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Regularized Two Level Algorithms for Model Problems

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(1)Regularized Two Level Algorithms for Model Problems Jichao Zhao, Jean-Antoine Desideri. To cite this version: Jichao Zhao, Jean-Antoine Desideri. Regularized Two Level Algorithms for Model Problems. [Research Report] RR-6382, INRIA. 2007, pp.29. �inria-00166639v2�. HAL Id: inria-00166639 https://hal.inria.fr/inria-00166639v2 Submitted on 5 Dec 2007. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published 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..

(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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(51) [.*jlpsrª­   σ 0 0 ··· 0·  ·  0 σ 0  · · · · · · · · · 0  ,  Σ=0  ·   ·  2. 1. 1. 2. 2. 1. 1. 2. 1. 2. 2. 1. 2. 2. 2. N. 1. i. 2. i. j. 0. i. j. j. j=1. N. T. T i i i. i=1. 1. 2. . 0. ···. 0 0. σN −1 0. 0  σN N ×N. CED F ` CHG.

(52) 

(53)  

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(55)  ÀZrumru‘@ƒ¹*mEjsnZ]qp2[˜ilÀ™]\]\‡ p\jl’Mruy{¤m‚XZi¦r¹¯²‡!yŒXPpr¹y{i~nZ¤p]\[^ƒªƒ¥yh] †h­h]qZƒÂƒ¹„‚ruplmZyŒ]LÀZ†¶ƒª]\ru[.m iq;a’hXZ{]\m‚plil]•]qm ¤¦ i€]•bPpl]L†h*jl]qƒ¹[^„Ày{m™†hi|]qjs[^ps„y™jl]~·•‚W#„ypljÀ™]Lpl]Li|j

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(57) r¹ilX‚‡qplŒ]qiHjlis]

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(65) . i. i. i=1.  

(66)  . uT i b σi. T i. i.  

(67) . −3. T i. i. . uT i b σi. . uT i b σi i.  

(68)  . −3. . . . ``ba"cIdefg.  

(69) . −3.

(70) Š.    

(71)   G. (a) without noise. 5. 10. 0. 10. −5. 10. σ. i T i. −10. |u b|. 10. T i. |u b|/σ −15. i. 10. −20. 10. 0. 5. 10. 15. 20. 25. 30. 35. 25. 30. 35. i. (b) with noise. 20. 10. σ. i T. |ui b|. 10. 10. T i. |u b|/σ. i. 0. 10. −10. 10. −20. 10. 0. 5. 10. 15. 20 i. Ê rª‘ŒnZps]S© A Ê yŒp N = 32 ’vZƒuy„j¦ilrumZ‘{nZƒ¹„p£*„ƒunZ]\i σ ’h‡ yE] V‡qrª]\mŒj!i u b {m‚† rumVjsXZ]am‚{rª£Œ]€ily{ƒunhjlruy{m y„¯¥jlXZ]

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(73) m‚y{r¹ik]•ƒu]q£{]\ƒ 10 · Y $&%'$&% * ž  8  {ŸZ Ä É 6 : WXZ]

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(75) [.*­hru[SnZ[ £*{ƒªnZ]

(76) ¤X‚ru‡!X%il„jlr¹Yi 9™]\i¦jlX‚]†hruis‡ ps] js]•w ru‡\„p!†@‡qy{m‚†hrªjlruy{mH· . T i. i. . uT i b σi. −3. . k.  . i=1. T i. i. i. CED F ` CHG.

(77) ”. 

(78)  

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(94) psy{ƒu]•y„¯HjsXZ]

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(122) jsXZ]®rªƒuƒ±° ‡ yŒm‚†hrªjlruy{m‚i\’5¤]®‡q„mµn™ik]SjlXZ]^N„ƒªjl]\plm‚„jlru£{]•¯²yŒpl[y„¯xWru§EXZy{m‚y*£@ps]q‘ŒnZƒu{plru³\„jlruy{m »|©L—{¾ ‡ yŒm‚ilru†h]\plrumZ‘˜jlXZ]

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(126) ‡!XZyEyŒil]. N. T i. λ. σi + λ.  . T i. i. i. i=1. σi + λ σi. i. D UIUIT  FIJADK T FHD O. y{m@jlX‚]‡ yŒ{psil] jƒu]q£{]\ƒ#A. n=4 Z 1 0 (BN (t)T (YK + EnN Y 0 − Y¯ ))2 dt . J(Y ) = γ 2. Õ» ‹Œ‹Œ¾ \Â]qj ’ZjsXZ]qm%¤¦]‡q{m]L{ilrªƒugVyŒÀhjs{rªmjlX‚]•¯²y{ƒuƒªy*¤rumZ‘^mZ]q¤ [.*jsplrª­@]LtŒn™*jlruy{m ¯²y{p2‡qyŒ{psYil]€=ƒu]q£ŒE]qƒÂ‡qYy{pspl+]L‡IYjlruy{m™−i A Y¯ »Õ‹ ^v¾ A Y =b , XZ]\pl]~jlX‚]‡ yE] V‡ ru]qmvj[.*jsplrª­ »Õ‹ "{¾ A = (E ) (A + λI)E , N n. 0. K. cy. 0. cy. . cy. N T n. N n. CED F ` CHG.

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(136) ƒª]\£{]qƒN’hrN· ]Œ·ª’ZrªjX‚{i¦jsXZps]q] ikjl]q™i¦¯²y{p]\Œ‡!X‡qgh‡ ƒu] A bcy = (EnN )T (A + λI)(−Y K + Y¯ ) = (EnN )T (b − (A + λI)Y K ) .. 0 0. 0j+1. K. N n. 0j. 0j. cy. cy. 0. Y j1 = Y j0 − τ1 (AY j0 − b) , Y j2 = Y j1 − τ2 (AY j1 − b) ,. ¤XZ]\pl] τ » i = 1, 2, 3¾„ps]•‘{ru£{]qmÀEg A. Y j3 = Y j2 − τ3 (AY j2 − b) ,. Õ» ‹ŒŒ¾ √ ¤XZ]\pl] r¹i~jlX‚]˜j!„ps‘{]qjkjl]L†¶rumvjl]qps£*„ƒrumPjsXZ]^]qru‘{]qmE£*„ƒunZ] y„¯ {m‚† BpsyEy„j•y„¯ jlX‚]_2W#y*‡!¤=XZ[a,]\ƒªÀE]qb]gEj ‡! i:X‚js]qnZ£.pl5m®y{jsƒuy•gvm‚jlXZy{[^]a{ru{m‚ƒK„y„ƒugh¯:ik†hr¹i¥]q‘Œ¯²y{pl]\p]~jsXZ‹‚]a· ‡qyŒ„p!il]ƒu]q£Œ]qƒ‚‡qy{psλpl]L‡IjlruAy{m™iq’¬¯²yŒpxrikru[^=Z0,ƒuru‡q±r±j|gŒ’{3/2 ¤¦]2‡q„m®jsXZrªm‚§ Z ’ N r · Œ ] ª · Z ’ l j ‚ X „   j l j Z X

(137) ] \ ] ª ƒ \ ] ¬ £ „  l j u r { y  m . [ *  l j s p ª r ­ ¹ r  i ¹ r h † q ] v m l j ª r | j B g . [ „  l j s p ± r K ­ : ·  W E X ‚ n  i  ¤ • ] ‚ X ¬  Œ £ ] N =n E »Õ‹{—v¾ A = A + λI , {‡\‡ yŒps†hrumZ‘˜jly®jlX‚]‡ yŒ{psil]•‡ y{psps]\‡IjsrªyŒm‚i~»Õ‹"^E¾¦y„¯¥[®]qjlXZyh† Y ’Z¤¦]•y{Àhj!„rum X σ w (b − (A + λI)Y ) X w (b − (A + λI)Y ) _» ^v“Œ¾ w = w , Y = i. b+a b−a 1 = + ri τi 2 2. i. N n. cy. 0. N. 0. ?> . T i. N. K. cy. σi + λ. i=1. T i. i. i. i=1. σi + λ. B/T DKO   DN U V!T(T.RTF  D UWUIT  FIJYDK. . K. cy. σi. T.FIHD?O. Ê yŒpjlXZ]‡qyŒ{psil]€ƒu]q£Œ]qƒÂ‡qy{pspl]L‡Ijlruy{mBy„¯ Z [^]qjlXZyh†K’Z¤¦]•n‚il]€jsXZ]•¯²y{ƒuƒuy*¤rªmZ‘.is‡!XZ]q[^] A 0. ¤XZ]\pl]. J(Z 0 ) = Q0 = W P N W T. Z. ’‚[^„jlpsr±­. γ. 1 (BN (t)T (Yk + Q0 EnN Z 0 − Y¯ ))2 dt , 2 PN. r¹i†h]9‚mZ]\†%Œi . PN. ``ba"cIdefg.    =  . ·· 1 1. 1. i.  1      . . N ×N. »_^™©L¾.

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(139)   G. \Â]qj ]\tvn‚„jlruy{m&ZA = (Q E 0. N 0 n )Z. + Y K − Y¯. XZ]\pl]~jlX‚]

(140) [^„jlpsr±­. · ¢ m y„jlX‚]qpB¤¦y{p!†Zi\’¤]¶X™¬£{]%jlX‚]¶¯²y{ƒuƒuy*¤rªmZ‘ m‚]q¤¸[.*jsplrª­ _» ^EŽ{¾ A Z =b , cz. 0. cz. »_^v‹Œ¾. „m™†VjsXZ]

(141) £{]L‡IjlyŒp. Acz = (EnN )T Q0 (A + λI)Q0 EnN = (EnN )T A1 EnN ,. »_^!^v¾. bcz = (EnN )T Q0 (b − (A + λI)Yk ) .. 2m™„ƒugEilr¹ix¯²yŒpjlX‚]•‡qyŒ{psil]€ƒu]q£Œ]qƒÂ‡qy{pspl]L‡Ijlruy{m™irumjlXZ]‡\{il]€y{¯ N = n ’ZjlXEn‚i¤¦]

(142) X‚¬£{] Acz. = Q0 (A + λI)Q0 = (W PN W T )W ΣN W T (W PN W T ) + λI = (W PN ΣN PN W T ) + λI = (W PN )ΣN (W PN )T + λI ,. ¯²psy{[ jsXZ]„À5y*£{]~pl]\ƒu„jlruy{m’h¤¦]•y{Àhj!„rum. A = (W PN )(ΣN + λI)(W PN )T =. N X. »_^ "{¾. T wN +1−i (σi + λ)wN +1−i .. Ê yŒpjlXZ]‡qyŒ{psil]~‡qy{pspl]L‡IjsrªyŒm‚ixy{¯¥[^]qjlXZyh† Z {m‚†@jlXZ]

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(147) [^„jlpsr±­. Z. γ. 1 (BN (t)T (Y K + ∆EnN L0 − Y¯ ))2 dt . 2. 0. cl. cl. »_^v—Œ¾. Acl = (EnN )T ∆(A + λI)∆EnN = (EnN )T A2 EnN ,. „m™†VjsXZ]

(148) £{]L‡IjlyŒp. bcl = (EnN )T ∆(bcl − (A + λI)Y K ) .. 2m™„ƒugEilr¹ix¯²yŒpjlX‚]•‡qyŒ{psil]€ƒu]q£Œ]qƒÂ‡qy{pspl]L‡Ijlruy{m™irumjlXZ]‡\{il]€y{¯ N = n ’ZjlXEn‚i¤¦]

(149) X‚¬£{] Acl. »#"{“Œ¾. = ∆(A + λI)∆ = ∆A∆ + λI = ∆W ΣN W T ∆ + λI = (∆W )ΣN (∆W )T + λI ,. ÀEg.jlX‚]†h]9‚m™*jlruy{m%y{¯HjlX‚]

(150) [^„jlpsr±­ ∆ ’‚¤]•X‚¬£Œ]. 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 yŒpjlXZ]‡qyŒ{psil]~‡qy{pspl]L‡IjsrªyŒm‚ixy{¯¥[^]qjlXZyh† L ’Z¤]•yŒÀhjs{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. »#"ŒŽ{¾.

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(175) À5] j|¤¦]q]\m={ZZpsy¬­Eru[.*jsrªyŒm £¬{ƒªn‚]\i „m™†^js{pl‘Œ] jx£*„ƒunZ]Lix*j¦[^]\ilXV‘{psru†‚i rum‚i|js]\Œ†^y„¯Âps]\ilru†hn™„ƒ‚]\plpsy{p!i\· ]€†h] 9‚mZ]2jsXZ]a[.„­Eru[Sn‚[ {À‚ily{ƒunhjl] £*„ƒunZ] e Œi¦¯²y{ƒuƒªy*¤27i A #» "{‹Œ¾ 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.

(176) ©\Š.    

(177)   G. (a) no regularization 2 1 0 0. 0.1. 0.2. 0.3. 0.4. 0.5. 0.6. 0.7. 0.8. 0.9. 1. 0.7. 0.8. 0.9. 1. 0.7. 0.8. 0.9. 1. −4. (b) with regularization λ = 10 1 0.5 0. 0. 0.1. 0.2. 0.3. 0.4. 0.5. 0.6. (c) true solution 1 0.5 0. 0. 0.1. 0.2. 0.3. 0.4. 0.5. 0.6. Ê rª‘ŒnZps]/^9A#_anZ[^]qpsru‡\„ƒZily{ƒunhjlruy{m‚i¥¯²yŒp ik]qjkjsrªmZ‘•jsXZ]ps]q‘{n‚ƒu{plru³\„jlruy{m‚{ps{[^] jl]\plru³\„jlruy{m λ ]\tvn‚„jl] 0 {m‚† ’™„m‚†BjlpsnZ]ily{ƒunhjlruy{m‚i¦ps]\il™]L‡Ijsrª£Œ]qƒug{· 10 XZ]\pl] „ps]

(178) „Z‚ply¬­hru[^„jlruy{m@£¬{ƒªn‚]\iy{ÀZjs„rumZ]L†@ÀEg†hr M]qps]qmvj2[^] jsXZyh†Zi\’‚„m‚† „ps]~jlpsnZ]

(179) £*„ƒunZ]\i\· ¢ m ƒ¹{ikj•]qY­h™]\plru[^]qmvjsi\’K¤]^£*„psg%jlX‚]®mEnZ[˜À™]\p•y„¯rªjl]\ps„jlruy{m‚i•{m‚†Ë‡qy{[^‚{pl]SjlXZ]LikY]^¯ [.*­hru[SnZ[d{À‚ily{ƒunhjl] £*„ƒunZ]\i{[^y{mZ‘®jlX‚]\il]~[^]qjlXZyh†Zi\· Ë]‡\„m@jl]\ƒªƒÂ]L{ilrªƒug.jsX‚*j2[^] jsXZyh† Z ‡ y{mE£Œ]qps‘{]\i£{]\plg^¯ÕŒi|js]qpL· −4. . i. i. CED F ` CHG.

(180) ©¬”. 

(181)  

(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 rª‘ŒnZps] "A XZ]qm N = 16 ’M¤]®Zƒuy„j

(183) ikrumZ‘{n‚ƒu{pa£*„ƒunZ]\i σ ’‡qyE] V‡ ru]qmvjsi u b {m‚† rum¶jsXZ]®m‚„ru£{] ily{ƒunhjlruy{m {m‚†Wrª§EXZyŒmZy*£.pl]\‘{nZƒ¹„psru³\*js]\†Vily{ƒunhjsrªyŒm y„¯#jlXZ]

(184) ƒurªm‚]\„pilX‚„5]•pl]L‡ yŒm‚i|jspln™‡Ijlruy{mBZplyŒÀh° ƒu]q['¤r±jsX Y mZyŒruil]¶ƒu]q£Œ]qƒ 10 ·,»7Œ¾VruiBjlXZ]Ëm‚„ru£{]ˁ„ZYZpsyŒŒ‡!X’»²À5¾.r¹iVjsXZ]{ZZpsyŒŒ‡!X¿y{¯SWrª§EXZyŒmZy*£ ps]q‘{n‚ƒu{plru³\„jlruy{m·  

(185) 

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(187) 1q2$ ]€‡q{mB„ƒ¹ilyS„Z‚ƒªg®jlX‚]aps]q‘{n‚ƒu{plru³q]L†˜[^]qjlXZyh†.jsyjsXZ]amZyŒmZƒurªmZ]L„p[®yh†h]\ƒ™‚plyŒÀZƒª]\[ {i ¤¦]€†Zru†^¯²yŒpxjsXZ] ƒurªmZ]L„p2ilX‚„5]•pl]L‡ yŒm‚i|jspln™‡Ijlruy{mZpsy{ÀZƒu]q[%·:WXZ]

(188) mZyŒmZƒurªmZ]L„p[^yh†h]qƒK‚plyŒÀZƒª]\[%’Z¤XZr¹‡!X@¤¦]

(189) †hruis‡ n™iliX‚]qps]{’ . i. λ. −7. . ``ba"cIdefg. T i. uT i b σi.

(190) ©\.    

(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 "!$#&%'"

(192) "*(:+,*(

(193)  % !31 "3#% u b . n = 6 / σi M. uT b 0   0! !,+6- (

(194) N*$= *,16 #-4B%'!

(195) #6"!( !(8 #6: " *$ %6)*( :6 !( % #6

(196) #6"!(B!(="i>  #6F)!(σ %' Yλ "+(!  #6) ! *(%6E"+(!( # ! 5A ( D !$8/#H>#6!74 . Y 0  .  b0 = b − (A + λI)Y K 10−7 . . .  

(197) . T 0 i. ". i. *$ 4.  %. λ = 7 · 10−5. #6H8G! "! . cy. ?.

(198) %6)*$;E#-%'#E!(=">.  ) >B!747-/ 0+  %'. min J = J (y(t)) = Y. pα , A.  3D A. IGJLKMION.

(199) ©\—. 

(200)  

(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).

(204) Ž„“.    

(205)   G. 1. 10. regularized single level regularized Y method regularized Z method 0. max error. 10. −1. 10. −2. 10. −3. 10. 50. ) ).  

(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

(207) >= -. !(8L #6-*$#6"!()%-?. . N X. i BN (t)Xi ,. y(t) =. i=0. +(. . 250. ! 3#6!( ! 0 !(. x(t) = "%B*. 200. 47-%6 . α=2 ω(t) = 1 3D "% ? J = 2π. N X i=0. +,*$ *$=" 8G! . ∀ t. Yi. 8G!(. *$ 4. . .  A. i BN (t)Yi , ?.  i! = 0,#1,B·>· "·,"N>

(208) > 0. L!. >. . D. * (F!

(209) B  8G-*(%6". +,*(

(210) 8G!(. #)!(" )-*$. IGJLKMION. .

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(214) ‚„p!„[^] js]qpy„¯¥pl]\‘{nZƒ¹„psru³\*jsrªyŒm‚i\· α. λ. Y. M>A@. 2 2. λ. J0KMT <  T.RSTF. P"FT UWN FIJADKV DK. ]~X™¬£{]~§vm‚y*¤m@rªjruiis„[^]~¯²y{p„ƒuƒMjlX‚]

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