Citation
Machine Learning Techniques for the Rational Design of Analytic Interatomic Potentials

Material Information

Title:
Machine Learning Techniques for the Rational Design of Analytic Interatomic Potentials
Creator:
Ragasa, Eugene J
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (199 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Materials Science and Engineering
Committee Chair:
Phillpot,Simon R
Committee Co-Chair:
Hennig,Richard
Committee Members:
Yang,Yong
Myers,Michele V
Hirschfeld,Peter J

Subjects

Subjects / Keywords:
atomistic-simulation -- molecular-dynamics -- pareto -- potential-development
Materials Science and Engineering -- Dissertations, Academic -- UF
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Materials Science and Engineering thesis, Ph.D.

Notes

Abstract:
In this work, we analyze the limitations to the current approaches to the parameterization of empirical potentials. Typically, a single objective function of the weighted sum of squared errors between the predictions of empirical potentials and reference targets is minimized. The choice of weights is subjective and an appropriate choice cannot be realistically determined a priori. Scalar optimization identifies local concave solutions which are dependent upon the choice of weights and initial conditions. This process requires the constant intervention of a skilled potential developer and is not amenable to an automated, algorithmic approach to potential development. These concerns are addressed by the presentation of a novel methodology. The problem is recast as a multi-objective optimization problem where the fidelity of prediction with respect to each material property is treated independently. This removes the need to communicate performance requirements a priori. In this way, the solution to parameterization expands from a single local solution to an ensemble of potentials through the use of the concept of Pareto optimality. Since each potential is optimal in some sense, a final potential can then be selected a posteriori, using the full knowledge of the performance tradeoffs between the candidate potentials. A solution scheme for this methodology is devised, combining Monte Carlo sampling with an iterative scheme to evolve a probability distribution function representing the epistemic uncertainty of Pareto optimal parameterizations. A Bayesian-like inference process is used to periodically update the distribution function. A software library pypospack, is developed to automate the simulation tasks, calculate material properties, and execute sampling based optimization routines. To demonstrate the flexibility and potency of this approach, applications to specific materials systems are presented. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2019.
Local:
Adviser: Phillpot,Simon R.
Local:
Co-adviser: Hennig,Richard.
Statement of Responsibility:
by Eugene J Ragasa.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
LD1780 2019 ( lcc )

UFDC Membership

Aggregations:
University of Florida Theses & Dissertations

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Full Text

PAGE 9

U(a;b)ab kEca0 kEca0 kk kk pkqN(k;q;k;q)

PAGE 10

i00ii6=1 12 q1q2C()ww0 x x

PAGE 11

ZZZ Z Z Z A jc12jjc44j B a0 Ec N=20

PAGE 12

Eca0 Eca0 K=4

PAGE 20

Md2R(t) dt2=F(R(t))tR=(r1;:::;rN)r2R3NF=(f1;:::;fN)fMVREPF f=rrV(R); r(t)r(t=0)v(t=0)tt r(t+t)=r(t)+dr(t) dtt+1 2d2r(t) dt2t2

PAGE 21

t<0 d2r(t) dt2=r(t+t))]TJ /F4 11.9552 Tf 11.955 0 Td[(2r(t)+r(t)]TJ /F4 11.9552 Tf 11.955 0 Td[(t) t2 r(t+t))]TJ /F4 11.9552 Tf 11.955 0 Td[(2r(t)+r(t)]TJ /F4 11.9552 Tf 11.956 0 Td[(t) t2=)]TJ /F4 11.9552 Tf 12.687 8.088 Td[(1 mdV(r(t)) dr r(t+t)=2r(t))]TJ /F7 11.9552 Tf 11.955 0 Td[(r(t)]TJ /F4 11.9552 Tf 11.955 0 Td[(t))]TJ /F4 11.9552 Tf 15.344 8.087 Td[(1 mdV(r(t)) drt2 RV(Ri)^V(Ri;)=(1;:::;n)V(Ri)

PAGE 22

000

PAGE 23

PV=NkBTPVTN

PAGE 24

dr dt=@H(r;p) @p;dp dt=@H(r;p) @rt!1NVE dH dt=@H(r;p) @rdr dt+@H(r;p) @pdp dt=0PT

PAGE 25

T0 TTKT K=hKi=1 2kBNdfTNdfKNdf=3N)]TJ /F3 11.9552 Tf 12.564 0 Td[(NcNNcNdf=3N)]TJ /F4 11.9552 Tf 12.409 0 Td[(3T T=2 kBNdfK;hTiTvii

PAGE 26

Kpi T=2 kBNdfNXi=11 2jpij2 mi=2 kBNdfNXi=11 2mijvij2 Ek=hKi=NXi=11 2mijvij2 tT(t) T(t)=T(t))]TJ /F3 11.9552 Tf 11.955 0 Td[(T(t)]TJ /F4 11.9552 Tf 11.955 0 Td[(1) =1 2NXi=12mi(vi)2 NdfkB)]TJ /F4 11.9552 Tf 13.15 8.087 Td[(1 2NXi=12miv2i NdfkB =(2)]TJ /F4 11.9552 Tf 11.955 0 Td[(1)T(t) =p T=T0T0

PAGE 27

dT(t) dt=1 (T0)]TJ /F3 11.9552 Tf 11.955 0 Td[(T(t)): T(t)=1 (T0)]TJ /F3 11.9552 Tf 11.955 0 Td[(T(t))t mid2ri dt2=)]TJ /F3 11.9552 Tf 9.299 0 Td[(dr dt+(t):(t) hi(t)j(t0)i=2kBT0ij(t)]TJ /F3 11.9552 Tf 11.955 0 Td[(t0);iii(t) B(t)B(t)tt>0B(t+t))]TJ /F3 11.9552 Tf 13.537 0 Td[(B(t)u0B(s)s
PAGE 28

Bt+t)]TJ /F3 11.9552 Tf 12.98 0 Td[(Bt0tBt+t)]TJ /F3 11.9552 Tf 12.887 0 Td[(BtN(0;t)B(t)tB(t)dB(t) B(t+t))]TJ /F3 11.9552 Tf 11.955 0 Td[(B(t)=Zt+ttdB();dB(t)=dt(t) (t)=dB(t) dt2=2miikBT0B(t)2R3Bi(t)Bj(t)i6=j(t)T0 i d2ri dt2=1 mifi)]TJ /F3 11.9552 Tf 16.6 8.088 Td[( midri dt+1 mii(t)i

PAGE 29

d2r dt2=fi mi)]TJ /F3 11.9552 Tf 11.955 0 Td[(ri; d dt=1 M(T(t))]TJ /F3 11.9552 Tf 11.955 0 Td[(NdfkBT0) TT0MM

PAGE 30

VPP NVi P=1 V 1 3NXi=1mivivi+1 3NXi=1riFi! P=NkBT V+*1 3VNXi=1riFi+ V PP=hPi !H=(a1;a2;a3)2R33a1a2a3 (r;p)^r;^pH ^ri=H)]TJ /F6 7.9701 Tf 6.587 0 Td[(1r:

PAGE 31

^riH ^pi=HTpiHH

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(MNP)MNP

PAGE 34

fRgV:fRg!EE2RRV(R)

PAGE 36

R3a1a2a3ai2R3s2Sr[^{;^|;^k](rx;ry;rz)(r1;r2;r3)[a1;a2;a3] rx^{+ry^|+rz^k=r1a1+r2a2+r3a3Tii2f1;2;3gTi(r)=r+niai=r T(r)=r+n1a1+n2a2+n3a3=r;8ni2ZNR=fa1;a2;a3;r1;:::;rNgri

PAGE 37

^VVV:fRg!R^V:fRg!R ^V(R)=V(R)+(R):V(R)R^V(R)R(R)iN ^V(R)=NXi=1^V(ri)^V(ri)iN

PAGE 38

^V ^V(R)^V(r1;:::;rN) =XiV1;i(ri)+XiXi
PAGE 39

=1:0=1 ^V(Rj)=(1;:::;n)^V(Rj)V(R)

PAGE 40

V(rij)=4" rij12)]TJ /F19 11.9552 Tf 11.955 16.857 Td[( rij6#:r)]TJ /F6 7.9701 Tf 6.587 0 Td[(12r)]TJ /F6 7.9701 Tf 6.587 0 Td[(6 ^V(Rj)RR^V(Rj)^V(Rj)V(Ri) =argmin2Xi^V(Rij))]TJ /F3 11.9552 Tf 11.955 0 Td[(V(R)2ifRig

PAGE 41

fRi:i
PAGE 42

q=(q1;:::;qNQ)NQ^q=(^q1;:::;^qNQ)qiiNC;i qi=qi(R1;:::;RNC;i):

PAGE 43

^qi()=^qi(jR1;:::;RNC;i):qi^qi() ^q 0KT>0K^q()

PAGE 44

qiq ()=^q())]TJ /F3 11.9552 Tf 11.955 0 Td[(q^q()q=(q1;q2;:::;qNQ)NQ 2i=(^qi())]TJ /F3 11.9552 Tf 12.216 0 Td[(qi)2 min2(^qi())]TJ /F3 11.9552 Tf 11.955 0 Td[(qi)2;i=1;:::;NQ

PAGE 45

^VV C()=NQXi=1wi(^qi())]TJ /F3 11.9552 Tf 11.956 0 Td[(qi)2=NQXi1wi2i()wiwi^V(Rj)wiw T>0T!00

PAGE 46

qi0ww wiqi0w0

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minL(()) g;j()0;j=1;2;:::;Ng h;k()=0;k=1;2;:::;Nh gQ;m(^q())0;m=1;2;:::;Nm NggNhhNmgQL(())=[L1(1());:::;LNQ(NQ())]

PAGE 48

Ljijjij0L(i)=0i=02RNPii=1;:::;NPL()2RNQ2L=fL1;:::LNQgNQ=1L =argmin2L():NQ>1f2g Li(ji()j)i

PAGE 49

L1()=ji()j=j^qi())]TJ /F3 11.9552 Tf 11.955 0 Td[(qijqiL1i=0 L2=()2=(^qi())]TJ /F3 11.9552 Tf 11.955 0 Td[(q)2:L2^q()L2^q()L1L2L2ji()j L1L2i()>0i()<0

PAGE 50

LLL1ijj(i)j
PAGE 51

ABBABA 2NQ12Li(1)Li(2)i=1;:::;NQLj(1)
PAGE 52

i00ii6=1 ijj1(i)jj1(j)jj2(i)jj2(j)jjk(i)j
PAGE 53

i 10232iij

PAGE 54

T>0

PAGE 55

C(;w)CLL>081;2 krC(1)-222(rC(2)k=Lkj1)]TJ /F7 11.9552 Tf 11.955 0 Td[(k0t C(t+1)
PAGE 56

12 w11w212121

PAGE 57

w2w31212w ww0

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q1q2C()ww0

PAGE 59

0(0;w)q

PAGE 60

2i^qi()1=p N N^Q=f^q()1);:::;^qNQ()g^Q^Q^Q ^Q

PAGE 61

minxf1(x)=2Xi=1)]TJ /F4 11.9552 Tf 9.299 0 Td[(10exp)]TJ /F4 11.9552 Tf 9.299 0 Td[(0:2q x2i+xi+12f2(x)=3Xi=1jjxijj0:8+5sin(x3i))]TJ /F4 11.9552 Tf 9.299 0 Td[(5xi)]TJ /F4 11.9552 Tf 21.918 0 Td[(51i3xx

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x x

PAGE 64

x

PAGE 68

NQ^qiqi^V()=(1;:::;NP)NP^q()=(^q1;::;:^qNQ)L()=(j^q1)]TJ /F3 11.9552 Tf 12.915 0 Td[(q1j;:::;j^qNQ)]TJ /F3 11.9552 Tf 12.915 0 Td[(qNQj)NQNPi^qjN=f1;:::Ng022j^qi())]TJ /F3 11.9552 Tf 12.079 0 Td[(qijj^qi()ji
PAGE 69

NQ

PAGE 72

[1;:::;N][^q(1);:::;^q(N)]

PAGE 73

(;F;P)P()=1(;F;P)FP!2FFPPP:F![0;1] XPXx2XX:!RX:!R(;F;P)Af!:X(!)=AgP(X=A)

PAGE 74

!2!x(!)2X()x2X (!i)2()!2()(!)!2P!ii=(!i)N()Nf(!1);:::;(!N)g=f1;:::;Ng=!!()f1;:::;Ngfp1;p2;:::pkgP()=p()p()=p()2()p()0 P(=0)=Z20p()d:

PAGE 75

P((!)) P((!))=0p(1)>p(2)12E E[X]=Xx2Xxp(x): E[X]=Z+1xp(x)dxfX E[f(X)]=Z+1f(x)p(x)dx

PAGE 76

()0T>0

PAGE 77

iii:i

PAGE 78

0C(jw)0000i=0ii2iEca0

PAGE 79

qEcqa0 p(^qj)p()p(^qj)^q()^Q()()^Q()^Q()(!1)2() Nf(!1);:::;(!N)g!i2 =f1;:::;Ngp(^qj) ^Q=^Q()=f^q(1);:::;^q(N)gQ^Q()i()!i2^q(i) ^q(i)=^q((!i))=^q(!i)2^Q()!i2

PAGE 80

g2p(i)=0igjijgihi(fj;i6=jg;A)fj;i6=jgAh ^q()

PAGE 81

U()[0;1](!)2()p() F()=P(<)=Zp(x)dx()U()()=F)]TJ /F6 7.9701 Tf 6.586 0 Td[(1(U()) y2N(;2)x2N(0;1) y=+xN(0;1)ij=ji pMVN(x)=1 (2)1=2jj1=2exp)]TJ /F4 11.9552 Tf 10.494 8.088 Td[(1 2(x)]TJ /F7 11.9552 Tf 11.955 0 Td[()T)]TJ /F6 7.9701 Tf 6.587 0 Td[(1(x)]TJ /F7 11.9552 Tf 11.955 0 Td[()

PAGE 82

xNMVN(0;I)00IxxxiN(0;1)yMVN(;)y=+SxSS Y=SST E[y]=E[+Sx]=+SE[x]=y Y=SXST=SIST=SSTy xN(;) Ax+bN(A+b;AAT)2 >0

PAGE 83

L=LLT 0i p(j^q) ()p()

PAGE 84

qp(j^q)p(^qj) ^Q=f^q(1);:::;^q(N)gM$^Q=f^q(1);:::;^q(M)gM
PAGE 85

Q() qiqjqiqj()()p()

PAGE 86

^q()q()=0)i()qi d()=0@NQXi=1 qi21A1=2d^q()q

PAGE 87

p() (1;:::;n)ppX(x1;:::;xN)x(!)2X()p(x) p(x)^p ^ph(x)=1 nNXi=1Kh(x)]TJ /F3 11.9552 Tf 11.955 0 Td[(xi)hKKh(u)=K(u=h)=h KK(u)=K()]TJ /F3 11.9552 Tf 9.299 0 Td[(u)

PAGE 88

RRK(u)du=1^ph(x)Kxin (x)=1 p 2exp)]TJ /F3 11.9552 Tf 14.848 8.087 Td[(t2 22h^ph(x) NNi1iNI(1;:::;N)W()=I+hW()h h

PAGE 89

DKL(1jj2)fgDKL D(fkg)=Zf(x)logf(x) g(x)dxD(fkf)=0D(fkg)=0f=gD(fkg)0fgp()DKLtDKL(pt)]TJ /F6 7.9701 Tf 6.586 0 Td[(1jjpt) xifEhlogf(x) g(x)i=D(fkg)NfxigNi=1 DMC(fkg)=1 NNXilogf(x) g(x)!D(fkg)N!1DMC(fkg)fxigNi=1f1iNf(xi)g(xi)DMCk<1

PAGE 90

p

PAGE 91

h

PAGE 95

^qi()^qi

PAGE 97

2(!)2()

PAGE 105

^V

PAGE 106

AMg;OA

PAGE 107

)]TJ /F1 11.9552 Tf 9.299 0 Td[<00520066006a0056002d>]TJ ET BT /F21 5.9776 Tf -13.615 -279.136 Td[<0033>]TJ /F1 11.9552 Tf 35.134 -.001 Td[<005501f6006100420061004200610042006e001b01f6002d0064005800790039004e003800380065006b006400640056002d>]TJ ET BT /F21 5.9776 Tf -13.615 -303.044 Td[<004e>]TJ /F1 11.9552 Tf 35.134 0 Td[<005501f6006100420061004200610042006e002201f6002d0079005800650079006b006b006b003900380038003300390056002d>]TJ ET BT /F21 5.9776 Tf -17.267 -326.952 Td[<00520079>]TJ /F1 11.9552 Tf 38.786 0 Td[<005501f6006100420061004200610042006e005401f6002d0039005800790056002d>]TJ ET BT /F21 5.9776 Tf -17.267 -350.86 Td[<00520052>]TJ /F1 11.9552 Tf 38.786 0 Td[<005501f6006100420061004200610042006e005b01f6002d0079005800790056002d>]TJ ET BT /F21 5.9776 Tf -17.267 -374.768 Td[<0052006b>]TJ /F1 11.9552 Tf 38.786 0 Td[<005501f6006100420061004200610042006e00690051004801f6002d0079005800790056>]TJ ET BT /F21 5.9776 Tf -17.267 -398.676 Td[<0052006a>]TJ /F1 11.9552 Tf 17.267 0 Td[<00290056>]TJ ET BT /F1 11.9552 Tf 23.384 -426.779 Td[<0041004b005400480032004b0032004d>26<00690032002f>-325<0054001c00420060>-326<0054>-27<005100690032004d>26<00690042001c00480062>-325<001c00600032>-326<0042004d002f0042002b001c00690032002f>-326<0042004d>-326<0068>81<001c002300480032>]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.9552 Tf 287.797 -426.779 Td[<003800400052>]TJ ET 0 G 0 g BT /F1 11.9552 Tf 303.41 -426.779 Td[<0058>]TJ ET BT /F1 11.9552 Tf 0 -449.593 Td[<0068>81<001c002300480032>-325<0038004000520058>]TJ ET BT /F1 11.9552 Tf 56.245 -449.593 Td[<0041004b005400480032004b0032004d>26<00690032002f>-325<0054001c00420060>-326<0054>-27<005100690032004d>26<00690042001c00480062>-325<0042004d>]TJ /F2 11.9552 Tf 161.993 0 Td[<005400760054>50<005100620054>50<001c002b0046>]TJ /F1 11.9552 Tf 49.912 0 Td[<0058>]TJ ET q .3985 w 34.689 -456.119 m 433.311 -456.119 l S Q BT /F1 11.9552 Tf 98.972 -466.43 Td[<002a0048001c00620062>-326<004c001c004b0032>]TJ 130.539 0 Td[<0047001b004a004a00530061>-999<003a006c00470053>-1000<0031001b004a>-1000<005f00320037003200600032>-1<004d002b0032>]TJ ET q .3985 w 34.689 -470.963 m 433.311 -470.963 l S Q BT /F1 11.9552 Tf 74.147 -481.274 Td[<0022006d002b>26<00460042004d003b003f001c004b0053>27<005100690032004d>27<00690042001c0048>]TJ 178.276 0 Td[<0074>]TJ 54.575 0 Td[<0074>]TJ 42.304 0 Td[<0074>-2888<0028>]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.9552 Tf 393.273 -481.274 Td[<0052>]TJ ET 0 G 0 g BT /F1 11.9552 Tf 399.131 -481.274 Td[<002d>]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.9552 Tf 406.28 -481.274 Td[<004e>]TJ ET 0 G 0 g BT /F1 11.9552 Tf 412.138 -481.274 Td[<0029>]TJ -321.678 -14.446 Td[<004a00510060006200320053>26<005100690032004d>27<00690042001c0048>]TJ 161.963 0 Td[<0074>]TJ 54.575 0 Td[<0074>]TJ 42.304 0 Td[<0074>-2942<0028>]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.9552 Tf 393.918 -495.72 Td[<0052006b0064>]TJ ET 0 G 0 g BT /F1 11.9552 Tf 411.492 -495.72 Td[<0029>]TJ -334.093 -14.446 Td[<002200510060004d004a001c>26<0076>27<003200600053>27<005100690032004d>27<00690042001c0048>]TJ 271.903 0 Td[<0074>-2942<0028>]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.9552 Tf 393.918 -510.166 Td[<0052006b0033>]TJ ET 0 G 0 g BT /F1 11.9552 Tf 411.492 -510.166 Td[<0029>]TJ -340.871 -14.446 Td[<00470032004d004d001c0060002f00430051004d003200620053>27<005100690032004d>27<00690042001c0048>]TJ 181.802 0 Td[<0074>]TJ 54.575 0 Td[<0074>]TJ 42.304 0 Td[<0074>]TJ 47.223 0 Td[<0028>]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.9552 Tf 399.776 -524.612 Td[<0033>]TJ ET 0 G 0 g BT /F1 11.9552 Tf 405.634 -524.612 Td[<0029>]TJ -364.967 -14.445 Td[<003a0032004d00320060001c0048004200780032002f00470032004d004d001c0060002f00430051004d003200620053>26<005100690032004d>27<00690042001c0048>]TJ 308.635 0 Td[<0074>-2942<0028>]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.9552 Tf 393.918 -539.057 Td[<0052006b004e>]TJ ET 0 G 0 g BT /F1 11.9552 Tf 411.492 -539.057 Td[<0029>]TJ ET q .3985 w 34.689 -543.59 m 433.311 -543.59 l S Q BT /F15 11.9552 Tf 0 -575.761 Td[<00380058006b005800390058006b>-1000<0068003f006000320032>-375<0023>-31<0051>-31<002f0076>-375<0054>-31<005100690032004d>30<00690042001c00480062>]TJ /F1 11.9552 Tf 23.384 -24.423 Td[<0068003f006000320032>-326<0023>-27<0051>-27<002f0076>-326<0054>-27<005100690032004d>26<00690042001c00480062>-325<001c00600032>-326<0042004b005400480032004b0032004d>26<00690032002f>-325<0042004d>-326<007200420069003f>-326<0069003f0032>-326<0023001c00620032>-326<002b0048001c00620062>]TJ /F16 11.9552 Tf 320.196 0 Td[<0068003f00600032003200220051002f00760053005100690032004d00690042001c0048>]TJ /F1 11.9552 Tf 110.824 0 Td[<002d>]TJ -454.404 -23.908 Td[<001c004d002f>-326<00480042006200690032002f>-326<0042004d>-326<0068>81<001c002300480032>]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.9552 Tf 99.873 -624.092 Td[<00380040006b>]TJ ET 0 G 0 g BT /F1 11.9552 Tf 115.487 -624.092 Td[<0058>-434<006c004d004800420046>26<0032>-325<0054001c0042006000400054>-27<005100690032004d>26<00690042001c00480062002d>-325<0069003f0032>-326<0054001c0060001c004b0032006900320060>-326<004d001c004b0042004d003b>-326<0062002b>26<003f0032004b00320062>-325<001c00600032>-326<004d00510069>]TJ -115.487 -23.908 Td[<0032005b006d00420070>53<001c00480032004d>27<00690058>]TJ ET BT /F1 11.9552 Tf 225.213 -687.851 Td[<005200790064>]TJ ET Q

PAGE 108

V2(rij)rijijV3(rij;rik;ijk)V2i ijk (rij)ji

PAGE 109

F(rhoi)iF(j) a0

PAGE 110

ai=[a11;a12;a13]2R3 266664a11a12a13a21a22a23a31a32a33377775=266664`x00xy`y0xzyz`z377775`x`y`z^ux^uy^uzxyxzyza1a2a3abcaia1

PAGE 111

a0 266664a000b000c377775=a02666641000b a000c a377775=a0266664h1h2h3377775=a0Hn1n2n3H a0H0=a0266664n1h1n2h2n3h3377775: 33r

PAGE 112

k

PAGE 113

q

PAGE 115

Ec;bulk

PAGE 116

Eca1a1a3

PAGE 117

a1

PAGE 121

a1a1

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a11a1a12a1a13a1a21a2a22a2a23a2a31a3a32a3a33a3 a1a2a2a11a3a12a3a13a3a21a3a22a3a23a3a31a3a32a3a33a3P11P12P13P21P22P23P31P32P33

PAGE 125

a1a2a2a11a3a12a3a13a3a21a3a22a3a23a3 a1a2a2a11a3a12a3a13a3a21a3a22a3a23a3 C11C12C13C22C23C22C44C55C66

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2 2

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^qi()i=^qi())]TJ /F3 11.9552 Tf 11.955 0 Td[(qi p()(!)2()!2

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N<1000

PAGE 134

00

PAGE 137

ijij V(rij)=ZZe2 rij+Aexprij )]TJ /F3 11.9552 Tf 13.151 8.088 Td[(C r6ij;ijACZip()

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p(i)iZ=)]TJ /F3 11.9552 Tf 9.298 0 Td[(ZA=0=0:5C=0ACAZ ZAC6 iii ZArhoAC6

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a0c11c12c44BG(100)BGB>0G>0c44>0 666k1110k991333k333800 qi q1a0q2c11q3c12q4c44q5Bq6Gq7Eq8Eq9Eq10h100i2

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^qi()qiLi()=j^qi())]TJ /F3 11.9552 Tf 11.955 0 Td[(qijqminL()=[L1();L2();:::;L9()] g1:Z>0:0 g2:A>0:0 g3:>0:0 g4:A>0:0 g5:>0:0 g6:C<0:0 g7:B>0 g8:G>0 g9:c44>0 h1:Z=)]TJ /F3 11.9552 Tf 9.298 0 Td[(Z h2:A=0:0 h3:=0:5 h5:C=0:0 h4:C=0:0 gihi

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N=955 pp()t)]TJ /F6 7.9701 Tf 6.587 0 Td[(1;1;:::;t)]TJ /F6 7.9701 Tf 6.586 0 Td[(1;NNNPt)]TJ /F4 11.9552 Tf 12.604 0 Td[(1^p(t) ^ft(jH)=1 NXi=1NKH()]TJ /F7 11.9552 Tf 11.955 0 Td[(i)KH=jHj)]TJ /F6 7.9701 Tf 6.587 0 Td[(1=2K(H)]TJ /F6 7.9701 Tf 6.586 0 Td[(1=2)KH2RNPNP H

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Ht)]TJ /F6 7.9701 Tf 6.587 0 Td[(1 ZZZZMg Z+1:8j^qi)]TJ /F3 11.9552 Tf 12.162 0 Td[(qij2

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ZZZ N=9 ji()=j^qi())]TJ /F3 11.9552 Tf 13.012 0 Td[(qij2tBa0NQBa0

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Z q()A A

PAGE 145

Z AN=7557

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Z ii D(f1jf2)f1f2D

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A D)]TJ /F19 11.9552 Tf 11.955 16.272 Td[(Zf1(x)logf1(x) f2(x)dxD0D(f1jf2)=0f1f2 iD(fi)]TJ /F6 7.9701 Tf 6.587 0 Td[(1jfi)0

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Dfi()andfi)]TJ /F6 7.9701 Tf 6.587 0 Td[(1() c11c44 jc12jjc44j

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iqi 10

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385)]TJ /F6 7.9701 Tf 6.587 0 Td[(1)1:0810)]TJ /F6 7.9701 Tf 6.587 0 Td[(5)]TJ /F6 7.9701 Tf 6.587 0 Td[(1

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ZAAC a0C11C12C44BGE()E()Eh100i2

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)]TJ /F3 11.9552 Tf 32.001 0 Td[((10)]TJ /F4 11.9552 Tf 9.298 0 Td[(5))]TJ /F6 7.9701 Tf 6.586 0 Td[(1

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223 23rijrikiijk E=Xi
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aABpq Eca0C11C12C44BGEvq=(q1;:::qNQ)NQ=7Ev333 =(1;:::;NP)=(1;:::;NP)2p() ^q()=(^q1();:::;^qNQ())()=(^q())]TJ /F7 11.9552 Tf 12.589 0 Td[(q)()=0

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q1Ecq2a0q3C11q4C12q5C44q6Bq7Ev L=(1();:::;NQ())L p;0()p()(!)2()q p=4:0q=0:0p=3:5q=0:5p=4:0q=0:0p=3:5q=0:5pq 0 0()

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U(a;b)ab p=4:0;q=0:0p=3:5;q=0:0p;q U(2:1;2:2)U(2:1;2:2)U(2:1;2:2)U(1:0;3:0)U(1:0;3:0)U(1:0;3:0)aU(1:5;2:0)U(1:5;2:0)U(1:5;2:0)U(20;32)U(20;32)U(20;32)U(1:0;2:0)U(1:0;2:0)U(1:0;2:0)AU(6:0;20:0)U(6:0;20:0)U(6:0;20:0)BU(0:5;1:5)U(0:5;1:5)U(0:5;1:5)pU(3:0;4:4)qU(0:0;1:0)cos(0ijk) Np()N()N=1 N>1p()^t)]TJ /F6 7.9701 Tf 6.587 0 Td[(1 pt()=^p(:^t)]TJ /F6 7.9701 Tf 6.587 0 Td[(1)^p(!)2() =((!1);:::;(!M)) =(i;:::;M) Mp()

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^q()(!)2^Q()()2 ^Q=(^q(1);:::;^q(M) ^Q^Q()2^q()^Q()F1F2F^q()2^Q()FFF(^Q())F(^Q())^Q()Fi(^Q())iF1F2 d()=24NQXi=1ji()j2 q2i351=2

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2 ^Q()=F1(^Q())\F2(^Q()): ^Q=(^q((!j));:::;^q((!0M))) =(^q(j);:::;^q(0M)) ^Q^QijQM0MQ^t^N p()N)]TJ /F6 7.9701 Tf 6.587 0 Td[(1QN)]TJ /F6 7.9701 Tf 6.586 0 Td[(1NQN10;000 N B

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B a0

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a0=5:435:0 a0

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Ec i=1i=2)]TJ /F4 11.9552 Tf 9.298 0 Td[(4:0)]TJ /F4 11.9552 Tf 9.298 0 Td[(4:63i=5)]TJ /F4 11.9552 Tf 9.299 0 Td[(4:5
PAGE 163

Ec X;Y=Y;XXYX0=a+bXY0=a+bYa;b;c;d2Rb;d>0()^q()ji()j2i()

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A;a=)]TJ /F4 11.9552 Tf 9.298 0 Td[(0:48Aaa;a=)]TJ /F4 11.9552 Tf 9.298 0 Td[(0:74B;B=)]TJ /F4 11.9552 Tf 9.298 0 Td[(0:85BaBaB;a=0:71 ^q(2N=20)^q(2N=20) p()i() =1=)]TJ /F4 11.9552 Tf 9.298 0 Td[(1C11C12C11BC12BB

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C11;BC12;BC11C12C11C12 q((!)2())EcEvEc;Ev=)]TJ /F4 11.9552 Tf 9.298 0 Td[(0:96C11C12C11;C12=0:05BC12C12;B=0:89 ^qi()NQNP

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N=20 Eca0 Eca0

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Eca0 Ec=)]TJ /F4 11.9552 Tf 9.298 0 Td[(3:86a0=5:22Ec=)]TJ /F4 11.9552 Tf 9.299 0 Td[(4:5a0=5:05Ec=)]TJ /F4 11.9552 Tf 9.298 0 Td[(4:1a0=5:45

PAGE 168

a0EcEca0 K

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x(!)2X()x2Rd p(xj1;:::;K;1;:::;K1;:::;K)=KXk=0kN(xjk;k)kkk0
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Eca0 k=1;2;3k=4kkkk

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kEca0 kEca0 K=4

PAGE 172

qqqqq

PAGE 173

x x=(1;:::;NP;^q1;:::^qNQ)kkx =(k;1;:::;k;NP;^qk;1;:::^qk;NQ)k k=0B@k;; k;;q k;^q; k;^q;^q1CA=0BBBBBBBBBBBBBB@k;1;1:::k;1;NP k;1;^q1:::k;1;^qNQ k;NP;1k;NP;NP k;NP;^q1k;NP;^qNQ k;^qNQ;NP:::k;^q1;NP k;^q1;^q1:::k;^q1;^qNQ k;^qNQ;1k;^q1;1 k;^qNQ;^q1k;^qNQ;^qNQ1CCCCCCCCCCCCCCAk;;k;^q;^qk;^q;^q1K100KK argminK(K)=7argminKBIC)K=57K=10

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kk;k;^qkkk;;k;^q;^q

PAGE 175

kEca0 kk kaABpq K

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kk kECa0c11c12c44BEv kk;;k

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k;^qtT2 qp1tqikiqii=k;qi=k;qi;qik;qik;qi;qik;q t pP[qi=2^Qi]iiP[q2^Q]=1)]TJ /F5 7.9701 Tf 13.095 16.383 Td[(NQYi1)]TJ /F18 11.9552 Tf 11.956 0 Td[(P[qi=2^Qi] 1)]TJ /F5 7.9701 Tf 13.095 16.383 Td[(NQYiP[qi2^Qi] P[q=2^Q]=NQYiP[qi2^Qi] ttki pttjepsilonji

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pkqN(k;q;k;q) kECa0c11c12c44BEv q2^Qjepsilonijic11c12c44Btp>0:01Ec57a008Ev579k=f1;2;3;4;6;8g

PAGE 182

tp

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z=2z=105