METHODS FOR THE CONSTRUCTION OF EXTRAPOLATION ...

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METHODS FOR THE. CONSTRUCTION OF. EXTRAPOLATION PROCESSES. Herbert H. H. Homeier, Regensburg. Plovdiv, 1996. • General Approach.
METHODS FOR THE CONSTRUCTION OF EXTRAPOLATION PROCESSES Herbert H. H. Homeier, Regensburg Plovdiv, 1996

• General Approach • Iterative Approach • Variational Approach • Perturbative Approach

1

MANY THANKS TO Prof. Dr. Drumi D. Bainov Prof. Dr. Otto Steinborn Priv.-Doz. Dr. Joachim Weniger Dr. Holger Meiner Dr. Johannes Dotterweich Prof. Dr. Bernhard Dick Prof. Dr. Hartmut Yersin Dipl.-Chem. Jens Decker Prof. Dr. Josef Barthel Prof. Dr. Hartmut Krienke Dipl.-Chem. Sebastian Rast Prof. Dr. Peter Otto (Erlangen) Prof. Dr. Claude Daul (Fribourg) Dr. Miloslav Znojil (Rez) Deutsche Forschungsgemeinschaft Auswrtiges Amt 2

Development of Methods: Extrapolation Methods Problem: Slow Convergence Example sn = 1 +

x

+

x2

+

x3

+ ··· +

xn

1 2 3 n s = lim sn = 1 − ln(1 − x) n→∞ 4.4

x = 0.9,

4.2+ 4 3.8 3.6

+

3.4

+

+

+

+

+

+

3

3

3

+ 3

7

8

9

10

3.2 3 2.8 2.6 2.4 2.2

sn 3 s0n + s

3

3

3

3

3 2

3

4

5

6

n Sequence transformation {s0n} = T ({sn}): Accelerate convergence 3

Development of Methods: Extrapolation Methods Problem: No Convergence Example: sn = 1 +

x 1

+

x2 2

+

x3 3

+ ··· +

xn n

s = lim sn = 1 − ln(1 − x) n→∞ 1+ 3

x = −1.3,

0.8 0.6

3 +

0.4 0.2

+

3

3

+

+

+

+

+

sn 3 3 0 3 sn + s

+

+

+

0 -0.2

3

3

3

3

-0.4

3

-0.6 0

2

4

6

8

10

n Sequence transformation {s0n} = T ({sn}): Achieve convergence 4

PROBLEMS • Extrapolation • Convergence acceleration • Summation of divergent series • No universal method available • Linear methods often not efficient enough • Accuracy of single method unknown in practice AIMS • problem adapted • nonlinear • efficient APPROACH • Model sequences • Iteration of simple methods • Comparison of results of various methods 5

BASIC PRINCIPLE

EXAMPLE

Model sequences σn = σ + Mn(~ c, p ~(n)) σn = σ + c ωn Exact Calculation of σ σ = Tn({σn}, p ~(n)) σ = σn+1 − ωn+1

σn+1 − σn ωn+1 − ωn

Sequence transformation for problem {sn} s0n = Tn({sn}, p ~(n)) sn+1 − sn 0 sn = sn+1 − ωn+1 ωn+1 − ωn Acceleration for {sn} ≈ {σn} sn − s 6

ωn

= O(1)

ANNIHILATION OPERATORS k X

σn = σ + ωn

cj φj (n)

j=0

with linear operator On(φj (n)) = 0 ,

j = 0, . . . , k

then  On

σn − σ



ωn

=0

or On(σn/ωn)

On(sn/ωn) 0 =⇒ sn = σ= On(1/ωn) On(1/ωn)

CONVERGENCE CLASSIFICATION lim

sn+1 − s



sn − s linearly convergent: 0 < |ρ| < 1 logarithmically convergent: ρ = 1 n→∞

7

REMAINDER ESTIMATES sn − s ωn

= O(1) ,

n→∞

Series sn =

n X

aj → s ,

n→∞

j=0

Levin tω = a , n n uω = (n + β)a , n n v ω = anan+1 . n an − an+1

Smith and Ford t˜ω = a n n+1

8

β > 0,

Asymptotically related series a ˆ n ∼ an sˆn =

n X

a ˆ j → sˆ ,

n→∞

j=0

Linear remainder estimates ltω = a ˆn , n luω = (n + β)ˆ an , n ˆ na ˆ n+1 lv ω = a , n a ˆn − a ˆ n+1 lt˜ω = a ˆ n+1 n

β > 0,

Kummer-type remainder estimate kω = s ˆn − sˆ n

Tails Tn = an+1 + an+2 + · · · = s − sn t˜T n = sn+1 − sn = an+1 + 0 + · · · , lt˜T ˆn+1 − sˆn = a ˆ n+1 + 0 + · · · , n = s kT ˆ − sˆn = a ˆ n+1 + a ˆ n+2 + · · · . n = s 9

Example Fm(z) =

∞ X

(−z)j /j!(2m + 2j + 1) ,

j=0 kω = n

(1 − e−z )/z −

n X

! (−z)j /(j + 1)! .

j=0

uω n sn n 5 –13.3 0.3120747 6 14.7 0.3132882 7 –13.1 0.3132779 8 11.4 0.3133089 9 –8.0 0.3133083 z = 8 ,m = 0

2J-Transformation

tω n

kω n

0.3143352 0.3131147 0.3133356 0.3133054 0.3133090

0.3132981 0.3133070 0.3133087 0.3133087 0.3133087

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ITERATIVE SEQUENCE TRANSFORMATIONS Idea: • Simple sequence transformation {s0n} = T ({sn}) • Iteration: 0({s0 }) , {s000} = T 00({s00 }) , . . . {s00 } = T n n n n

• Example: Aitken ∆2 method (sn+1 − sn)2

(1)

sn = sn −

sn+2 − 2sn+1 + sn

.

Iteration (n) A0 = sn , (n)

(n)

Ak+1 = Ak

(n+1)



(Ak

(n)

− Ak )2

(n+1) (n) (n+2) Ak − 2Ak + Ak

11

.

Problem: • Iteration not unique, for example (k+1)

sn

(k)

= T (k)({sn })

with T (0) = T . • Example (n) A0 =sn , (n) (n) Ak+1 =Ak

(n+1) (n) 2 − Ak ) (Ak −Xk (n+2) (n+1) (n) Ak − 2Ak + Ak

with X0 = 1 “allowed”, i.e., Xk = (2k + 1)/(k + 1).

12

HIERARCHICAL CONSISTENCY Example: • Simple transformation: sn+1 − sn 0 sn = sn+1 − ωn+1 ωn+1 − ωn 0 ? • Problem: ωn

• For σn = σ + ωn(c0 + c1rn) we have 0 =σ−c σn 1

ωnωn+1 ωn+1 − ωn

(rn+1 − rn)

• Hence ωnωn+1 0 ωn = − (rn+1 − rn) ωn+1 − ωn

13

• Iteration =⇒ J transformation (0) sn = sn ,

(0) ωn = ωn , (k) (k) − s s n (k+1) (k) (k) n+1 sn = sn − ωn , (k) (k) ωn+1 − ωn (k) (k) ω n ωn+1 (k+1) (k) ωn =− δn , (k) (k) ωn+1 − ωn (k) (k) (k) Jn ({sn}, {ωn}, {rn }) = sn

• for power series, • very versatile and powerful. • related to E algorithm with model σn = σ +

k X j=0

14

cj gj (n)

• Analysed in H. H. H. Homeier, A hierarchically consistent, iterative sequence transformation, Numer. Algo. 8 (1994) 47-81. —, Analytical and numerical studies of the convergence behavior of the J transformation, J. Comput. Appl. Math. 69 (1996) 81-112. —, Determinantal representations for the J transformation, Numer. Math. 71 (1995) 275-288.

15

Numerical example: Gn =

Z

1

0

Z

Γ/π



−∞ (x − ∆ω)2 + Γ2

"

Y q 2Γ/π exp − 2 x + Γ2

#

! −1

) dx

qn d q

Taylor series in Y =⇒ Power series Acceleration with J transformation

n 15 16 17 18 19 20 ∞

sn -3047434. 5412146. -9099655. 14525645. -22070655. 31994427. -0.16361729 (k)

(n) s0

-0.16361565 -0.16361782 -0.16361716 -0.16361732 -0.16361728 -0.16361729 -0.16361729

n = 2, ∆ω = 5, Γ = 1, Y = 100, r k = 1/(n + 1 + k), k variant 16

More general: • Hierarchy of model sequences: The higher, the more parameters • Iteration of simple transformation T is consistent, if – T is exact for lowest level sequences – Variant of T maps higher to lower level EXAMPLE: Hierarchy for J transformation σn = σ + ωn(c0 + c1

n−1 X

(0)

δ n1

n1=0 n −1 n−1 X (1) (0) 1X δ n2 δ n1 + c2 n1=0 n2=0

+ · · ·) at level k.

17

FORMAL DESCRIPTION • Simple model {σn(~ c, p ~)} → σ(~ p) • Simple transformation ∞ T (~ p ) : {σn(~ c, p ~ )}∞ −→ {σ(~ p )} n=0 n=0

• Hierarchy of model sequences (`) ∞ (`) (`) (`) (`) a }n=0}L {{σn (~ c ,p ~ )|~ c ∈C `=0 0) (`) (` with a > a for ` > `0

• Mapping between levels (`) (`) (`) T (~ p ) : {σn (~ c ,p ~ (`))}∞ n=0 (`−1) (`−1) −→ {σn (~ c ,p ~ (`−1))}∞ n=0

• Hierarchical consistent transformation T (`) = T (~ p (0)) ◦ T (~ p (1)) ◦ . . . ◦ T (~ p (`))

18

METHODS FOR ORTHOGONAL SERIES • I transformation: – Simple model sequence: σn = σ+ωn(c exp(inν)+d exp(−inν)) – Simple sequence transformation sn+2 sn+1 sn − 2 cos(ν) + ωn+2 ωn+1 ωn 0 sn = 1 1 1 − 2 cos(ν) + ωn+2 ωn+1 ωn 0 ? – Compute ωn – More complicated model sequence:

σn = σ + ωn(einν (c0 + c1rn) +e−inν (d0 + d1rn)) yields 0 ≈ σ+ω 0 (c0 exp(inν)+d0 exp(−inν)) σn n 0 = ωn

−(rn+1 − rn) 1 1 1 − 2 cos(ν) + ωn+2 ωn+1 ωn 19

– Iteration =⇒ I transformation s(0) n = sn ,

ωn(0) = ωn (k)

sn+2 sn(k+1) =

(k) ωn+2

1 (k) ωn+2

ωn(k+1) =

1 (k) ωn+2

(k)

− 2 cos(ν) − 2 cos(ν)

sn+1 (k) ωn+1

1

(k) ωn+1 (k) −∆rn+1

− 2 cos(ν)

1 (k) ωn+1

+ +

s(k) n (k)

ωn 1

(k)

ωn

+

1 (k)

ωn

– for Fourier series – Notice three-term recurrence un+2 − 2 cos(ν)un+1 + un = 0 satisfied by exp(±inν) (or cos(nν), sin(nν)

20

• K transformation: – Simple model sequence : σn = σ + cωnPn(cos(ν)) – Three-term recurrence (0)

(1)

(2)

ζn Pn + ζn Pn+1 + ζn Pn+2 = 0 . (ν dependent) – Algorithm (analog to I transformation) s(0) n = sn ,

ωn(0) = ωn , (0)

sn(k)

(0)

ωn 1

ζn+k s(k+1) = n

ζn+k ωn(k+1) =

(0) ζn+k

(1)

(1)

ωn+1 1

(k)

+ ζn+k

(k)

+ ζn+k

ωn

1 (k) ωn

(k)

(k)

sn+1

(2)

ωn+1 1

(k)

+ ζn+k

(k)

+ ζn+k

ωn+1 δn(k) 1 (1)

+ ζn+k

(2)

sn+1

(k) ωn+1

+

(k)

(k)

ωn+1

(2) ζn+k

1 (k)

ωn+1

Kn(k)({δn(k)}, {ζn(j)}, {sn}, {ωn}) = s(k) n

– ν dependent, for orthogonal series. 21

MULTIPOLE EXPANSIONS r /r) m∗ 1 Y`m(~ UQ(~ r ) = 4π Q` `+1 2` + 1 `m r X

Qm ` =

Z

r 0`Y`m(~ r 0/r 0)ρ(~ r 0) d 3r 0 ,

Rotational symmetry UQ(~ r) =

∞ X

P`

`=0

Legendre Expansion

22

~ ~ r·R rR

!

q` r `+1

Example ~ ρ(~ r ) = exp(−αr) exp(−β|~ r − R|)

` − lg |1 − s`/s| − lg |1 − s0`/s| 2 2.6 5.1 4 4.3 9.7 6 5.9 11.5 8 7.6 16.0 10 9.2 16.0 K transformation, r = 12, θ = 60o

14 12 10 8 6 4 2

++ + s` 3 ++ s0` + + + 33 3 +++ 333 33 3 ++ 33 3 + ++ 33 3 3 3 3 ++ 33 3 + 3 3 3 ++33 3 3

0 5

10

15

20

25

`

Exact digits (r = 4, θ = 60o) 23

MANYFOLD FREQUENCIES • Important near singularities • Increases stability • Instead of s0, s1, s2, . . . , ω0, ω1, ω2, . . . take sτ ·0, sτ ·1, sτ ·2, . . . , ωτ ·0, ωτ ·1, ωτ ·2, . . . • for Fourier and orthogonal series put ν →τ ·ν i.e., for x = cos ν x → xτ = cos(τ · arccos x)

24

∞ X

1

`=0 ` + 1



s

P`(x) = ln 1 +

2 1−x

 

x = 0.9 K transformation τ =1 n − lg |1 − sn/s| 16 2.07 18 1.75 20 1.91 22 3.59 24 2.01 τ =3 m n 48 16 54 18 60 20 66 22 72 24

0 − lg 1 − sn/s 5.24 6.88 6.58 6.91 6.80

− lg |1 − sm/s| 2.51 2.45 2.48 2.59 2.80 25

0 − lg 1 − sn/s 9.31 10.40 11.63 13.18 14.47

ASSOCIATED POWER SERIES Example: ∞ X

1 + in

n=0

n2

∞ 1 + in 1 X inν cos(nν) = e 2 n=0 n2 ∞ 1 + in 1 X −inν + e 2 n=0 n2

Accelerate power series separately and add Adaptable to more complicated examples: X

cos((n + 1/2)ν)Pn(cos ν 0)

(singular at ν = ν 0) is sum of 4 power series ∞ 1 X 0) e±i(n+1/2)ν ρ± (ν n 4 n=0 2 ± 0 0 ρn (ν ) =Pn(cos ν ) ± i Qn(cos ν 0) π 0 exp(±inν ) ∼



× const.

n

26

τ = 10,

ν = 6π/10,

ν 0 = 2π/3

near singularity. n − lg |(sτ n − s)/s| 8 12 16 20 24 28

1.3 1.2 1.0 1.5 1.3 1.2

(τ ) − lg (Gn − s)/s 7.7 14.0 18.1 22.6 27.3 31.2

P4 (τ ) (0) Gn = j=1 Ln (1, [pj,τ n]|n=0, [(τ n+1)(pj,τ n−

pj,(τ n)−1)]|n=0)

27

VARIATIONAL METHODS Problem: Exact limit invariant under addition of null sequences Nonlinear sequence transformation usually not ! Idea: Restore invariance variationally for certain null sequences xn s0n = f ({sn + αxn}) ∂s0n ∂α

28

=0

Example: Aitken ∆2 method (1)

sn = sn −

(∆sn)2 ∆2s

.

n

Use sn → sn + αxn with lim xn = 0 n→∞ (1)

(1)

sn → sn (α) = sn+αxn−

(∆sn + α∆xn)2 ∆2s

n + α∆xn

(1) Choose α such that sn (α) is stationary:

(1) ∂sn ∂α

=0 α=α0

Result: For sn = 10 + 1/n2 new method accelerates (O(n−3) error), but Aitken does not(!) accelerate convergence.

29

.

Example s0n =

k X

cj sn+j ,

k X

cj = 1

j=0

j=0

Put k X

sn → sn +

(ν) αν xn

ν=1

Saddle point =⇒ Linear system ∂s0n ∂αµ

=

k X j=0

Result identical to E s n (1) xn .. (k) xn s0n = 1 (1) xn .. (k) xn

(µ) cj xn+j = 0

30

algorithm ! . . . sn+k (1) . . . xn+k .. ... (k) . . . xn+k . . . 1 (1) . . . xn+k .. ... (k) . . . xn+k

PERTURBATIONAL METHODS Rayleigh-Schrdinger Perturbation Theory H = H + βV yields E (n) = E0 + βE1 + . . . + β nEn Goldhammer-Feenberg H = (1 − α)H0 + [V + αH0] yields E (n)(α) = E0(α)+βE1(α)+. . .+β nEn(α) Choose α variationally (True E is α-independent) ∂E (n)(α) ∂α

=0

For n = 3 solution is α = E3/E2 −→ Feenberg series Fn = E (n)(E3/E2) 31

EFFECTIVE CHARACTERISTIC POLYNOMIALS ˇ ıˇ C´ zek

Pn(E)= det φj |H|φk − E δj,k =

n X j=0

Ej

n−j X

fn,j,k β k

k=0

Obtain f ’s from perturbation series Pn(E0 +βE1 +β 2E2 +. . .) = O(β n(n+3)/2) Zero of P2: Π2= E0 + E1 E22 E2 − E3 + 2 2 E E − E 3 q2 4 2 − 4 (E E − E 2) 2 (E − E ) E2 2 3 2 4 3 + 2 E2 E4 − E32 Invariant under Feenberg scaling Π2(E0, . . . , E4) = Π2(E0(α), . . . , E4(α)) . Scaling property Π2(c E0, . . . , c E4) = c Π2(E0, . . . , E4) . 32

MANY-BODY PERTURBATION THEORY Dissociation barrier (kJ/mol) for H2CO−→H2 + CO Method E0 + E1 E2 E3 E4 F4 [2/2] Π2 Best Estimate

Minimum -113.912879 -114.329202 -114.334186 -114.359894 -114.360838 -114.362267 -114.364840

Transition state -113.748693 -114.182435 -114.185375 -114.219892 -114.220603 -114.223409 -114.227767

Barr. 431.1 385.3 390.7 367.6 368.2 364.6 359.9 360

(TZ2P Basis at MP2 Geometries)

33

ITERATION SEQUENCES

Fixed-point equation x = Ψ(x)

Direct Iteration x0, x1 = Ψ(x0), . . . , xn+1 = Ψ(xn), . . .

Cycling s0 = xStart, s1 = Ψ(s0), . . . , sk = Ψ(sk−1) xStart = T (s0, . . . , sk )

Corresponds to new iteration function: yn+1 = T (yn, Ψ(yn), . . . , Ψ(Ψ(. . . Ψ(yn)))

34

ORNSTEIN-ZERNIKE-EQUATION • Classical many-particle systems (fluids) • Pair distribution function g(r) = 1 + h(r)

• Integral equation h = c + ρh ∗ c g(r) = exp(−βu(r)+h(r)−c(r)+E(r)) • Bridge diagrams E(r) =⇒ Closure relations

35

• Solution on lattice with FFT: Γi = (h(ri) − c(ri))ri, ri = i∆r ~ Γ = Ψ(~ Γ) via – direct iteration – direct iteration + vector extrapolation • Extrapolation reduces CPU time by up to 50 % • Extrapolation useful to achieve convergence

2

directit 0 -2

ln ζ

-4 -6 -8 -10 0

200

400

600

800

1000

N Figure 1: Unstable Fixed-point of Direct Iteration (Hard spheres, high density) 36

m2vj 0 -5

ln ζ

-10 -15 -20

200

300

400

500

600

700

N Figure 2: Convergence of the Cycling Method

37

800

900