Transcript A Skyrme QRPA code for deformed nuclei J. Engel and J.T. Univ.

A Skyrme QRPA code for deformed nuclei
J. Engel and J.T.
Univ. of North Carolina at Chapel Hill
We now have a working and tested code. We are speeding
it up prior to starting systematic calculations.
Assumed symmetry:
parity and axial symmetry (2D)
HFB gs: time-rev. invariant
Volume-type pairing interaction used for tests
Density-dependent pairing also implemented
What are the aspects of our science that require
high-performance computing?
All our science requires high-performance computing, e.g.:
• Calculating energies and transition strengths of
excited states in many nuclei to test predictive power of
energy functionals.
• Calculating beta-decay for r-process and understanding
resonances near neutron drip line.

What has been done after the last Pack Forest meeting:
1. Integration with coordinate-space HFB code
2. Parallelization
3. Test of separation of translational spurious component
4. Improvement of efficiency (still in progress, nearing
completion)
5. Distribution of spherical QRPA code to Livermore group
for reaction studies
What are the main accomplishments?
We expect an efficiency-improved version to be completed
in a month. Should be able to begin systematic calculations
shortly thereafter.
Integration with coordinate-space HFB code
(previous version of code used HFBTHO)
For enabling description of nuclei near drip lines efficiently,
and for making separation of translational spurious state
easier than the harmonic-oscillator basis.
Symmetry of wave function space is important.
Parallelization
a. Speeding up
• Division of interaction matrix elements
among individual processors
• ScaLAPACK diagonalization
b. Memory management
• distribution of quasiparticle wave functions F a ( z, r ),
• distribution of Hamiltonian matrix
Our QRPA code can handle arbitrarily large basis
as long as more than a few processors are available.
Parallelization
quasiparticle wave functions are read by processor 1
processor 2
… processor 4
 
Interaction matrix elements Vi j
processor 1
V1,…,Vn
4 qp states
a,b,c, and d
processor 2
i , j 1, 2,
 V1 ,V2 ,
processor 3
processor 4
Separation of spurious translational mode
Absolute value of transition matrix elements of
A
e  ri3Y10 ( i , i ) between ground and Kπ=0– states
i 1
(efm3)
A correction operator subtracts translational spurious
good
components from the matrix elements
test of code
without correction operator
with correction operator
26Mg
(Energies of two
impulses are slightly
shifted horizontally)
(efm3)
Higher-energy region
without correction operator
with correction operator
26Mg
Computational Issues (before speedup)
https://www.nersc.gov/nusers/status/jobs/?hostname=franklin
4x108 interaction matrix elements
(efm3)
(Size of H = 40000 by 40000)
without a correction operator
with a correction operator
.72 processor sec/me
26Mg
In test on one processor,
a typical matrix element
takes .36 sec. About half
the time not in computation.
Improvement of efficiency of algorithm
1. B-spline + Gauss-Legendre → factor of 2 speedup
first version
100x100 mesh for wf
new version
70x70 B-splines for wf
4-point formula for differentiation
analytical formula of B-spline
for differentiation
integration by Simpson rule
integration by Gauss-Legendre
using 70x70 mesh
Improvement of efficiency of algorithm
2. Canonical basis (being implemented) → factor of 4 speedup
Definition: eigen functions ψ(r) of density matrix
eigen value = v2 : occupation probability
Conversion of QRPA formulation using quasiparticle basis
to another one using canonical-BCS basis is obtained by

 U (r ) 
Re placequasiparti
cle wave function  
 V (r ) 

 u ( r ) 
by 
, and

 v ( r ) 
Extendquasiparti
cleenergyEa to Eaa .
A
 B

B  X   E  X 
Y 
A  Y 


Aab ,cd   E d  ad  cb   bd  ca   E c  ac  db   bc  da 
 
 T 
3
3
T 
 
ˆ
  d r1 d r2 Vd ( r1 )U a ( r2 )W ( r1 , r2 )U c ( r1 )Vb ( r2 )
 
 T 
T 
 
ˆ
 Vd ( r1 )U b ( r2 )W ( r1 , r2 )U c ( r1 )Va ( r2 )
 
 T 
T 
 
ˆ
 Vc ( r1 )U a ( r2 )W ( r1 , r2 )U d ( r1 )Vb ( r2 )
 
 T 
T 
 
ˆ
 Vc ( r1 )U b ( r2 )W ( r1 , r2 )U d ( r1 )Va ( r2 )




 U ( r )   u ( r ) 
 V ( r )    v ( r ) 

 

The 4 integrals are all closely related
in canonical basis. Only need one integral.

More on speedup
1. B-spline + Gauss-Legendre → factor of 2
2. Canonical basis (being implemented)→ factor of 4
total factor of 8
Further speedup will come (we hope) from ADLB
Expected:
reduction of dead time and number of
communications between processors
Access to resources?
For a Kπ=1– calculation of A≈30, estimated charge is
5,000 Mpp·h. (without ADLB on franklin.nersc.gov)
--- We need more resources to calculate lots of nuclei.
What nuclei should be chosen for testing energy
functionals if resources are limited?
INCITE?
Other options?
Summary
What has been done since the last Pack Forest meeting:
1. Integration with coordinate-space HFB code
2. Parallelization
3. Test of separation of translational spurious component
4. Two improvements of efficiency
5. Distribution of spherical QRPA code to Livermore group
6. Comparison with J-scheme calculation for a spherical
nucleus.
7. A few other Kπ calculations
The code is very close to completion, though we’d hoped to
be done by now.