Stephen Grace and Xiangwei Tang

californiamandrillΛογισμικό & κατασκευή λογ/κού

13 Δεκ 2013 (πριν από 3 χρόνια και 6 μήνες)

67 εμφανίσεις

Stephen Grace and
Xiangwei

Tang


Background of Numerical Relativity


Numerical Relativity Equation


Interesting Characteristics


Parallel Computing Field


Solving Algorithms Used


Binary Black Hole Grand Challenge


Results


Current Issues




Why Black Hole


Black hole collisions should be strong generators
of gravitational radiation.


This is an extremely difficult problem for
numerical relativist.


To detect gravitational radiation and waves


Binary Black Holes (BBHs) are potentially a good
source of data


Europe and USA have mounted experiments to show
this


More topics than just BBHs


Binary Neutron Stars


Collapsing gravity fields, etc.


Uses Einstein’s General Relativity Equation as a
basis of study


There are hundreds of linear, nonlinear variations of
his original equation



G
uv

= 8πGT
uv


G
uv



Einstein Tensor (Space
-
time curvature)


T
uv



Stress
-
energy tensor (mass distribution)


G


Gravitational Constant


Can be quite small on the outside of the black hole, and quite large
on the interior


Indices
u,v



four index values


Corresponding to time and three spatial directions


Spatial directions are not specified in Polar, Cartesian, or Cylindrical
coordinates


This is the most basic form of the equation


All information from here on is with knowledge of the
nonlinear derivations


Total freedom in choosing the coordinate
system


Each coordinate system can provide different results
that are completely different from each other


Validity with different coordinate systems are
questionable


There are a variety of formulations of Einstein’s
equations


One potential formulation is a
constrained
Hamiltonian system (Hamiltonian Mechanics)


Forces are momentum invariant


12 equations: 6 spatial, 6
momenta


More sensitive sensors are requested to study
black holes and gravity


Modeling BBHs in parallel computing will help
refine sensors and look for anomalies


Simulating and handling the curvature
singularity inside the black hole


Interior of the black hole cannot affect the
exterior


Differing velocities of BBH collisions/mergers



Binary Black Hole Grand Challenge Alliance


Was started to solve Einstein’s Equation


Rules:


Develop problem solving for Nonlinear Einstein equations


Including dynamical adaptive multilevel parallel infrastructure


Provide controllable, convergent algorithms


The BBHGCA concluded with some major results


Actual simulation of a BBH collision


Organizations and groups still trying to solve the
problem based upon the initial findings of the
challenge


Algorithms used


Finite Difference


Most commonly found in potential solutions of the problem


Finite Element


Pseudo
-
spectral/spectral


Fourier Transforms used as an example


Can be highly accurate and use lower memory


Parallelization Techniques


Adaptive Mesh
-
Refinement


Multi
-
Grid


There are quite a few papers on each algorithm and
parallelization


Sadly, each paper differs so much that is hard to find a common
underlying equation to use as a basis


To solve the problem of the 3D spiraling
coalescence of two black holes


10 non
-
linear PDEs


4 Initial Value Equations (Elliptic)


6 Coupled Hyperbolic equations (Hyperbolic)


AMR is used to solve these equations.




Built by
Megware

for the
Albert Einstein Institute
(AEI) Numerical
Relativity Group (NRG)


Simulation video on next
slide was done using this
super computer


Ranked 192 on the
TOP500 list in 2007


Delivers efficiency of 80%


Frontends

2

Fileservers

17

Nodes

262

Primary Interconnect

Infiniband

Secondary Interconnect

Gigabit Ethernet

Storage Network

Gigabit Ethernet

Management Network

Fast Ethernet

Network Storage (Total)

140TB

CPUs (total) / Cores

524/1048 (Compute Nodes)

Processor Type

Intel Xeon 5160 Woodcrest
4MB Shared L2
-
Cache

RAM (total)

2096GB (Compute Nodes)


Video is of two black
holes of equal mass and
size colliding.


The various colors of red,
orange, and yellow are
gravitational waves
observed.


The white edges of the
box are to be the
boundaries of the
simulation



More development is proposed for gravitational sensors like the
LISA G.R.S.


Laser Interferometer Space Antenna with Gravitational Reference
Sensor


More info on LISA and GRS:
http://lisa.stanford.edu/LISAOverview.html


Scaling of the model


The simulation environment is not large enough to show all valid data


If the scaling is increased, supercomputers of today might not have enough
memory to run


Harmonic (Wave) evolution Scheme


Wave coordinates that satisfy the covariant scalar wave equations


Can simplify the equations greatly


More specific information:
http://arxiv.org/PS_cache/gr
-
qc/pdf/0512/0512093v3.pdf


G. Fox explains that by using the general
Einstein equation, tensors are used


High Performance Fortran is an attractive
language to use


Downside: need a Perl interface to adapt the
hierarchy to the language


Whenever the Fortran code is modified, the Perl
interface has to be re
-
written


Jack
Dongarra
, Ian Foster, Geoffrey Fox, William
Gropp

Ken
Kennedy Linda
Torczon
, Andy White. "Sourcebook of Parallel
Computing." San Francisco: Morgan Kaufmann Publishers,
2003. 195
-
199.


Supercomputers of the Max Planck Institute for Gravitational
Physics (
Damiana
)


http://supercomputers.aei.mpg.de/damiana/technical
-
specifications
-
1


http://supercomputers.aei.mpg.de/


AEI Numerical Relativity Group (BBHs Video)


http://numrel.aei.mpg.de/Visualisations/Archive/BinaryBla
ckHoles/GrazingBlackHoles/GrazingBH.html



The Hamiltonian.
OpenCourseWare
.
http://ocw.mit.edu/ans7870/18/18.013a/textb
ook/chapter16/section03.html


Binary Black Hole Grand Challenge.
http://www.npac.syr.edu/users/gcf/bbhklasky
/bbh.html