Slides

CAA: Covariant
Approximation Averaging
a new class of error
reduction techniques
Taku Izubuchi,
with T. Blum E. Shintani,
C. Lehner, T. Kawanai, T. Ishikawa, J.Yu,
R. Arthur, P. Bolye, ….
RBC/UKQCD in preparation
RIKEN BNL
Research Center
INT La'ce QCD Summer School, August 21
will be posted just aAer this lecture !
RBRC-967
A new class of variance reduction techniques using lattice symmetries
Thomas Blum,1, 2 Taku Izubuchi,3, 2 and Eigo Shintani2
2
1
Physics Department, University of Connecticut, Storrs, CT 06269-3046, USA
RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA
3
Brookhaven National Laboratory, Upton, NY 11973, USA
We present a general class of unbiased improved estimators for physical observables in lattice gauge
theory computations which significantly reduces statistical errors at modest computational cost. The
error reduction techniques, referred to as covariant approximation averaging, utilize approximations
which are covariant under lattice symmetry transformations. We observed cost reductions from the
new method compared to the traditional one, for fixed statistical error, of 16 times for the nucleon
mass at Mπ ∼ 330 MeV (Domain-Wall quark) and 2.6-20 times for the hadronic vacuum polarization
at Mπ ∼ 480 MeV (Asqtad quark). These cost reductions should improve with decreasing quark
mass and increasing lattice sizes.
PACS numbers: 11.15.Ha,12.38.Gc,07.05.Tp
n 
As non-perturbative computations using lattice gauge
theory are applied to a wider range of physically interesting observables, it is increasingly important to find numerical strategies that provide precise results. In Monte
Carlo simulations our reach to important physics is still
often limited by statistical uncertainties. Examples include hadronic contributions to the muon’s anomalous
magnetic moment [1], nucleon form factors and structure
functions [2], including nucleon electric dipole moments
[3–6], hadron matrix elements relevant to flavor physics
(e.g., K → ππ amplitudes) [7], and multi-hadron state
physics [8], to name only a few.
As a generalization of low-mode averaging (LMA) [9,
10], we present a class of unbiased statistical error reduction techniques, utilizing approximations that are covariant under lattice symmetry transformations. LMA has
worked well in cases where low eigenmodes of the Dirac
operator dominate [11]: low energy constants in the εregime [9, 12–15], pseudoscalar meson masses and decay
constants [16–18], an so on. With a modest increase in
computational cost, the generalized method can reduce
statistical errors by an order of magnitude, or more, even
in cases where LMA fails.
Unlike LMA, we account for all modes of the Dirac
In lattice gauge theory simulations an ensemble of
gauge field configurations {U1 , · · · , UNconf } is generated
randomly, according to the Boltzmann weight, e−S[U ] ,
where S[U ] is the lattice-regularized action. The expectation value of a primary, covariant observable, O,
Precise theoretical calculation becomes even more
important to confirm or reject the standard model
More than half of CPU cycles of lattice QCD are for
valence calculations
• 
• 
�O� =
1
Nconf
N
conf
�
i=1
O[Ui ] + O
�
�
1
√
,
Nconf
(1)
is estimated as the ensemble average, over a large number
of configurations, Nconf ∼ O(100 − 1000). Here, we primarily consider observables made of fermion propagators
SF [U ] computed on the background gauge configuration
U.
By exploiting lattice symmetry transformations g ∈ G,
that transform U → U g , a general class of variance reduction techniques is introduced. First construct an approximation O(appx) to O which must fulfill the following
conditions,
on physics point QCD simulation
multi hadron simulation
It’s shame to be limited by statistical error
appx-1: O(appx) should fluctuate closely with O,
(appx)
�
r ≡ Corr(O, O(appx) ) = √ �∆O∆O
≈ 1,
(appx) 2
2
�(∆O) ��(∆O
) �
and �(∆O)2 � ≈ �(∆O(appx) )2 � , where ∆X = X −
statistical noise reduction techniques
n 
LMA
L. Giusti, P. Hernandez, M. Laine, P. Weisz and H. Wittig, JHEP 0404, 013 (2004)
see also H. Neff, N. Eicker, T. Lippert, J. W. Negele and K. Schilling, Phys. Rev. D 64 (2001) 114509 and T. DeGrand and S. Schaefer, Comput. Phys.
Commun. 159 (2004) 185
works for low mode dominant quantities
n 
Truncated Solver Method (TSM)
G. Bali, S. Collins, A. Schaefer, Comput. Phys. Commun. 181 (2010) 1570
uses stochastic noise to avoid systematic error
n 
All-to-all propagator [S. Ryan’s lecture ]
J.Foley, K.Juge, A. O’Cais, M. Peardon, S. Ryan, J-I. Skullerud, Comput.Phys.Commun. 172 (2005) 145
uses stochastic noise
could use CAA as a part of A2A
n 
Also closely related to the improved solvers
Deflations, EigCG, Domain decomposition, MultiGrid, …..
n 
Other application specific reductions
multi-hit (pure Gauge)
multi level: Luscher, M. and Weisz, P. (2001). J. High Energy Phys., 09, 010
Multiple timestep in HMC
n 
Multiple time steps in MD integrators
n 
n 
Sexton & Weingarten trick
Hasenbusch trick : introduce intermediate mass
n 
Clark & Kennedy RHMC (quotient force term)
A. Kennedy 06
Berlin Wall was torn down by Smart Work Sharings Similar tricks for valence ? 100%
75%
Residue (")
L! Force
"/(!+0.125)
CG iterations
50%
25%
0%
-12.6 -10.1 -8.5 -7.1 -5.8 -4.4 -3.1 -1.7 -0.3 1.5
Shift [ln(!)]
State of Obvious n 
Many interesting physics are limited by statistical
error
n 
Do more number of measurements, Nmeas
n 
Change to observable with smaller fluctuation, C
n 
Covariant Approximation Averaging (CAA)
Combine the above using
•  symmetries of the lattice action
•  (crude) approximations
Covariant Approximation Averaging
( CAA )
n 
Original observable
n 
Covariant approximation of the observable
under a lattice symmetry
n 
Unbiased improved estimator
Covariant approximation
n 
O(appx) needs to be precisely (to the numerical
accuracy required) covariant under the
symmetry of lattice action to avoid systematic
errors.
Delta x = 4
O^g(x,y), y=4
O(x,y), y=1
U^g(x)
U(x)
0
1
2
3
4
5
X
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
X
g
Figure c
1:heck Transformed
linkcode field Uµu
(x)
and e
a xplicitly bi-local observable
Og (x,
y). If OcisonfiguraRon
covariant
One should i
n t
he sing s
hiAed g
auge g
observable, the shape of O (x, y) are exactly same as O(x, y).
Unbiasness proof
n 
Consider a element g of lattice symmetry G e.g.
transformation of fields
n 
Observable (and its approximation) is called to have covariance under g iff
n 
or, more explicitly,
n 
When g is a symmetry of lattice, and O(appx) is covariant
Why expect improvements ?
n 
O(imp) has smaller error, smaller C
<= accuracy of approximation controls error,
need not to be too accurate (0.1% is good enough)
n 
NG suppresses the bulk part of noise cheaply
Valence version of Hasenbushing in HMC
AMA : a smart work sharing
n 
Ideal approximation
Oappx is strongly correlated with original one.
ε
ensemble
R(corr) b/w O and O(appx) needs to be larger than 0.5 [C. Lehner]
ε
ensemble •  ε, accuracy of approximaRon should be smaller than Oave appx •  ΔΟrest which is staRsRcal error of Orest depends on the strength of correlaRon.
•  The computaRonal cost of Oappx should be much smaller than original.
10
AMA : not working
n 
Nightmare case
•  Anti-correlated or bad approximation
ensemble
ensemble
ensemble ensemble 11
Examples of covariant approximations
n 
Low mode approximation used in the Low Mode
Averaging ( LMA )
L. Giusti et al (2004), see also T. DeGrand et al. (2004)
accuracy control : # of eigen mode
1000
100
10
1
0
0.5
1
1.5
2
2.5
Deflation using low eigenmodes from
Lanczos
[alsoNeff
et
al,
JLQCD
] asfermion
We
describe
the
Dirac
operator
of domain-wall
fermion
4D even-odd
b
Wecan
can
also
describe
the Dirac
operator
of domain-wall
as 4D
n 
4D even/odd preconditioning
[ R. Arthur ]
DDW
!
"
"
M!
K(M4 )eo
5
=
, 4 )eo
M5M K(M
DDWK(M
= 4 )oe
,
5
K(M4 )oe
and the inverse of Dirac operator is given as 4D even-odd preconditioned form,
and the inverse!of Dirac operator is given
preconditio
" ! as 4D "even-odd
!
"
−1
−1
−1
DDW
=
Dee 0
1 −K(M4 )eo M5
"!
"!
−10
11 −K(M
0
0 0 1 Dee
1
0
! −1
−KM5 (M4 )1oe M5−1
−1
D
=
2
−1 −1 (M )
−1
DW
Dee = M5 − K−KM
(M4 )eo
4 oe M5
5 4 )oe
5 M(M
n 
n 
0
1
0
2
Polynomial accelerated
= solve
M5 the
−K
(M4 )ofeoD
M5−1by(M
and then we D
need
inverse
CG
method. The inverse of M is
ee to
4 )oe
Pn( H_DWF)
M (s,
= to
A solve
P +B
+ δ , of Dee by CG method. The inv
and then
wet)need
theP
inverse


With shift
m κ
m κ
m κ
··· m κ



 Bs,t
M5−1 (s, t) = As,t PR +
P
+
δ
,
−κ
m
κ
m
κ
·
·
·
m
κ
L
st
H-> H-c



1

 −κ
−κ
m
κ
L
Lκ−1 · · · Lm−2

A
= −

m
κ
m
κ
m
κ
·
f
f
f


1+m κ 
..
..
..
.. 
eigen Compression

L
L
−1
.
.
.
.



−κ
m
κ
m
κ
·
f
f
/ decompression
1 −κ 
−κ2
−κ
·m
·· κ
mL κ ·
 −κ
−κ
ee
−1
5
s,t
R
s,t
L
f
st
Ls
5
Ls −1
f
f
Ls
2
n 
M5
s,t
s
Ls
f
Ls −2
f
f
Ls −1
f
f s
f
Ls
sf
s
Ls −1
Ls −2
As,t = −

..
Bs,t = ATs,t , 1 + mf κLs 

.

1
L
s −1
κ =
.
−κ
5 − M5
ψ
=
v1 +
v2
H (ψ) = λ1 v1+λ2 v2
Bs,t = ATs,t ,
It has important property that the Γ
5
−κLs −2
3
s
Ls −3
..
.
2
f
fs
Ls
..
.
−κLs −3
·
(not γ5 ) Hermicity of Dee does not flip the
1 bases, and therefore Hermitian matrix is given
bases in contrast to 5D even-odd
κ =
.
multiplication of Γ5 ,
5 − M5
†
!"#$%"&'(&')"%*"+,-,".(
‡ /0('1'.$"&&(&')"%*"+,-,".(
(
(
(
(
(
‡ !"#(%"&'(&')"%*"+,-,".(
(
Examples of Covariant Approximations
(contd.)
n 
All Mode Averaging
AMA
Sloppy CG or
Polynomial
approximations
1000
100
10
1
0
1
0.5
1.5
2
2.5
accuracy control : •  low mode part #10,
of theemini-max
ig-­‐mode Figure 3: Polynomial
approximation
of 1/λ,
N : =
approximati
the relative error, for λ ∈ [0.05 , 1.67 ].
•  mid-­‐high mode : degree of poly.
poly
2
2
All mode approximation via sloppy CG
poly(lam) * lam vs lam
poly(lam) * lam vs lam
16x32x16, ml=0.01, raj 2200, nev=0
16x32x16, ml=0.01, traj 2200, nev=100
1.05
rsd = 1e-­‐2
1.05
rsd = 1e-­‐3
1
1
rsd = 10.95
e-­‐4
inv_poly.nev0_rsd1e-2_k14
inv_poly.nev0_rsd1e-3_k73
inv_poly.nev0_rsd1e-4_k296
inv_poly.nev0_rsd1e-6_k863
inv_poly.nev0_rsd1e-8_k1514
inv_poly.nev0_rsd1e-10_k2061
0.95
inv_poly.nev_rsd1e-2_k14
inv_poly.nev_rsd1e-3_k46
inv_poly.nev_rsd1e-4_k84
inv_poly.nev_rsd1e-6_k158
inv_poly.nev_rsd1e-8_k234
inv_poly.nev_rsd1e-10_k310
rsd = 1e-­‐6
0.9
0.9
0.001
0.0001
0.01
no eigenvector assists
1
0.01
0.0001
1e-06
1e-08
0.01
0.1
1
1
100 eigenvector assists
| poly(lam) * lam - 1 | vs lam
•  Conjugate residual ith sloppy convergence criteria, wml=0.01,
hich trajis enev=100
quivalent to | poly(lam) * lam
- 1 | vs w
lam
16x32x16,
2200,
16x32x16, ml=0.01, traj 2200, nev=0
construct a polynomial approximaRng 11/λ •  The starRng vector needs to be translaRon invariant to be a covariant approx. 0.01
•  low eigenvectors reduces the size of the dynamic range of 1/λ 0.0001
→ Beher approximaRon with smaller polynomial degrees err-inv_poly.nev_rsd1e-10_k310
err-inv_poly.nev_rsd1e-2_k14
1e-06
•  low λ region has larger r
elaRve e
rrors err-inv_poly.nev_rsd1e-3_k46
err-inv_poly.nev0_rsd1e-10_k2061
err-inv_poly.nev_rsd1e-4_k84
err-inv_poly.nev0_rsd1e-2_k14
err-inv_poly.nev_rsd1e-6_k158
•  One could employ other c
onstrucRon o
f p
olynomial a
pproximaRon f
or 1/λ, 1e-08
err-inv_poly.nev0_rsd1e-3_k73
err-inv_poly.nev_rsd1e-8_k234
err-inv_poly.nev0_rsd1e-4_k296
such as min-­‐max, conjugate rasidual1e-10
err-inv_poly.nev0_rsd1e-6_k863
Correlation
n 
NN propagator at short time-slice 18
Correlation
n 
NN propagator (LMA) at short time-slice 19
Correlation
n 
NN propagator (AMA) at short time-slice 20
AMA results for hadron 2pt functions
[ E. Shintani ]
Nucleon effective mass
using DWF
imp
O
O
(LMA,NG=32)
imp
O
(AMA,NG=32)
Effective mass
0.8
0.7
0.6
0
Point sink
Smeared sink
5
10
Time slice
15 0
5
10
Time slice
15 0
5
10
Time slice
15
Hadronic vacuum polarization( AsqTad )
0.2
0.2
0.18
0.18
0.16
2
-Π(Q )
0.16
0.14
0.14
0.12
0
0.2
0.4
0.1
0.08
0.06
0
2
4
6
2
2
Q (GeV )
8
10
TABLE IV. Computational cost. The unit of cost is one quark
propagator without deflated CG, per configuration. NG = 32
for nucleon masses and 708 for HVP. The last column gives
the cost to achieve the same error for each method, normalized
to [2] (nucleon mass mN ) and [1] (HVP) and scaled by the errors in Tab. III. HVP scaled costs are maximum and minimum
2
n  x 16 for
Nucleon
inDWF
the range
Q2 = mass
0 − 1 (M
GeV
. For m = 3fm)
0.005, in [2], nonPS=330MeV,
relativistic
spinors(MPS=470
were used which
scaled costs in
n  x 20 for
AsqTad HVP
MeV, means
5 fm) the(appx)
this case were increased by two. The cost of OG
for AMA
n  should be better for lighter mass & larger volume ?
is split to show the sloppy CG and low-mode costs separately.
Aoki, Rudy A
man Christ, T
Hashimoto, T
Kaneko, Chris
Ruth Van de
Jianglei Yu an
collaborations
CPS QCD lib
are used, wh
DOE SciDAC
the Japanese
22540301 (TI)
DOE grants D
92ER40716 (T
Research Cen
necessary for c
Cost comparison for test cases
(appx)
mN
AMA
LMA
Ref. [2]
AMA
LMA
Ref. [2]
HVP
AMA
LMA
Ref. [1]
a
Nconf Nmeas LM O OG
m = 0.005, 400 LM
110
1
213 18 91+23
110
1
213 18
23
932
4
- 3728
m = 0.01, 180 LM
158
1
297 74 300+22
158
1
297 74
22
356
4
- 1424
m = 0.0036, 1400 LM
20
1
96 11 504+420
20
1
96 11
420
292
2
- 584
-
Tot. scaled cost
gauss pt
350 0.063 0.065
254 0.279 0.265
3728a 1
1
693 0.203 0.214
393 0.699 0.937
1424
1
1
max min
1031 0.387 0.050
527 10.3 3.56
584
1
1
[1] C. Aubin,
arXiv:1205
[2] T. Yamaza
et al., Phy
[hep-lat].
[3] E. Shinta
Variants of CAA
n 
CAA (Covariant Approximation Averaging)
•  Name
approximation,
approximation accuracy control
•  LMA (Low Mode Averaging)
• 
• 
• 
low mode approx of propagator,
# of eigen vectors
AMA (All Mode Averaging),
low mode (optional)+Polynomial approx,
(# of eigenV) Polynomial degree
(also other type of minimization)
Heavy quark averaging [T. Kawanai]
heavier mass quark prop as an approx of light prop
quark mass
?????
Other Examples of Covariant
Approximations
n 
Less expensive (parameters of) fermions :
• 
• 
• 
• 
n 
n 
Larger mf
Smaller Ls DWF
Mobius
even staggered or Wilson …..
Different boundary conditions
More than one kinds of approximation
(c.f. multi mass Hasenbushing)
Strongly depends on Observables / Physics (YMMV)
Would work better for EXPENSIVE observables and/or
fermion, potentially a game changer ?
reduces the dispersion of rest part. Also I plot the parameter t and q. These parameters satisfy the
relation r = q + 1. We need r > 1/2 to reduce the statistical error in final analysis.
Finally I show the results for the effective mass in the Fig.3. and Table 1. The results for improvement are consistent with original ones within error bars and its statustical error bars are slightly
smaller than original ones. Actually approximation plotted as green points is corresponding to Bs
meson in this test. The ratio of dispersion for improvement to original one is also plotted as gray line
on the right axis. Fitting results are shown in Table 1. As a result, the improvement finaly gives 82 %
statistical error of original. This improvement rate is corresponding to statistics increased by half. As
24^3x64x16, 20 config , an experiment, I have tried to extrapolate the data by using the improvement and approximate one.
moff=0.01 (target) mtechniques
f=0.04 “approximaRon”
The advantage
this error
reduction
using heavier quark propagator is what approximate
is used to extrapolate to physical point. Finally I show the results for the effective mass in the Fig.3.
Larger mass as CAA
[ Taichi Kawanai ]
3.14
100
3.1
75
3.08
50
3.06
25
3.04
0
3.02
3.04
4
6
8
10
12
time slice [lattice unit]
14
16
improvement
original
3.1
3.08
mass
effective mass
3.12
3.12
original
improvement
approxmation
3.06
18
0
0.5
1
1.5
2
Figure 3: (left) The effective mass plots. The ratio of dispersion for improvement to original one is
also plotted as gray line. (right)Extrapolation
and Table 1.
3
Estimation for error reduction per cost.
Summary
n 
CAA , LMA, AMA, …. : Class of Statistical error reduction technique
•  AMA is a valence version of the Hasenbush trick
•  AMA could improve existing data easily
1.  Do Full CG for selected config / source
(existing data : This expensive part is already done )
2. 
3. 
Find a good approximation (accuracy of sloppiness / number of eigenvalue) that reproduce your
exact CG result by, say, 95%
(mathematically find a strongly correlated approximation, R(corr) > 0.5 )
Subtract the approx obs with same source location as full CG
4. 
Perform many source location using approx obs, average, add back
You could use other config.
l 
l 
Your Millage May Varies….
Home Work : find a good / cheap / funny approximations
AMA in USQCD Static-light
[ PI Tomomi Ishikawa ] 16^3x64x16, 20 conf, 100 eigenvectors
1
40
O
O - Oapprox
0.5
Oapprox src summed
20
10
O - Oapprox
0.5
0
Oapprox
30
Oapprox src summed
20
10
Oapprox
Oapprox
0
O - Oapprox
Oapprox
error normalized by O [%]
O - Oapprox
1
portion of 2pt function
error normalized by <O> [%]
portion of 2pt function
30
5
10
15
t
0
0
0
5
10
t
LMA
15
0
5
10
15
t
0
0
5
10
t
AMA
15
3pt function [ E. Shintani ]
n 
Application to the form factor measurement
•  CP-even and CP-odd nucleon EM form factor
•  Complicated structure in the ratio method
Cf. Yamazaki et al., PRD79, 114505 (2009) Ratio has complicated combination of both low and high
mode,
so AMA has more advantage than LMA even if AMA need larger
cost.
30
LMA
AMA
q2 GeV2
Ge (LMA)
Ge (AMA)
0.0
0.96(11)
0.98(3)
0.198
0.72(12)
0.73(3)
0.382
0.58(10)
0.56(3)
0.574
0.48(10)
0.45(2)
0.733
0.52(12)
0.44(3)
StaRsRcal error of AMA is about 3-­‐-­‐5 Rmes smaller than LMA.
31
Comparison of isovector F1,2
[ E. Shintani ]
m=0.01
m=0.01
•  Results are well consistent with full staRsRcs. •  StaRsRcal error is much reduced in AMA rather than LMA. •  Compared to full staRsRcs, AMA results (m=0.01) have sRll 1.2 -­‐-­‐ 1.5 Rmes larger staRsRcal error (except for F1(0)). •  This may be due to correlaRon between different source points. 32
CP-odd part
n 
Nucleon 2pt function with θ reweighting
•  Q is topological charge.
•  α which is CP-odd phase is necessary to extract EDM form
• 
• 
factor.
It is good check of applicability of LMA/AMA to CP-odd sector.
Effective mass plot shows the consistency of the above formula
33
CP-odd part [ E. Shintani ]
•  There is good plateau in AMA, and this figure actually shows CP-­‐odd part has consistent exponent with CP-­‐even(nucleon mass) part as expected. •  CP-­‐odd part has both contribuRon from high and low lying mode. •  AMA works well even in CP-­‐odd sector ! 34
-1
-1.2
Nucleon Magnetic formfactor
-1.4
-1.6
0
2
4
6
8
10
12
t
Gm at m=0.01 for N
Gm at m=0.01 for N
0
q22=0.202
q =0.395
q22=0.583
-0.2
0
GeV22
GeV
GeV22
-0.2
q =0.753 GeV
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
-1
-1
-1.2
-1.2
-1.4
-1.4
-1.6
-1.6
0
2
4
6
8
10
t
q22=0.200 GeV22
q =0.388 GeV
q22=0.569 GeV22
q =0.721 GeV
12
0
2
4
6
8
10
12
t
AMA
Original C
G
Figure
7:
G
for
neutron
at
m
=
0.01.
(Top)
Original, (Middle) LMA
m
G at m=0.01 for N
m
0
-0.2
q22=0.199 GeV22
q =0.389 GeV
q22=0.576 GeV22
q =0.731 GeV
N, m=0.01, point sink
N, m=0.005, point sink
Full staRsRcs
LMA [7,15]
m = 0.01
AMA [7,15]
staRsRcs
Full staRsRcs (Gaussian sink)
0.712(16) 0.710(5) Nconf=80, N’mes=32 0.703(4), Nconf = 356, Nmes=4
m = 0.005 0.673(22) 0.666(13) Nconf=26, N’mes=32 0.663(4), Nconf = 932, Nmes=4
Yamazaki et al., PRD79, 114505 (2009) 36
Cost (in the case of 24cube m=0.01)
Use of unit of quark propagator “prop” in full CG w/o
deflation
Yamazaki et al., PRD79, 114505 (2009) n  Case of full statistics
In Nconf = 356, Nmes=4,
Total : 356×4 = 1424 prop
n 
Case of AMA w/o deflation
Since calculation of Oappx need 1/50 prop, then in Nconf=81,
N’mes=32
Total : 80 + 80×32/50 = 131 prop ⇒ 10 times fast
n 
Case of AMA w/ deflation
When using 180 eigenmode, calculation of Oappx need 1/80 prop,
but in this case the calculation of lowmode is ~1 prop/configs.
Deflated CG makes reduction of full CG to 1/3 prop, then
Total : 80/3 + 80×32/80 + 80 = 138 prop ⇒ 10 times fast
Note that stored eigehmode is useful for other works.
37
Correlation
n 
NN propagator at long time-slice 38
Correlation
n 
NN propagator (LMA) at long time-slice 39
Correlation
n 
NN propagator (AMA) at long time-slice 40
Other technical details
n 
n 
n 
n 
Implicitly Restarted Lanczos with Polynomial
acceleration and spectrum shifts for DWF and
staggered in CPS++ [ E. Shintani, T. Blum, TI ].
Eigen Vector compression / decompression
Sea Electric Charge is now controlled by QED
reweighting
[ T. Ishikawa et. al. arXiv:1202.6018 ]
Aslash-SeqSrc method