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β π°π+1; π°π, Ξπ‘ = 0
Chose Ξπ‘ Perform Newton iterations
If failed, chop Ξπ‘ and try again!
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3
ππΊ
ππ+ππ
ππ= π
v
π‘ = πΊπ+π β πΊπ +βπ
π½π(πΊπ+π) β ππππ
π βπ‘πππππ
| β |
0
ππππ πΉ(π)
π
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Failed
ππΊ
ππ+ππ
ππ= π
v
π‘ = πΊπ+π β πΊπ +βπ
π½π(πΊπ+π) β ππππ
π βπ‘πππππ
| β |
0
ππππ πΉ(π)
π
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Converged
π βπ‘πππππ
| β |
0
Ξπ‘
v
v
ππΊ
ππ+ππ
ππ= π
π‘ = πΊπ+π β πΊπ +βπ
π½π(πΊπ+π) β ππππ
ππππ πΉ(π)
π
Safeguards
Classical
Trust Region
Line Search
Heuristic
Eclipse Appleyard (EA)
Modified Appleyard (MA)
Modified Trust-region (X. Wang)
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π° Ξπ‘
β
v
v
v
vv
π
Zero level curve: β π° π ,Ξπ‘ π = 0
Parameterized curve: πβ
ππ= π₯
dπ°
ππ+πβ π°π+1,π°π;Ξπ‘
πΞπ‘
dΞπ‘
ππ= 0
π‘ =ππ°
ππ= βπ₯β1
πβ
πΞπ‘
πΞπ‘
ππ
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Drawbacks:
1. Convergence neighborhood
2. Too many residual evaluations
3. No optimal steplength, πΌ
Ξπ‘ parameterization:
-Requires no additional equation
Parameterization
unknown
Residual Equation Parameterization
π β π° π , Ξπ‘ π = 0 πβ
ππ=πβ
ππ°
dπ°
ππ+πβ
πΞπ‘
dΞπ‘
ππ= 0
Ξπ‘ β π° Ξπ‘ , Ξπ‘ = 0 πβ
dΞπ‘=πβ
ππ°
dπ°
πΞπ‘+πβ
πΞπ‘= 0
π°
Ξπ‘
β
π12
Taylor series expansion :
π°pred = π°0 + Ξπ‘ππ°
πΞπ‘+Ξπ‘2
2!
π2π°
πΞπ‘2+Ξπ‘3
3!
π3π°
πΞπ‘3+Ξπ‘4
4!
π4π°
πΞπ‘4β¦
Order of
approximation
Terms
(1) Zero Order π°pred = π°0
(2) First Orderπ°pred = π°0 + Ξπ‘
ππ°
πΞπ‘
(3) Second Orderπ°pred = π°0 + Ξπ‘
ππ°
πΞπ‘+Ξπ‘2
2!
π2π°
πΞπ‘2
Ξπ‘
(1) (2) (3)
π° 1 π° 2 π° 3
Solution
path
π°π+1
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Parameterization
unknown
First order
approximationSecond order approximation
Ξπ‘ ππ°
πΞπ‘= βπ₯β1
πβ
πΞπ‘
π2π°
πΞπ‘2= βπ₯β1
ππ₯
ππ°β¨ππ°
πΞπ‘+ππ₯
πΞπ‘
ππ°
πΞπ‘+ππΊ
ππ°
ππ°
πΞπ‘+ππΊ
πΞπ‘
πΊ =πβ
πΞπ‘β¨ - Tensor-vector multiplication
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Ξπ‘ = 0.8
(1) (2) (3)
π° 1 π° 2 π° 3
Solution
path
π°π+1
Order of
approximationTerms
Predicted
valueError, πΊπ
(1) Zero Order Spred = π0 0.0 -0.6255
(2) First Order πpred = π0 + Ξπ‘ππ
πΞπ‘0.80 0.1745
(3) Second Order πpred = π0 + Ξπ‘ππ
πΞπ‘+Ξπ‘2
2!
π2π
πΞπ‘20.7488 0.1233
ππππ πΉ(π)
πππππ‘ = 0.0
ππππ’π‘πππ ππ+1 = 0.6255
π0
π π‘ + Ξπ‘ = π π‘ + πΌππ π‘
πΞπ‘+πΌ2
2!π π‘ π π‘ + π πΌ3
πππππ = π π‘ + πΌππ π‘
πΞπ‘
β°0 = ππ β πππππ βπΌ2
2!π π‘ π π‘
Take πΌ2
2!π π‘ π π‘ β€ πΌ
ππ π‘
πΞπ‘
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πΌ
ππ β πβ π0
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π π =π + 10 β π2
5 β π2
π π =π2
3 β 2π
π π = ππ
Assume Quadratic convergence
(Newton-Kantorovich):
SuperLinear error model:
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ππ’πππππ‘ππ ππππ£πππππππ πππππ: π π =π + 10 β π2
5 β π2
ππ+1 β€ π(ππ)
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ππ’ππππΏπππππ ππππ£πππππππ πππππ: π π = ππ
ππ+1 β€ π(ππ)
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π π = π01 + exp π΅π
1 + exp(βπ΅(π βπ)), π€βπππ π΅ πππ π πππ πππ‘π‘πππ πππππππππππ‘π
π΅ ππππππππππ¦ π ππππππππππ¦
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π π = π01 + exp π΅π
1 + exp(βπ΅(π βπ))
ππ+1 β€ ππ Ξ²(πΎ)
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π‘1
π‘2
π‘3Ξπ‘π‘πππππ‘
Ξt = 0
π°n+1π0 = (π°n, 0)
π2
π3
Solution path
π1Tiny initial
steplength
Steplength from
Adaptation
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π»ππ‘πππππππππ’π ππππ π€ππ‘βπ·πππππ πππ ππ 60 Γ 220 Γ 50
6 πππππ’ππ‘πππ πππππ ππ€π = 1500 ππ π.
2 πΌπππππ‘πππ πππππ ππππ = 5000 ππ π.
πΌππππ’πππ πΊπππ£ππ‘π¦ ππππΆππππππππ¦ ππππππ‘π
P1
P2
P3
P4
P5
P6
Inj1
Inj1
http://www.spe.org/
P1
P2
P3
P4
P5
P6
Inj1
Inj1
Modified Continuation-Newton:
Never cuts the timestep due to lack of convergence;
Devised a robust and automatic steplength selection strategy;
Showed that high-order CN methods are not competitive;
Future work:
Test on complex physics;
Implement in GENSOL.
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