GPU-Accelerated Computation for Statistical Analysis of the Next-Generation Sequencing Data
GPU computation
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Transcript of GPU computation
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Efficient simulation of quantum dynamics system using GPU
Lipeng Chen
Division of Materials Science
Nanyang Technological University, Singapore
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Schrodinger Equation Liouville equation
1. MCTDH
2. propagator
1. QUAPI
2. HEOM
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Exciton diffusion in chlorosome antennas
Single molecule spectra
Dissipative effect on the dynamics at a conical intersection
Polaron dynamics of Holstein model
Dynamics of sub-Ohmic spin boson model
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Natural Light Harvesting Complexes
Fenna-Matthews-Olson (FMO)Trimer complex in green sulfur bacteria 4
Light harvesting system in purple bacteriaLHCII in green plants
Chlorosomes antennas in green sulfur bacteria
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5
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Exciton diffusion in chlorosomes antennas
6S. Ganapathy et al, PNAS. 2009, 106 , 8525
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Variational Parameters
7
Quantum state of the system can be approximated, by the Davydov D1 Ansatz
SchrÖdinger Equation
Dirac-Frenkel time-dependent variational method
Equations of motion for the variational parameters
( ) ( )i t H tt
( ) ( )2iL t H t
t
�
( ) 0m
m
d L Ldt
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SMX: Higher performance and efficiency increase processing cores reduce control logic
Dynamic parallelism: spawns new threads without going back to the CPU
Hyper-Q: increases GPU utilization provide streams access to 32 independent hardware work queues or MPI ranks.
KEPLER - The World‘s fastest, most efficient HPC architecture
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30 rings of 18 sites each540 sites in total
Phonon bandwidth: W=0.5Coupling: g=0.4
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18 rings with 18 site each (324 sites)
Coherence size: 175 sites
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1
( )
,
20
( ) ( , ) 0 0
( )( ) [coth (1 cos ) (sin )]2
0 ( )
S
S
H tg tph ph
n m
B
iH tph D
R t e C n m n e m
Dg t d t i t tk T
e n t
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Single molecule spectra
R Hildner, D Brinks, J. B. Nieder, R. J. Cogdell, N F van Hulst, Science 340, 1448 (2013)
Ultrafast phase coherent excitation of individual LH2 complexes
Emission of single LH2 complexes as a function of delay time
Histogram of the oscillation period T
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Single molecule spectra
1
2
†
1 2
1
( )2
[ ( ) ( ) ]( ) ( ) ( )
( ) ( )
( ) ( )
F
i t
i t
H t X t Xt t t
t E t e
t E t e
Field-matter interaction Hamiltonian
Liouville Equation
( ) [ , ]Fd t i H Hdt
( ) { ( )}
( ) 2 ( ) ( )
P t Tr X t
S Im dt t P t
2
1
( ) ( )kk
k
S S
2
1, 1
1 1 10
( )
,
20
( ) ( )
2Re ( ) ( ) ( )
( ) ( , ) 0 0
( )( ) [coth (1 cos ) (sin )]2
S
aba b
ab a b
H tg tph ph
n m
B
S
dt dt t t t R t
R t e C n m n e m
Dg t d t i t tk T
1
1
†
0 ( )
( ) ( ) exp [ ( ) . .] 0
SiH tph D
D n nq q phn q
e n t
t t n t b H c
D1 dynamics
1 1( ) ( ) ( )
2
( ) 0
( ) 0
D S D
nn
nqnq
iL t H tt t
d L Ldt
d L Ldt
� � � � � � � � � � � � � �
Pump pulse 1Pump pulse 2
Signal
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Room temperature LHCII absorption spectra
Single molecule signal for 3 values of the exciton-phononcoupling strength (decreasing)
same signals but calculated without the heat bath
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the effect of the single molecule orientation The effect of disorder
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Effect of a dissipative environment on the dynamics at a conical intersection
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Polaron dynamics of Holstein model
J: transfer integralN: number of sitesϕ : off-diagonal couplingg: diagonal couplingW: phonon band width
A linear dispersion phonon band is assumed as
0
21 1q
qW
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Multitude of Davydov D1 and D2 trial states
1D t 2D t
† †2 ,
0
0 ( )exp 0M
Mn i n iq q iq qex ph
n i q
D a t b b
† †1 ,
0
0 ( )exp 0M
Mn i n inq q inq qex ph
n i q
D a t b b
Multi-D2 and multi-D1 Ansatz with the multiplicity number M
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Measurement of the validity for the trial state
1,2( ) ( )
Mtt i H D
t t t
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Multi-D1 results in the diagonal coupling case
Case of J=0.5. W=0.5, g=0.1 and ϕ=0
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Multi-D2 results in the diagonal coupling case
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In the off-diagonal coupling case with g=W=J=0 and ϕ≠0
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1,2( ) ( )
( )=max ( )
Mt
ph
t i H D
t t t
tE t
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Reduced density matrix
Feynman-Vernon influence functional
HEOM treatment of Holstein model
Auxiliary density matrix
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1
N
q trunq
m N
Terminator
Ntrun
N
3 4 5 6 7 8 9 10
6 455 1820 6188 18564 50388 125970 293930
8 969 4845 20349 74613 245157 735471
9 1330 7315 33649 134596
10 1771 10626 53130 230230
12 2925 20475 118750 593770
16 6545 58905 435897
Total number of hierarchy
1 !! 1 !
trun trun
trun
N N NN N N
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0.2, 0.5, 0.1, 0J W g
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0.5, 0.5, 0.1, 0J W g
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0.2, 0.5, 0.2, 0J W g
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0.5, 0.5, 0.2, 0J W g
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Dynamics of sub-Ohmic spin-boson model
Δ: the tunneling constantε: the spin bias λl: the coupling strength of the spin to the bath
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Multi-D1 results v.s D1 results Sub-Ohmic bath s=0.25 Δ=0.1 Relative error
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Multi-D1 compare with HEOM
Factorized initial condition bath
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Multi-D1 compare with HEOM
Factorized initial condition bath
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Multi-D1 compare with PIMC
Polarized initial condition bath
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Entropy of the system
Factorized initial condition
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Some useful libraries
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