Preconditioned low-rank Riemannian optimization for linear systems with tensor product structure

The numerical solution of partial differential equations on high-dimensional domains gives rise to computationally challenging linear systems. When using standard discretization techniques, the size of the linear system grows exponentially with the number of dimensions, making the use of classic iterative solvers infeasible. During the last few years, low-rank tensor approaches have been developed that allow to mitigate this curse of dimensionality by exploiting the underlying structure of the linear operator. In this work, we focus on tensors represented in the Tucker and tensor train formats. We propose two preconditioned gradient methods on the corresponding low-rank tensor manifolds: A Riemannian version of the preconditioned Richardson method as well as an approximate Newton scheme based on the Riemannian Hessian. For the latter, considerable attention is given to the efficient solution of the resulting Newton equation. In numerical experiments, we compare the efficiency of our Riemannian algorithms with other established tensor-based approaches such as a truncated preconditioned Richardson method and the alternating linear scheme. The results show that our approximate Riemannian Newton scheme is significantly faster in cases when the application of the linear operator is expensive.Comment: 24 pages, 8 figure

Riemannian Optimization for Solving High-Dimensional Problems with Low-Rank Tensor Structure

In this thesis, we present a Riemannian framework for the solution of high-dimensional optimization problems with an underlying low-rank tensor structure. Here, the high-dimensionality refers to the size of the search space, while the cost function is scalar-valued. Such problems arise, for example, in the reconstruction of high-dimensional data sets and in the solution of parameter dependent partial differential equations. As the degrees of freedom grow exponentially with the number of dimensions, the so-called curse of dimensionality, directly solving the optimization problem is computationally unfeasible even for moderately high-dimensional problems. We constrain the optimization problem by assuming a low-rank tensor structure of the solution; drastically reducing the degrees of freedom. We reformulate this constrained optimization as an optimization problem on a manifold using the smooth embedded Riemannian manifold structure of the low-rank representations of the Tucker and tensor train formats. Exploiting this smooth structure, we derive efficient gradient-based optimization algorithms. In particular, we propose Riemannian conjugate gradient schemes for the solution of the tensor completion problem, where we aim to reconstruct a high-dimensional data set for which the vast majority of entries is unknown. For the solution of linear systems, we show how we can precondition the Riemannian gradient and leverage second-order information in an approximate Newton scheme. Finally, we describe a preconditioned alternating optimization scheme with subspace correction for the solution of high-dimensional symmetric eigenvalue problems

Low-rank tensor completion by Riemannian optimization

In tensor completion, the goal is to fill in missing entries of a partially known tensor under a low-rank constraint. We propose a new algorithm that performs Riemannian optimization techniques on the manifold of tensors of fixed multilinear rank. More specifically, a variant of the nonlinear conjugate gradient method is developed. Paying particular attention to efficient implementation, our algorithm scales linearly in the size of the tensor. Examples with synthetic data demonstrate good recovery even if the vast majority of the entries are unknown. We illustrate the use of the developed algorithm for the recovery of multidimensional images and for the approximation of multivariate functions

All-sky search for periodic gravitational waves in the full S5 LIGO data

We report on an all-sky search for periodic gravitational waves in the frequency band 50–800 Hz and with the frequency time derivative in the range of 0 through −6×10−9  Hz/s. Such a signal could be produced by a nearby spinning and slightly nonaxisymmetric isolated neutron star in our Galaxy. After recent improvements in the search program that yielded a 10× increase in computational efficiency, we have searched in two years of data collected during LIGO’s fifth science run and have obtained the most sensitive all-sky upper limits on gravitational-wave strain to date. Near 150 Hz our upper limit on worst-case linearly polarized strain amplitude h0 is 1×10−24, while at the high end of our frequency range we achieve a worst-case upper limit of 3.8×10−24 for all polarizations and sky locations. These results constitute a factor of 2 improvement upon previously published data. A new detection pipeline utilizing a loosely coherent algorithm was able to follow up weaker outliers, increasing the volume of space where signals can be detected by a factor of 10, but has not revealed any gravitational-wave signals. The pipeline has been tested for robustness with respect to deviations from the model of an isolated neutron star, such as caused by a low-mass or long-period binary companion. © 2012 The American Physical Societ

Search for gravitational waves from compact binary coalescence in LIGO and Virgo data from S5 and VSR1

We report the results of the first search for gravitational waves from compact binary coalescence using data from the Laser Interferometer Gravitational-Wave Observatory and Virgo detectors. Five months of data were collected during the Laser Interferometer Gravitational-Wave Observatory’s S5 and Virgo’s VSR1 science runs. The search focused on signals from binary mergers with a total mass between 2 and 35M⊙. No gravitational waves are identified. The cumulative 90%-confidence upper limits on the rate of compact binary coalescence are calculated for nonspinning binary neutron stars, black hole-neutron star systems, and binary black holes to be 8.7×10−3  yr−1 L10−1, 2.2×10−3  yr−1 L10−1, and 4.4×10−4  yr−1 L10−1, respectively, where L10 is 1010 times the blue solar luminosity. These upper limits are compared with astrophysical expectations. © 2010 The American Physical Societ

Search for gravitational waves from intermediate mass binary black holes

We present the results of a weakly modeled burst search for gravitational waves from mergers of nonspinning intermediate mass black holes in the total mass range 100–450  M⊙ and with the component mass ratios between 1∶1 and 4∶1. The search was conducted on data collected by the LIGO and Virgo detectors between November of 2005 and October of 2007. No plausible signals were observed by the search which constrains the astrophysical rates of the intermediate mass black holes mergers as a function of the component masses. In the most efficiently detected bin centered on 88+88  M⊙, for nonspinning sources, the rate density upper limit is 0.13 per Mpc3 per Myr at the 90% confidence level. © 2012 The American Physical Societ

Search for gravitational waves from low mass compact binary coalescence in LIGO's sixth science run and Virgo's science runs 2 and 3

We report on a search for gravitational waves from coalescing compact binaries using LIGO and Virgo observations between July 7, 2009, and October 20, 2010. We searched for signals from binaries with total mass between 2 and 25M⊙; this includes binary neutron stars, binary black holes, and binaries consisting of a black hole and neutron star. The detectors were sensitive to systems up to 40 Mpc distant for binary neutron stars, and further for higher mass systems. No gravitational-wave signals were detected. We report upper limits on the rate of compact binary coalescence as a function of total mass, including the results from previous LIGO and Virgo observations. The cumulative 90% confidence rate upper limits of the binary coalescence of binary neutron star, neutron star-black hole, and binary black hole systems are 1.3×10−4, 3.1×10−5, and 6.4×10−6  Mpc−3 yr−1, respectively. These upper limits are up to a factor 1.4 lower than previously derived limits. We also report on results from a blind injection challenge. © 2012 The American Physical Societ

Upper limits on a stochastic gravitational-wave background using LIGO and Virgo interferometers at 600-1000 Hz

A stochastic background of gravitational waves is expected to arise from a superposition of many incoherent sources of gravitational waves, of either cosmological or astrophysical origin. This background is a target for the current generation of ground-based detectors. In this article we present the first joint search for a stochastic background using data from the LIGO and Virgo interferometers. In a frequency band of 600–1000 Hz, we obtained a 95% upper limit on the amplitude of ΩGW(f)=Ω3(f/900  Hz)3, of Ω3<0.32, assuming a value of the Hubble parameter of h100=0.71. These new limits are a factor of seven better than the previous best in this frequency band. © 2012 The American Physical Societ

All-sky search for gravitational-wave bursts in the second joint LIGO-Virgo run

We present results from a search for gravitational-wave bursts in the data collected by the LIGO and Virgo detectors between July 7, 2009 and October 20, 2010: data are analyzed when at least two of the three LIGO-Virgo detectors are in coincident operation, with a total observation time of 207 days. The analysis searches for transients of duration ≲1  s over the frequency band 64–5000 Hz, without other assumptions on the signal waveform, polarization, direction or occurrence time. All identified events are consistent with the expected accidental background. We set frequentist upper limits on the rate of gravitational-wave bursts by combining this search with the previous LIGO-Virgo search on the data collected between November 2005 and October 2007. The upper limit on the rate of strong gravitational-wave bursts at the Earth is 1.3 events per year at 90% confidence. We also present upper limits on source rate density per year and Mpc3 for sample populations of standard-candle sources. As in the previous joint run, typical sensitivities of the search in terms of the root-sum-squared strain amplitude for these waveforms lie in the range ∼5×10−22  Hz−1/2 to ∼1×10−20  Hz−1/2. The combination of the two joint runs entails the most sensitive all-sky search for generic gravitational-wave bursts and synthesizes the results achieved by the initial generation of interferometric detectors. © 2012 The American Physical Societ

Multi-messenger observations of a binary neutron star merger

On 2017 August 17 a binary neutron star coalescence candidate (later designated GW170817) with merger time 12:41:04 UTC was observed through gravitational waves by the Advanced LIGO and Advanced Virgo detectors. The Fermi Gamma-ray Burst Monitor independently detected a gamma-ray burst (GRB 170817A) with a time delay of ~1.7 s with respect to the merger time. From the gravitational-wave signal, the source was initially localized to a sky region of 31 deg2 at a luminosity distance of 40+8-8 Mpc and with component masses consistent with neutron stars. The component masses were later measured to be in the range 0.86 to 2.26 Mo. An extensive observing campaign was launched across the electromagnetic spectrum leading to the discovery of a bright optical transient (SSS17a, now with the IAU identification of AT 2017gfo) in NGC 4993 (at ~40 Mpc) less than 11 hours after the merger by the One- Meter, Two Hemisphere (1M2H) team using the 1 m Swope Telescope. The optical transient was independently detected by multiple teams within an hour. Subsequent observations targeted the object and its environment. Early ultraviolet observations revealed a blue transient that faded within 48 hours. Optical and infrared observations showed a redward evolution over ~10 days. Following early non-detections, X-ray and radio emission were discovered at the transient’s position ~9 and ~16 days, respectively, after the merger. Both the X-ray and radio emission likely arise from a physical process that is distinct from the one that generates the UV/optical/near-infrared emission. No ultra-high-energy gamma-rays and no neutrino candidates consistent with the source were found in follow-up searches. These observations support the hypothesis that GW170817 was produced by the merger of two neutron stars in NGC4993 followed by a short gamma-ray burst (GRB 170817A) and a kilonova/macronova powered by the radioactive decay of r-process nuclei synthesized in the ejecta