=Numerical Linear Algebra with Applications|language=en|volume=22|issue=6|pages=1102–1114|doi=10.1002/nla.2003|issn=1099-1506|arxiv=1409.7852}}</ref> The key observation in the Generalized Rybicki Press (GPP) algorithm is that the matrix <math>A</math> is a semi-separable matrix with rank <math>p</math>. More precisely, if the matrix <math>A \in \mathbb{R}^{n\times n}</math> has a semi-separable rank is <math>p</math>, the cost for solving the linear system <math>Ax=b</math> and obtaining the determinant of the matrix scales as <math>\mathcal{O}\left(p^2n \right)</math>, thereby making it extremely attractive for large matrices. This implementation of the GPP algorithm can be found here.<ref>{{Cite web|url=https://github.com/sivaramambikasaran/ESS|title=sivaramambikasaran/ESS|website=GitHub|language=en|access-date=2018-04-05}}</ref> The key idea is that the dense matrix <math>A</math> can be converted into a sparser matrix of a larger size (see figure on the right), whose sparsity structure can be leveraged to reduce the computational complexity.
The fact that matrix <math>A</math> is a semi-separable matrix also forms the basis for {{proper name|celerite}}<ref>{{Cite web|url=https://celerite.readthedocs.io/en/stable/|title=celerite — celerite 0.3.0 documentation|website=celerite.readthedocs.io|language=en|access-date=2018-04-05}}</ref> library, which is a library for fast and scalable Gaussian Process (GP) Regression in one dimension<ref name=":1">{{Cite journal|last=Foreman-Mackey|first=Daniel|last2=Agol|first2=Eric|last3=Ambikasaran|first3=Sivaram|last4=Angus|first4=Ruth|date=2017|title=Fast and Scalable Gaussian Process Modeling with Applications to Astronomical Time Series|url=http://stacks.iop.org/1538-3881/154/i=6/a=220|journal=The Astronomical Journal|language=en|volume=154|issue=6|pages=220|doi=10.3847/1538-3881/aa9332|issn=1538-3881|arxiv=1703.09710|bibcode=2017AJ....154..220F}}</ref> with implementations in [[C++]], [[Python (programming language)|Python]], and [[Julia (programming language)|Julia]]. The {{proper name|celerite}} method<ref name=":1" /> also provides an algorithm for generating samples from a high-dimensional distribution. The method has found attractive applications in a wide range of fields, especially in astronomical data analysis.<ref>{{Cite journal|last=Foreman-Mackey|first=Daniel|date=2018|title=Scalable Backpropagation for Gaussian Processes using Celerite|url=http://stacks.iop.org/2515-5172/2/i=1/a=31|journal=Research Notes of the AAS|language=en|volume=2|issue=1|pages=31|doi=10.3847/2515-5172/aaaf6c|issn=2515-5172|arxiv=1801.10156|bibcode=2018RNAAS...2a..31F}}</ref><ref>{{Cite book|title=Handbook of Exoplanets|last=Parviainen|first=Hannu|date=2018|publisher=Springer, Cham|isbn=9783319306483|pages=1–24|language=en|doi=10.1007/978-3-319-30648-3_149-1|chapter = Bayesian Methods for Exoplanet Science|arxiv = 1711.03329}}</ref>
==References==
| |||