Analyse numérique d’équations aux dérivées aléatoires, applications à l’hydrogéologie

(1)

HAL Id: tel-00625092

https://tel.archives-ouvertes.fr/tel-00625092v2

Submitted on 7 Jun 2012

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

applications à l’hydrogéologie

Julia Charrier

To cite this version:

Julia Charrier. Analyse numérique d’équations aux dérivées aléatoires, applications à l’hydrogéologie.

Mathématiques générales [math.GM]. École normale supérieure de Cachan - ENS Cachan, 2011.

Français. �NNT : 2011DENS0030�. �tel-00625092v2�

(2)

THÈSE / ENS CACHAN - BRETAGNE

sous le sceau de l’Université européenne de Bretagne

pour obtenir le titre de

DOCTEUR DE L’ÉCOLE NORMALE SUPÉRIEURE DE CACHAN

Mention : Mathématiques

École doctorale MATISSE

présentée par

Julia Charrier

Préparée à l’INRIA Rennes - Bretagne Atlantique

Analyse numérique

d’équations aux dérivées

aléatoires, applications à

l’hydrogéologie.

Thèse soutenue le 12 juillet 2011

devant le jury composé de :

Fabio Nobile

Professeur à Politecnico di Milano / rapporteur

Denis Talay

Directeur de recherche à l’INRIA / rapporteur

Olivier Le Maître

Chargé de recherche CNRS au LIMSI / examinateur

Florent Malrieu

Maître de conférences à l’université de Rennes 1 /

examinateur

Arnaud Debussche

Professeur à l’ENS Cachan-Bretagne / directeur de thèse

Jocelyne Erhel

(3)

(4)

!"#$!%&!$"$!%!$ &!%!()"*+())+,)(-),$!..!%!()/%!"0&$! )!,$"0!)12"!3$(-,0!)4+ !.5(!#+,$

-*+&$- !#)607!( -0$!$%-)12"!89+,)-,.+(:0! !")$+&"-(";&."+()",%!:,&0!$!)%! +("!&..!$)+,)!(

%-.-&""-(),(!:$-(0!-,)+(+%&!!)!(%!<-&"-() +(=-( !84!$!%!$ &!)+,)#-$)& ,.&2$!%!()3$(-,0#+,$

"-:$-(0!0&"#+(&>&.&)6;"+( "+,)&!("-(" <-&..!;)+,)!".!" +((-&""-( !"?,7&.%7-)$-("%&"!"; #+,$ %7-*+&$

<-&)#$+=)!$0! "+( !@#6$&!( !;#+,$ "->&!(*!&..-( ! 0-(".!" %+%!()"0&A &.!"; #+,$ -*+&$ $,!(%+& !) #+,$%7-*+&$<-&)-*-( !$",$.! 1!%&(?,&%2(!076),0&-()!/ 1!$ 1!,"!8

4!$!%!$ &!B!(&"9-.-5!)C->&+D+>&.!07-*+&$- !#)60!$-##+$)!$ !))!)12"!;%-.:$6.!,$"!%#.+&"0,

)!%#")$2" 1-$:6"8E7!") ,(1+((!,$ #+,$%+&?,7&."-&!().,-*! -))!()&+(%+()$-*-&.!)?,7&."0+((!()

-&("&,(#+&()0!*,!"#6 &-.&"6",$ !.,&F &8

4!$!%!$ &!C.+$&!() -.$&!,!)G.&*!$H! -I)$!07-*+&$- !#)60!<-&$!#-$)&!0!%+(J,$584!.!,$",&"

)$2"$! +((-&""-()!#+,$.7&()6$K)?,7&."#+$)!()/%+()$-*-&.8

4!$!%!$ &!L+>!$) M 1!& 1.;N*-( O$-1-%!)3$!)1-9! P!()$,##+,$ .!,$- ,!&.)$2" 1-.!,$!,@.+$"

0!%+("6J+,$/Q-)1;!)-*! ?,&J7-&)$-*-&..6-*! #.-&"&$!)>!-, +,#-##$&"8

4!)&!("6:-.!%!()/$!%!$ &!$)+,".!"%!%>$!"0,06#-$)!%!()0!%-)1"0!R!$H-((;#+,$.!,$"?,-.&)6" 1,%-&(!";#60-:+:&?,!"!)" &!()&=?,!"847-&*$-&%!()>!-, +,# -##$6 &6.7-%>&-( !0!)$-*-&. 1-.!,$!,"!

?,& 5 $2:(!84!)&!(" !( #-$)& ,.&!$/!@#$&%!$)+,)! %-:$-)&),0! / & 1!.S&!$$!; ?,& -6)6 ,(!@ !..!()

#$+<!""!,$#+,$ %+&;!)/ O$6:+$5T&-.;0+() .-:!()&..!""!!).!"?,-.&)6" #60-:+:&?,!""+() $!%-$?,->.!";

#+,$ .-0&"#+(&>&.&)6 !).7!A - &)6 0+() &."+())+,"0!,@ <-&)#$!,*!/ 1-?,!<+&"?,!J!",&" -..6!.!"*+&$

-*! 0!" ?,!")&+(" -, +,$" 0! %- )12"!8 !$ & / U$V-( C-+, !) S1&.&##! E1-$)&!$ #+,$ .!,$ :!()&..!""!

!) .!,$1,%!,$ J+*&-.!-&("& ?,7/ O$6:+$5 T&-.; J7-& *$-&%!() !, #.-&"&$/ )$-*-&..!$ -*! !,@)$+&"0-(" .!

-0$!0!%!"!("!&:(!%!()"8 !$ &/T&$:&(&!Q+((-&..&!FD+W. #+,$ "-:!()&..!""!!)"+( "+,)&!(;/L+X!(( 9!@&!$FS& -$0#+,$-*+&$6)6%-),)$& !0!%+(&)+$-)!)#+,$ "-:!()&..!""!;/Y-((& PS$&*-)-*! ?,&+(

-#-""60!""+&$6!""5%#-"!)/Q!(+I)E-0$!8 !$ &/%!" +F>,$!-,@0!R!$H-((;- ),!."!)-( &!(";:$Z !

/?,&.7-%>&-( !!")*$-&%!()*&*-()!!)-:$6->.![O,&..-,%!;\,!()&(#+,$"->+((!1,%!,$!).!""+&$6!"

"5%#-" #-""6!"!("!%>.!;E1-$.&!#+,$ "-0&"#+(&>&.&)6<- ! /%!"?,!")&+(" !).!""+&$6!" "5%#-"#-""6!"

!("!%>.!;H,0+*& #+,$"!" +("!&.";3:(2";M1-("1-(!) -$)&(-#+,$.!,$:!()&..!""!!).!,$>+((!1,%!,$;

91&>-,);4&%%5;C-((5;M6>-")&!(8G(-/.-<+&">&!($&:+.6!)!,0!"0&" ,""&+("0!%-)1")$2"&()6$!""-()!"

",$.-#.-)!-,8

4!$!%!$ &!)+,".!" %!%>$!"0! .76?,&#! M3OU / .7NLNM3;?,& %7+() - ,!&..&! 1-.!,$!,"!%!()89+,) 07->+$0 J! $!%!$ &! .!" #!$%-(!()" [ 4+ !.5(!; Q!$(-$0 S1&.&##! !) U0+,-$0 E-(+); !) #.," $6 !%%!()

O6$-.0&(! S& 1+) #+,$ .!,$:!()&..!""!8 4! $!%!$ &! 6:-.!%!() Q-#)&")!; D-0&$ !) O6$-.0&(! #+,$ %7-*+&$

-&06! !A - !%!() ?,-(0J7!( -& !, >!"+&(8 4!$!%!$ &!B6"&$6 #+,$ -*+&$6)6 ,( +F>,$!-, )$2" -:$6->.!;

?,& %7- <-&) $&$! !) !") - +,$, / %+( "! +,$" ?,-(0 J7-*-&"0!" #$+>.2%!" 07&(<+$%-)&?,!; .! )+,) 0-("

,(! -%>&-( ! "),0&!,"! !) "5%#-)1&?,!8 4! $!%!$ &! +1-%-0 #+,$ "- :!()&..!""!8 U) !(=( J! $!%!$ &!

+..! )&*!%!() !,@?,!J7-&%+&("*,"!)0+( %+&(" +((,"8

U(=(J7!(#$+=)!#+,$$!%!$ &!$ !,@!) !..!"?,&%7+()-##+$)60!.-J+&!!)%7+() "+,)!(,!0-("

%-*&!#!$"+((!..! !"0!$(&2$!-((6!";!(#-$)& ,.&!$0-(".!"%+%!()""+%>$!"89+,)07->+$0%!$ &/%-"+!,$

(5)

!"#$%&'()%&*+,-,&%""&).&-,%&$&/)%*,-%0+&0-0+/")*$&1%02+-0*'&+()%',3,&*%#0&*++.4*$%56.&0+&*'

&*+ "7/#). ,86 #)+&.' ,&*#)+&0%.' *-0* /#.+&.$& :)2&,&0+ $&/)%*+#)+ & +&,/*;<8&*/4.&$& +#)+ #&).

()8#0 .6*)**%.- 7 =-%.& $).&. &+ ,>.%. &++& ?&""& -,%+%6; @&. % 62-"&,&0+ 7 @-)$ /#). *#0 -,%+%6' *#0

*#)+%&0'*-2&0+%""&**&&+*-:#%&$&1%1.&;<8&0/.#A+&/#). .&,&. %&.62-"&,&0+B-/C-D"/#).*-2&0+%""&**&' &+7+#)*"&*$&)E/#).,8- )&%""%.7F-.%*.62)"%4.&,&0+G@&. %7@-.%&/#). *#0-,%+%6'"-1%&0#)*-)0

/&)6"#%206&*'/#).-1#%.6+6"7/")*%&).*=#%*()-0$:8-1-%* .)&""&,&0+?&*#%0$8-%$&'/#).,8-1#%..&$#006

"&2#>+$&"- )%*%0&'&+/#)."&*1- -0 &*/-**6&*&0*&,?"&'()&:8-%1.-%,&0+-//.6 %6&*;@&. %7H6.-"$%0&

/#). *#0 -,%+%6' /#). "&* .-0$#* &0I.&+-20& =-%+&*&+ 7 =-%.& $#0+ :& 2-.$&)0 *)/&. *#)1&0%.' /#). *#0

*#)+%&0&+*-*&.1%-?%"%+6; @&. %7 H%""&*/#). +#)+&* &**)/&. $%* )**%#0*/&0$-0+ 0#* /#*&*+C6J "#/&*7

"8KBKLM&+*#0*#)+%&0&N - &()-0$:&$#)+-%*?&-) #)//#).,-+C4*&;@&. %7<%,,O/#).-1#%..6/#0$)

7,&*()&*+%#0*$&,-+C*',8-1#%.*#)+&0)&-)$6?)+$&,-+C4*&'&+&0 #.&,-%0+&0-0+;@&. %7H)%""-),&

&+PC%?-)+ /#). -1#%..&0$) "& /"-+&-) /")* 1%1-0+' ,-%* *).+#)+/#). -1#%.6+6 ,&*-,%*; Q#)* -1&R6+6 ?&-) #)//.6*&0+*$-0*"&*,#,&0+*"&*/")*$%N %"&*&+&0/-.+% )"%&.:&0&*-%*/-* #,,&0+:8-).-%*A0%

,-+C4*&*-0*1#)*;@&. %-)**%/#).+#)+&*"&**#%.6&*?%&0*O,/-*()8#0-=-%+&*&0*&,?"&$-0* "&*,#,&0+*

,#%0* *#,?.&*; @&. %7 @#)+#0 /#). "&* *#%.6&* *O,/-* ()8#0 - /-**6&* &0*&,?"& &+/#). *-2&0+%""&**&;

@&. %7SO.%""&/#). +#)+&*"&*$%* )**%#0**O,/-*&++.4*%0+6.&**-0+&*()8#0-&) &*$&.0%&.*+&,/*'-%0*%

()&/#).*-?%&01&%""-0 &;@&. %7T)$#&+L-.-C/#)."&*U+.#//&)0#,?.&)EV?#0*,#,&0+*/-.+-26*-1&

&)E &++& -006&' /#). "&). *#)+%&0&+ "&).?%&01&%""-0 &' &0 &*/6.-0+ &0 /-.+-2&. $8-)+.&* 7 "8-1&0%.; F")*

/#0 +)&""&,&0+' ,&. %7M0+#%0& /#). *- #,/.6C&0*%#0&+*#0 *#)+%&0',&. %7F</#).*-2.W & &+/#). ,8-1#%.$#006&01%&$&,&,&++.&7"-2)%+-.&',&. %7@-0)/#)."- ).&$&.%.&&+,&. %7X.-0Y#%*/#).

*-$#) &).'*-A0&**&&+*#0*#)+%&0;

F#). A0%.' ,&. %7 +#)*"&*-)+&).*' C-0+&).*&+ .6-"%*-+&).*'()% #0+ "-%**6*$&* +.6*#.*Z "%1.&*'A",*'

C-0*#0*()%#0+6+6/#).,#%$&*6+#%"&*',&/&.,&++-0+$&0&/-*,&/&.$.&'&+,8-//#.+-0+$&*,#,&0+*

$&"),%4.& &+$& 2.W &$-0* "&*0)%+*"&*/")* 0#%.&*;<&/&0*&/-. &E&,/"&7S"-.%**-F%0[#"-!*+&*' M"% &

(6)

!"#$%&' #)%" *

! "#$%&'&()&#*+,-.)(/&0 )/&2(3*+&( *#/&/.3*+ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4

! ! 5.6*+/ *-.*%)-.)(/&0 )/&2(3*+&( *#/&/.3*+7 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4

! !8 9(:#2;%<'*3*-.)(/&0 )/&2(3*+&( *#/&/.3*+*(=>3#2?$2%2?&* ! ! ! ! ! ! ! ! ! ! ! ! 4

! !@ A$/=23*+*/*+/&')/&2(+36*##*.#+ 2((.*+:2.#%)-.)(/&0 )/&2(3*+&( *#/&/.3*+ ! ! ! B

!8 C()%>+*(.'$#&-.*3*%6$-.)/&2(36$ 2.%*'*(/ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 8

!8! "#$+*(/)/&2(3.:#2;%<'**/#$+.%/)/+:#$%&'&()&#*+! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 8

!8!8 D##*.#3*/#2( )/.#*3.3$E*%2::*'*(/3*F)#=.(*(GH2<E* ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! @

!8!@ D##*.#$%$'*(/+0(&+ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! I

!8!J A$/=23*3* 2%%2 )/&2(+/2 =)+/&-.* ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 8K

!8!4 A$/=23*3*A2(/*GL)#%2'.%/&G(&E*).M ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 8

!@ C()%>+*(.'$#&-.*3. 2.:%)?*3*%6$-.)/&2(36$ 2.%*'*(/*/3*%6$-.)/&2(36)3E* /&2(G3&N.+&2( 8@

!@! 9(*'$/=23*3*A2(/*GL)#%2:)#/& .%)&#*:#2;);&%&+/* ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 8@

!@!8 C()%>+*(.'$#&-.*3*%)'$/=23* ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 8J !"#$%&#$'(" )*+,,+-," .',#+" +# .&$/0+" 1'-, )+" !23 +00$1#$4-+" 5 '+78 $+(#" &09&#'$,+" :; + ,#$%"-."&/0.10$$%$02#)3.#024%$#502%6'#)%"2%4066)7#) 7.$#).6&)80$0"#).609'.#)%"2 /)#5 $."&%3 %0: )0"#2 +; 8! O(/#23. /&2(! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! @K

8!8 D-.)/&2(P*M&+/*( *)(3.(&-.*(*++2Q/=*+2%./&2( ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! @

8!@ R/#2(? 2(E*#?*( *2Q

a

N

/2

a

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! @8 8!J R/#2(? 2(E*#?*( *2Q

u

N

/2

u

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! @B 8!4 S*)T 2(E*#?*( *2Q

u

N

/2

u

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! @U 8!I C(*+/&')/*2Q/=*/2/)%*##2#Q2#/=* 2%%2 )/&2('*/=23! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! JJ

8!B DM)':%*+ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4 8!B! V=**M:2(*(/&)%T*#(*% )+*2();2M! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4 8!B!8 V=*)()%>/& 2E)#&)( *T*#(*% )+* ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4J 8!U L2( %.+&2(+ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 44 8!W C::*(3&M ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 44 8!W! V=* )+*

D = R

d

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4I 8!W!8 V=* )+*

D = R

d

+

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4I 8!W!@ V=* )+*

D

;2.(3*3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4U < =>2'6#.#2 "'3>$)9'02 ?< @! D##*.#3*/#2( )/.#* ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! IJ @! ! X$ #2&++)( *3*+E)%*.#+:#2:#*+ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! IJ @! !8 D##*.#3*/#2( )/.#*+.#%* 2*Y &*(/ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! IJ @! !@ D##*.#Q)&;%*3*/#2( )/.#*+.#%)+2%./&2( ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! I4 @!8 D##*.#$%$'*(/+0(&+ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! IU

(7)

!"#"$ %&&'(&)*)+',-./,0.-&12' -4&0'**' " " " " " " " " " " " " " " " " " " " " " " " " " " " " " 56

!"#"# %&&'(&)*)+',-./,0..(&*1*40 " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " 78

!"! %&&'(& 4**4 1-04, " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " 7$

!"#$%&' "()*+,-(' .' $/'++'(+ *$*)'"0& 1",& 23(+ .'& 456 '$$,20,-('&

7 3'9 ,'"0& #$*#03,+'&: !22$, #0,3" #(; )*0<3.'& .' =3"0'>?#+$3 )($0,>

",@'#(; AB

!"#"$% &'%(%#$ &))*) +#,'-.". */ &''"0$" 23&. 4"$5 6,#7*(8*%9 "%#$. ,#7 "$. +0:

0'" ,$"*#$* ;<'$"'%=%';*#$%8,)'*;%$5*7. >? "$ 9,-&4:( -04," " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " 75 "# ;&'*0+0,1&0'. " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " 77 "#"$ <4-1-04," " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " 77 "#"# ;&4=*'+>'--0,? " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " 7@ "#"! A4?B,4&+1*C1,:4+D0'*:. " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " 76 "#" E&(, 1-':F1&G(,',BA4HI'%JK1,.04,. " " " " " " " " " " " " " " " " " " " " " " " " " " @8 "! D0,0-' %*'+',-%&&4&L,1*M.0." " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " @$ "!"$ C'?(*1&0-M4N-G' >4*(-04, " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " @$ "!"# D0,0-' %*'+',-LKK&4J0+1-04, " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " @! "!"! O(1:&1-(&'%&&4&" " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " @5 "!" E&(, 1-':D0'*:. " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " @7

" P4,I'&?', 'L,1*M.0.4NQ(*-0*'I'*Q4,-'P1&*4Q'-G4:. " " " " " " " " " " " " " " " " " " " " @@

"R <(+'&0 1*C'.(*-. " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " 68

"R"$ P4,I'&?', 'S0-GC'.K' --4

K

" " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " 68 "R"# P4,I'&?', 'S0-GC'.K' --4

h

" " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " 6$ "R"! Q(*-0*'I'*Q4,-'P1&*4P4,I'&?', ' " " " " " " " " " " " " " " " " " " " " " " " " " " " " 6$

"5 P4, *(.04,.1,:D(&-G'&T4&U " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " 6# "7 LKK',:0J " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " 6! "7"$ >-'K$VEG'P1.'

D = R

d

" " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " 6! "7"# >-'K#VEG'P1.'

D = R

d

+

" " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " 6R "7"! >-'K!VEG'P1.'

D

W4(,:': " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " 6@

!"#$%&' "()*+,-(' .' $/#.@' 0,3">.,C(&,3" ./("&3$(0* .#"& .'& ),$,'(;

#$*#03,+'& DED ? @<(%)" ,' ,#,'-.".*/$5%,7=% $"*#:7"A<."*#*/ ,.*'<$%"#),#7*((%7", BCD R"$ 9,-&4:( -04," " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " $8 R"# ;GM.0 1*+4:'* " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " $8R R"#"$ >-'1:MX4S'Y(1-04, " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " $8R R"#"# L:I' -04,B:0Z(.04,'Y(1-04, " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " $8R R"#"! >K&'1:1,::0.K'&.04, " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " $8R R"! ['. &0K-04,4N-G',(+'&0 1*+'-G4: " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " $85 R"!"$ L Q4,-'BP1&*4+'-G4:-4:'1*S0-G(, '&-10,-M " " " " " " " " " " " " " " " " " " " " " $85 R"!"# LKK&4J0+1-04,4N-G'X4SI'*4 0-M " " " " " " " " " " " " " " " " " " " " " " " " " " " " " $85 R"!"! L K&4=1=0*0.-0 K1&-0 (*1&+'-G4:" " " " " " " " " " " " " " " " " " " " " " " " " " " " " " $85 R" <(+'&0 1*1,1*M.0.4N-G'+'-G4:S0-G1::0-04,1*1..(+K-04,. " " " " " " " " " " " " " " " " " $87 R" "$ >4*(-04,4N-G'X4S'Y(1-04,1,:0-.1KK&4J0+1-04,(.0,?/,0-''*'+',-. " " " " " " " $8@

R" "# EG'1:I' -04,B:0Z(.04,'Y(1-04, " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " $8@

R" "! T'1U'&&4&4N-0+':0. &'-0\1-04, " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " $86

R" " >K1 ':0. &'-0\1-04,'&&4&4,-G'.4*(-04,4N-G' >[%" " " " " " " " " " " " " " " " " " " $$$ R" "R E4-1*'&&4&4,-G' .K&'1: " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " $$#

(8)

!"#$%& "($!

! "#$%&'&()&#*+ , -.)(/&0 )/&2( 3*+ &( *#/&/.3*+

! ! "#$%&' % )#% *+ )#+,'-. +'-/, 0%& -, %1'-'#0%&2

!"# $% "&'()%*+ $&'!,"%# $-!../, !1,&"#2 &" %#1 &"3)&"14 5 $%# ," %)1,1*$%# #*) /%# $&""4%#2 .!)

%+%'./% %)1!,"%#.)&.),414#.67#,8*%#$-*"'!14),!*2$-*"',/,%*9 :%#," %)1,1*$%#.%*;%"1.)&;%",)$-,'< .)4 ,#,&"##*)/%#'%#*)%#&*$-*""&'()%$%'%#*)%#,"#*=#!"1217.,8*%'%"1 $!"#/% !#$-*"',/,%* 1)>#

6414)&?>"%.&*)/%8*%/$%1)>#"&'()%*#%#'%#*)%##%)!,%"1"4 %##!,)%#9@"$&,1!/&)#," /*)% %#," %)1,1*$%#

$!"#/%#'&$>/%#9@".%*1.&*) %/!'&$4/,#%) %#$&""4%#.!)$%# 6!'.#!/4!1&,)%#9@"%#1!/&)#!'%"45

)4#&*$)%$%#48*!1,&"# !*+$4),;4%#.!)1,%//%#5 &%= ,%"1#!/4!1&,)%#9A-," &""*%%#1!/&)# *" 6!'.!/4!<

1&,)%2$&"1&"#&*6!,1% &""!B1)%/!/&,9C".)!1,8*%2&"#% &"1%"1%#&*;%"1$%/-%#.4)!" %%1$%/-4 !)1<17.%9

@"#&*6!,1% &""!B1)% &''%"1 /%#," %)1,1*$%##*)/%#$&""4%##%.)&.!?%"1#*) /!#&/*1,&"9 :%.)&(/>'%

$%8*!"1,D !1,&"$%#," %)1,1*$%# !..!)!B1$!"#$% "&'()%*+$&'!,"%#1%/# 8*%/-41*$% $%/! $43&)'!1,&"

$%'!14),!*+,"6&'&?>"%# E(&,#2 '&*##%#2(,&'!14),!*+F2 /-41*$%$%/-! 1,&" $*;%"1 #*)$%##1)* 1*)%#2/! #,#'&/&?,%2/-41*$%$%#'!14),!*+ &'.&#,1%#2/%#;,()!1,&"#!/4!1&,)%#2/!?%#1,&"$%#)%##&*) %#%".41)&/%G

;&,).!)%+%'./%HI2JK2LM2IJ2NO2PJ℄%1/%#)434)%" %#8*,#-71)&*;%"19A%#'416&$%#%1)4#*/1!1#8*%/-&"

.)4#%"1%, ,&"1*"%.&)14%?4"4)!/%2"4!"'&,"# %1)!;!,/%#1?*,$4.!)$%#!../, !1,&"#%"67$)&?4/&?,%8*%

/-&";!'!,"1%"!"1.)4 ,#%)9

! !3 4, 51/6*78% 0% )#+,'-. +'-/, 0%& -, %1'-'#0%& %, 9:01/;</*/;-%

A-!../, !1,&" 8*, #&*#<1%"$ %1)!;!,/ %#1 /!'&$4/,#!1,&" "*'4),8*% $%#4 &*/%'%"1# $!"#/%# "!..%#

.6)4!1,8*%#%1$%/!'!",>)%$&"1/%#.&//*!"1##-7)4.!"$%"19R&*#.)4#%"1&"#$&" , , %11%!../, !1,&"9

"#$%&%'(

@" &"#,$>)%$&" *"',/,%*.&)%*+#!1*)42$!"#/%8*%/#-4 &*/%*"S*,$%'&"&.6!#,8*%," &'.)%##,(/%2

%")4?,'%#1!1,&""!,)%9 %'!",>)% /!##,8*%2&"*1,/,#%.&*)'&$4/,#%) %14 &*/%'%"1/!/&,$% !) 7

v(x) =

−a(x)∇u(x),

&T

u

%#1 /! .)%##,&" 67$)!*/,8*%2

a

/% 1%"#%*) $% .%)'4!(,/,14 %1

v

/! ;,1%##% $% !) 79 U&*) #,'./,D%)2 &"#*..&#%)!#&*;%"1/% 1%"#%*)$%.%)'4!(,/,14 # !/!,)%2 %8*, &))%#.&"$5 &"#,$4)%)*"',/,%* .&)%*+

,#&1)&.%9A!.%)'4!(,/,14

a

1)!$*,1/!3! ,/,14!;% /!8*%//%/-%!*.%*1#-4 &*/%).&*)*"?)!$,%"1$%.)%##,&" $&""49@" &'./>1% %11%48*!1,&"!;% /-48*!1,&"$-," &'.)%##,(,/,14

div(v(x)) = 0,

.&*)&(1%",)/-48*!1,&" $-4 &*/%'%"1*#*%//%

−div(a(x)∇u(x)) = 0,

EJ9JF

8*-&" &'./>1%!;% $%# &"$,1,&"#!*+(&)$#9V&*;%"12/%#3&)'!1,&"#?4&/&?,8*%#"!1*)%//%#.)4#%"1%"1$%

(9)

!"#$!%&'()*+# '#)(-./%'$ '/.-#)&0% #!"'$ #%('(0!#)!.$)-# .- 0-!#)1 &0-#2#$())&0(#%%.'$)*3&0%2&4

!1-')#% #)'$ #%('(0!#)5!#)2&!6-#))(& 7.)('80#)&$(1(1/%&/&)1)!6)-#).$$1#)9:/.%!#)7;!%&<1&-&<0#)5

=&'%>? 5?@5?A5B9℄/.%#D#2/-#*E# 7.2/!#/#%21.F'-'(1

a

!#='#$(.-&%)0$ 7.2/.-1.(&'%#5 "#)(4G4!'%# 0$# H&$ ('&$!# -. /&)'('&$

x

#( !"0$ /.%.26(%# .-1.(&'%#

ω

5 #(&$ %#2/-. #.-&%) -"180.('&$ !1(#%2'$')(# I?*?J/.%-"180.('&$.0D!1%'=1#)/.%('#--#)G &#K '#$().-1.(&'%#)L/&0%/%#)80#(&0(

ω

−div(a(ω, x)∇u(ω, x)) = 0,

I?*BJ

80# -"&$ &2/-6(# G $&0=#.0 /.% !#) &$!'('&$) .0D F&%!)* M$ 2&!6-# (%6) H%180#22#$( 0('-')1 /.% -#) 7;!%&<1&-&<0#)#)(!#/%#$!%#/&0%

a

0$ 7.2/-&<$&%2.-7&2&<6$#5 "#)(4G4!'%#

a(ω, x) = e

g(ω,x)

5&N

g

#)( 0$ 7.2/<.0))'#$5!&$(-.-&'#)( .%. (1%')1#/.%)&$ #)/1%.$ ##().H&$ ('&$!# &=.%'.$ #

cov[g](x, y) =

E[(g(ω, x) − E[g(ω, x)])(g(ω, y) − E[g(ω, y))]

*O$)0//&)#80#

cov[g](x, y)

$#!1/#$!80#!#-.!')(.$ #!#

x

G

y

5 "#)(4G4!'%#80#-# 7.2/#)(7&2&<6$#*+#2&!6-#)'2/-#<.%.$('(80#-"180.('&$I?*BJ.!2#(0$#0$'80# )&-0('&$/%#)80#/.%(&0(5#('-!&$$#!#)%1.-').('&$)!#

a

80'=.%'#$()0%/-0)'#0%)&%!%#)!#<%.$!#0%5 #80' #)((;/'80#!.$)-#)1 &0-#2#$())&0(#%%.'$)*P.$)-# .!%#!#).//-' .('&$)50$ 7&'DH%180#$( !#H&$ ('&$

!# &=.%'.$ ##)(-. &=.%'.$ ##D/&$#$('#--#>BQ5AB℄

cov[g](x, y) = σ

2 e

−

kx−yk

ℓ

_,

I?*RJ

&N

σ

#)( -"1 .%(4(;/# #(

l

-. -&$<0#0% !# &%%1-.('&$*+&22# $&0) -# =#%%&$)#$ !1(.'- /.% -. )0'(#5 #((# H&$ ('&$!# &=.%'.$ #1(.$(/#0%1<0-'6%#5-#)%1.-').('&$)!0 7.2/!#/#%21.F'-'(1.))& '1)&$(1<.-#2#$(

/#0 %1<0-'6%#)5 # 80' #$ H.'( 0$F&$ 2&!6-# .% !.$) -#) H&%2.('&$)<1&-&<'80#)$.(0%#--#)5 -# 7.2/ !#

/#%21.F-'(1.(#$!.$ #GS(%#/#0%1<0-'#%G .0)#!#)/%& #))0)!#)1!'2#$(.('&$*P#)H&$ ('&$)!# &=.%'.$ # /-0)<1$1%.-#)!#-.H&%2#

cov[g](x, y) = σ

2 e

−

(

kx−yk

ℓ

)

δ

I?*AJ

)&$(0('-')1#)* P.$)-# .!%#!#).//-' .('&$) &$)'!1%1#)' '5&$)"'$(1%#))##)#$('#--#2#$(.0 .)&N

σ

2 _{≥ 1}

5 80' &%%#)/&$!G !#)'$ #%('(0!#) '2/&%(.$(#) #( &N

ℓ

≤ 1

580' &%%#)/&$!G 0$# -&$<0#0% !# &%%1-.('&$ '$H1%'#0%#G-.(.'--# !#-.T&$#G-.80#--#&$)"'$(1%#))#5=&'%#2S2#

ℓ

≪ 1

*

3&0% &$ -0%# # /.%.<%./7#5 '- #)( '2/&%(.$( !"1=&80#% !6) 2.'$(#$.$( !#0D /.%(' 0-.%'(1) !"0$

(#- 7&'D!# 7.2/!#/#%21.F'-'(1580"&$%#=#%%./-0)#$!1(.'-0-(1%'#0%#2#$(#(80'#D -0#$(-".//-' .('&$!#

$&2F%#0D%1)0-(.()80"&$/#0((%&0=#%!.$)-.-'((1%.(0%#*U&0(!".F&%!0$ 7.2/-&<$&%2.-$"#)( -.'%#2#$(

$'0$'H&%212#$(F&%$1/.%%.//&%(G

ω

5$'0$'H&%212#$( &#% 'H/.%%.//&%(G

ω

*P#/-0)5.=# -# 7&'D!#-. &=.%'.$ ##D/&$#$('#--#I?*RJ5-#)(%.V# (&'%#)!#

a

)&$(/#0%1<0-'6%#)5#--#)$#)&$(#$/.%(' 0-'#%/.)

C

1

&0

W

1,∞

*3&0%/-0)!#!1(.'-))0%-#)/%&/%'1(1)!0 7.2/!1 %'( '4!#))0)5&$)#%#/&%(#%..0/.%.<%./7#?*B*?5 #(/&0%/-0)!#!1(.'-))0%-#)%1)0-(.()80"&$/#0((%&0=#%!.$)-.-'((1%.(0%#5&$)#%#/&%(#%..0/.%.<%./7#

?*?*R#(.0D%1H1%#$ #)80')";(%&0=#$(*

!"# %&'()!&*+,&'(!# -'..+/(%,

O$)"'$(1%#))#2.'$(#$.$(G -".!=# ('&$4!'W0)'&$ !"0$ )&-0(1I(;/'80#2#$( 0$/&--0.$(J !.$)-# 2'-'#0

!1 %'( '4!#))0)*3-0)/%1 ')12#$(5&$ &$)'!6%#0$)&-0(1'$#%(#'$V# (1G-"'$)(.$('$'('.-!.$)-#2'-'#0/&4

%#0D !1 %'( '4!#))0)#(&$ )0//&)# 80"'- #)( (%.$)/&%(1 /.% -"1 &0-#2#$( !# -"#.0 #(!'W0)1* O$ &$)'!6%#

0$'80#2#$(-.!'W0)'&$2&-1 0-.'%#5$1<-'<#.$(-.!')/#%)'&$ '$12.('80#*O$)0//&)#80#-.!'W0)'&$2&-14

0-.'%##)(7&2&<6$##(')&(%&/##(80#-./&%&)'(1#)( &$)(.$(#51<.-#G

1

*+#(;/#!#2'<%.('&$!"0$)&-0(1 #)(!1 %'(/.%-"180.('&$!".!=# ('&$4!'W0)'&$)0'=.$(#L

∂c(ω, x, t)

∂t

+ v(ω, x).∇c(ω, x, t) − D∆c(ω, x, t) = 0,

I?*QJ

&N

D

#)(-# &#K '#$( !#!'W0)'&$2&-1 0-.'%#5

v

-.='(#))#!#P.% ;!1X$'#!.$)-# /.%.<%./7#/%1 1!#$(5 #(

c

-. &$ #$(%.('&$!#)&-0(1*O$)"'$(1%#))#.0 .)&N-".!=# ('&$#)(!&2'$.$(#5 "#)(4G4!'%#.0 .)&N-# $&2F%#!#31 -#(

ℓkvk

mean

D

#)( '2/&%(.$(I(;/'80#2#$(

≥ 100

J*E. &$!'('&$'$'('.-#G

t = 0

#)( -"'$V# ('&$ !0)&-0(15 "#)(4G4!'%#

c(t = 0) = 1

R

5&N

R

#)(0$%# (.$<-#'$ -0)!.$)-#!&2.'$#

O

*E"180.('&$I?*QJ!&'(

(10)

!"#$ &'()*"*$+,$ -$. &/-0"0&/.+123&#-.4561/(&0/"-$,1$+(()0 +"078&/.&19+0"$.+,&0# &''$/"1/

(&))1+/" 0/:$ "* ; 1/0/."+/" -&//*-$ '+/0<#$ )& +)0.*$ -+/. 1/$ /+(($ (9#*+"0=1$ ,+ .$(#&(+>$# -+/.

$))$? 08&/($1"$/(+#"0 1)0$#.$-$'+/-$#=1$))$"+0))$,++""$0/-#$)+@&/$(&))1*$8;=1$))$,0"$..$$))$,+

.$-*()+ $#8 &'30$/-$"$'(.)+(&))1"0&/,+'$""#$(&1#+""$0/-#$1/$@&/$-$(&'(+>$4

A/-*B/0"'+0/"$/+/")$.-$12=1+/"0"*.=1$)6&/.&19+0"$0 0 +) 1)$#C)6$2"$/.0&/'&D$//$$")+E'+ #&F?

-0.($#.0&/'&D$//$4G&1# $)+&/ &''$/ $(+#-*B/0#)$ $/"#$-$'+..$-1.&)1"*C

G(ω, t) =

Z

O

c(ω, x, t)xdx.

H6$2"$/.0&/-1.&)1"*

S(ω, t)

$."+)&#.-*B/0$(+#

S(ω, t) =

Z

O

c(ω, x, t)(x

− G(ω, t))(x − G(ω, t))

t

_dx,

$")+-0.($#.0&/

D(ω, t)

&''$)+-*#0,*$"$'(&#$))$-$)6$2"$/.0&/C

D(ω, t) =

dS(ω, t)

dt

.

A/-*B/0"$/B/)6$2"$/.0&/ '&D$//$$")+-0.($#.0&/'&D$//$4

S(t) = E[S(ω, t)]

$"

D(t) = E[D(ω, t)].

H+'+ #&?-0.($#.0&/"#+-10")+,0"$..$;)+=1$))$)$(+/+ 9$-$.&)1"*.6*"$/-4H+-*"$#'0/+"0&/-1 &$I 0$/"

-$'+ #&?-0.($#.0&/8$" $/(+#"0 1)0$#-$)60/J1$/ $-$ )69*"*#&>*/*0"*-1'0)0$1 .1# $ &$I 0$/"8 +*"*)$

.1:$"-$/&'3#$12"#+,+12 $.KL-$#/0<#$.+//*$.8,&0#(+#$2$'()$ML8NO8NP8KP8QL8Q 8QK℄4S*+/'&0/.8

-+/.)+()1(+#"-$ $."#+,+128)+-0T1.0&/+*"*/*>)0>*$8 &/.0-*#+/"1/0=1$'$/")$"$#'$-6+-,$ "0&/4G+#

&/.*=1$/"8&/$."$/"#$+1"#$.0/"*#$..*.(+#)60/J1$/ $=1+)0"+"0,$$"=1+/"0"+"0,$-$)+-0T1.0&/'&)* 1)+0#$

.1#)+'+ #&?-0.($#.0&/E,&0#MNL℄(&1#-$.#*.1)"+"./1'*#0=1$.F4

! !" #$%&'()* )% )*%+,-%+'.* (/)00)10* '..1)* 3'10 4- 51-.%+6 -%+'. ()* +.7

)0%+%1()*

5+/. $""$(+#"0$8&/(#*.$/"$()1.0$1#.-$.(#0/ 0(+)$.'*"9&-$./1'*#0=1$.1"0)0.*$.(&1#)+=1+/"0B +?

"0&/-$.0/ $#"0"1-$.8+0/.0=161/&1"0)0'(&#"+/"-6+((#&20'+"0&/-+/.1/$.(+ $."& 9+."0=1$-$-0'$/.0&/

B/0$C)$ -*,$)&(($'$/" -$U+#91/$/?H&<,$8 $""$ +((#&20'+"0&/:&1+/"$/"#$ +1"#$.1/#)$7&/-+'$/"+) -+/.)$.'*"9&-$..($ "#+)$.."& 9+."0=1$.4G&1#.0'()0B$# $"$2(&.*8&/(#*.$/"$ $.'*"9&-$.-+/.)$ +.

-$)6*=1+"0&/'&-<)$.10,+/"$C(&1#(#$.=1$"&1"

ω

∈ Ω

−div(a(ω, x)∇u(ω, x)) = f(x)

x

∈ D,

u(ω, x) = 0

x

∈ ∂D,

EN4QF

/*+/'&0/.)$.'*"9&-$.-* #0"$. 0?-$..&1.&/"1/$(&#"*$()1.>*/*#+)$8 $#"+0/$. ()1.=1$-6+1"#$.4WB/

=1$)$ (#&3)<'$ .&0"30$/(&.*8&/.1((&.$-+/. $(+#+>#+(9$=1$

f

∈ L

2 _(D)

8 $"=1$(&1# (#$.=1$"&1"

ω

8

x

7→ a(ω, x)

$." '$.1#+3)$8$" =160) $20."$

a

min

_{(ω) > 0}

$"

a

max

_{(ω) < +∞}

"$). =1$ (&1# (#$.=1$ "&1"

x

∈ D

&/+

a

min

_(ω)

_{≤ a(ω, x) ≤ a}

max

_(ω)

4A/.1((&.$=1$

D

$."1/&1,$#"3&#/* &/,$2$0/ )1. -+/.

R

d

4

A/ &/.0-<#$

V

h

1/$.(+ $."+/-+#--6*)*'$/".B/0..1#

D

-$7&/ "0&/. &/"0/1$.8)0/*+0#$.(+# '&# $+128 +..& 0* ;1/$ 7+'0))$ #*>1)0<#$-$ "#0+/>1)+"0&/.-$

D

-&/" &/ /&"$

h

)$ -0+'<"#$ '+20'1' -1 '+0))+>$4 G&1# +))*>$#)6$2(&.*8&/ /$-&//$#+ =1$)$.$."0'+"0&/.-6$##$1#

H

1

8'+0. 30$/.X#-+/. 9+=1$ +.&/+ *>+)$'$/"-$.$."0'+"0&/.-6$##$1#

L

2

4

Y/$9D(&"9<.$ 7+0"$(+#-$/&'3#$12+1"$1#.(&1#&3"$/0#-$.$."0'+"0&/.-6$##$1#.$.")+.10,+/"$C

!"#$%&'()*)*)* !"#$$%"& '#&

a

∈ L

∞

_(Ω,

_C

1 _{( ¯}

_D))

&( '#)*+ &,*"(&

a

min

_{> 0}

(&+'#& $%#-$-&"'#&(%#(

ω

%!.*(

a

min

_{≤ a(ω, x)}

(11)

!"##"$%&'#$()" '+#,"+#-./0',)1+"$%&'#$()"2"3451./3,#42")34/.,)/#,'+)2"

a

6&'13/&&.,71"3./#$4'3," )#/+2/3228")#,9/#,'+)28"33"134.49"+#):+,);"#1+"$%&'#$()"28"<,)#"+ "281+"='3+""#281+" '+)#/+#"

2" '"3 ,>,#41+,0'39"&/33/&&'3#-

ω

6&'13&'1>',3'=#"+,32")")#,9/#,'+)28"33"13)1+,0'39")&/33/&&'3# -

ω

;?@++'#"3/2()-&34)"+#71".") $/9&).'5+'39/1<24:+,)/1&/3/53/&$"AB '1."9"+#C 2"./)" #,'+ D?D?E+">43,:"+#&/)." 2"1<,(9"&',+#F"#71"2/+)." /)281+" '>/3,/+ ""<&'+"+#,".."6D?G;F." &3"9,"3

&',+#+8")#&/)>43,:4+'+&.1)?

!"#$%&'%& "()& $*"&+,-./$

H" &3,+ ,&" 2" ./ 94#$'2" 2" I'+#"J!/3.' ")# ),9&."F '+ '+),2(3"

M

34/.,)/#,'+) ,+24&"+2/+#") 21 $/9& 2" &"394/=,.,#4

a

1 _{, ...a}

M

F &'13 $/ 1+" 2" ") 34/.,)/#,'+)'+ /. 1." 1+"/&&3'<,9/#,'+

u

i

h

2"./ )'.1#,'+

u

i

2".8BKL 24#"39,+,)#" '33")&'+2/+#"

−div(a

i

(x)

∇u

i

(x)) = f (x),

2/+).8")&/ "284.49"+#):+,)

V

h

6'+ &"1#1#,.,)"3 +8,9&'3#"71".."/1#3"94#$'2"+1943,71"&'13 /. 1."3 ./)'.1#,'+2" .8BKL 24#"39,+,)#"F'++" '+),2(3"3/, ,71"2")94#$'2"284.49"+#):+,) 9/,)." &3,+ ,&"

")# #'1#- 0/,#54+43/.;?L'131+" 0'+ #,'+

ϕ : R

→ R

'1

ϕ : H

1 _(D)

_{→ R}

F'+ /&&3' $" /.'3)

E[ϕ(u)]

&/3

1 M

M

X

i=1

ϕ(u

i

h

)

? @+ &"1# /.'3) 24 '9&')"3 .8"33"13 '99,)" "+ 1+ #"39" '33")&'+2/+# - ./ 2,) 34#,)/#,'+ )&/#,/.""#1+#"39" '33")&'+2/+#/1I'+#"J!/3.'F71,")# "+

1/

√

M

?L.1)&34 ,)49"+#F'+/&/3"<"9&." ."34)1.#/#)1,>/+#ME℄&'13." /)21 /. 1.2".8")&43/+ "?

0.$)$'1"1$*232343 !"#$$%"&'()*$%+),"&-.-.-/01230&4%!55'%1"'(&"+265+2%!7(&11&#1"#2/5!+&82'&92"+&

#!& %!"+5!+&

C

+&''&;#& $%#1+%#+

h > 0

&++%#+

M

%!52+

E[u] −

1 M

M

X

i=1

u

i

h

L

2 _(Ω,H

1

0 (D))

≤ C

h +

√

1 M

kfk

L

2 _(D)

.

6D?O;

H") 94#$'2") 2" #%&" I'+#"J!/3.' '+# .8/>/+#/5" 28P#3" 0/ ,.") - 9"##3" "+ '"1>3" "# &/3/..4.,)/=.")F

/,+),71"28/>',31+ /23"28/&&., /#,'+#3()54+43/.Q"..")&"1>"+#)8/&&.,71"3/>" +8,9&'3#" 71".."BKL

&'13./71".."'+2,)&')"281+"94#$'2"+1943,71"624#"39,+,)#";F71".8'+&"1#/.'3)1#,.,)"3"+='R#"+',3"?

H/ >,#"))" 2" '+>"35"+ "")# 2" &.1) ,+24&"+2/+#" 2" ./A2,9"+),'+ )#' $/)#,71" 2"

a

C 6>',3 &/3/53/&$" AI4#$'2"))&" #3/."))#' $/)#,71")C ,J2"))'1);?S4/+9',+)"..")'+#&'13,+ '+>4+,"+# 28/>',31+">,#"))"

2" '+>"35"+ "/))"T."+#"6

1/

√

M

;F." 1#2" /. 1.4#/+#

M

0',) ".1,21&3'=.(9"24#"39,+,)#"6-)/>',3 ".1,2"./34)'.1#,'+281+)%)#(9".,+4/,3"2"#/,.."

dim(V

h

)

;? K")/94.,'3/#,'+)2" "##"94#$'2"2"=/)")'+#&')),=."F"+&/3#, 1.,"353V "-2")#" $+,71")2"3421 J #,'+2">/3,/+ "F'1"+1#,.,)/+#2")94#$'2")2"#%&"71/),I'+#"J!/3.'MG ℄?

5* $6"1/%7-)).$819-"1$* %-*' 6*&')- & '"$ #-'"1;6& %& %19&*'1$*<*1& =/&%!>&/$))&9&*"

%& ?-.#6*&*+@$A>&

H"&',+#2"24&/3#2")94#$'2"))&" #3/."))#' $/)#,71")718'+>/&34)"+#"32/+)."&/3/53/&$"AI4#$'2")

)&" #3/.") )#' $/)#,71")C ,J2"))'1)F71, ")# 1+&',+# 31 ,/.F")# .8/&&3'<,9/#,'+ 21 '"W ,"+#

a

2/+) 1+ ")&/ " /.4/#',3"2" 2,9"+),'+ :+,"F 8")#J-J2,3" &/3 1+" 0'+ #,'+281+ +'9=3" :+, 2" >/3,/=.") /.4/#',3")?

H")2"1<94#$'2")03471"99"+#1#,.,)4")&'13'=#"+,31+"#".."/&&3'<,9/#,'+)'+#.")24>".'&&"9"+#)"+

&'.%+9")2"!$/')"#.")24>".'&&"9"+#)2"X/3$1+"+JH'(>"?@+"<&')", , "##"2"3+,(3"94#$'2"MEFGD℄

71,")# "..")13./71".."'+)" '+ "+#3"3/&/3./)1,#"F /3".."")#=,"+/2/&#4"/1 /)281+ $/9&.'5'+'39/.

$'9'5(+" "<&')4 2/+) ." &/3/53/&$" D?D?EF &'13 2") 3/,)'+) 71" .8'+ "<&.,71"3/? H" 24>".'&&"9"+# 2" X/3$1+"+JH'(>"")#45/."9"+#1#,.,)4&'13 "3#/,+")94#$'2")2"#%&"&"#,#")&"3#13=/#,'+)6>',3&/3/53/&$"

AI4#$'2")2"#%&"&"#,#")&"3#13=/#,'+)C;?!8")#"+:+1+"2")94#$'2")1#,.,)4")&'13),91."3." $/9&

a

F 2/+)." /23"2")94#$'2")2"#%&"I'+#"J!/3.'&/3"<"9&."?

(12)

!"#$

b : Ω

× D → R

%& '(" *++%+ +$" ,-+$#.%* /"&$ 0- 1"& $#"& /* "2-(#-& *

cov[b]

*+$ "&$#&%* +%(

¯

D

× ¯

D

34& "&+#/5(*06"'7(-$*%(/*8#09*($:! ,;#/$-%$":-/<"#&$'"+#$#1-++" #7/7=&#'-(

h

∈ L

2 _(D)

_7→

Z

D

cov[b](x,

·)h(x)dx.

4&&"$*

(λ

n

, b

n

)

n∈N

∗

0- +%#$*/* "%'0*+'("'(*+/* *$ "'7(-$*%(>"? 0*+2-0*%(+'("'(*+

λ

n

+"&$ (-&@7*+ /-&+06"(/(*/7 ("#++-&$34&(-''*00*.%6*00*+27(#=*&$

X

n≥1

λ

n

=

Z

D

cov[b](x, x)dx

*$.%*0*+2* $*%(+'("'(*+

b

n

1"(;*&$%&*1-;#00*"($,"&"(;-0*34&--0"(+

b(ω, x)

L

2 _(Ω×D)

=

E[b](x) +

X

n≥1

p

λ

n

b

n

(x)Y

n

(ω),

"?0*+2-(#-90*+-07-$"#(*+

Y

n

+"&$ *&$(7*+(7/%#$*+*$/7 "((707*+A*$/"& 1"(;*&$%&*1-;#00*"($,"&"(;-0*B> /7=&#*+'"%(

λ

n

> 0

/*;-&#5(*%&#.%*'-(

Y

n

(ω) =

√

1 λ

n

Z

D

(b(ω, x)

− E[b](x))b

n

(x)dx.

C6-'(5+0*$,7"(5;*/*D*( *(EFG℄>+#"&/7=&#$0-+";;*'-($#*00*

b

N

(ω, x) = E[b](x) +

N

X

n≥1

p

λ

n

b

n

(x)Y

n

(ω)

> "&

-sup

x∈D

E[(b − b

N

)

2 ](x)

→

N →+∞

0.

AI3JB

K0*+$#;'"($-&$/*&"$*(.%*/-&+0* -+"?

b

*+$%& ,-;'@-%++#*&>"&-/*+'("'(#7$7++%''07;*&$-#(*+$(5+ #&$7(*++-&$*+3L&*M*$+#

b

*+$%& ,-;'@-%++#*&>"& "&&-N$0-0"#/*+

Y

n

O *+"&$/*+@-%++#*&&*+ *&$(7*+ (7/%#$*+> *.%#*+$$(5+%$#0**&'(-$#.%*>/*'0%+/-&+ * -+0*+2-(#-90*+-07-$"#(*+

Y

n

+"&$#&/7'*&/-&$*+> *$'-++*%0*;*&$/7 "((707*+3C*'0%+>/-&+0* -+/6%& ,-;'@-%++#*&-2* "((70-$#"&*P'"&*&$#*00*A/7=&#*

'-(AI3QBB>"&-/*+*P'(*++#"&+.%-+#:*P'0# #$*+'"%(0*+2-0*%(+'("'(*+/-&+0* -+"?"&'(*&/0-&"(;*

1

*$+*'0- *+%(%&/";-#&*(* $-&@%0-#(*A'"%(%&/";-#&*.%*0 "&.%*>"&'*%$%$#0#+*(0*/72*0"''*;*&$/*

R-(,%&*&:S"52*+%(%&/";-#&*(* $-&@%0-#(* "&$*&-&$ */";-#&*B3L&*M*$>*&/#;*&+#"&

1

> "&+#/7(-&$ 067.%-$#"& -(- $7(#+$#.%*

(ℓ

2 _w

2 _{− 1) sin(w) = 2ℓw cos(w)}

*$&"$-&$

(w

n

)

n≥1

0-+%#$*/*+*+(- #&*+'"+#$#2*+"(/"&&7*+/-&+06"(/(* ("#++-&$>-0"(+0*+2-0*%(+'("'(*+ /%/72*0"''*;*&$/*R-(,%&*&:S"52*/*

b

+"&$/7=&#*+'-(

λ

n

=

2ℓσ

2 ℓ

2 _w

2 n

+ 1

AI3 B *$0*+2* $*%(+'("'(*+'-(

b

n

(x) = α

n

(sin(w

n

x) + ℓw

n

cos(w

n

x)),

AI3IGB

"?

α

n

=

1 √

(ℓ

2 _w

2 n

+1)/2+ℓ

T '"%( 0- '(*%2*> 2"#(EQI> UF℄ '-( *P*;'0*3L& /#;*&+#"& +%'7(#*%(*>0*+ "%'0*+

'("'(*++"&$"9$*&%+*&$*&+"(#+-&$0*+ "%'0*+'("'(*+;"&":/#;*&+#"&&*0+3C*+;7$,"/*+&%;7(#.%*+*V:

- *+"&$7$7/72*0"''7*+'"%( -0 %0*(0*+ "%'0*+'("'(*+/%/72*0"''*;*&$/*R-(,%*&:S"52*/-&+0* -+

/6%&* "2-(#-& *.%*0 "&.%*>2"#('-(*P*;'0*EFW>XX℄3

4&%$#0#+*=&-0*;*&$0* /72*0"''*;*&$ /*R-(,%&*&:S"52* '"%(-''(" ,*(0*'(" *++%+ +$" ,-+$#.%*

b

/-&+%&*+'- *+$" ,-+$#.%*/*/#;*&+#"&=&#*> 6*+$:Y:/#(*'-( %&*1"& $#"&/6%&&";9(*=&# /*2-(#-90*+

-07-$"#(*+>*&06" %(*& *'-(

b

N

(ω, x) = E[b](x) +

N

X

n=1

p

λ

n

b

n

(x)Y

n

(ω)

34&'*%$%$#0#+*(0*/72*0"''*;*&$/* R-(,%&*&:S"52*'"%(0* "*V #*&$

a

/*06LCZAI3FB>"%/-&+0* -+/6%& ,-;'0"@&"(;-02%-%'-(-@(-',* [L "%0*;*&$\ /* 0- +* $#"& I3I3X '"%( 0* ,-;' @-%++#*&

g = log(a)

3 C-&+ * -+> 0* /72*0"''*;*&$ /*

(13)

"#$%&'(')*+,-((./&'+&/0120('#3#4/56 #$

g

5/#'/&' %#849#&..0('6.+'35-(1+44(8('/3("#$%&'(') *+,-(:#0/#44#$#;/$(3(.-#$0#21(.#15#/+0$(.9#&..0(''(. ('/$5(.$53&0/(.0'354('3#'/(.6:# 01(.<.08&1($6

(/3(41&.63#'.1( #.3(1# +-#$0#' ((=4+'('/0(11(> ?@A61(.:+' /0+'.4$+4$(..+'/:# 01(. < #1 &1($?B'

#44$+ %(3+' 1( +(C 0('/1+9'+$8#1

a = e

g

4#$

a

N

(ω, x) = e

g

N

(ω,x)

,

+D

g

N

(ω, x) = E[g](x) +

N

X

n=1

p

λ

n

b

n

(x)Y

n

(ω),

> ? A

+D

(λ

n

, b

n

)

n≥1

(./ 1# .&0/(3(. +&41(.4$+4$(.3( 1E+45$#/(&$3( F012($/)G %083/ 3('+H#&1# :+' /0+'3( +$$51#/0+'3(

g = log(a)

?

!"#$%&''(& "*+,&''"$ #+'"-./&'

*(.85/%+3(..4( /$#1(../+ %#./0I&(.6I&0 +84$(''('/1(.85/%+3(.3(J#1($K0'./+ %#./0I&(.LM6NO6@ 6

PN6MN6QM℄(/1(.85/%+3(.3( +11+ #/0+'./+ %#./0I&(.L 6QP6Q@℄+'/.&. 0/52(#& +&43E0'/5$S/ (.3($'0,$(. #''5(.?T(.85/%+3(..E#4410I&('/4+&$3(. +(C 0('/.

a

3(308('.0+'./+ %#./0I&(U'0(6+' +88(' (3+' 4#$ #44$+ %($1( %#84

a

4#$ &'(#44$+=08#/0+'

a

N

6:+' /0+'3(

N

-#$0#21(.#15#/+0$(.6 E(./)<)30$(3#'. &'(.4# (./+ %#./0I&( 3(308('.0+'

N

?T+88(+'-0('/3( 1(-+0$61( 35-(1+44(8('/3( "#$%&'(')*+,-( 4($8(/3E+2/('0$&'(/(11(#44$+=08#/0+'>1(.4+1H'8(.3(T%#+..+'/59#1(8('/:$5I&(88('/&/010.5.6-+0$

4#$(=(841(L@ 6QM6@!6MW6QM6NO6NQ℄A?*(.85/%+3(..4( /$#1(../+ %#./0I&(.4($8(//('/#1+$.3( #1 &1($

1#.+1&/0+'

u

N

3(1E5I&#/0+'X#44$+ %5(Y.&0-#'/(Z

−div(a

N

(ω, x)

∇u

N

(ω, x))

= f (x)

x

∈ D,

u

N

(ω, x)

= 0

x

∈ ∂D.

!"!#$

%&'()'*+,-& ,./-0+1&+' '*+12-0* '22'4' 21()1&+,5 1+,-&4' 26'00')0 -..,*' '& 1//0- 71&+

u

/10

u

N

" 8'++'()'*+,-&'*+461)+1&+/2)*,./-0+1&+'9()' -..'&-)*122-&*2':-,092' -;+4'*.<+7-4'**/' +012'*

*+- 71*+,()'* 0-=+4'.1&,>0',./-0+1&+'1:' 214,.'&*,-&*+- 71*+,()'4) -'? ,'&+

N

'++' 0-,**1& ' '*+ +@/,()'.'&+ 'A/-&'&+,'22' /-)0 2'* :'0*,-&* B1*,()'* 4'* .<+7-4'* */' +012'* *+- 71*+,()'*9 :-,0 ,C

4'**-)*$"%&'/0'.,>0'0</-&*'D '++'()'*+,-&'*+4-&&<'41&*E#℄*-)*G-0.'46'*+,.1+,-&46'00')0G-0+'9

*-)*4'*7@/-+7>*'*+0>*G-0+'* 4'4< 0-,**1& '4'*:12')0*/0-/0'*'+46'A,*+'& '46)&'B-0&')&,G-0.'/-)0

2'*

Y

n

$'+:,1)&''*+,.1+,-&

L

∞

_(Ω

_{× D)}

4'26'00')0 -..,*'*)0

a

9'&/10+, )2,'0 '0<*)2+1+&'*61//2,()' /1*/-)0)& 71./2-H&-0.12"

I-)* /0<*'&+-&*.1,&+'&1&+ 2'*.<+7-4'**/' +012'* *+- 71*+,()'* D /0-/0'.'&+ /102'0"

a

N

<+1&+ )&' G-& +,-&4'

N

:10,1B2'*12<1+-,0'*

(Y

1 , ...Y

N

)

9-&)+,2,*'0121&-+1+,-&

a

N

(ω, x) = ˜

a

N

(Y

1 (ω), ..., Y

N

(ω), x)

"J& *)//-*' ()'2' :' +')012<1+-,0'

Y = (Y

1 , ..., Y

N

)

14.'+ )&'4'&*,+< K-,&+'$

ρ

/10 01//-0+D 21.'*)0'4' L'B'*H)'*)0

R

N

6'*+/10'A'./2'2' 1*/-)0)& 71./2-H&-0.12$9-&:1/-):-,01,&*,01.'&'021G1.,22' 46MNO !"!#$/101.<+0<'/10

ω

D )&'G1.,22'46MNO /101.<+0<'41&*

R

N

"M& 'P'+9*,/-)0

y

1//10+'&1&+ 1)*)//-0+

Γ

⊂ R

N

4'

ρ

-&&-+'

u

˜

N

(y,

·)

21*-2)+,-&4'26MNO

−div(˜a

N

(y, x)∇˜u

N

(y, x))

= f (x)

x

∈ D,

˜

u

N

(y, x)

= 0

x

∈ ∂D,

!"!Q$

-&112-0*

u

N

(ω, x) = ˜

u

N

(Y (ω), x).

L'*.<+7-4'*4'R102'0S,&*+- 71*+,()'**'B1*'&+ 12-0**)02'G1,+()'2'/0-B2>.'4'4</10+ !"!#$'*+

<(),:12'&+D21G-0.)21+,-&:10,1+,-&&'22'H2-B12'*),:1&+'T

u

˜

N

'*+21G-& +,-&1//10+'&1&+D

L

2 ρ

(Γ, H

0

1 (D))

:<0,51&+/-)0+-)+

v

∈ L

2 ρ

(Γ, H

0

1 (D))

Z

Γ

Z

D

˜

a

N

(y, x)

∇˜u

N

(y, x)

∇v(y, x)ρ(y)dxdy =

Z

Γ

Z

D

f (x)v(y, x)ρ(y)dxdy.

!"!U$

M& *' B1*1&+ *)0 '++' 0'G-0.)21+,-&9 -& 1//0- 7'

u

N

41&* 26'*/1 ' 4' 4,.'&*,-& 5&,'

P

⊗ V

h

/10 )&' .<+7-4'4'R12'0S,&9-V

P

'*+)&'*/1 '4'/-2@&.'*-)4'/-2@&.'*/10.-0 '1)A'&

N

:10,1B2'*'+

V

h

26'*/1 '46<2<.'&+*5&,*4<5&,/0< <4'..'&+"N1&*2' 1*-V

P

'*+26'*/1 '4'*/-2@&.'*4'4'H0<,&G<0,')0 -)<H12 D

p

<+1&+)&.)2+,C,&4, '4'2-&H)')0

N

$9-&&-+'

u

˜

h,p

(14)

!"#$%&'()$*+(,-.('*

u

h,p

N

(ω, x) = ˜

u

h,p

N

(Y (ω), x)

/0+*"(*

P

p

#1$23" $,$23+#5(7$2$(

N

8"%'"9#$2,$

,$:%-p

)

u

˜

h,p

N

$2*,-.(' +77$#1;('<;$=+( *'+(,$

P

p

⊗ V

h

8-%'."(*3+;%*+;*

v

∈ P

p

⊗ V

h

Z

Γ

Z

D

˜

a

N

(y, x)

∇˜u

h,p

N

(y, x)

∇v(y, x)ρ(y)dxdy =

Z

Γ

Z

D

f (x)v(y, x)ρ(y)dxdy.

> / ?@ A+;2,$2B53+*BC2$2#-:C%$7$(*3#;2=+%*$2<;$#1D53+*BC2$ / / )<;$#1+(($,-*"'##$%"3"2' ')#1$2*'7"*'+( ,1$%%$;%2;'8"(*$$2*3%+;8-$,"(2EF℄H

!"#$%&'()()*) !"#$%&"'(" *(%&+(&"

C > 0

&"!!",'"-*'.&*'&

τ

∈]0, 1[

/$!"#$%&"

(r

n

)

n

∈]0, 1[

N

&"!%,'" -*'.&*'&

h > 0

"&

p

∈ N

N

*(+$&

kE[u

N

− u

h,p

N

]k

H

1 _(D)

≤ C

h +

1 τ

N

X

n=1

(r

n

)

p

n

+1

!

.

I$2 7-*B+,$2 ,$ +##+ "*'+(2*+ B"2*'<;$2 2$9"2$(* <;"(* J $##$2 2;% #" %$=+%7;#"*'+( ,$ > / F@ 2+;2 =+%7$,1;($ ="7'##$,1KLM 3"%"7-*%-$ 3"%

y

∈ R

N

)$* 2;%#$ ="'*<;$3+;% 3%$2<;$*+;*

x

∈ D

)#" #+',$

ω

∈ Ω 7→ u

N

(ω, x)

$2* $##$ ,$

y

∈ Γ 7→ ˜u

N

(y, x)

) +N +( " 7;('

Γ

,$ #" 7$2;%$ "5"(* 3+;% ,$(2'*-

ρ

3"% %"33+%*J#" 7$2;%$,$I$9$2:;$/O($2*,+( %"7$(-J ;(3%+9#C7$ ,1'(*-:%"*'+((;7-%'<;$2;%

R

N

"8$ 3+',2

ρ

/ I$ 3%'( '3$ :-(-%"# ,$2 7-*B+,$2 ,$ +##+ "*'+(2*+ B"2*'<;$2 $2* ,$ "# ;#$% ;($ 2+#;*'+( "33%+ B-$

u

˜

h

N

(y,

·)

,"(2#1$23" $,1-#-7$(*2.('2

V

h

,$#"2+#;*'+(

u

˜

N

(y,

·)

,$> / F@3+;% $%*"'($28"#$;%2 ,$

y

H

y

1 , ...y

N

p

)"33$#-23+'(*2,$ +##+ "*'+(/O($(,-,;'*"#+%2;($2+#;*'+("33%+ B-$

˜

u

h,p

N

)J3"%*'%,$2 8"#$;%2+9*$(;$2$(#$23+'(*2,$ +##+ "*'+($(;*'#'2"(*;($'(*$%3+#"*'+(,$I":%"(:$+;,$23+#5(7$2,$ PB"+2:-(-%"#'2-2/ O(,-*"'##$' '#"3%$7'C%$3+22'9'#'*-)2;'8"(*#"7-*B+,$3%+3+2-$,"(2E ℄/O(2;33+2$' '#$28"%'"9#$2 "#-"*+'%$2

(Y

n

)

n=1...N

'(,-3$(,"(*$2)+(3$;*,+( - %'%$

ρ

2+;2#"=+%7$

ρ(y) = ρ

1 (y

1 )...ρ

N

(y

N

)

/A+'*

p

;( 7;#*'Q'(,' $ ,$#+(:;$;%

N

)3+;%

1 ≤ n ≤ N

)+( (+*$

(y

n,k

n

)

1≤k

n

≤p

n

+1

#1$(2$79#$,$2

p

n

+ 1

%" '($2,; 3#5(+7$ +%*B+:+("# ,$ ,$:%-

p

n

+ 1

"22+ '- "; 3+',2

ρ

n

/ O(*$(2+%'2$ $(2;'*$ #$23+'(*2 ,$ <;",%"*;%$ B+'2'2,"(2 B"<;$,'%$ *'+(/K("22+ '"(*J*+;*8$ *$;%,1'(,' $2

[k

1 , ..., k

N

]

;('(,' $:#+9"#

k

)+(+9*'$(* ;($(2$79#$,$3+'(*2,1'(*$%3+#"*'+(

y

k

= [y

1,k

1 , ..., y

N,k

N

]

3+;%

1 ≤ k ≤ N

p

.

P$23+'(*22+(*#$2(+$;,2,$ #"7-*B+,$,$<;",%"*;%$"22+ '-$";3+',2

ρ

/O("33%+ B$.("#$7$(*

u

˜

N

(y,

·)

3"%#1'(*$%3+#-$,$I":%"(:$

˜

u

h,p

_N

,$

y

7→ ˜u

h

N

(y,

·)

";R3+'(*2

(y

k

)

k=1...N

p

)$*+(,-.('*("*;%$##$7$(*

u

h,p

N

(ω, x) = ˜

u

h,p

N

(Y (ω), x).

A+;2#1B53+*BC2$ / / )#1$2*'7"*'+(,1$%%$;%2;'8"(*$$2*3%+;8-$,"(2E ℄H

!"#$%&'()()+) !"#$%&"0"% *(%&+(&"%-*%$&$1"%

r

1 , ..., r

N

"&'(" *(%&+(&"

C

&"!!"%,'"-*'.&*'&

h > 0

"&-*'.&*'&

p

∈ N

N

kE[u

N

− u

h,p

N

]

k

H

1 _(D)

≤ C

h +

N

X

n=1

√

_p

n

e

−r

n

√

_p

n

!

.

> / S@ K( ="'*)+( *%+;8$ ,"(2 E ℄ ;($ $2*'7"*'+( ,1$%%$;% $( (+%7$

L

2 _{(Ω, H}

1 _(D))

,+(* #$ %-2;#*"* 'Q,$22;2 $2*;($ +(2-<;$( $)"'(2'<;1;($7$'##$;%$$2*'7"*'+(,1$%%$;%,"(2 #$ "2+N

Γ

$2* 9+%(-/L$ (+79%$;2$2 "7-#'+%"*'+(2,$ $**$7-*B+,$,$ +##+ "*'+(+(*-*-3%+3+2-$23$%7$**"(*,$%-,;'%$*%C2($**$7$(*#$ +T* ,$ "# ;#)*$##$2<;$#1;*'#'2"*'+(,$:%'##$2,$3+'(*2,$ +##+ "*'+( %$;2$2U8+'%3"%$R$73#$E?V)?W℄/ L"(2 #$ "2,$2 7-*B+,$2,$ !"#$%&'( 2*+ B"2*'<;$2) +77$ ,"(2 #$ "2 ,$2 7-*B+,$2 ,$ +##+ "*'+( 2*+ B"2*'<;$2) +( +9*'$(* ;($ +(8$%:$( $ $R3+($(*'$##$ 3"% %"33+%* "; ,$:%- ,; 3+#5(7$ ,"(2 B"<;$ ,'%$ *'+()$* ,+( *%C2%"3',$/ L"(2 #$2,$;R "2>XB-+%C7$ / /Y $*XB-+%C7$ / /Z@) #" 3%$;8$ $2* 9"2-$ 2;% ;( %-2;*"* ,1"("#5 '*- ,$

y

∈ R

N

_{7→ u(y, ·) ∈ H}

1

0 (D)

/ I$ +T* ,$ "# ;# ,$ #" 7-*B+,$ ,$ !"#$%&'( 2*+ B"2*'<;$$2* $#;',$#"%-2+#;*'+(,1;(252*C7$#'(-"'%$,$*"'##$

dim(P

p

)dim(V

h

)

)2+'*

(1 + q)

N

_dim(V

h

)

,"(2#$ "2+N+(3%$(,,$23#5(+7$2,$7[7$,$:%-

q

3"%%"33+%*J B"<;$8"%'"9#$)#$ +T*,$ "# ;#,$ #"7-*B+,$,$ +##+ "*'+(2*+ B"2*'<;$$2*<;"(*J#;' $#;',$#"%-2+#;*'+(,$

N

p

252*C7$2#'(-"'%$2,$*"'##$

dim(V

h

)

)"8$

N

p

= q

N

,"(2 #$ "2+N+(3%$(,#$ 7[7$(+79%$,$3+'(*2

q

,"(2 B"<;$,'%$ *'+(/L"(2 #$2,$;R "2) $ +T*";:7$(*$,$7"('C%$'73+%*"(*$"8$

N

/P$ +T*$2*7+'(,%$,"(2 #$2"7-#'+%"*'+(2 ,$ $27-*B+,$2)7"'2%$2*$'73+%*"(*3+;%

N

:%"(,/M"% +(2-<;$(*) $27-*B+,$22+(**%C2$\ " $23+;% ,$2 8"#$;%2 ,$

N

%$#"*'8$7$(* ="'9#$2) 7"'2 '("33#' "9#$2 <;"(,

N

,$8'$(* *%+3 '73+%*"(*/ I" <;$2*'+(

(15)

"#$%&''()*+,&-+)."#

a

"&./0.#/'& #/-) 2&/-+30#"# "+,#./+).4.+##/-"). 0.#30#/-+). (0 +&$#56. .)-#(&"%0.#'&(-30#$#/,7-2)"#/"# )$$) &-+)./-) 2&/-+30#//%7-#."#.-.&-0(#$$#,#.-80.#9(&."# $&//#

"%:;<=> ),'(+/.).$+.7&+(#/=&$)(/30# %#/-,)+./7?+"#.-')0( $#/,7-2)"#/"#@&$#(A+./-) 2&/-+30#/5

;%&0-(# '&(- $#/ ,7-2)"#/ "# )$$) &-+). /-) 2&/-+30#/ '#(,#--#.- "%0-+$+/#( "#/ )"#/ '(7#*+/-&.-/ ')0( $%:;<"7-#(,+.+/-#= ).-(&+(#,#.-&0*,7-2)"#/"#@&$#(A+./-) 2&/-+30#/5

!"#$%&'%& "()& )&"*"&')&+",+-."*$/'

B#/,7-2)"#/"#->'#'#-+-#/'#(-0(C&-+)./ )./+/-#.-8"7?#$)''#($# 2&,'&$7&-)+(##./7(+#D"#E&>$)(

)0"#F&(20.#.GB)H?#I&0-)0("#/&,)>#..#=#-"#-().30#( #--#/7(+#80. #(-&+.)("(#J?)+('&(#*#,'$#

KLM=LN=M ℄5P#/,7-2)"#//).-"). (7/#(?7#/&0* &/)Q).&"#'#-+-#/+. #(-+-0"#/5

6.?&,&+.-#.&.-'(7/#.-#($#/'(+. +'&0*(7/0$-&-/"# #--#-2H/#="&./0.#'(#,+H(#'&(-+#)./%+.-7(#//#

8"#/,7-2)"#/.0,7(+30#/')0($%730&-+)."%7 )0$#,#.-'(7/#.-7#"&./$#'&(&9(&'2#R: )0$#,#.-S "#$&

/# -+). 5 5!="&./0.#/# )."#'&(-+#)./%+.-7(#//#8"#/,7-2)"#/.0,7(+30#/')0($# )0'$&9#"#$%730&G

-+)."%7 )0$#,#.-#-"#$%730&-+)."%&"?# -+).G"+T0/+).'(7/#.-7"&./$#'&(&9(&'2#U"?# -+).G"+T0/+)."#

')$$0&.-/"#$&/# -+). 5 5!5

!" #$%&'() $*+,-./*) 0) &1,/*%2.3$ 01, 3*&)+)$2

;&./ #--#/# -+). )./%+.-7(#//#8$%&.&$>/#.0,7(+30#"# "+T7(#.-#/ ,7-2)"#/')0( $%730&-+)."%7 )0G

$#,#.-D 5!I&?# )#V +#.-$)9.)(,&$= '(7/#.-7#"&./ $#'&(&9(&'2#R: )0$#,#.-S "#$& /# -+). 5 5!5B#/

(7/0$-&-/"# #--#/# -+)./).-"7-&+$$7/"&./ $#/P2&'+-(#/!=W#-L5U?&.-"#/%+.-7(#//#(8$%&.&$>/#.0,7G

(+30#=). ),,#. # '&( "74.+(0. &"(# '(7 +/=#-'&( ")..#( "#0*(7/0$-&-/'(7$+,+.&+(#/ 30+/#().- -(H/ 0-+$#/'&($&/0+-#=#-30+'#(,#--#.- #.'&(-+ 0$+#("%)C-#.+(0.(7/0$-&-"%#*+/-#. ##-"%0.+ +-7"#/)$0-+).

"&./

L

p

_{(Ω, H}

1

0 (D))

5B#/(7/0$-&-/"# #--#'&(-+#/).-"7-&+$$7/"&./$#/P2&'+-(#/!=W#-L5

!"! #$%&'()*)+,( -. /$,0123' ') $%&.1)*)& /$%1+3+(*+$'&

X)+-

D

0."),&+.#C)(.7"#

R

d

"# $&//#

C

2

#-

(Ω,

F, P)

0.#/'& # "#'()C&C+$+-75X)+-

f

∈ L

2 _(D)

#-

a

0. 2&,'$)9.)(,&$2),)9H.#/0(

D

¯

= %#/-G8G"+(#

a(ω, x) = e

g(ω,x)

=)Q

g

#/-0. 2&,'9&0//+#.2),)9H.#= "#,)>#..#/0'')/7#.0$$#')0(/+,'$+4#($%#*')/7=#-"# )?&(+&. #

cov[g](x, y) = k(

kx − yk),

D 5 YI )Q

k

∈ C

0,1

_{(R, R).}

6./%+.-7(#//#8 $%730&-+).&0*"7(+?7#/'&(-+#$$#/#$$+'-+30#8 )#V +#.-/&$7&-)+(#//0+G

?&.-#Z')0('(#/30#-)0-

ω

"&./

Ω

−div(a(ω, x)∇u(ω, x)) = f(x)

x

∈ D,

u(ω, x) = 0

x

∈ ∂D.

D 5 NI

6.)C-+#.- "). 0.#[&,+$$# "%730&-+)./&0* "7(+?7#/ '&(-+#$$#/'&(&,7-(7#'&( 0.'&(&,H-(#&$7&-)+(#

ω

5 P%#/- ')0(30)+= "&./ # 30+ /0+-= ')0( ,).-(#( "#/ '()'(+7-/ "%#*+/-#. # #- "%0.+ +-7 #- '$0/ -&(" "%&'G

'()*+,&-+)./'&-+&$#=).(&+/)..#(&/0($#'()C$H,#"7-#(,+.+/-#8

ω

4*7='0+/)./%+.-7(#//#(&8#*'$+ +-#($& "7'#."&. ##.

ω

56. )./+"H(#0./# ).",#,C(#"7-#(,+.+/-#')0(/+,'$+4#($%#*')/7=,&+/$#/(7/0$-&-/30+ /0+?#.-'#0?#.-\-(#[& +$#,#.-7-#."0/&0 &/)Q

f

#/-79&$#,#.-&$7&-)+(#=/)0/$#/2>')-2H/#/&"730&-#/5 <)0( ),,#. #(=).")..#"#0*(7/0$-&-/'(7$+,+.&+(#/[)."&,#.-&0*')0($&/0+-#56.&-)0-"%&C)("0.

(7/0$-&-"#(790$&(+-7"#/(7&$+/&-+)./"#

g

#-"). "#

a

= )./730#. #"0-27)(H,#"#F)$,)9)()?5 0+$)$'*"*$/123212

g

!"#$%&#'#()*+&!+&$,#)$( -# $+*(#))+&$

α

/01,!#(*#&&#)2 3+%($+%$

α < 1/2

4

;&./ #30+/0+-=).+"#.-+4#

g

8 #--#?#(/+).56.#."7"0+-#.'&(-+ 0$+#(30%+$#*+/-#')0('(#/30#

-)0-ω

= '&( ).-+.0+-7= "#/ )./-&.-#/

a

min

_{(ω) > 0}

#-

a

max

_{(ω) < +}

_∞

-#$$#/ 30#

a

min

_(ω)

_{≤ a(ω, x) ≤ a}

max

_(ω)

(16)

"#$"

ω

%&'()*+ ,-"./#)*0- %- 1'2345,6)'071- .'0(

a

/"'9" ,#69#)0',: 5,9&-+" 95$95;#)0/0-9" <#)9/ 95 #-) 5;(') )'((#)"=

ω

:9/'90#59+:#9 %/%$5"%$"./#)*0- %->-)95?$- ,'()#()5/"/$9(-$ (,$+ ;'5<,-+$5@'9"-A !"#"$%&%"'()*)*)

1 a

min

_(ω)

∈ L

p

(Ω)

!

a

max

_(ω)

_{∈ L}

p

_(Ω)

"#$%!#$!

p > 0

& B-)/+$,"'":'59+5?$-%-+)/+$,"'"++505,'5)-+(#$)%-+'(()#250'"5#9+%-

a

?$-,&#9@-))'('),'+$5"-:?$5 ()#@5-99-9"%&$9-'((,5 '"5#9%$"./#)*0-%- >-)95?$-:9#$+ +-)#9"")*+$"5,-+(#$) #<"-95)%-+<#)9-+

L

p

(#$),'+#,$"5#9-"('),'+$5"-(#$),-+-+"50'"5#9+%&-))-$)%&/,/0-9"+C95+:'D'9"'$()/','<,-#<"-9$%-+ 0'E#)'"5#9+=

ω

C2/:-2()50/-+-9 ;#9 "5#9+%-

a

max

_(ω)

-"

1/a

min

_(ω)

7F$5@'9" -""-%/0') .-:#9%/%$5" (#$) #00-9 -),-)/+$,"'"%&-25+"-9 --"%&$95 5"/+$5@'9"7 !"#"$%&%"' ()*)+) '()*$+!,#- G 7 HI +./ !$- $-,*$ 0#1$!,#-

u

2 *$,+""+%!, -! 3

L

p

_{(Ω, H}

1

0 (D))

2 "#$% !#$!

p > 0

& J99#"-)'?$-,- ')' "*)-<5-9(#+/-",'?$-+"5#9%-,&'(()#250'"5#99$0/)5?$-(#$)$9-KLM -,,5("5?$-= #-N 5-9" ,#69#)0',#9" /"/ /"$%5/+ 59%/(-9%'00-9" %'9+ O!P℄:%'9+ ORP℄ '@- $9- '(()# .- (') <)$5" <,'9 -")/ -00-9"%'9+OPP℄:?$5")'5"-%$()#<,*0-59@-)+-7

!"!" #$$%&$ '% ($)* ,(&$% '& '-.%/)00%1%*( '% 2,$3&*%*45)6.%

B#00-#9,&'@$%'9+,-(')'6)'(.-S4/".#%-++(- ")',-++"# .'+"5?$-+T%-,'+- "5#9 7 7!:,'?$-+"5#9

%- ,&'(()#250'"5#9 %$ #-N 5-9"

a

%'9+ $9 -+(' - %- ;'5<,- %50-9+5#9 +"# .'+"5?$- -+" )$ 5',- (#$) ,-+ 0/".#%-+ +(- ")',-+ +"# .'+"5?$-+7 J9 @' %#9 .-) .-)= ?$'9"5C-) ,&-))-$) #005+- +$) ,' +#,$"5#9

%-,&KLM -9'(()# .'9"

a

(')

a

N

:#U

a

N

-+" #<"-9$-9 ")#9?$'9" ,- %/@-,#((-0-9" %- V').$9-931#*@-

%-log(a)

=,&#)%)-

N

: %/C95-9G 7 I7W'((-,'9" ?$-,'+#,$"5#9%-,&KLM G 7 HI'@- #-N 5-9"

a

-+"

9#"/-u

: -" -,,- %- ,&KLM '@- ,- #-N 5-9" '(()# ./

a

N

G 7 RI -+" 9#"/

u

N

: #9 #00-9 - (') %#99-) $9--+"50'"5#9%&-))-$);#)"-G &-+"3=3%5)-?$&#9 #0(')-

u

N

(ω, x)

-"

u(ω, x)

(#$) S,-0X0-

ω

TI($5+ #9 %#99-$9--+"50'"5#9%&-))-$);'5<,-G &-+"3=3%5)-?$&#9 #0(')-,',#5%-

u

N

-",',#5%-

u

:-9-+"50'9"%-+?$'9"5"/+ %$"D(-

E[ϕ(u

N

)]

− E[ϕ(u)]

I7 M#$) #<"-95) -+-+"50'"5#9+ #9@' ;'5)-,&.D(#".*+-+$5@'9"-+$),-+ #$(,-+ ()#()-+'++# 5/+'$%/@-,#((-0-9"%-V').$9-931#*@-%-

g = log(a)

7 ,-#"&./$0()*)1) 4-0$""#0 *$ ,5 1 06#- !,#-0"%#"% 0

b

n

+""+%!, -- -!3

C

1 _{( ¯}

_D)

2 ,,5 1+0)%,

X

n≥0

λ

n

kb

n

k

2 _∞

0! #-8 %9 -! 2 ,,,5 *$(,1 :,0!

0 < α < 1

! 1*$ 1+0)%,

X

n≥0

λ

n

kb

n

k

2(1−α)

_∞

k∇b

n

k

2α

_∞

0#,! #-8 %9 -! & J959")#%$5"',#)+,'9#"'"5#9+$5@'9"-A 234'%&%"' ()*)5) ;#$%

0 ≤ α < 1

8)%,<+-!1+ #-.,!,#-. 1(=>"#!?@0 A&B&C !

N

∈ N

2#--#!

R

α

N

= max

X

n>N

λ

n

kb

n

k

2 ∞

,

X

n>N

λ

n

kb

n

k

2(1−α)

∞

k∇b

n

k

2α

∞

!

.

M#$)#<"-95)%-,' #9@-)6-9 -()-+?$-+Y)-:#9;'5",&.D(#".*+- +$5@'9"-7