Regularized Two Level Algorithms for Model Problems
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Regularized Two Level Algorithms for Model Problems Jichao ZHAO — Jean-Antoine DÉSIDÉRI. N° 6382 August 8, 2007. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--6382--FR+ENG. Thème NUM.
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(150) [^jlpsr± ∆ ¤]X¬£]. A = (∆W )(ΣN + λI)(∆W )T =. ¤XZ]\pl]. N X. »#"Z©L¾. w ˜i (σi + λ)w˜iT ,. i=1. . . w1,i −w2,i w3,i −w4,i. w ˜i = , (−1)N −1 wN −1,i (−1)N wN,i. ··. mZy{jl]jsX*j. Ê ypjlXZ]qy{psil]~qy{pspl]LIjsrªymixy{¯¥[^]qjlXZyh L Z¤]yÀhjs{rªm. . w1,i w2,i w3,i w4,i. . wi = , wN −1,i wN,i. ··. 0. L0 =. N X w ˜T (bcl − Acl Y K ) i. i=1. ``ba"cIdefg. σi + λ. w ˜i =. N X i=1. σi w ˜iT (bcl − Acl Y K ) w ˜i . σi + λ σi. »#"{¾.
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(153). TUWJACETKSFIV. Ê rªp!ikjlugu] j iaªyEy§j~hr¹ilqpl]qjl]wxru\p!¶ ymhrªjlruy{mi\rN· ]{·u5ilrªm{nZ¹p£¬{ªn]\i σ Kqyv] V ru]qmvjsi u b m rªm^jsXZ]am{rª£]2ikyªnZjlruy{m y¯MjlX]2ªrumZ]LpilX{]ps]\qy{mikjlpsnIjsrªymplyÀZª]\[ ¤rªjlX · ¢ m Ê rª· P»7¾2X{imZyBmy{r¹ik]5¤Xrªu]@Y»ÕÀ¾2X{iajlXZ]mZy{r¹ik]ª]\£{]\ 10 M{mµrªjil]q]\[.i2jsX*j~[^Nhrª=[nZ16[ k = 15 ¯²y{pjsXZ]qik]y{¯#mZy^mZyruil]{{m k = 12 ¯²y{pjlX]q{il]y¯»²À¾ · . T i. i. uT i b σi. −7. (a) without noise. 5. 10. 0. 10. σi. −5. 10. b| |uT i T |ui b|/σi. −10. 10. −15. 10. 0. 2. 4. 6. 8. 10. 12. 14. 16. 18. 12. 14. 16. 18. i. (b) with noise level 10− 7. 5. 10. 0. 10. σi. −5. 10. T |ui b| T |ui b|/σi. −10. 10. −15. 10. 0. 2. 4. 6. 8. 10 i.
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(167) ¹*jkjs]qpqik]·:̦ik]L.jsXZ]\il]~*jsh¤]~X¬£]ajsXZ]~¯²y{uªy*¤rumZ®pl]\[^{pl§hi b = b − (A + λI)Y mByÀik]\pl£*jlruy{m7i A ©·2WXZ]~Zruis ps] js]w r¹q{ps ymhrªjlruy{mi¯²ypqy{psil]au]q£]qK yplps]\ jlruy{miikXZynZ¹BÀ5]
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(175) À5] j|¤¦]q]\m={ZZpsy¬Eru[.*jsrªym £¬{ªn]\i m^js{pl] jx£*unZ]Lix*j¦[^]\ilXV{psrui rumi|js]\^y¯Âps]\ilruhn]\plpsy{p!i\· ]h] 9mZ]2jsXZ]a[.Eru[Sn[ {Àily{unhjl] £*unZ] e i¦¯²y{uªy*¤27i A #» "{¾ e= max |Y − Y¯ | , 6 18. F%&%)( . . . . −7. . uT i b σi. T i. i. λ. λ. . . . . . i. 0 uT i b σi. T 0 i. λ. −7. 0. 0. −5. 0. 0. K. cy. K. cz. . . 0 uT i b σi. T 0 i. i. 0. 0. −5. 0. 0. −5. −6. . i∈{0,1,··· ,N }. ``ba"cIdefg. i. i.
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(182) !" $#$%&('*)+,!.-/ 0!1 &2'. −7. (a) before regularization with noise 10. 5. 10. 0. 10. σi. −5. 10. |uTb| i. T. |ui b|/σi. −10. 10. −15. 10. 0. 2. 4. 6. 8. 10. 12. 14. 16. 18. 12. 14. 16. 18. i. −4. (b) after regularization with λ = 10. 2. 10. 0. 10. −2. 10. −4. 10. σi |uTb| i. −6. 10. |uTb|/σ i. i. −8. 10. −10. 10. 0. 2. 4. 6. 8. 10 i. Ê rªnZps] "A XZ]qm N = 16 M¤]®Zuyj
(183) ikrumZ{nu{pa£*unZ]\i σ ÂqyE] V ru]qmvjsi u b {m rum¶jsXZ]®mru£{] ily{unhjlruy{m {mWrª§EXZymZy*£.pl]\{nZ¹psru³\*js]\Vily{unhjsrªym y¯#jlXZ]
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(191) G. (a) After 10 iterations. 5. (b) After 50 iterations. 0. 10. 10. 0. 10. −5. 10 −5. 10. −10. 10 10. i T i. i. |u b|. T |u b| i T i. −15. T i. −15. |u b|/σ. 10. |u b|/σ. 10. −20. 0. 5. 10 i. 15. 20. (c) After 100 iterations. 0. 10. 0. 5. 10 i. 15. 20. (d) After 300 iterations. 0. 10. 10. −5. −5. 10. 10. −10. −10. 10. 10 σ. σ. i. i. T i. |u b| −15. 10. T i. |u b|. −15. 10. T |u b|/σ i i. T i. |u b|/σ. −20. 10. i. i. −20. 10. σ. σ. −10. i. −20. 0. 5. 10 i. 15. 20. 10. 0. 5. 10 i. 15. 20. )- *$)4 *$ 4 "!$#&%'"
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(193) % !31 "3#% u b . n = 6 / σi M. uT b 0 0! !,+6- (
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(197) . T 0 i. ". i. *$ 4. %. λ = 7 · 10−5. #6H8G! "! . cy. ?.
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(201) !" $#$%&('*)+,!.-/ 0!1 &2'. (a) After 3 iterations. 0. (b) After 10 iterations. 0. 10. 10. −5. −5. 10. 10. σi. −10. 10. −10. 10. |uTi b| T. |ui b|/σi −15. 10. −15. 0. 5. 10 i. 15. 20. (c) After 20 iterations. 0. 0. 5. 10 i. 15. 20. 15. 20. (d) After 40 iterations. 0. 10. 10. −5. −5. 10. 10. −10. −10. 10. 10. −15. −15. 10. 10. −20. 10. 10. −20. 0. 5. 10 i. 15. 20. 10. 0. 5. 10 i.
(202) !#"$ &% '" * ,+ ,* " T 0 - u b ( 5<u ,i= b 4/5σ>"$ /.0123%4 5N =6 16. 7 "$n45= 6 5894 & :"4 - & 4 5-σ";i )4( 457 ( ( Yλ ,%5 −7 ? "4 @ ,% , 4 45BADCE5F:!8)4 G=7 H 0 , 0 K ( ( ( 10 Z b = b cz − (A + λI)Y - −5 I λ = 7 · 10 T 0 i i. . 4 p=. Z. 1 0. p. x0 (t)2 + y 0 (t)2 ω(t) dt ,. A=. Z. 1 0. x0 (t)y(t)ω(t) dt ,. 4 8L5M"$N PO- . Q"$ 5RK ,54
(203) 58N p. <NA , 4 ( ω(t) > 0>"& 5%5 ,α >"41= 254 S=7554 HRT . ( ( x(t) (. U)UWV!XY[Z[\[]. AKJ5JF. 7 "4 ,-!R "4!( S<N! x(t).
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(206) . *$ 4. y(t). 100. . *(6&47. N = 16 3 .. . 47-4=. .. .. 150 iterations. ! > *$E#6)-%6. >B#6!74%. Y . . .. !0!3%'. . !(=">. #6! . A. .. !(3#*( ". *$ 4. 300. . #6+,*$ 3" #6H0
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In this chapter, we prove an a posteriori error estimates for the generalized overlapping domain decomposition method with Dirichlet boundary conditions on the boundaries for
Revue française d’héraldique et de sigillographie – Études en ligne – 2020-10 © Société française d’héraldique et de sigillographie, Paris,
Dans ce cas, comme les coulées du Bras de Sainte-Suzanne appartiennent au massif de La Montagne daté à plus de 2 Ma (McDougall, 1971), l’ensemble des coulées pahoehoe