Optimization of projection schemes of digitization of ill-posed problems
We construct new projection schemes of digitization of ill-posed problems, which are optimal in the sense of the amount of discrete information used. We establish that the application of self-adjoint projection schemes to digitization of equations with self-adjoint operators is not optimal.
Збережено в:
| Дата: | 1999 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
1999
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/4739 |
| Теги: |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510908371959808 |
|---|---|
| author | Solodkii, S. G. Солодкий, С. Г. Солодкий, С. Г. |
| author_facet | Solodkii, S. G. Солодкий, С. Г. Солодкий, С. Г. |
| author_sort | Solodkii, S. G. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:12:54Z |
| description | We construct new projection schemes of digitization of ill-posed problems, which are optimal in the sense of the amount of discrete information used. We establish that the application of self-adjoint projection schemes to digitization of equations with self-adjoint operators is not optimal. |
| first_indexed | 2026-03-24T03:04:28Z |
| format | Article |
| fulltext |
Y~K 517.968
Co r . CoJIo~{Kil~ (Hn-'r Ma'roMa'H4Ktl HAH YKpamn~, KxeB)
OIITHMH3AIH4H IIPOEKI]HOHHbIX CXEM
~HCKPETH3AI.[HH HEKOPPEKTHbIX 3A~IAq
New projection schemes of digitization of ill-posed problems are constructed which are optimal in the
sense of amount of used discrete information. The fact is established that the application o f self-adjoint
projection schemes to digitization of equations with self-adjoint operators is not optimal.
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�9 C. F. CO~O~KH~. 1999
1398 I$SN 0041-6053. Yr, p. ~tam. ~'ypR., 1999, m. M, N e ]0
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b2 ..... bin}, T. e. Pa,mg = ~'~-~l (bj'g)bi" Torna npoH3sO~mrnall oneparop A e
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{(bi, Abj)}~j= l s c.ne/~ylomeH BH~e:
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(3)
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sup inf Ilu- P (A)AIIx = 0,
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r ) = sup inf 8HXo - Xdisc]] x .
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> 0 ypa~HeHHe (1) ~L~eeT pCIIIeHHC X 0 E Mp,p(A). EC~IH TCnCpb BBCCTH B paccMo-
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ISSN 0041-6053. Yrp. ~rn. :.~'y. pu.. 1999. m. 51, N'-' 10
OITrHMH3AHH~ ['IPOEKI.[HOHI-II:)IX CXEM LIHCKPETH3AHHH ... 1401
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[1 0], ~ e r z o nozazaTb, wro
ea.se(a, An, Yr R a) > r.o'-62 (A a, ~7r
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ueM, IIP.II~ ~ 8
1 _ l .
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= (8 2 / p ) ~/(P+ ~) 6y~;eT n p a a a ~ e ~ a T ~ cne~Tpy onepaTopa IA 1. Ec~H ~ ~ co6-
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c~zeayeT
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[llal"vll -<
r~e ~= I + 81/82 H I < ~ 2 .
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1 -- I , , l / (p+l ) ~pl(p+l)
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pa A ~ ~a~c~ctrra ~ .
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ZjaczpeT~atm~, pacc~aTp~Ba~cs cJ~ezlymm~t aaa~o r ycJIomm (7):
I I ( ~ A - An)xol lx -~ 0 npn 8 - * 0 .
qepea r l ~ p , p ( H ) , d = (d I , de), 06OaHaqHM bIHO2KeCTBO BCeX BO3MO~IFIMX npo-
eztmonn~rx cxe~ ( f~, B), a~a ZOTOpUX sunonnJnovca coovnomenna
~gs,t),p(H, ~, B)< d~ p ~/(t'+])8Pl(t'+~), d I _> I ,.
r~ Zl2~ ~no6oro y p a s n e r n ~ (1), r/xe A e ~/', f = A x o, x 0 e MI,,p(A ),
I$SN 0041-6053, Yxp. ,,am. xypu., 1999, m. 5i, 1~ 10
1402 C.r. COJIO/1KH~I
II (PleA - Ata)xollx < d25. (9)
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3fro 6y~teT B~ano:m~rr~ca, HanpHMep, ecn~
(d 2 + 1 ) 2~+1 _< ( d l ( 2 d 2 + 1)) ~+1.
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KpeTH3albVlH (5) ypaBHeHH~I (2). HacToamaa CTaT~a rlOCBalIIeHa BblqHCJIeHHIO BeJ'IH-
HHHbl
Card~,t~(Y/) = rain {card(O): ( ~ B) ~ l'Id~,p (Y-/')},
KOTOpa~ xapaKTepH3yeT MHm4Ma~'mH~IR o6~eM/IrlcKpeTHO~ HH~I3opMatl[HH (4), rapaa-
TrlpYmmnfl onTnr~aYmHtafl rlopa/IOK T0qHOCTH 0 (5 p[(P+I)) Ha Knacce ypaBaeHHi~ (1)
c oncpaxopa~a A ~ ~c a Hop~an~H~H pememaa~H x 0 H3 MH03KCCTBa M p . p ( A ) .
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CTBO X TaKOC, qTO/]~Jt rlpOH3BOJIbHOrO ~JICMCHTa g ~ X r crlpaBc/]JIHBO COOTHO-
ulerlHe II g IIx -< II g Ilx, H ~ ~TOM rlalrlReTc~t TaKOIt 6a314c B, qTO g.rl~l n loeoro k =
= 1, 2 . . . . BtaIIOJIHMIOTC.q RBOI~ICTBeHHble COOTHOIIIeHH~I ~H/Ia
I[I - PB,klIx'--,X <- f~r ~-r, IIP~.klIx-~x ~ ~ ~ ,
r~te I ~ TOJTK~eCTBeHH~ onepaTop B X, a KOHCTaHTI~ ~ r n .. ~r He 3aBncffr OT k.
COBOKyIIHOCTb 6a3HCOB B, y~0BneTBOpammHX ~THM yC$10BHJlM, OSO3HaqHM
tlep~3 B r. B ~anbHe~ttue~ orpaHH~n~ca pacc~oTpeHae~ 6azncoB TOJIbKO H3 ~r"
-nycr~ Y/~ = { a : a ~ L ( X , X ) , Ilallx.-,x" <- T, IIa*llx.-,x "<- ~}. ~Ierxo ~a-
]~e'r~, wro X r aBnaeTcz o6o6menne~ CO60~CBCKOrO npocTpaHCTBa Wf qbyHKUnfl,
rr~e~ott~x c y ~ H p y c m a e B Kaa/I~aTC r-c IIpOH3BO/~,H/~tC, a MHOT,(.CCTBO ff'~.~ o6o6ma-
e~ ~'~ar m r r e r p a . r ~ x onepaTopo~ ~Hlla A x (t) = h (t, x) x(x) dx , a ~ p a h ( t , "~)
~o'mp~x ~ e m ' r , ~ a c r ~ e n l X m ~ B o ~ e ~
O<i+jgr 0 [ Ot'~'~J
~ a a X r = W~ B KaqoerBc npHMepa 6a3uca a3 B r Mo~ao HaaBaTb TpUroHO~0Tpa-
q0CKytO CaCTOMy (8 n0pHo~a~0CKOM cJlyqao) Hna opToHopuapoaanHym CaCTOMy
dpyuKmat, nocTpoeHmaX Ha 6ase TaK uas~aeaeM~ix scnnecKos (noapo6aee 06 3TOM
cM. [!2]).
B pmaKax npe/Lnarae~oR neana npoeetmonHo~t CXeMU ~tHCKpeTH3amm non f~ 6y-
~tcM nOaHMaTb qbHrypy Koop/I~HaTHOIt n~IOCKOCTa cncay~mero Brl~a:
n
r. LI ok,
k=0
Q 0 - { 1 } • , Qk = ( 2 k - t , 2 k ] • k = l , 2 . . . . . n,
r gc b > a _> 0. 3aMeTHM, wro eoop~;HHaTta ( i , j ) sepXHHX npaBux y r n o s npa~o-
yrozmtmKos Qt ne~KavaarHnep6one iaj = 2 bn, a caMa dpHrypa Fn a'b IlpC/ICTaB-
naCT co6olt Knatlparrr TaK ~aabmaeMoro cTyneHqaToro r r raep6ont~ecKoro KpecTa.
Bnepmae rHncp6o.rn~ecK.~ KpCCT 6~n npH~eHeH n TPA)pHH llpH~JIH~KOHH~ K. I/I. Ba-
6emr n pa6oTe [13], r~e ~ He~oTopux xnaccoe nepao/m~ecKHx qbyHKtm~ atone-
ISSN 0041-6053, Y~p. l~am. ~.Tpa.. 1999. m. 51. N e 10
OrlTHMH3AI.LI4,q FIPOEKIjHOHHblX CXEM ~,ICKPETH3AI.[HH ... 1403
/~osancJ~ sonpoc 06 orrmMa~bHOM npH6na~KmOmCM no~anpocTpancTac B CMbICJIC no-
nepcqHnKa KonMoroposa. HpH /IHcKpeTHzauHH HOK0ppeKTHMX 3a/Iaq KBa~paHT
cTyneHqaToro rnncp6onHqCCKOrO KpccTa F : b , a = l , b = 2, yacc rlcnom,3onanca
B pa6oTe [5]. IIpM aTOM y a~tep COOTBeTCTBy~twaX mrrerpanbmax onepaTopon npe~-
no~aranocb aa~aqHe cyMMapyeMo/~ S KsaRpaTe noMaanpymme~ cMemanaott ~aCT-
HOll IIpOtI3BO/IHOIt. Mra 3KO cr riOKa~KeM, Wl'O Ha caMoM/~CJlO riO/J~O~, CB$13aI-IHI~
c ncnon~3osaHneM rnnep6o~naecKoro KpecTa, MO~KeT 6blTl, yClletlIHO npnMeneri
tinct 3Haqrrre~u, no 6o:Iee ttmpoKax KnaCCOB ypanaemn~ (1), r~e A ~ H ~ .
C:Ic/IyJ~ [14], paccMoTpriM Knacc peryn~pHaaTopon, 3 a ~ a n a e ~ x napaMcTpriqe-
CKH. A HMeHH0, HyCTb q m { gc~ }, 0 < 1~ < 1, - - HeKOTOp0e IlapaMeTpItqeCKOe ce-
~e~CT~0 dpynXLU~la, n3~cpn~ux no ]30pe:no aa 07peaxe [0 , 72], t]A 11 -< 7, a npn
0 < p < p , , y~oszcT~opa~max yCJIOBHJIM
sup Z/'I1- Xga(X)l < gp~xt', sup ~)/2lga(~.)l -< g , a -t/2,
O~;k<7 2 0 <3.<-i ,2
F/~e p . , ~vp H Z. --HeKOTOpMe He 3aBHCJIII.IHO OT 0t nO~IO~KHTO~bHMe KOHCTaHTM.
�9 CoBoKyrlHOCTb perynJ~pri3aTopos, 3a/~aBaeMHx COOTHOmeriHeM R a =
= g a ( A * A ) A*, gc~ ~ G' 06o3riaanM R ' C -~.. Hpri~tepoM perynapri3aTopa ri3 ~ '
aB~ZeTca o6o6meHnu~t MeTO~ TnxoHosa [4]=
CornacHo cxe~e (F a'b, B), Ka.~oMy onepaTopy A r .9/~ CTaBrrrca B COOTBeT-
CTB~e onepaTop
n
A n = Aa, b, n := ~_~ A~AP2b. -~ ,
k=0
rAe A o = P=, A t = P2 ~ -P2~-~ , k = 1 ,2 . . . . . n, a Pn = Pa,n~OpTOnpOeKTOp,
~teltcT~y~oturi~t Ha nriHe~Hy~O 060~IOqKy rIepsux n 331CMCHTOB 6a3rica B E ~r" Ecnr~
bn - ak He RBd/~CTC~ IJe-nHM qriCJ'lOM, TO rlO~ P2bn-,,t 6yneM riOriribiaTb P2l~n~t},
r~e { g } ~ 6nriacaitmee cBepxy K g ttenoe aricno. B Ka~eCTBe npri6~H~tceririoro
pemerinz Xdisc (6)6yaCM paccMarpnsaT~ ane~errr
Xdisc = ga (a*a.b,n aa, b,n) A~,b.n P2" f~"
r~e ga ~ rrpori3BOZt,naz qbyHKtmZ ri3 G, p . >-- p / 2.
Cne~y~mri~t pe3y~bTaT ~toKa3uBaeTca ana~orriaHO TeopeMe 2 I43 [6].
Teope~a 2. l'lycmb a -~ 5 2/(p+l) U B ~npou3eonbnb ta 5a~uc uz &r" ~ n ~
d (Y[~) , Oocmamouno e, anonuenu.~ycnoeua: moeo ~mo6bt (rn a'b, B) ~ I'ls, p, p
a) 2 - r t P + l ) n - - 8 , a = O , b = l npu O < p < l ;
6) 2 - r ~ + l ) n ~ 5, a = 3 - .p , b = 2 npu l < p < 2 ;
2
a) 2 -2r(p+l)n/p ~ 5, a = p - . 1 , b = 2 npu p>_2.
P
Onmu~taabntatl nop.,~OK mo~nocmu 0 (5 p/(p+ 1 )) cgocmuzaemc.~ npu ucnon~ao6anuu
npouz~on~ozo peo, n.~pu3amopa R a ~ ~ ' . l lpu 9mo~t
Cards, p.p(~f';) < card(r# "b) ~ ~-.Y
I$SN 0041-6053. Y~o. ~tam. ~.'vpn. 1999. m. 51. f i l e ! 0
14~ C.F. CO~O~KH~
2 p + 3 2 p + l
zOe y = npu 0 < p < l , y = npu l _ < p < 2 , y =
r(p + I) 2r(p + 1) 2r(p + 1)
npu p>_2.
3aMeTHM, wro yTBep~.aenHe TeopeMta 2 npa 0 < p <: 1 CHe/I~j'OT I13 peay.rlbTaTOB
pa6OT~ [4].
3. PaCCMOTpHM Tenepb cayqa~t, Korea onepaTop A n (1) anHaeTca caMocorrpa-
~KeHHHbl rl HeOTpHRaTeHbHHM. I'IoHo3KHM
9~. t = { A : A E L ( X , X ) , IIAll_<y, A = A * _ 0 } ,
=
1-IOCKOJ-IbKy npH pemeHrm 3a/IaqH (1), rz~e A = A* > 0, annpoKcHM~py~omaa
onepaTop At~ npHHaTO CTpOaTb (CM., HarIpHMep, [10, C. 31]) TaK~e caMoconp~-
.,~KeHHI:,IM (He06Jt3aTeJIr.HO HeOTpHUaTeJIbHHM), TO 0FpaHHqHMCSI HI, DKe MHO~KeCTBOM
l-is, p , p ^ d (ff'f.~) C I'[ dS,p,p ( f f ~ ) HpOeKUHOHHHX cxeM C CHMMCTpHHHMMH OTHOCHTeJIb-
HO 6HCCeKTpHCH o6HaC'r.qMH ~ Koop/~rlHaTHOIa rlYIOCKOCTH. HpH 3TOM :3CI~C13eKTHB-
HOCTb ]~HCKpeTH3alIHH ypaBHeHHI~ (1), (2) B paMKaX PHM 6y~eM accHel~o~aTb c no-
MOI/~bIO BeJIHqHHH
C-"ards, p,p(9~/'~) = min{card(s (n.B)~.,.o(.~7['~) }.
Kaacc pcry.m~pn3aTopos, 3a/IaBaeM~ax cl:)ynKl.~He~I 0r onepaTopa pemaeMoro ypaB-
Herin.a, onpeReaaM C~e/IyU)mHM o6pa3oM. IlycTh G = { ga}, 0 < a < 1, - - aeKo-
Topoe napaMeTpa~ecKoe ceMe~ICTBO qbynKttafl, a3MepaMtaX no 13ope~o Ha oTpe3Ke
[--70 a , C.7] , Y0 > 0, c . > 1 (70, soo6me roBopa, 3aBHCnT OT KOHKpeTHOfl
qbynKtmH get) H npa 0 < p < /3, y~aoBZeTBOpammHX c~e~;y~ottmM yC~OBnaM:
sup I~.IPll- ~.get(X)l <- ~eta ~',
-yoet <X<c,y
00)
sup Iget (X)I -< ~, r -~,
-yoet~X<c,y
r/Ic /3,, ~p H ~, ~ KaK H IIpeY, cJ:le, HeKOTOpble He 3aBHC~IU.IHC OT 17. IIO3I[O~KHTeJlb-
n e e KOHCTaHT~. COSOKyHHOCT~ perynapa3aTopo~ anna Ret = g e t ( A ) , get e G ,
0603HaqHM ~epe3 R ' .
Hpu:nep. IIycTb q = 0, 1, 2 . . . . . CoraacHo 0606meHHO~y MeTo~y J'IaspeHTbeBa
[3] ypaBHeHHaM (1), (2) CTaBHTCa n COOTBeTC'rBHe peryHapHaoBaHHOe ypaaHeHae
(0~q+l/ + Aq+t)x = A q f s . ~TOT MeTOZ~ perynapn3atmH Ha R ' c qbynKuHeit
~q
get(~') = r + ~q+l ' /IH~t KOTOpOfl s~anoJraema yCHOBHZ (10) npH /3. = q + 1,
c . = 1 H nm6~aX 0 < Y0 < 1, y > 0. B c~ay~ae q = 0 nonyaacM o6taama~ MeTO~
J'Iaspem'besa. B Ka~ecTae ~pyrHx rlpHMepOB peryazpnaaTopon H3 R" MO~.HO Ha-
3BaTh aaBeCTHtae HTepaUX~OHHMe rtpotle/Iyp ra YlaHllBe6epa, ~ a x e e B a - - J l a p / m n ~p.
( n o ~ 6 H e e cM. [3, 10]).
AHaHOrHqHO TP~pc~MO 1 [7] yCTaHaBHHBaeTCJI CHe/IylOUlHfl pe~yHbTaT.
Teope~a 3. l"D'cmb O~ -'- ILl 811(p+I), IIA-AnII ~ Yoa u B --npou3oonbo.t,ta
5azuc ~ ~r" TozOa Ha x.nacce ypaenenutl(1)c onepamopa~m A r Y~u u x o
r Mp,p(A) #nn n~oSoa npoext4uonnoa c x e ~ ( f L B) u npou~oonbnozo peey,a~puza-
ISSN 0041-6053. Ylq~. ~aam. ~y. pn., 1999, m. 51, IV ~ 10
OITTHMH3AI.[H,q HPOEKHHOHHHX CXEM J1HCKPETH3AHHH ... 1405
4
r~e ap = -
~ , > 0 .
mopa R a uz ~ ' , ~. > p, cnpa~ea.auea o~enra
+ ~, ~?~ S -~/(p§ II(a- Aft) x 0 IIx. (11)
B Kaqeerae ~ BOabMeM crynenqaruf l rHnep6omi~ecratfl Kpecr BH/Ia
2n
l~n = {l} x [l, 22n] U ( 2 k-t , 2 k] X [I, 22n-k]. (12)
k=l
Tor~la Ka.,~V.~oMy oilepaTopy A ~ ~/'~ 6yReT COOTBeTCTBOBaTb KOHe'-IHOMepHI~n~ oIIe-
paTop Aft = A n := Z~nO AkAP22.-, , r~e, Kax . paaee, Pn= P s , n - - OpTOnpo-
eKTop Ha nepBHe n 3JIeMertTOB 6aanca B a &r" Hpn6zne~enHoe petnerme Xdisr 6y-
/~eT 3a~aaaT~,CJ~ COOTHOmenHeH Xdise = ga (An) P22*f~, r ~ e ga ~ rrportaaom.aaz
di3yHKRHJ/Ha G, P* -> P "
FIpHae~eM Tenepb pa~1 a c n o H o r a r e z s n u x peay~braToa, KOTOpme no rpe6ymrca a
^ r za~mHeameM. IIyca~ A, H ~ H v . ~ a ~m6oro p > 0 cnpaae / I~aa onteara [4] �9
IIA p - H"ll -< %113 - HII mintp'~}, (13)
rrprlp _< 1 H qbyHKUaJl p "-4 ap orpaurtqerla Ha (0, ~o] ~ z s acex
dIenHa 1. Ilycmb A ~ ff~[~ u p > 1. TozOa npu n > 3
II ( A - A,,)APi[ < vp2-2~4-~, (14)
IIA2-A~II ~ v,E-2rnn, ( 1 5 )
[](A - An)l] < 2T~r 2-rn, 0 6 )
zDe v , = (1 + 2-vr-~)2rr2~ 2, Vp = (2 + 2 r )Tp+I~ 2.
~]{osa3ame~u,cmeo. 1 - l p c ~ c accro aanHmcH paa~o~ccrmc
2n
A - A n= ( I - P22.)A + ~ A ~ A ( I - P22._~).
k=0
C yqeTOH onpe/~e/ieHrdt ~/'~ H X r lI.rlz ~ 6 o r o p > 1 H~eeH
_ p+l 2 -2rn II ( t - P22.)AP+tll < V fi r 2 ,
I[ n n AkA(I_ P22"-*)Ap <_ ~,~ I I ( t - P22,-,)AII211AII p-L~
k = 0
2 r < ~,p+l ~2 9:--2rn
- - r ' r - , 2 r - 1
<
Ilsllx s ~ I I~ = .+ t x
ISSN 0041-6053. Yr, p. .~#am. ~. pn., 1999, m, 31~ bl ~ 10
1406 C.F. COSIO]~KHI;I
sup IIA~gll 2 IIAkAll211(/- P:._k)AII ~ / 2 <_ IIAllV-lllsIIx ~l~.k:,,+l ) ~,~=,.,+l
<:
< 2,~,,,+~1~ 2-2-,4~.
O6"Be]IHHJt#I rloJlyqeHHr~e OI~eHKPI, n0~y~aeM (I 4).
~J1a Hax0m/IeHH~ OUeHKH (15) BocnoJ1s3yeMca npeacTa~J1eHHeM An 2 =
2n
= ~ k - - 0 P2 ~#-k AAk A P:z2n_k. Tor~a HMee~d
2n
AP22~A - A2n = ~.~ GIc, Gk = AA/e4 - P~2~-kaAt.Ap2z~-k.
k=0
3aMeTHM ]~a.rl~, tlTO aJ-l~l J-IK)6OFO g e X 1,0 ( II g IIx < 1 ) sunoJmae ' rca
II Gkzllx <- IIaAka(/- P:.-~)gllx +
+ II ( / - P22.-k)A,X: P:.-~g IIx ---
II aka II II ( / - P:.-k)A I1(11P:.:~g IIx + II ( / - P:.-~)g IIx),
o'rzy~a aertocpellCTSeHHO c~te]ZyeT
2n
IIAP:.A-A~,II <- sup ~ IIGkgllx -< 4~2r~'Zl~2-~'~(2n+1).
II#llx < = ~ =0
IIo~c'rasstaa nata~Zcrn-Iym oraenzy ~ cooTaomcrme
I[A z -A~,[[ < [ I A ( ! - Pzz.)A[I + [[APz~, A - A ~ I [ ,
noz3rqae~ (15). Hcpaseac 'rso (16)aozazsmae-rca ~a .noranao .
TeopeMa 4. 17ycmb cc = p.~5 ~/r 2-~rn4'-n = I.t~5 u B ~npousso.abnb~a
I~a3uc U3 ~r" ~11.11 moeo ~tmo6bt ( fn , B) e FI d ( f[~) p > 1 cgocmamotmo ~,- ~,p.9 , .
nohnenu.e ycnoou~t
~,l . t l I + p ~ p ~ l p + p~oapl2(V,P.2) min{p/2"l} + p ~ . [ t l l V p ~ 2 <
< di 19 l/(p+l), (17)
pvp~t~ _< d~. (18)
Onmu~tanbna~ no nop.~cgtcy ot~entc.a mo,~nocmu d~ p 1 If t,+ 1) ~pl(p+ 1 ) c~ocmueaemc~z
Hpu ~mo~t card(f 'n) = 2~n(n + 1) ~- 5-~/~ log~+~l(~r)(5-~).
f l{o~asamenbcm~o. O-rMcvaM. n p e ~ e acero, qrO n c~ay (16) npH n a ~ ,
y~tomuersopJnottmx yCJIOBH.~M TCOpeMbl, s~anoJmaexca IIA-A.I] < Yo~. 21azee,
crony (13) rt (15) a~eeM
~a t' - I A . I ~ I I _< a t , / ~ l l a ~ - A~llm~"~,/~,~) __. ap/~(v.Z-Z'~n)m~P/Z,l).
IIo~tc'rasaaa s (11) l.la~eanyIO OIJ.CHKy, coozHOtttemte (14), a TalK}Ke mapa)zem~a
a a a napa~e'rpos r a n, no,uy~ae~ yc~osHe (17). H, na nony~emta (18) a o c r a r o a -
no s yc~onue (9) noncraswrs (14).
3a,~e,~anue 2. B czy~ae 0 < p < 1 coornerca-sytottmlt pezyns ra ' r (noay,~ea n
ISSN 0041-6053. Yr4~, ~ara. ~. peL, ! 999, m. 51, iV ~ i 0
OHTHMH3AIIH~I HPOEKI2HOHHblX CXEM/]J4CKPETH3AI2HH ... 1407
[3, 4 ] ) ~tocrriraeTca s paMKaX Tpa/~HIIHOHHOfl rlpOeKtlHOltHOlt cxeblra (A n =
= P2 "A P2") H coaep~r r rca B reopeMe 2 aacToamelt pa6oTU.
4. B 3aKmoqeHne noKa~eM, qTO nalt;aeHaaa n TeopeHe 4 o~erma mtrmManbHoro
A
ql, ICJ'la qbyHKl.g4OHaJlOB (4) ,qB.rDleTC,q S CMblCJ'Ir Br Cards, p,p TOqHOlrt 110 rlo-
pa~aKy.
TeopeHa 5. Ilycmb ~tno.anenbt yc.noou,~ (17), (18). ToeOa Onn p = 2, 3 . . . .
cnpasea.ausbt otlet-tru
" r t l r |og l+l / (2r) (~- I ) C l S - I l r l o g ( 5 -1) < Cards .p .p(H v) < c25-
Onmu.ata.abnbtO nopaOor oS'be~ta aucrpemnot~ unqbop~ta~luu Oocmas.aaem cxe,~ta
(f2, B), eSe ~ -- ' ~'n (12), a B ~ npoussonbnbn~ 6asuc us B,.
]~oKn3ameJu,cmso. B e p x H ~ ORCHKa C~r H3 TcopCMbl 4. ~rLq HaXO~KL~CHH~I
H~CHefl O~eHK~ paccHOTpnH c~y~a~ p = 2. Hp~ p = 3, 4 . . . . p a c c y ~ e ~ H a a~a-
nor~qH~. ~ n a ynpomeHHa ~K~a~ot< 6y~eM CqnTaT~ p = 1. HTa~, 3adpnKcapye~
" d . ~r npoHasom~H~fl anc i enT ( ~ , B) r13 MHO:~KeCTBa H~.2.1 (~(~) . d'IerKo I~rl/ZeTb, qTO B
cnny (9) cnpaBeRnnB0 ( 1 , 1 ) ~ ~2.
~ n ~ 6 o r o x ~ X n L = 1, 2 . . . . nO~O~Ka~
HLX = b l ( b I + L-rbL, x) + L-rbL(bi +bL, X ).
r o r ~ ; a n p n JIIO60M V!-< V ( 2 ~ r ) - l O~ICBH~HO BKnlo~IeHI~e TIHL ~ ~r 3 a -
~CTnM, wro 3aMelia KOHe~HoHepHoro o~epaTopa TIHL oncpaTopoM y! H a + H ,
r]~e H - - n p o H a l O ~ b H b I ~ 3JIeMeHT KJIacca f/ '~ TaKOJt, qTO H b L = H b I = O, S
/IadlbHel~IIIHX paccy~/IeHH~iX HHqeFo He H3MeHPIT.
FlpeacJIe Bcero ot~ennH BemtqnHy M = rain { i : i e co }, rzte, KaX ~ npex~ae, CO =
= { i : ( i , j ) e g2}. l i n k zToro paccHoTprlH ypasHeHne A 1 x = f t , Petuen~er~ KOTO-
poroaBJl~leTcaaneMeHT X 1 = A 2 b t , r/te a I = YIHM n f l = a3bI I YqrtT~Baa
oqeauztnoe COOTUOmem~e Al,t~x = T~ bl (b~, x), naxotatH
a~b~ = Tt (b~ + M-rbM), x~ = T2(1 + M - 2 r ) b l + T2(M -r + M - 2 r ) b M ,
Al , t~x I = y ~ ( l + M - 2 r ) b l , Pf l f l = T~(1 + 2 M -2r + M-3r)bl �9
OTClO/~a cneayeT II ( P n A I - A~.n)x~ IIx = T31( M-2r + M-3r) �9 T o r a a a c rmy (9)
nonyqae~ ,wro M yZtoBzeTaopaeTycnoamo M -2 r + M -3r < d 2 5 / T 3.
IIono:~caH r I = max { 2 d 2, 1 }. PaccHoTpr~M eme o~ano ypaBneune A 2x = f2, pe-
tuenr~eM KOTOpOro annaeTCa aneMeHT x = x 2 = al2f~ b 1/rl = T 2 b I/T1, rzte a 2 =
= A 1, fl, f2 = T~ bl / rl. HMOeM
, - x2 >- ' T~ "q
l 'Iocro~a~ry cnpaae~zmBO
H -:211-,: + : , n x "q
(19)
< 5, (20)
TO n cnyqac f2,8 = PoYl /TI Ha6OpH c~yHKI!I4oHa.rlon (4)/Isla ypaBHelmfl 'AI x I /T I =
= : l / ' q I,l A2x 2 = f2 ,8 COBlla/IaIOT. CJIC/IOBaTr B[JLq Jllo6oro R~G ~ HHeeT
MCCTO COOTHOIIICHHC
ISSN 0041.6053. Yrp, ~tam. ~. pn.. 1999, m. 5 !, N f 10
1408 C.F. CO/[O/][KH~
Xdisc (Rc~,F~ B, Al,-~ ) = Xdisc(Ra,~Z,B, A2,f2.8) : = R~(AI.~)Pf2 f--I .
Tor~a, y n n ~ r a a a (19), (20), a cHny n p o n a a o ~ n o C ~ R~ H aKmOqenaa (~2, B)
r l~I~.2.,(.~r;), no.nyqae~
�9 I1
72 M - r < - x 2 _<
~1 x
- {IX +'ll x= -
_< sup [{x~- ~(A~)P~y~IIx +
fs: {IA -f8 II <
+ sup II~2 - ~(A,,n)P~fsllx
/s: {lf2-fdlx < s
< sup sup fix, - xd~o (Ra, n, B, A,, fs)llx +
x! r M2., (A l ) fs: {[AI x! -fsllX ~ 8
+ s u p sup {{x 2 - xdisc(Rc~,~,B, A2, fs){lx <_
x2 ~M2.1(A2) fs: IlA2x2-fallx ~; 8
< 2 sup sup sup 11~o - X~sc(e~,n,a,A, fDllx <-
A E .~f'.~ XO r M2,I(A) f,~: IlA. xo-fsllx a s
< 2~&2,t( / [~, ,P~,f2, B) < 2dl8 2/3.
= (~-2/(3 r), ( Y r c ~ a n p H TI{ = (72/(2dl'q)) l/r c n e ~ y o T o u e n r a M >_. M l + 1 : Tll
r~r [ 1 , M I ] C co.
~Ia~ee, Ha cne/~ylomHx npnMepax o n p e / x e n ~ T 0 ~ , rlpHHa/~C~KHOCTb KOT0pUx
MHO,~XeCTBy ~ HeO6XO/IHMa/IJLq BKJIIOHeHHR (~'~, B ) C I"[8.2.1" d (if{y).^ r
1. ~oKameM,~r0 npH n ~ 6 u x L, 1 < L < M{, ace TOqKH (L , 1), (1 , L ) npH-
Ha~e.~KaT hrS. l'[pe/InO~OT, O~M HpoTHBHOr a ~4eHHO, rlyCT~ Ha~/IeTC.~ TaKOe 3Haqc-
rote L, 1 < L <_. M l , ~rro (L, 1 ), ( 1, L) e ~2 (HanO~H~, qTO ma paccMaTpHBar
o6nacT~ G, CHMMC, TpHqHyIO 0THOCHT~JI~H0 ~HSFOHK/IH Koop~HHaTHOfl rIn0CKOCTH).
PaccMoTpHM ypaaHeHHe A 3 x = f3 , HMe~omee pemeHHr x 3 = A 2 b I , rne A 3 =
= 71Ht ~ f3 = (7i Hz) 3 b~. Hevpy~Ho aH~ea~, qTO
X3 m 72 (1 + t -2r) b i + 72 (L-r + L-2r) bL '
PfIA3x3 = f3 = 713( 1 + 2L-2r + L-3r)bl + 73(L-r + L-2r + 2L-3r)bL"
B TO me BI~MR 3aCMCHT A 3 ,t~X3 MO2KOT HM~'rb/Iaa aHaa:
A3,ax3 = 7~(l+L-2r)bl, ec.rm ( L , L ) ~ ~ ,
H
-3r A3,ax3 - y{3(l+L-2r)/~ + y~(L -2r + I'. )bl. , e, csm (L,L)E ~.
B o6onx cnyqaaxcnpaae / Immo I[(PaA3- A3,~)x311x >-. 7,3L -" > ~',3 M/-" x 6 2/3,
wro rip. ~ocTaTO,mO Masrux 8 npommmopeq~r (9).
ISSN 0041-605J, Y ~ . ~am. ~. p . . . 1999. m. 51,1r ~ 10
OHTHMH3ALLHJ] FIPOEKllHOHHblX CXEM/~HCKPETH3ALIHH ... 1409
~ - i 1(2 r) /'.,3 / .4 ~lt(2r)
2. ~OKa~KCMTCIICpb, qTO CCJIH 1 < L < TI2 r ~ e r i2 = ~ r l , - 2 s ,
TO
(L, L) ~ n . (21)
KaK ~ s u m e , ]~oxaaaTenbCTBO npoBcl~eM OT npoTnsHoro. BocnonbaycMca ~ n a aToro
ypaBHeHHeM (20) H3 npe~ l~Iymcro npnMepa. A nMcmto, B c ~ / q a c ( L , L ) ~ ~ ~iMC-
eM (PnA3 - A3,n)X 3 = Tl L-rbL(bz, x3) = T~( L-2r + L-3r)bt OTcm~a cne~ycT
II(PnAa - Aa.a)x3llx > T31L-2r > y31~22r~ = d25, qTO B cH~y (9) /IOKaSbl-
BaeT (2 I ) .
3. PaccMoTpHM onepaTop A 4 H3 Y~/'~ cne~ylomero Brt~a:
- - r
A4x = y 2 b l ( b t + L-rba + J by, x) +
+ Y2bL(L-rbl + L-rbL ,+ j - rby , x) + y2J-rbj (b l + b L + b j , x) ,
r a e T2 < Y ( 3 " f 3 ~ r ) - I , a L ~ J ~ nponaaonbnste uemae q a c n a TaKrle, qTO 1 <
3 llr < L < J < M 1 H L J < 1"13 5-11r, 113 = (~/2/'d2) . H a n p e ~ a y m n x npHMepoa cne-
ZyeT, qTO ( 1 , L ) , (L, 1) ( 1 , J ) , ( J , 1), ( L , L ) e ~ . C.ny,aalt ( J , J ) ~t I2 B0aMo-
hKeH dlI, llllb npn J > T~-~5 - l / ( 2 r ) . . ~OKa3s aTO (L , J ) , ( J , L) ~ f2. H p e a n o n o -
xriM npoamaaoe H paccHoTpnM ypaBrtenHe A4x = f4, petueaae KOTOpOro ec rb x 4 =
= A~ b t . BhlqHCJII4M
" ?
x 4 = y [ (1 + L -2r + j -2r)b l + ,y~(L-r + L-2r + j -2r )bL +
2 -r L - r j - r + T2 (J + + j -2r) b j .
~ a n e e Haxog~M (c0OTBeTCTBeHHO npH (J , L) ~ ~ H B npoTrmHOM cJ1yqae)
( b j , (PnA4 - A 4 . n ) x 4 ) = T32J-r(L -r + L -2r + j - 2 r ) ,
(b j , (PflA 4 - A4.fl)x4) 3 -r j - r L-r j - r L-2J" = Y2 J ( L-~ + + + + 2 J - 2 r ) �9
B o6oHx c n y a a a x crlpaBegnrlBO II (P~A4 - a4 . f l )x4 IIx > T3 (Lj),~ >_ y~,113 r ~ =
= . n~,2.t ( H r ) . d.~ 5, qTo IIpOTHaopeqHT BKJ'IIOqeHHIO (~"~, B) ~ ^ d ~ r
TaKHM o6pa3oM, Ha OCHOBaHHH paCCMOTpeHH~X npHMcpoB MO3KHO cnenaTb Bta-
BO~;
z z ~ r o r o a r o 6 ~ ( n , B ) e rid ~ 5,2.1 ( ) , HeO6XOnrlMO, qTO6bl Bce TOqKH (L , J ) ,
1 < L , J < M t = r l l ~ - 2 / ( 3 r ) - 1, LJ < ('f13/~) 1/r, s x o / m n a s M n o ~ e c r B O ~ .
Ocra.rlOCb rlo~cqI, iTaTb 06111e-..e qHc.rlo N TaKax TOqeK:
M~
N = a2~) -fIr f d.x = ~-I/rXog(~-l).
I x
T~M caM~M TeopeMa noJmocTmo ~oKa3aHa.
AHaROFHqHO TeoI~Me 5 yCTaHaBnHBa0TC~ cRe/ly~oItIe~ yTaep~c~leHIaC.
Teope,a 6. ]/a~ Oocmamonno ~an~X 8 abmoanaemc~ Card&Lo( )
5 "I/r. Onmu~tam, m,t~ no nopmgh'y o6"be~ unqbop~mtcuu (4)cgocmueaemca ~ pa~-
r.nx eanepxunc~cot~ cxe~tbt Ouc~cpemusa~uu npu s = [ I. 2 n] • [ I ,~ 2n], 2 n
~- I/(2r).
ISSN 0041-6053i YKp. :*tam. &'vpn:, 1999. m, 51. N ~ 10
1410 C.F. CO.J'IO/],KH~
3 a ~ e , ~ a u u e 3. C p a a H e m i e TeopeM 2, 5, 6 no3BoJI.,qeT c~e~aTb c .ne/~y~o~ee
aaga~o,-iemie. O g a a u a a e T c a , qTO n p n ~ I n c g p e r n a a t m r i ypaanemtJ~ (2 ) aaMena
c a ~ o c o n p z a c e H n o r o onepaTopa A Ha c a M o c o n p a x e H u u l t g o H eq n o Mep ma~ o n ep aT o p
Af~ onpaa~Iana : m t u b n c.nyqae p < 1 (nanpriMep, ~ paMgax c ' r a n ~ a p T n o r o MeTozta
d " I a B p e n T h e a a ) . B TO )Ke BpeM~ nprl 6 0 ~ e e BblCOKHX 3HaqOHHJtX p ,,BIIOJIne
e e r e c ' r a e r m a a " CHMMe'rpriaatBiX/IHCKpeTHO~ CXeMr~ npHBO/LHT m yBe.rrdqeHrnO o6a, e ~ a
rtcno~mayeMolt , . q b o p M a t ~ a , (4).
3 a ~ t e r 4. HeTpyztno BH~eTb, qTO Bce rlpHBe/ieHHr-ae Bl~llle pe3y.,qbTaTbI
crlpaBeR.r~tBH H B c.nyqae, KOr'/la BMec'ro TOqHOVO o n e p a T o p a A aa~ta~o HeKOTOpOe
e r o np r i6~n~en r~e A h r ~ ( X , X r) Taroe , wro IIA-Ah[[ < h, r/~e h <~ c ~ .
1. Tpay6]b~., Bo~rbn~Koocruti X. O6maa "reopm~ onTnMa~n,max aJn'opwl~ol;. - M.: Mrtp, 1983. -
382 c.
2. Hepesep3eo C. B. Oml4Mnaa~Ha MeTOlton npti6Jm~emloro petuenHa onepawoplmlx ypannenHtt. -
Krlen: Hn-T ~,la'reMa'rngn HAH Ygpamn,I, 1996. - 252 c.
3. Plato R., Vainikko G. On the regularization of the Ritz-Galerkin method for solving III-posed
problems//Yqen, nan. TapT. yu-Ta. -- 1989. -- Blan. 863. - C. 3 - 17.
4. Plato R., Vainikko G. On the regularization of projection methods for solving III-posed problems//
Namer. Math. - 1 9 9 0 . - 5 7 . - P . 6 3 - 7 0 .
5. Pereverzev S. V. Optimization of projection methods for solving Ill-posed problems//Computing.
- 1995 . -55 . -P . 113-124.
6 . ConoOrua C. F. 0 ilncKpe'n4aalma aegoppewrmax 3altaq//)Kypn. al~qttCamT. Ma'rema'rnrtl tt MaT.
0pnatlgn. - 1996. - 36, N ~ 8. - C. 15-22.
7. Co,wOru~ C. F. HuqbopMal~lomma cJloa<uOCl~ npoeglmomn,lx a~n'opwrMoa petuelma ypaaneHHlt
Ope/wo:n, Ma I po/ta. I/I Ygp. MaT. ~ypu. -- 1998. --50, N ~ 5. - C . 699-711.
8. [/[oaltoo B. K., 8acult B. B., Tallalta B. I'!. Teopna Jmuegmax neKoppeK'rHrax aa/aa,4 r l e e
npnJio~envta. - M.: Hayga, 1978. - 206 c.
9. Tuxouoa A. H., Apcenutt B. ~. 'Me'rolrLbl pememta nexoppeK'rnlax aa/taq. - M.: HayKa, 1979. -
285 c.
10. Ba~nurro F. M., 8epemetuturoa A. I0. H'repaltrlomltae npolteltypu a neKoppega'max aaltaqax. -
M.: HayKa, 1986. - 182 c.
11. Font~apcxutl A. B., Jleonoo A. C., Reona A. F. KoneqHopa'a|locTnaa annpogcm, laRrta JIHnefltlblX
lleKoppcg'rlihix ~a/Laq//~Kypn. BblqHedlwr. MaTeMaTI, IKI, I tl MaT. qbt4aUgl4. -- 1974. -- 14, N ~ 1. - C.
15-24.
12. Dahmen W., Kunoth A., Schneider 17. Operator equations, multiscale concepts and complexity//
Leet. Appl. Math. - 1996.- 32. - P . 225-261.
13. Eafenro K. 14. 0 npa6Jmxxenrm nepaoaaqecgax qbyng~tHtt MIIOI'HX nepeMemnax
Tpi, lrOllOMeTpntleCKl, IMit MllOl'Oq.IleilaMtt//~OKJI. AH CCCP.- 1960.- 132, No- 2. - C . 247-250.
14. 6ar A. E. O~Hll O611/,l, lfl npHeM noc'rpoemta peryJlapaaylomnx aJiropa'rMoa ItJm
JmHetmo|-o llegoppeK'rnoro ypaane|ma B l'i,I.;ih6epTOnOM npocTpancTne H )Kypn. BMtiHCJIHT.
MaTeMaTt~gn ~ MaT. c~t~a~gn. -- 1967. --7, N ~ 3. - C . 672-677.
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ISSN 0041-6033. Y~p. stare. ~pn . . /999, rn. 51. ye 10
|
| id | umjimathkievua-article-4739 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:04:28Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/5c/7ec4dea181ff022537141b1b154d035c.pdf |
| spelling | umjimathkievua-article-47392020-03-18T21:12:54Z Optimization of projection schemes of digitization of ill-posed problems Оптимизация проекционных схем дискретизации некорректных задач Solodkii, S. G. Солодкий, С. Г. Солодкий, С. Г. We construct new projection schemes of digitization of ill-posed problems, which are optimal in the sense of the amount of discrete information used. We establish that the application of self-adjoint projection schemes to digitization of equations with self-adjoint operators is not optimal. Побудовано нові проекційні схеми дискретизації некоректних задач, що є оптимальними у сенсі обсягу використовуваної дискретної інформації. Встановлено, що при дискретизації рівнянь з самоспряженими операторами використання самоспряжених проекційних схем не є оптимальним. Institute of Mathematics, NAS of Ukraine 1999-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4739 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 10 (1999); 1398–1410 Український математичний журнал; Том 51 № 10 (1999); 1398–1410 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/4739/6179 https://umj.imath.kiev.ua/index.php/umj/article/view/4739/6180 Copyright (c) 1999 Solodkii S. G. |
| spellingShingle | Solodkii, S. G. Солодкий, С. Г. Солодкий, С. Г. Optimization of projection schemes of digitization of ill-posed problems |
| title | Optimization of projection schemes of digitization of ill-posed problems |
| title_alt | Оптимизация проекционных схем дискретизации некорректных задач |
| title_full | Optimization of projection schemes of digitization of ill-posed problems |
| title_fullStr | Optimization of projection schemes of digitization of ill-posed problems |
| title_full_unstemmed | Optimization of projection schemes of digitization of ill-posed problems |
| title_short | Optimization of projection schemes of digitization of ill-posed problems |
| title_sort | optimization of projection schemes of digitization of ill-posed problems |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4739 |
| work_keys_str_mv | AT solodkiisg optimizationofprojectionschemesofdigitizationofillposedproblems AT solodkijsg optimizationofprojectionschemesofdigitizationofillposedproblems AT solodkijsg optimizationofprojectionschemesofdigitizationofillposedproblems AT solodkiisg optimizaciâproekcionnyhshemdiskretizaciinekorrektnyhzadač AT solodkijsg optimizaciâproekcionnyhshemdiskretizaciinekorrektnyhzadač AT solodkijsg optimizaciâproekcionnyhshemdiskretizaciinekorrektnyhzadač |