Numerical studies of space filling designs: optimization algorithms and subprojection properties

Transcription

1 umercal studes of sace fllng desgns: otmzaton algorthms and subroecton roertes Bertrand Iooss wth Gullaume Dambln & Matheu Coulet CEMRACS 03 July, 30th, 03

2 Motvatng eamle: Uncertantes management n smulaton of thermal-hydraulc accdent Scenaro : Loss of rmary coolant accdent due to a large brea n cold leg [ De Crecy et al., ED, 008 ] ~ 0-50 nut random varables X: geometry, materal roertes, envronmental condtons, Comuter code Y =f X Tme cost ~ -0 h - ~ Pressurzed water nuclear reactor Interest outut varable Y : Pea of claddng temerature Goal: numercal model eloraton va sace fllng desgn, then metamodel B. Iooss CEMRACS 30/07/3 Lumny - Source: CEA

3 Model eloraton goal GOAL : elore as best as ossble the behavour of the code Put some onts n the whole nut sace n order to «mamze» the amount of nformaton on the model outut Contrary to an uncertanty roagaton ste, t deends on Regular mesh wth n levels =n smulatons To mnmze, needs to have some technques ensurng good «coverage» of the nut sace Smle random samlng Monte Carlo does not ensure ths E: =, n =3 =9 = 0, n=3 = E: = = 0 Monte Carlo B. Iooss CEMRACS 30/07/3 Lumny - 3 Otmzed desgn

4 Obectves When the obectves s to dscover what haens nsde a numercal model e.g. non lneartes of the model outut, we want to buld the desgn whle resectng the constrants:...,.... To «regularly» sread the onts over the -dmensonal nut sace c. To ensure that ths nut sace coverage s robust wth resect to dmenson reducton because most of the tmes, only a small number of nuts are nfluent low effectve dmenson Therefore, we loo for some desgn whch nsures the «best coverage» of the nut sace and ts sub-roectons The class of Sace fllng Desgn SFD s adequate. It can be: - Based on an nter-ont dstance crteron mnma, mamn, - Based on a crteron of unform dstrbuton of the onts entroy, varous dscreancy measures, L² dscreances, B. Iooss CEMRACS 30/07/3 Lumny - 4

5 . Two classcal sace fllng crtera Mndst dstance: mn d,, Mamn desgn Mm : ma mn d,, B. Iooss CEMRACS 30/07/3 Lumny - 5

6 . Two classcal sace fllng crtera Mndst dstance: mn d,, Mamn desgn Mm : ma mn d,, Dscreancy measure: Devaton of the samle onts dstrbuton from the unformty D * su Volume Q t t[0,[ Q t L dscreancy allows to obtan analytcal formulas * D Volume Q t Q t [ 0,[ dt / B. Iooss CEMRACS 30/07/3 Lumny - 6

7 B. Iooss CEMRACS 30/07/3 Lumny - 7 Eamle of dscreancy Varous analytcal formulatons whle consderng L² dscreancy and dfferent nd of ntervals Centered L -dscreancy ntervals wth boundary one verte of the unt cube C, 3 [ Hcernell 998 ] Modfed L dscreancy allows to tae nto account onts unformty on subsaces of [0,[ coordnates n of cube unty on the roecton of and,..., wth Volume Ø u C Q Q u d Q D u u u C u Q u u u t t t t t

8 . Undm.-roecton robustness va Latn Hyercube Samle Class of LHS ensures unform roecton on margns LHS,: - Dvde each dmenson n ntervals - Tae one ont n each stratum - Random LHS: erturb each ont n each stratum Fndng an otmal SFD LHS: mossble ehaustve eloraton:! dfferent LHS E: =, =4 Methods va otmzaton algo e: mnmzaton of. va smulated annealng :. Intalsaton of a desgn LHS ntal and a temerature T. Whle T > 0 :. Produce a neghbor new of ermutaton of comonents n a column. relace by new wth roba 3. decrease T new mn e, T 3. Sto crteron => s the otmal soluton B. Iooss CEMRACS 30/07/3 Lumny - 8 [ Par 993; Morrs & Mtchell 995 ]

9 LHS mamn: regularzaton of the crteron Mndst crteron : to be mamzed mn d,, Eamle : Mamn LHS,6 Regularzed mndst crteron : to be mnmzed / q q q d,,, [ Morrs & Mtchell 95 ] umercal test: = 00, = 0 These crtera are equvalent for the otmzaton when q [Pronzato & Müller] q s easer to otmze than mndst In ractce, we tae q = 50 B. Iooss CEMRACS 30/07/3 Lumny - 9

10 B. Iooss CEMRACS 30/07/3 Lumny - 0 Udatng crtera after a LHS erturbaton Between and, ont coordnates and are modfed Regularzed mndst crteron -/ dstances Only recalculate the - dstances of these onts to other onts L² dscreancy crtera cost n O² Cost n O q q q d /,,, C, 3 c C c C,, ' 3 ' ; 3 c c ' then, and, If ' ' ' ' ',, c c c c c c c c C C [ Jn et al. 005 ]

11 Two dfferent otmzaton algorthms Morrs & Mtchell Smulated Annealng MMSA [ Morrs & Mtchell 995 ] Lnear rofle for the temerature decrease geometrcal alternatve: T = c T 0 Temerature decreases when B new LHS do not mrove the crteron Slow convergence but large eloraton sace Enhanced Stochastc Evolutonary ESE [ Jn et al. 005 ] Inner loo I teratons: Prooston of M new erturbed LHS at each ste Outer loo to manage the temerature can decrease or ncrease B. Iooss CEMRACS 30/07/3 Lumny -

12 Comarson of otmzaton algorthms convergence umercal tests: = 50, = 5 MMSA - lnear rofle T 0 = 0., B = 300, c = 0.9 ESE M = 00, I = 50 Both algorthms converge slowly to the same value, after the same teraton numbers ESE shows a faster convergence at the frst teratons than MMSA It s ossble to mrove ths result, but at a rohbtve cost MMSA: T 0 =0.0, B=000, c=0.98; ESE: M=300 B. Iooss CEMRACS 30/07/3 Lumny -

13 C-dsc. C-dsc. Robustness tests n D subroectons of otmal LHS /3 3 tyes of LHS n = 00 wth ncreasng ; 0 relcates for each dmenson All D subroectons are taen nto account Standard LHS reference Low C-dscreancy LHS C = L -centered Mamn LHS 0.05 From dmenson =0, the mamn LHS behaves le a standard LHS From dmenson =40, the low C-dscreancy LHS behaves le a standard LHS Another test for the low L²-star dscreancy: convergence for =0 It confrms the relevance of C-dscreancy crteron n terms of subroectons B. Iooss CEMRACS 30/07/3 Lumny - 3

14 Another sace-fllng crtera based on Mnmal Sannng Tree M[,] M[,] M[,] MST for random desgn M[,] MST for mamn LHS umercal tests on varous SFD: s = 0, = Random LHS Sobol sequence C-dscreancy Bad D roectons otmzed LHS Strauss desgn Ponts algnments 0.8 Mamn LHS B. Iooss CEMRACS 30/07/3 Lumny - 4 Usng the Mnmal Sannng Tree MST [ Franco et al., Chem. Lab., 009 ] m Concluson Ths MST-based grah s a tool to comare desgns n terms of regularty n the -dmensonal sace Comlementarty wth mndst

15 Robustness tests n D subroectons of otmal LHS 3/3 MST crtera = 00 Mamn LHS Low C-dscreancy LHS B. Iooss CEMRACS 30/07/3 Lumny - 5

16 R²test R²test Eamle: fttng a rgng metamodel X Comuter code on monotonc test functon = 5 : g-functon of Sobol g X X 4X a 5,..., X 5 avec a et X ~ U0; our... 5 a Smle LHS Low W-dscre. LHS Metamodel rgng s bult on a learnng samle of szes =,,40 [ Marrel 008 ] B. Iooss CEMRACS 30/07/3 Lumny - 6

17 Conclusons SFD are useful n an ntal eloraton ste, small, large Algorthms for LHS otmzaton: ESE seems referable faster convergence Tunng arameters are dffcult to ft; some recommendatons are made n refs. 3 Modfed L² dscreances tae nto account unformty of the ont roectons on lower-dmensonal subsaces of [0,[ In our tests, low L²-centered dscreancy LHS have shown the best sace fllng robustness on the roectons over D subsaces same effects on 3D subroectons Imortant roerty for metamodel fttng and senstvty ndces comutaton 3 Dstance-based desgns show stronger sace fllng regularty but no D robustness Challenge: Buldng good & robust SFD outsde the LHS class B. Iooss CEMRACS 30/07/3 Lumny - 7

18 Bblograhe G. Dambln, M. Coulet & B. Iooss, umercal studes of sace fllng desgns: otmzaton algorthms and subroecton roertes, submtted Pacage n R software: DceDesgn D. Duuy, C. Helbert, J. Franco, O. Roustant, G. Dambln, B. Iooss K-T. Fang, R. L & A. Sudanto, Desgn and modelng for comuter eerments, Chaman & Hall, 006 F.J. Hcernell. A generalzed dscreancy and quadrature error bound. Mathematcs of Comutaton, 67:99-3, 998. B. Iooss, L. Boussouf, V. Feullard & A. Marrel. umercal studes of the metamodel fttng and valdaton rocesses. Internatonal Journal of Advances n Systems and Measurements, 3:-, 00. R. Jn, W. Chen & A. Sudanto. An effcent algorthm for constructng otmal desgn of comuter eerments. Journal of Statstcal Plannng and Inference, 34:68-87, 005. M.E. Johnson, L.M. Moore & D. Ylvsaer. Mnma and mamn dstance desgn. Journal of Statstcal Plannng and Inference, 6:3-48, 990. M. Morrs & T. Mtchell. Eloratory desgns for comutatonal eerments. Journal of Statstcal Plannng and Inference, 43:38-40, 995. J-S. Par. Otmal Latn-hyercube desgns for comuter eerments. Journal of Statstcal Plannng and Inference, 39:95-, 994. L. Pronzato & W. Müller. Desgn of comuter eerments: sace fllng and beyond. Statstcs and Comutng, :68-70, 0. B. Iooss CEMRACS 30/07/3 Lumny - 8

19 Annees B. Iooss CEMRACS 30/07/3 Lumny - 9

20 Robustness tests n D subroectons of otmal LHS /3 tyes of LHS n = 00 wth ncreasng ; 0 relcates for each dmenson All D subroectons are taen nto account Mamn LHS Low C-dscreancy LHS It confrms the non-relevance of mndst dstance n terms of subroectons B. Iooss CEMRACS 30/07/3 Lumny - 0