IEE/CEEM Seminar: Vidvuds Ozolins and Sparse Physics and its Applications to Energy Materials

Vidvuds Ozolins Department of Materials Science & Engineering

University of California, Los Angeles

UCLA Laboratory for the Quantum Prediction of Advanced Materials

Supported by:

Sparse physics via compressive sensing

DMR-1106024

Gus Hart (BYU Provo)

Lance Nelson (BYU Idaho)

Fei Zhou (LLNL) Weston Nielson

(UCLA)

Rogjie Lai (UC Irvine)

Russ Caflisch (UCLA)

Stan Osher (UCLA)

Co-conspirators

Outline

Premise: •  Most physics models are approximately sparse

(i.e., have a few terms) in some basis

Questions: •  How do we pick the basis? •  How do we determine which terms to pick? •  How should we generate fitting data? •  How do we systematically improve accuracy?

High-rate charge storage in Nb2O5

V. Augustyn, J. Come, M. A. Lowe, J. W. Kim, P.-L. Taberna, S. H. Tolbert, H. D. Abruña, P. Simon, B. Dunn, Nature Materials 12, 518–522 (2013)

Structure of orthorhombic Nb2O5

Acta Cryst. B31, 673 (1975)

Layered structure (stacked along c) 001 layer with octahedra and pentagonal bipyramids

Atomic disorder in Nb2O5

•  Nb (8i): 16 Nb off-center in the c direction

Nb layer

•  Nb (4g) partial occupancy: 0.8 Nb between Nb layers

•  O: distorted close packing within {001} layers

Lithium energy landscape in the (001) plane

Oxygen

C.-P. Liu, F. Zhou, and V. Ozolins (2013)

Li intralayer diffusion map

50 meV

> 700 meV


Configurational problem for Nb2O5

Occupation Si: Vacancy, Li, or Nb


Configurational Ordering in Alloys

Cu3Au CuAu CuAu3

All these structures are based on the face-centered cubic (FCC) lattice.

Cluster Expansion

Rewrite as an expansion in clusters of lattice sites:

= + +

+ + ...

+

ADVANTAGES: ü Very fast – can be used in Monte Carlo simulations ü Works for any structure based on a given lattice

E = E0 + J1c + J f Sii∈f∏

f

Pairs, Triplets, ...

∑

Cluster selection is difficult

0.5 1.0 1.5 2.0 2.5 3.01.

10.

100

1000

10 000

cluster radius

pairs

triplets

4-bodies5-bodies

Compressive sensing CE

=

?0

BBBBBBBBBB@

JnnJnnnJtrip.....

1

CCCCCCCCCCA

. . . .0

BB@

�1 1 . . . . . .0 �1 . . . . . .. . . . . . . .. . . . . . . .

1

CCA

0

BBBBBB@

E( )E( )E( )

.

.

.

1

CCCCCCA

JCS = argminJ

J 1 : ΠJ − E 2 ≤ ε{ }

Clusters

Stru

ctur

es Si

i∈f∏

L. J. Nelson, F. Zhou, G.L.W. Hart, and V. Ozolins, Phys. Rev. B 87, 035125 (2013)

Why compressive sensing works

x* = argmin x: Ax=b x 1

x 1 = xii=1

n

∑

x1

x2

a11x1 + a12x2 = b1

Why Euclidean distance is worse

x* = argmin x: Ax=b x 2

x1

x2

a11x1 + a12x2 = b1

x 2 = xi2

i=1

n

∑

Theorem

E. Candès, J. Romberg, and T. Tao, Comm. Pure Appl. Math., vol. 59, no. 8, pp. 1207–1223 (2006).

M ≥C ⋅µ2 ⋅S ⋅ log N δ( )Coherence

1≤µ≤√N Sparsity Size of basis set

P >1−δ

Probability to find the correct S-sparse solution is

if the number of data points satisfies

Ag-Pt cluster expansion

Pairs Triplets Quadruplets

Cluster radius

L. J. Nelson, F. Zhou, G.L.W. Hart, and V. Ozolins, Phys. Rev. B 87, 035125 (2013).

Ag-Pt cluster expansion

Pairs Triplets Quadruplets

Cluster radius


Performance of CSCE

Training set size

RMSError(meV)(solidline)

kJfitk

1(D

ashed

Line)

Compressive sensing

Discrete Opt.


Next step – Bayesian CSCE

L. J. Nelson, G. L. W. Hart, S. Reese, F. Zhou, and V. Ozolins, Physical Review B 88, 155105 (2013).

CSCE results for zinc-finger protein

-30 -20 -10 0 10

-30

-20

-10

0

10

Direct energy HkcalêmolL

PredictedenergyHkca

lêmolL


Direct conversion of heat to electricity

Up to 30% conversion efficiency with right materials

Thermopower S = ΔV/ΔT

hot cold

ZT =σ ⋅ S2

κ total

•T

σ ⋅ S 2Power factor

Total thermal conductivity

electrical conductivity thermopower

ZT =σ ⋅ S2

κ total

•T

σ ⋅ S 2Power factor

Total thermal conductivity

electrical conductivity thermopower

PbTe: Standard of excellence

From Wei & Zunger, Physical Review B 55, 13605–13610 (1997)

(direct gaps at L point) result from the occurrence of the Pb s band below the top of the valence band, setting up coupling and level repulsion at the L point.

Image from Pei et al. Nature 473, 66–69 (2011)

Giant anharmonicity of PbTe TO mode

Chemical compound space

PbTe IV-VI

Group I + V Group VI

I-V-VI2

AgSbTe2, NaSbSe2, …

M. Nielsen, V. Ozolins, and J. P. Heremans, Energy & Environ. Sci. 6, 570-578 (2013).

New materials: Search space

ABX2 in cubic D4 (AF-IIb) for A=Cu, Ag, Au ABX2 in rhombohedral R-3m (AF-II) for A=Na, K, Rb, Cs, Tl Screened a total of 8 × 3 × 3 = 72 compounds


4

3

2

1

0

Grun

eise

n pa

ram

eter

a

AuAs

Te2

CuBi

Se2

AuBi

Te2

CuAs

Te2

AgAs

Te2

AgSb

Se2

CuBi

Te2

AuSb

Te2

NaSb

S2KA

sSe2

AgBi

S2Cu

SbTe

2Ag

BiSe

2Ag

BiTe

2Ag

SbTe

2Na

AsSe

2Cs

SbSe

2Rb

SbSe

2Na

SbSe

2KS

bSe2

CsBi

S2Rb

BiS2

NaAs

Te2

NaBi

S2KB

iS2

NaSb

Te2

KAsT

e2Na

BiSe

2Rb

AsTe

2Cs

AsTe

2Rb

BiSe

2Cs

BiSe

2Na

BiTe

2KB

iSe2

RbSb

Te2

KSbT

e2Cs

BiTe

2Rb

BiTe

2KB

iTe2

CsSb

Te2

Calculated Gruneisen parameters


Experimental results from OSU


Cu12Sb4S13: Thermoelectric mineral with ZT=1

Cu

S

Sb

X. Lu, D. T. Morelli, Y. Xia, F. Zhou, V. Ozolins, H. Chi, and C. Uher. Advanced Energy Materials (2013)

Natural tetrahedrite mineral

!!!!!!!!!!!!!!!!!!!!!!!!!! !

!!!1 cm

Figure of Merit

0

0.2

0.4

0.6

0.8

1

1.2

300 400 500 600 700 800

Temperature (K)

Figu

re o

f Mer

it zT

a)

!

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Brillouin Zone Occupation Fraction

Figu

re o

f Mer

it zT

b)

f =1 for Zn (2+) f =2 for Fe (3+)

Lu, Morelli, Xia, Zhou, Ozolins, Chi, Zhou, and Uher, Advanced Energy Materials (2013).

Frustrated Cu(1) bonding environment

Cu(1) (3-fold)


Anharmonic optical mode

-0.4 -0.2 0.0 0.2 0.4

-10

0

10

20

30

40

50

uCu HfiL

DEHme

VL


CS lattice dynamics

Taylor expansion of the total energy in terms of the atomic displacements ua=Ra-R0

a:

� = �0 + �aua +1

2�abuaub +

1

6�abcuaubuc + · · ·

Fa = �@�/@a = �(�a + �abub +1

2�abcubuc + · · · )

Forces:

Use for: Thermal transport, free energies, phase transformations, etc.

Expansion in cluster series

Φaua, Φaaua2, Φaaaua

3, …

Φabuaub, Φaabua2ub, …

Φabcuaubuc , Φaabcua2ubuc , …

Φabcduaubucud, …

…

=

…

How to calculate FCT’s from DFT?

Calculate forces Fa Displace atoms ua

Fa = �@�/@a = �(�a + �abub +1

2�abcubuc + · · · )Fa = �@�/@a = �(�a + �abub +

1

2�abcubuc + · · · )

Compressive sensing

ΦCS = argminΦ

µ Φ 1 +12

ϒΦ + F 22

1 ub1 ub

1uc1

1 ub2 ub

2uc2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

Φa

Φab

Φabc

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟=

−Fa1

−Fa2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

We need to solve an underdetermined linear system of equations:

•  Commutativity of derivatives

•  Translational invariance (Noether’s theorem)

Symmetry constraints

Dynamical properties of solids, eds. G.K. Horton & A.A. Maradudin (1974)

�abc = �acb = . . .

X

a

�I({a, b, c · · · }) = 0

�(aa) = �X

b 6=a

�(ab) Pair ASR

Multibodies

•  Space group symmetry

•  Transformation matrix:

•  Reduces ## of FCT’s by a factor of 10 in cibic crystals

Symmetry constraints

Dynamical properties of solids, eds. G.K. Horton & A.A. Maradudin (1974)

F2 F3

u1

�I(s↵) = �IJ(s)�J(↵)

�IJ(s) = �i1j⇡(1)

· · · �inj⇡(n)

γ =cosα −sinα 0sinα cosα 00 0 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

The final CSLD L1 problem

ΦCS = argminΦS

µ Φ 1 +12

ϒΦ + F 22⎛

⎝⎜⎞⎠⎟

DFT data constraints are imposed via a regular least-squares norm:

where Φ = BTTrans BS

Rot BP

PermΦS.

are symmetry-distinct FCT components. ΦS

100 200 300 400 500 600

-0.10

-0.05

0.05

0.10

0.15

F

•  Solution is sparse •  Force prediction error = 3%

2 3 4 5 6

DFT tests: NaCl

Independent FC’s by order, then by distance

-15 -10 -5 0 5 10 15 20-15

-10

-5

0

5

10

15

20

FDFT @eVêfiD

F CSLD@eVêfiD

Δu/r0=25%

Applications

Performance of CSLD

Applications

Sparsity for PDEs

Classically, sparsity is limited to discrete objects. [Questions] •  What is an analog of the “sparsity” for continuous

functions? •  How to design a tractable method to create

“sparse” continuous functions?

L1 regularization of DFT

E = minψ i{ }i=1

Nψ j H ψ j

j=1

N

∑ s.t. ψ i ψ j = δ ij

Conventional Kohn-Sham problem:

This method converges to spatially localized wave f-ns with compact support. O(N) becomes possible.

′E = minψ i{ }i=1

Nψ j H ψ j

j=1

N

∑ + 1µ

ψ j 1j=1

N

∑⎛⎝⎜

⎞⎠⎟

s.t. ψ i ψ j = δ ij,

“Sparse” = localized or short-ranged

ψ 1 ≡ ψ (x) dx, x ∈Rd∫

V. Ozolins, R. Lai, R. E. Caflisch, and S. Osher, “Compressed modes … ,“ PNAS (2013)

Results

Results


Dependence on µ

0 5 10 15 20 25 30 35 40 45 50

0

0.2

0.4

0.6

!1

!2

!3

!4

!5

µ = 300 5 10 15 20 25 30 35 40 45 50

0

0.2

0.4

0.6

!1

!2

!3

!4

!5

µ = 50

0 5 10 15 20 25 30 35 40 45 50

0

0.2

0.4

0.6

!1

!2

!3

!4

!5

µ = 500 5 10 15 20 25 30 35 40 45 50

0

0.2

0.4

0.6

!1

!2

!3

!4

!5

µ = 300

0 5 10 15 20 25 30 35 40 45 50

0

0.2

0.4

0.6

!1

!2

!3

!4

!5

µ = 5000 5 10 15 20 25 30 35 40 45 50

0

0.2

0.4

0.6

!1

!2

!3

!4

!5

µ = 5000

Fig. 4. Computation results of CMs with di↵erent values of µ. The first column: the first 5 CMs of the 1D free-electron model. The second column: the first 5 CMs of the1D Kronig-Penny model.

electrons in a one-dimensional crystal, where the potentialfunction V (x) consists of a periodic array of rectangular po-tential wells. For simplicity, in our experiments we replace therectangular wells with inverted gaussians so that the poten-

tial is given by V (x) = �V0P

Nelj=1 exp

h� (x�xj)

2

2�2

i. We choose

Nel = 5, V0 = 1, � = 3 and xj

= 10j in our discussion be-low and, in spite of the di↵erent potential, continue to referto this case as the 1D KP model. This model exhibits twolow-energy bands separated by finite gaps from the rest of the(continuous) eigenvalue spectrum, and the Wannier functionscorresponding to these bands are exponentially localized.

In our experiments, we choose ⌦ = [0, 50] and discretize⌦ with 128 equally spaced nodes. The proposed variationalmodel Eq. [4] is solved using algorithm 1, where parametersare chosen as � = µN/20 and r = µN/5. We report the com-putational results of the first 5 CMs of the 1D free-electronmodel (the first column) and the 1D KP model (the secondcolumn) in Figure 4, where we use 5 di↵erent colors to dif-ferentiate these CMs. To compare all results more clearly, weuse the same initial input for di↵erent values of µ in the free-electron model and the 1D KP model. We flip the CMs ifnecessary such that most values of CMs on their support arepositive, since sign ambiguities do not a↵ect minimal valuesof the objective function in Eq. [4]. For comparison, Figure 3plots the first 5 eigenfunctions of the Schrodinger operatorused in the free-electron model and KP model. It is clearthat all these eigenfunctions are spatially extended withoutany compact support. However, as we can observe from Fig-ure 4, the proposed variational model does provide a series ofcompactly supported functions. Furthermore, all numericalresults in Figure 4 clearly show the dependence of the size of

compact support on µ, as suggested by general considerationsbased on the variational formula Eq. [4]. In other words, themodel with smaller µ will create CMs with smaller compactsupport, and the model with larger µ will create CMs withlarger compact support. In addition, we find that the result-ing compressed modes are not interacting for small µ (the firstrow of Figure 4). By increasing µ to a moderate value, themodes start to interact each other via a small amount of over-lap (the second row of Figure 4). Significant overlap can beobserved using a big value of µ (the third row of Figure 4).

0 5 10 15 20 25 30 35 40 45 50

!0.2

!0.1

0

0.1

0.2

0.3

!1

!2

!3

!4

!5

0 5 10 15 20 25 30 35 40 45 50

!0.2

!0.1

0

0.1

0.2

0.3

!1

!2

!3

!4

!5

Fig. 3. The first 5 egienfunctions of the Schrodinger operator H in the free-electron model (top) and the KP model (bottom).

4 www.pnas.org/ Footline Author


Total energy convergence

µ = 10,M = 50, N = 50 µ = 10,M = 50, N = 60 µ = 10,M = 50, N = 128

0 10 20 30 40 50!0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Eigs of !T H !

Eigs of H

0 10 20 30 40 50!0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Eigs of !T H !

Eigs of H

0 10 20 30 40 50!0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Eigs of !T H !

Eigs of H

0 10 20 30 40 50!1

!0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Eigs of !T H !

Eigs of H

0 10 20 30 40 50!1

!0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Eigs of !T H !

Eigs of H

0 10 20 30 40 50!1

!0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Eigs of !T H !

Eigs of H

Fig. 5. Comparisons of the first 50 eigenvalues of the 1D free electron model (the first row) and the 1D Kronig-Penney model (the second row).

10 50 100 200 300 400 5000

0.02

0.04

0.06

0.08

0.1

µ

Re

lative

Err

or

M = N = 50

Potential free

Kronig!Penney

50 60 70 80 90 100 1280

0.02

0.04

0.06

0.08

0.1

N

Rela

tive E

rror

M = 50, µ = 10

Potential free

Kronig!Penney

Fig. 6. Relative eigenvalue error of the 1D free-electron model (red dots) and 1DKP model (blue circles). Top: relation of the relative error via di↵erent values of µfor fixed M = N = 50. Bottom: relation of the relative error via di↵erent valuesof N for fixed µ = 10 and M = 50.

We further test Conjecture 1 numerically, i.e., uni-tary transformation of the derived compactly supported

compressed modes can represented the eigenmodes of theSchrodinger operator. We compare the first M eigenvalues(�1, · · · ,�M

) of the matrix h T

N

H N

i obtained by the 1DKP model and 1D free-electron model with the first M eigen-values (�1, · · · ,�M

) of the corresponding Schrodinger oper-ators. Figure 5 illustrates the comparisons with a relativesmall value µ = 10, when the CMs are highly localized. Wecan clearly see that {�

i

} gradually converges to {�i

} withincreasing number N of CMs. In addition, we also plot therelative error E =

PM

i=1(�i

��i

)2/P

M

i=1(�i

)2 in Figure 4. Aswe speculated in conjecture 1, the relative error will convergeto zero as µ ! 1 for fixed M = N = 50, which is illustratedin the top panel of Figure 4. The relative error will also con-verge to zero as N ! 1 for fixed µ = 10 and M = 50, whichis illustrated in the bottom panel of Figure 4.

Moreover, the proposed model and numerical algorithmalso work on domains in high dimensional space. As an ex-ample, Figure. 7 shows computational results of the first 25CMs of the free-electron case on a 2D domain [0, 10]2 withµ = 30. All the above discussions of 1D model are also truefor 2D cases. In addition, our approach can also be naturallyextended to irregular domains, manifolds as well as graphs,which will be investigated in our future work.

In conclusion, the above numerical experiments validatethe conjecture that the proposed CMs provide a series of com-pactly supported orthonormal functions, which approximatelyspan the low-energy eigenspace of the Schrodinger operator(i.e., the space of linear combinations of its first few lowesteigenmodes).

Footline Author PNAS Issue Date Volume Issue Number 5


Higher dimensions, etc.

Discussions and ConclusionsIn conclusion, we have presented a method for producing com-pressed modes (i.e., modes that are sparse and spatially local-ized with a compact support) for the Laplace operator plus apotential V , using a variational principle with an L1 penal-ization term that promotes sparsity. The trade-o↵ betweenthe degree of localization and the accuracy of the variationalenergy is controlled by one numerical parameter, µ, withoutthe need for physical intuition-informed spatial cuto↵s. TheSOC algorithm of Ref. [13] has been used to numerically con-struct these modes. Our tests indicate that the CMs can beused as an e�cient, systematically improvable orthonormalbasis to represent the low-energy eigenfunctions, energy spec-trum of the Schrodinger operator. Due to the fact the CMsare compactly supported, the computational e↵ort of total en-ergy calculations increases linearly with the number of modesN , overcoming the O(N3) orthogonalization bottleneck lim-iting the performance of methods that work by finding theeigenfunctions of the Schrdinger operator.

In addition, note that the discretized variational principlein Eq. [13] is related to sparse principal component analysis(SPCA) (Ref. [18, 19]). SPCA, however, does not involvean underlying continuum variational principle and the sparseprincipal components are not localized, since the componentnumber does not correspond to a continuum variable.

These results are only the beginning. We expect that CM-related techniques will be useful in a variety of applicationsin solid state physics, chemistry, materials science, and otherfields. Future studies could explore the following directions:

1. Use CMs to develop spatially localized basis sets that spanthe eigenspace of a di↵erential operator, for instance, theLaplace operator, generalizing the concept of plane wavesto an orthogonal real-space basis with multi-resolution ca-pabilities. More details will be discussed in Ref. [20].

2. Use the CMs to construct an accelerated (i.e., O(N)) sim-ulation method for density-functional theory (DFT) elec-tronic structure calculations.

3. Construct CMs for a variety of potentials and develop CMsas the modes for a Galerkin method for PDEs, such asMaxwell’s equations.

4. Generalize CMs for use in PDEs (such as heat type equa-tions) that come from the gradient descent of a variationalprinciple.

5. Extend CMs to higher dimensions and di↵erent geometries,including the Laplace-Beltrami equation on a manifold anda discrete Laplacian on a network.

Finally, we plan to perform an investigation of the formalproperties of CMs to rigorously analyze their existence andcompleteness, including the conjecture that was hypothesizedand numerically tested here.

Fig. 7. The first 25 CMs of freel-electron case on a 2D domain [0, 10]2 withµ = 30. Each CM is color-coded by its height function.

ACKNOWLEDGMENTS. V.O. gratefully acknowledges financial support from theNational Science Foundation under Award Number DMR-1106024 and use of com-puting resources at the National Energy Research Scientific Computing Center, whichis supported by the US DOE under Contract No. DE-AC02-05CH11231. The re-search of R.C. is partially supported by the US DOE under Contract No. DE-FG02-05ER25710. The research of S.O. was supported by the O�ce of Naval Research(Grant N00014-11-1-719).

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6 www.pnas.org/ Footline Author


Compressed plane waves

ψ CPW (x) = argminψ

− 12ψ Δψ + 1

µψ 1

⎛⎝⎜

⎞⎠⎟

s.t. ψ (x)ψ (x − jw) = δ0 j

Set V=0 to derive general compactly supported basis for the Laplacian:

V. Ozolins, R. Lai, R. E. Caflisch, and S. Osher, “Compressed plane waves … ,“ UCLA CAM Report (2013)



Introduction Compressed Modes (CMs) Compressed Plane Waves (CPWs)

Numerical algorithm and results

The first six basic CPWs

0 10 20 30 40 50 60 70 80 90 100!2

0

2

4

6

1

0 10 20 30 40 50 60 70 80 90 100

!5

0

5

2

0 10 20 30 40 50 60 70 80 90 100

!5

0

5

3

0 10 20 30 40 50 60 70 80 90 100

!5

0

5

4

0 10 20 30 40 50 60 70 80 90 100

!5

0

5

5

0 10 20 30 40 50 60 70 80 90 100

!5

0

5

6

Figure: From top to the bottom, the first six modes 1, 2, 3, 4, 5, 6 obtained by Eq. (10 -11)using L = 100, µ = 5, w = 5.

V. Ozolins & R. Lai & R. Caflisch & S. Osher Compressed Modes for Variational Problems in Math. & Phy.



Introduction Compressed Modes (CMs) Compressed Plane Waves (CPWs)

Numerical algorithm and results

Spectral density distribution

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

0.2

0.4

0.6

0.8

1

Wave vector G

|!i (G

)|2

!1

!2

!3

!4

!5

!6

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

0.2

0.4

0.6

0.8

1

Wave vector G

!i=

1

6|"

i (G)|

2

Figure: Spectral density distribution of CPWs. Top: the spectral density distribution of 1, 2, 3, 4, 5, 6. Bottom: The total spectral density distribution of the first four modes.

V. Ozolins & R. Lai & R. Caflisch & S. Osher Compressed Modes for Variational Problems in Math. & Phy.

Summary

✔  CS-based sparse physics is based on rigorous math, not intuition

✔  Accuracy can be improved systematically ✔  Constraints (symmetry and translational

invariance) can be straightforwardly incorporated ✔  CS automatically picks out important terms ✔  A simple prescription for gathering data

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IEE/CEEM Seminar: Vidvuds Ozolins and Sparse Physics and its Applications to Energy Materials

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