Kriging_Dcmp - – iDOT The Center of Innovative Design...

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Unformatted text preview: – iDOT The Center of Innovative Design Optimization Technology Kriging 1 2006. 08. 11 Ø Ò é ª ¶ Ø Ò é ª ¶ * Ø é ª ¶ AMOD Lab.– Applied Mechanics & Optimal Design Lab. Build Kriging Model General Kriging Maximum Likelihood Estimation (MLE) Find maximize ˆ ( −0.5) nexp ln ( σ 2 ) + ln R 0 ≤ θi , 0 ≤ pi ≤ 2, i = 1,L , ndv where, ˆ σ 2 θi , pi ( yexp − Fβ ) = T R −1 ( yexp − Fβ ) nexp x12 M xnexp 2 L O L x1ndv M xnexp ndv nexp×ndv yexp1 yexp = M yexpnexp nexp×1 xexp1 x11 xexp = M = M xexp nexp xnexp1 2 2 f T ( xexp ) 1 x11 x12 x11 x11 x12 x12 1 F= M M = 2 2 T 1 x xnexp 2 xnexp1 xnexp1 xnexp 2 xnexp 2 nexp×nsat f ( xexp nexp ) nexp1 β = ( FT R −1F ) FT R −1yexp −1 R ( xexp1 , xexp1 ) L R ( xexp1 , xexp nexp ) R= M O M R ( xexp nexp , xexp1 ) L R ( xexp nexp , xexp nexp ) nexp×nexp iDOT 2 /12 The Center of Innovative Design Optimization Technology http://idot.hanyang.ac.kr/ Applied Mechanics & Optimal Design Lab. http://amod.hanyang.ac.kr/ Evaluate Kriging Model General Kriging Best Linear Unbiased Predictor (BLUP) % y = f T ( xβ + )r T xR () −1 yexp − Fβ ( ) 2 f ( x ) = 1 x1 x2 x12 x1 x2 x2 T β = ( FT R −1F ) FT R −1yexp −1 yexp1 yexp = M yexpnexp x12 M xnexp 2 L O L x1ndv M xnexp ndv xexp1 x11 xexp = M = M xexp nexp xnexp1 2 2 f T ( xexp ) 1 x11 x12 x11 x11 x12 x12 1 F= M M = 2 2 T 1 x xnexp 2 xnexp1 xnexp1 xnexp 2 xnexp 2 nexp×nsat f ( xexp nexp ) nexp1 R ( xexp1 , xexp1 ) L R ( xexp1 , xexp nexp ) R= M O M R ( xexp nexp , xexp1 ) L R ( xexp nexp , xexp nexp ) nexp×nexp iDOT 3 /12 The Center of Innovative Design Optimization Technology http://idot.hanyang.ac.kr/ R ( x, xexp ) 1 r ( x) = M R ( x, xexp nexp ) Applied Mechanics & Optimal Design Lab. http://amod.hanyang.ac.kr/ Correlation Function General Kriging R ( x1 , x 2 ) = R ( d ) , where d = x1 − x 2 Exponential Gaussian ndv ndv R ( d ) = ∏ exp ( −θi di ) = exp −∑ θi di i =1 i =1 R ( d ) = ∏ exp −θi di i =1 ndv i =1 ndv ( ( 2 ) ) ndv 2 = exp −∑ θi di i =1 ndv = exp −∑ θi di i =1 pi Exponential General R ( d ) = ∏ exp −θi di pi where, θi ≥ 0 and 0 < pi ≤ 2, i = 1,L , ndv iDOT 4 /12 The Center of Innovative Design Optimization Technology http://idot.hanyang.ac.kr/ Applied Mechanics & Optimal Design Lab. http://amod.hanyang.ac.kr/ Decomposition ˆ Ò ª ¶ K rigng Model È Ò *¶ ª , -1 1) MLE : R det(R) 2) beta : R-1 3) BLUP : R-1 È Ò ä Cholesky ¸* Ø * ª + K riging Decomposed Kriging K riging Model @ + • 7 @ QR decomposition ˆ Ò *¶ ª matrix È Òª ¶* ! i nverse determinant È Ò ä ªR * 2001, Jeong-Soo Park, Efficient computation of maximum likelihood estimators in a spatial linear model with power exponential covariogram, Computers and Geosciences 27 (2001) 1-7 iDOT 5 /12 The Center of Innovative Design Optimization Technology http://idot.hanyang.ac.kr/ Applied Mechanics & Optimal Design Lab. http://amod.hanyang.ac.kr/ Decomposition ¸ Ò ª ¶ Cholesky Decomposition LINPACK Positive Definite K riging Decomposed Kriging dchdc.for A¸ ¶ * ,A Ò m Symmetric ø ¶ Ò C ø ª¶ *Ò . A m× m = C T C C¸ Ò : ¶ a. C upper triangular matrixé ˆ ˜ Ø * °Ò à * ª @ p‘ b. C ˜ ¸ Ø * Ò à° ª * p‘ C= m ª ¶ * i nverse Ø ¸ i nverse matrix @ A m ! . determinant 2 ! R = ∏ ( Ci , i ) i =1 inverse matrix ˜ iDOT 6 /12 determinant Ø ¸* *@ . , Ø * Ò é¶ The Center of Innovative Design Optimization Technology http://idot.hanyang.ac.kr/ Applied Mechanics & Optimal Design Lab. http://amod.hanyang.ac.kr/ Decomposition Ø ªq ¹ QR Decomposition (Factorization) Aø ¶ Ò * ,A Q K riging Decomposed Kriging LAPACK . m dgeqrf.for n m n RØq * ª ¹ A n = m Q m X m R n A m×n = Q m×m R m×n m R A = m Q X m a. Q orthogonal matrix QT = Q −1 ! QT Q = Q −1Q = I a. R iDOT 7 /12 upper triangular matrix ° i nverse matrix ˜ Ø ¸ Ò ì¶ @* ! The Center of Innovative Design Optimization Technology http://idot.hanyang.ac.kr/ Applied Mechanics & Optimal Design Lab. http://amod.hanyang.ac.kr/ Decomposition ( Ò ª ¶ Kriging I. M LE a. b. Determinant : K riging Decomposed Kriging Cholesky QR decomposition R nexp×nexp = CT ×nexp C nexp×nexp nexp R = ∏ Ci2,i nexp i =1 R Xq Ý determinant ª! * ¨ ˆ2 SigmaSquare : σ = = = 2 = (Y nexp − Fnexp×nsat β nsat ) R −1 ( Ynexp − Fnexp×nsat β nsat ) T ( Y − Fβ ) T ( CT C) −1 ( Y − Fβ ) nexp nexp ( Y − Fβ ) ( Y − Fβ ) −1 T C −1C −T ( Y − Fβ ) nexp 2 , D −T D −1 ( Y − Fβ ) nexp T −1 D = CT 2 T { D ( Y − Fβ ) } { D ( Y − Fβ ) } = Y − D −1Fβ D −1Y − D −1Fβ nexp ~ ~ T~ ~ Y − Fβ Y − Fβ = nexp ~ ~ 2 Ynexp = D −1 ×nexp Ynexp , Fnexp×nsat = D −1 ×nexp Fnexp×nsat nexp nexp (D = nexp T −1 )( ) ( )( ) D-1 I nverse Ø ¸ R-1 ! iDOT 8 /12 The Center of Innovative Design Optimization Technology http://idot.hanyang.ac.kr/ Applied Mechanics & Optimal Design Lab. http://amod.hanyang.ac.kr/ Decomposition ¨ Ò ª ¶ Kriging I. BETA β nsat = ( F R F ) F R Y T −1 −1 T −1 K riging Decomposed Kriging Cholesky QR decomposition I. −1 = FT ( DDT ) F −1 { ( } −1 BLUP T T % y = f ( xβ r+ x ( R ) ) T = fβ + r T −1 Y( Fβ − − Fβ ) FT ( DDT ) Y DD ( −T −T T ( )Y −1 ) = ( FT D−T D−1F ) FT D−T D−1Y −1 T = fβ + D −D rT T −1 −1 = { ( D F) −1 T T ( D F)} ( D F) ( D Y) −1 −1 −1 T T T = fβ + D rT T = fβ + D rT D −1 − −1 ( Y D Fβ Y% − % Fβ Y− ( Fβ ) %% = FF * ظ ) −1 %% FY . . (R −1 ( ) ) % F = QP = ( QP ) P ) { T ( QP ) } %% FT Y M D-1 I nverse Ø ¸ R-1 ! −1 %% = ( PT QT QP ) FT Y M QT Q = I ( Q −1 orthogonal matrix ) %% = ( PT P ) F T Y (PTP)-1 I nverse Ø ¸ R-1 ! iDOT 9 /12 The Center of Innovative Design Optimization Technology http://idot.hanyang.ac.kr/ Applied Mechanics & Optimal Design Lab. http://amod.hanyang.ac.kr/ ...
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This note was uploaded on 04/12/2010 for the course ME master taught by Professor Mon during the Spring '09 term at Hanyang University.

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