PhD thesis o ff er: Metamodelling under inequality constraints Chair in Applied Mathematics OQUAIDO
Context
This PhD thesis offer is associated to the Chair in Applied Mathematics OQUAIDO of Mines Saint-Étienne (2016− 2019). The Chair, which follows the ReDICE consortium (www.redice-project.org), gathers partners from academia, technological research and industry around innovative mathematical methods for the Design and Analysis of Computer Experiments (DACE). The partners of the Chair are:
• Mines Saint-Etienne, Ecole Centrale de Lyon, Univ. of Nice, Univ. of Toulouse (academia)
• BRGM, CEA, IFPEN, IRSN, Safran, Storengy (technological research and industry)
The DACE research domain is devoted to exploit industrial computer codes that are costly to evaluate. Such computer codes typically model complex physical phenomena such as encountered in the development of new technologies or risk studies. The partners of the Chair use them in various sectors such as energy, environment and transport.
The Chair proposes an original and dynamical framework to develop innovative research, and fosters interaction between all its members: researchers, engineers, post-doctoral and PhD students. Expected developments are methodological, but guided by case studies from real-life problems. The Chair is funding1post-doc and2PhD thesis starting in2016. This offer is one of them.
Research lines and aim
Among the research lines with strong methodological interest and promising application potential, the Chair scientific committee identified the case of models with inequality constraints, such as boundary or monotonicity. Indeed in practice, underlying physical phenomena are often known to be monotonic with respect to one or several variables, or to belong to a range of possible values. Taking into account that information may lead to more accurate results.
In addition to this main line, another axis regarding constraints can be brought by the PhD candidate, depending on its master work or past experience in DACE.
The aim of this3-year research project is twofold:
• Methodology: To improve existing works for real-life problems, involving tens of input variables.
• Application: To apply the proposed new methodology to at least 1case study of the Chair.
Among starting points on inequality constraints, the PhD works should investigate the extension of a promising Gaussian process-based model (Maatouk, Bay, 2014), which satisfies inequality constraints (boundary, inequality, convexity) everywhere in the space, contrarily to other approaches based on spatial discretization. At this stage, the model has two limitations: Its application is limited to a small number of constraints (1, 2 or 3) because of the tensorization technique used, and parameter estimation is based on a time-consuming cross-validation. More generally, the PhD methodological developments may develop Gaussian process-based models in3 main directions:
1. Estimation under inequality constraints.
2. Extension to higher dimensions.
3. Case of a high number of observations (consequence of the previous point).
1
Supervising team
The supervising team gathers researchers from 2 sites :
• Olivier Roustant and Nicolas Durrande, Mines Saint-Etienne
• François Bachoc, Institut de Mathématiques de Toulouse
Applicant profile
Candidates should have completed a Master in applied mathematics, statistics, machine learning, or related disciplines.
The applicant should demonstrate both theoretical and computational skills. Implementations in R are expected.
CV and motivation letter in English or French should be sent to the coordinators of the Chair (Olivier Roustant and Nicolas Durrande) using the e-mail address: [email protected]
Conditions
•Date/duration: The position is a 3-year contract, expected to start in September 2016.
• Location: The main location is Mines Saint-Étienne, France. Visits at the Institut de Mathématiques de Toulouse are expected.
•Participation to teaching: 40 hours / year (in French or English)
•Net salary: 1630e/month
References
• S. Da Veiga and A. Marrel (2012), “Gaussian process modeling with inequality constraints”, Annales de la Faculté des Sciences de Toulouse,21, p. 529–555.
• S. Golchi, D. R. Bingham, H. Chipman, and D. A. Campbell (2015), “Monotone Emulation of Computer Exper- iments”, SIAM/ASA Journal on Uncertainty Quantification, 3(1), p. 370-392
• H. Maatouk and X. Bay (2014), “Gaussian Process emulators for computer experiments with inequality con- straints”, in revision for SIAM/ASA Journal on Uncertainty Quantification.
• Robert B. Gramacy & Daniel W. Apley (2015), “Local Gaussian process approximation for large computer experiments”, Journal of Computational and Graphical Statistics,24 (2), p. 561-578.
• Deisenroth, Marc Peter and Ng, Jun Wei (2015), “Distributed Gaussian Processes”, Proceedings of the 32nd International Conference on Machine Learning, Lille, France. JMLR: W&CP volume 37.
• François Bachoc (2014), “Asymptotic analysis of the role of spatial sampling for covariance parameter estimation of Gaussian processes”, Journal of Multivariate Analysis,125, p. 1-35.
2
!
!"#$%&&'()
*+,"-.+/'01+%023$%45+/+625+%0$%&$5"'$+05'(023$20.$'75'(023$2'(%.802/+91$%&$2$:'"+93'
!"#$%&"!#$%&'()!*+,-!.!#$%&'()!*+,/
'%()*$%+"!0123%.4%2(33(5!6)(78(3%!9%1:9!23!%;(!<1)29!13=!>&8?&89(!1)(19@
,-+&$+."!
! ABCD4! E)&8F(! <0G! HB3=89%)21?! GI)((J(3%! 6&)! >)12323I! %;)&8I;! D(9(1)$;K! .! >;(! ABCD4 6(??&L!L2??!92I3!1!M!:(1)9!68??!%2J(!L&)N!$&3%)1$%!L2%;!E)&8F(!<0G@
! O(1)!I)&99!91?1):"!&6!%;(!&)=()!&6!M+P(8)&9@
/&0$1%#1"!D&=&?F;(!Q(!D2$;(!HARD0!13=!4$&?(!=(9!S23(9!=(!0%.4%2(33(K5!T2$%&)!<2$;(3:!HBRDG
>&8?&89(K5!U(3&V%!4318W!13=!A?XJ(3%!Y8J13=!HE)&8F(!<(8I(&%!0&$2X%X!G3&3:J(K
2%+*)(*1"!D&=&?F;(!Q(!D2$;(!H?()2$;(Z(J9(@6)K5!T2$%&)!<2$;(3:!HT2$%&)@<2$;(3:Z%&8?&89(@23)1@6)K5 U(3&V%!4318W!H'(3&2%@(318WZJF91@$&JK@
/334$()+*5 !#%6$4""! A13=2=1%(9! 9;&8?=! ;1[(! $&JF?(%(=! 1! S19%()! 23! 1FF?2(=! J1%;(J1%2$95! 9%1%29%2$95 J1$;23(! ?(1)323I5! &)! (3I23(()23I! L2%;! 1! J1%;(J1%2$1?! '1$NI)&83=@! >;(! 1FF?2$13%! 9;&8?=
=(J&39%)1%(!'&%;!%;(&)(%2$1?!13=!$&JF8%1%2&31?!9N2??9@!BJF?(J(3%1%2&39!23!D!1)(!(WF($%(=@
2%+*"7*"!A&JJ2%%(=!%&!)(=8$23I!%;(!68(?!$&398JF%2&3!13=!(J2992&39!&6!2%92%()21!&F%2J2\1%2&3!J(%;&=9!1)(!3((=(=@
89:"(*$0"15 %65 *;"5 !;<"! >;(! &']($%2[(! &6! %;29! <;Y! 29! %&! =([29(! 13=! [1?2=1%(! 1! J8?%2.&']($%2[(
&F%2J2\1%2&3! J(%;&=! &6! MY! 1()&=:31J2$! 92J8?1%2&39@! B3! %;29! $&3%(W%5! &F%2J2\1%2&3! F)&'?(J9! 1)(
F1)%2$8?1)?:! $;1??(3I23I! %&! %1$N?(5! =8(! %&! %;(! 3&3.?23(1)2%:! &6! %;(! 92J8?1%2&3! &8%F8%95! %;(
$&JF8%1%2&31?!$&9%!&6!1!923I?(!92J8?1%2&3!H6)&J!,*!;&8)9!%&!%;)((!=1:9K5!%;(!3((=!%&!;13=?(!$&JF?(W
$&39%)123%95!&)!%;(!?1)I(!38J'()!&6!F1)1J(%()9!%&!&F%2J2\(!&[()!H*+!%&!^+!83N3&L39K@
D($(3%! 1FF)&1$;(9! '19(=! &3! E1899213! F)&$(99(9! ;1[(! 9;&L3! F)&J2923I! )(98?%95! '8%! 1==)(99! %;29 F)&'?(J! &3?:! 23! F1)%"! 9((! _,5*5M5^`! 6&)! (W1JF?(9@! <)&]($%2&39! 23! 98'9F1$(9! ;1[(! '((3! F)&F&9(=! %&
;13=?(!?1)I(!38J'()!&6!83N3&L39!_a`@!C&)!%;29!<;Y5!%;(!23[(9%2I1%(=!J(%;&=&?&I:!L2??!6&??&L!%;29
?23(!&6!L&)N5!13=!L2??!6&$89!23!F1)%2$8?1)!&3"
! (662$2(3%!91JF?23I!9%)1%(I:!23!%;(!F)(9(3$(!&6!9([()1?!&']($%2[(9!13=!$&39%)123%95
! 13!18%&J1%(=!)(=8$%2&3!23!=2J(392&3!23!&)=()!%&!=(1?!L2%;!J13:!83N3&L39!L2%;!1!)(9%)2$%(=
38J'()!&6!92J8?1%2&395
! %;(!F&992'2?2%:!%&!=29%)2'8%(!$1?$8?1%2&39!23!13!19:3$;)&3&89!619;2&3!&3!1'&8%!,+!$&JF8%23I 3&=(95!13=
! =(1?23I!L2%;!%;(!([(3%!&6!92J8?1%2&3!612?8)(9@
!"#$%&&'(%)"$*#+#,(&#*$-.,,$/#$+%,.*%0#*$(1$0-($.1*230'.%,$)%3#3$*#%,.14$-.0"$3"%&#$(&0.5.6%0.(17
! (&0.5.6%0.(1$(8$0"#$)"%5/#'$*#3.41$(8$%1$.10#'1%,$)(5/230.(1$#14.1#9
! (&0.5.6%0.(1$(8$0"#$#:0#'1%,$3"%&#$(8$%$+#".),#$;<9=9>?@
A1$ 0"#3#$ )%3#39$ 0"#$ 21B1(-13$ %'#$ *#3.41$ &%'%5#0#'3$ C+.%$ DEF$ 0((,3G9$ %1*$ *'%4$ )(#88.).#10$ (' )(5/230.(1$#88.).#1)H$%'#$#:%5&,#3$(8$(/I#)0.+#3$%1*$)(130'%.103@$!"#$5#0"(*$-.,,$/#$&'(4'%55#*
.1$0"#$J$,%142%4#@
!"#$"%&'()*+
;K?$L@$M.1(.3$%1*$N@$O.)"#1H9$PQO%'#0(7$E1$J$O%)B%4#$8('$Q%233.%1RO'()#33$M%3#*$L2,0.RS/I#)0.+#$S&0.5.6%0.(1$%1*
E1%,H3.3P9$TUK<9$"00&37VV)'%1@'R&'(I#)0@('4V-#/V&%)B%4#3VQO%'#0(V+.41#00#3VQO%'#0(W+.41#00#@&*8
;T?$ N@$ O.)"#1H9$ PL2,0.(/I#)0.+#$ (&0.5.6%0.(1$ 23.14$ Q%233.%1$ &'()#33$ #52,%0('3$ +.%$ 30#&-.3#$ 21)#'0%.10H$ '#*2)0.(1P9 X0%0.30.)3$%1*$D(5&20.149$N(,25#$TY9$A332#$<9$&&@$KT<YRKT>U9$TUKY
;Z?$ O%2,$ [#,.(09$ \2,.#1$ M#)09$ ]55%12#,$ N%6^2#69$ PE$ M%H#3.%1$ %&&'(%)"$ 0($ )(130'%.1#*$ 3.14,#R$ %1*$ 52,0.R(/I#)0.+#
(&0.5.6%0.(1P9$_#%'1.14$%1*$A10#,,.4#10$S&0.5.6%0.(17$`0"$A10#'1%0.(1%,$D(18#'#1)#9$_ASa$`9$_.,,#9$['%1)#9$TUKY
;b?$c%41#'9$!(/.%39$ #0$ %,@$ PS1$ #:&#)0#*R.5&'(+#5#10$ )'.0#'.%$ 8('$ 5(*#,R/%3#*$ 52,0.R(/I#)0.+#$ (&0.5.6%0.(1@P$ O%'%,,#, O'(/,#5$X(,+.14$8'(5$a%02'#9$OOXa$dA@$X&'.14#'$M#',.1$e#.*#,/#'49$TUKU@$=K>R=T=@
;Y?$ $f@$c%149$L@$f(4".9$[@$e200#'9$F@$L%0"#3(1$ $%1*$a@$F#$['#.0%39$PM%H#3.%1$S&0.5.6%0.(1$.1$e.4"$F.5#13.(13$+.%
J%1*(5$]5/#**.143P9$A\DEA$)(18@9$TUKZ@
;<?$_@$D2.9$!@$c%149$f@$_29$L@$\.%$%1*$g@$X219$P[2,,RO%'%5#0#'$E&&'(%)"$8('$0"#$A10%B#$O('0$F#3.41$(8$%$[(2'RN%,+#
F.'#)0RA1I#)0.(1$Q%3(,.1#$]14.1#P@$EXL]@$,-./0&-.1(2.34'#"052.6%75'-.TUKYhKZ=C`G@
;=?$ L@$ g%4(2/.9$ _@$ !"(/(.39$ L@$ X)"(#1%2#'9$ PE1$ %3H1)"'(1(23$ 30#%*HR30%0#$ aXQERAA$ %,4('.0"5$ 8('$ 52,0.R(/I#)0.+#
(&0.5.6%0.(1$(8$F.#3#,$)(5/230.(1P9$A10@$D(18#'#1)#$(8$]14.1##'.14$S&0.5.6%0.(19$TUKU@
;>?$\("1$M@$e#H-((*@$PA10#'1%,$D(5/230.(1$]14.1#$[21*%5#10%,3P@$891'(7:;"$$9$K`>>@
