On the conditions of convergence for one class of methods used for the solution of ill-posed problems
We propose a new class of projection methods for the solution of ill-posed problems with inaccurately specified coefficients. For methods from this class, we establish the conditions of convergence to the normal solution of an operator equation of the first kind. We also present additional condition...
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| Дата: | 2008 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2008
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3201 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509248640778240 |
|---|---|
| author | Lebedeva, E. V. Лебедева, Є. В. |
| author_facet | Lebedeva, E. V. Лебедева, Є. В. |
| author_sort | Lebedeva, E. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:48:06Z |
| description | We propose a new class of projection methods for the solution of ill-posed problems with inaccurately specified coefficients. For methods from this class, we establish the conditions of convergence to the normal solution of an operator equation of the first kind. We also present additional conditions for these methods guaranteeing the convergence with a given rate to normal solutions from a certain set. |
| first_indexed | 2026-03-24T02:38:05Z |
| format | Article |
| fulltext |
UDK 519.642
{. V. Lebed[va (In-t matematyky NAN Ukra]ny, Ky]v)
PRO UMOVY ZBIÛNOSTI DLQ ODNOHO KLASU
METODIV ROZV’QZANNQ NEKOREKTNYX ZADAÇ
A new class of projection methods for the solving of ill-posed problems with pertubated coefficients is
constructed. For these methods, the conditions for convergence to a normal solution of operator
equation of the first kind are established. Moreover, additional conditions for these methods are given
which guarantee the convergence at a given rate to normal solutions that belong to some set.
PredloΩen nov¥j klass proekcyonn¥x metodov dlq reßenyq nekorrektn¥x zadaç s netoçno za-
dann¥my koπffycyentamy. Dlq metodov yz πtoho klassa ustanovlen¥ uslovyq sxodymosty k
normal\nomu reßenyg operatornoho uravnenyq I roda. TakΩe pryveden¥ dopolnytel\n¥e us-
lovyq na πty metod¥, pry v¥polnenyy kotor¥x obespeçyvaetsq sxodymost\ s zadannoj skoro-
st\g k normal\n¥m reßenyqm, prynadleΩawym opredelennomu mnoΩestvu.
1. Postanovka zadaçi. Ostannim çasom intensyvnoho rozvytku nabuly doslid-
Ωennq z problem konstrugvannq ekonomiçnyx skinçennovymirnyx nablyΩen\ do
rozv’qzkiv nekorektnyx zadaç iz zadanog ßvydkistg zbiΩnosti (dyv., napryklad,
robotu [1] ta bibliohrafig do ne]). Vidomo, wo sered alhorytmiv nablyΩenoho
rozv’qzannq vkazanyx zadaç najbil\ß poßyrenymy [ metody, wo pobudovani na
zasadax proekcijno] sxemy Hal\orkina. Qk vyqvlq[t\sq, bil\ß ekonomiçnog [
modyfikaciq metodu Hal\orkina, wo pobudovana za dopomohog sxidçastoho hi-
perboliçnoho xresta. Inßymy slovamy, cq modyfikaciq dozvolq[ dosqhaty na-
pered zadano] toçnosti nablyΩennq, vykorystovugçy pry c\omu znaçno menßyj
obsqh dyskretno] informaci], niΩ u standartnomu metodi Hal\orkina. Perßyj
rezul\tat z efektyvnoho zastosuvannq tako] modyfikovano] proekcijno] sxemy
pry rozv’qzanni nekorektnyx zadaç otrymano v [2]. Cq robota znajßla svo[ pro-
dovΩennq v bahat\ox publikaciqx, sered qkyx vidmitymo roboty [3 – 5]. Tut na
pidstavi ide] hiperboliçnoho xresta bulo zaproponovano novi proekcijni metody
dlq rozv’qzannq nekorektnyx zadaç, v ramkax qkyx harantuvalysq nablyΩennq
iz zadanog ßvydkistg zbiΩnosti v prypuwenni, wo ßukanyj rozv’qzok naleΩyt\
pevnij kompaktnij mnoΩyni. Vodnoças inßym, ne menß vaΩlyvym, aspektom v
ob©runtuvanni bud\-qkoho nablyΩenoho metodu [ vstanovlennq umov joho zbiΩ-
nosti do toçnoho rozv’qzku poçatkovo] zadaçi bez bud\-qkyx dodatkovyx prypu-
wen\ wodo c\oho rozv’qzku. U c\omu sensi dana robota [ dopovnennqm ta pro-
dovΩennqm [5]. A same, v meΩax provedenyx nyΩçe doslidΩen\ zaproponovanyj
u [5] pidxid do rozv’qzannq nekorektnyx zadaç bude uzahal\neno na vypadok, ko-
ly koefici[nty poçatkovoho rivnqnnq zadano z poxybkog, a takoΩ budut\ vsta-
novleni dostatni umovy zbiΩnosti pobudovanyx aproksymacij do ßukanoho roz-
v’qzku.
Rozhlqnemo operatorne rivnqnnq perßoho rodu
Ax = f (1)
u separabel\nomu hil\bertovomu prostori X zi skalqrnym dobutkom ( , )⋅ ⋅ i
normog x = ( , )x x . VvaΩa[mo, wo A X∈L ( ) , Rang( )A ≠ Rang( )A ta
f A∈Rang( ), de L ( )X — prostir linijnyx neperervnyx operatoriv, qki digt\ v
X. Norma v L ( )X vyznaça[t\sq standartnym çynom: A = sup
x
Ax
≤1
.
Prypustymo, wo zamist\ toçnyx koefici[ntiv A ta f rivnqnnq (1) vidomo ly-
ße deqki ]xni nablyΩennq A Xh ∈L ( ) i f Xδ ∈ taki, wo
A Ah− ≤ h, f f− δ ≤ δ,
de h > 0 i δ > 0 — vidomi ocinky poxybky poçatkovyx danyx.
Zafiksu[mo u prostori X deqkyj ortonormovanyj bazys E = { }ei i =
∞
1. Vvede-
© {. V. LEBED{VA, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6 843
844 {. V. LEBED{VA
mo do rozhlqdu sukupnist\ operatoriv A . Çerez H r
poznaçymo mnoΩynu ope-
ratoriv A X∈L ( ) , A ≤ 1, dlq qkyx vykonugt\sq nerivnosti
( )I P Am− ≤ βr
rm− , ( )I P Am− ∗ ≤ βr
rm− (2)
pry bud\-qkomu m = 1, 2, … ta fiksovanyx r = 1, 2, … i βr > 0. Tut A∗
—
operator, sprqΩenyj do A , a Pm — ortoproektor na linijnu obolonku perßyx
m elementiv bazysu E , tobto P fm = ( , )f e ek kk
m
=∑ 1
. Pryklad operatora z H r
navedeno v p.I4 ci[] roboty, inßi pryklady takyx operatoriv dyv. u [6]. U po-
dal\ßomu budemo vvaΩaty, wo A , Ah ∈ H r
.
Vzahali kaΩuçy, Ker( )A moΩe skladatysq ne lyße z nul\ovoho elementa, i
v c\omu vypadku zadaça (1) ma[ v X neskinçennu kil\kist\ rozv’qzkiv. Budemo
buduvaty nablyΩennq do rozv’qzku (1) z minimal\nog normog v X , tobto
x A† ( )∈ ⊥Ker , qkyj, zazvyçaj, nazyva[t\sq normal\nym rozv’qzkom rivnqnnq (1).
Meta dano] roboty polqha[ v pobudovi novoho klasu proekcijnyx metodiv dlq
rozv’qzannq operatornyx rivnqn\ (1) z A ∈ H r
. U vypadku, koly zamist\ toçnyx
koefici[ntiv vidomo lyße ]x zburennq Ah ∈ H r
j fδ ∈ X , dlq koΩnoho takoho
metodu budut\ vstanovleni dostatni umovy, wo zabezpeçugt\ zbiΩnist\ do nor-
mal\noho rozv’qzku x†
u metryci X pry δ → 0, h → 0.
2. Klas metodiv rozv’qzannq. Vidomo, wo najbil\ß poßyrenog proekcij-
nog sxemog dyskretyzaci] rivnqn\ (1) [ metod Hal\orkina, vidpovidno do qkoho
skinçennovymirni nablyΩennq operatora Ah magt\ vyhlqd
Ah m n, , = P A Pm h n . (3)
Metody rozv’qzannq rivnqnnq (1), wo zasnovani na sxemi (3), buly pobudovani j
ob©runtovani v [6, 7] dlq vypadku A ∈ H r
. U danij roboti dlq dyskretyzaci]
koefici[ntiv Ah i fδ budemo vykorystovuvaty modyfikacig sxemy (3) taku, wo
Ah n, = ( )P P A P P A Pk k n k nh
k
n
h2 2 2
1
1 21 2 2− +− −
=
∑ ,
(4)
P fn2 δ = ( )f e ek k
k
n
δ −
=
∑
1
2
.
Pid dyskretnog informaci[g pro rivnqnnq (1) rozumi[mo nabir znaçen\ skalqr-
nyx dobutkiv
( ),A e eh i j , ( ),f ejδ . (5)
Todi, vidpovidno do sxemy (4), sukupnist\ nomeriv ( , )i j skalqrnyx dobutkiv
( ),A e eh i j , wo zadiqni pry pobudovi operatora Ah n, , utvorg[ na koordynatnij
plowyni mnoΩynu, qka nabyra[ vyhlqdu
Γn = ( ] [ ] [ ], , { } ,2 2 1 2 1 1 21
1
2 2k k
k
n
n k n−
=
−× ×∪ ∪
i nazyva[t\sq sxidçastym hiperboliçnym xrestom.
Oskil\ky zadaça (1) za prypuwen\ p.I1 [ nekorektno postavlenog, to dlq po-
budovy stijkyx nablyΩen\ potribno zastosuvaty special\ni metody rehulqryza-
ci]. Rozhlqnemo operator Rα = R Ah nα( ), : X → X , wo ma[ vyhlqd
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
PRO UMOVY ZBIÛNOSTI DLQ ODNOHO KLASU METODIV … 845
R Ah nα( ), = g A A Ah n h n h nα( ), , ,
∗ ∗ , (6)
de parametr α > 0. Tut funkciq gα λ( ) [ vymirnog za Borelem na [ 0, ∞ ) ta
operatorna funkciq g A Ah n h nα( ), ,
∗ = g dPα λ λ( ) ( )
0
∞
∫ , de P( )λ — spektral\na
sim’q proektoriv operatora A Ah n h n, ,
∗
. Nexaj gα λ( ) zadovol\nq[ umovy
sup ( )
0
1
≤ <∞
−
λ
αλ λg ≤ 1, (7)
sup ( )
0
1
≤ <∞
−
λ
ν
αλ λ λg ≤ χ αν
ν, 0 < ν ≤ ν∗
, (8)
sup ( )/
0
1 2
≤ <∞λ
αλ λg ≤ χ α∗
−1 2/ . (9)
Tut χν , χ∗ — deqki nezaleΩni vid α dodatni konstanty, a ν∗ — parametr kva-
lifikaci] (tobto maksymal\ne ν, pry qkomu vykonu[t\sq (8)). Todi, zhidno z [8],
Rα [ rehulqryzugçym operatorom z parametrom rehulqryzaci] α . Za nablyΩe-
nyj rozv’qzok viz\memo element
xα = R A P fh n nα δ( ), 2
, (10)
de Ah n, ma[ vyhlqd (4) i Rα — rehulqryzator z funkci[g, qka zadovol\nq[ (7)
– (9). Sukupnist\ usix proekcijnyx metodiv (4), (6) – (10) poznaçymo çerez A .
Takym çynom, v osnovu doslidΩuvanoho pidxodu do rozv’qzannq (1) pokladeno
proekcijnu sxemu dyskretyzaci] (4) dlq koefici[ntiv Ah , fδ i opysanyj vywe
pidxid do rehulqryzaci] nekorektnyx zadaç.
Navedemo pryklady vidomyx metodiv rehulqryzaci] Rα (6) – (9).
Pryklad01. Metod Tyxonova polqha[ v perexodi vid rivnqnnq (1) do rehu-
lqryzovanoho rivnqnnq II rodu
α α αx A A xh n h n+ ∗
, , = A P fh n n,
∗
2 δ . (11)
Cej metod utvorg[t\sq funkci[g gα λ( ) = ( )α λ+ −1
z parametramy χ∗ = 1 / 2 ,
χν = ν νν ν( )1 1− −
. Kvalifikaciq metodu Tyxonova dorivng[ ν∗ = 1.
Pryklad02. Metod asymptotyçno] rehulqryzaci] (abo metod vstanovlen-
nq). Sut\ c\oho metodu polqha[ v tomu, wo funkciq gα λ( ) z (6) dlq dovil\nyx
t = 1 / α ma[ vyhlqd
gt ( )λ =
0
t
t se ds∫ − −( )λ = λ λ− −−1 1( )e t ,
todi 1 − λ λgt ( ) = e t− λ . Vidomo, wo x x tt = ( ) moΩna rozhlqdaty qk toçnyj roz-
v’qzok zadaçi Koßi dlq operatornoho dyferencial\noho rivnqnnq
′ + ∗x t A A x th n h n( ) ( ), , = A P fh n n,
∗
2
, x ( 0 ) = 0.
Umovy (8) ta (9) vykonugt\sq pry χ∗ = 0,6382 ta χν = ( )/ν νe z kvalifikaci[g
metodu ν∗ = ∞ .
Pryklad03. Qvna iteracijna sxema (metod Landvebera). Poklademo x0 =
= 0 i poslidovno znajdemo
xl = x A A x P fl h n h n l n−
∗
−− −1 1 2
µ δ, ,( ), (12)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
846 {. V. LEBED{VA
de l = 1, 2, … , a stala µ zadovol\nq[ spivvidnoßennq 0 < µ < 2
2
Ah n, .
Metod (12) porodΩu[t\sq funkci[g
gα λ( ) = 1 1 1 1
λ
µλ α− −( )( ) / , λ ≠ 0,
de parametr rehulqryzaci] α takyj, wo 1 / α = l = 1, 2, … . Umovy (8), (9)
vykonugt\sq pry χ∗ = µ1 2/ , χν = ( / )( )ν µ νe . Kvalifikaciq metodu ν∗ = ∞ .
3. Osnovnyj rezul\tat. Prypustymo, wo dlq metodiv iz klasu A pry δ ,
h → 0 vykonugt\sq nastupni umovy na vybir parametriv dyskretyzaci] n i rehu-
lqryzaci] α :
α δ−1 2/ → 0,
α−1 2/ h → 0, α → 0, (13)
α− −1 2 2/ rn = O( )1 .
Todi ma[ misce taka teorema.
Teorema01. Nexaj A, Ah ∈ H r , x A† ( )∈ ⊥Ker . Todi dlq dovil\noho metodu
z A pry vykonanni umov (13) x xα − † → 0, qkwo δ , h → 0.
Dovedennq. Dlq vstanovlennq spravedlyvosti teoremyI1 zastosu[mo sxemu
dovedennq, qka bula vykorystana v robotax [6] (teoremaI3.1), [3] (teoremaI1).
OtΩe, poznaçymo
R nα, = g A A Ah n h n h nα( ), , ,
∗ ∗ ,
S nα, = I g A A A Ah n h n h n h n− ∗ ∗
α( ), , , , .
Todi poxybku bud\-qkoho metodu z A moΩna zapysaty u vyhlqdi
x x† − α = R f f S x R A A xn n n h nα δ α α, ,
†
, ,
†( ) ( )− + + − . (14)
Perevirymo zbiΩnist\ koΩnoho z dodankiv u pravij çastyni rivnosti (14).
Analohiçno [6] (dyv. teoremuI3.1), dlq perßoho dodanka z uraxuvannqm vlas-
tyvosti rehulqryzatoriv (9) oderΩymo
R nα, = g A A Ah n h n h nα( ), , ,
∗ ∗ ≤ χ α∗
−1 2/ .
Oçevydno,
R f fnα δ, ( )− ≤ χ α δ∗
−1 2/ → 0.
Dlq dovedennq zbiΩnosti druhoho dodanka nam budut\ potribni dopomiΩni
spivvidnoßennq, qki vstanovleni raniße. A same, vidpovidno do lemyI1 [3] ma[
misce
A A A Ah h h n h n
∗ ∗− , , ≤ c n rn
1
22− (15)
ta z lemyI4.1 [6] vyplyva[
A A A Ah h
∗ ∗− = A Ah
2 2− ≤ c A Ah2 − , (16)
de c c r r1 1= ( , )β i c2 — deqki vidomi konstanty.
Nexaj Y = Ker⊥( )A = Rang( )A A∗ . Dlq bud\-qkoho x Y† ∈ poklademo
x
Y
† : = x
X
† . Todi vnaslidok (7) ma[mo sup
,
,
x Y x
nS x
∈ ≤1
α ≤ S nα, ≤ 1.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
PRO UMOVY ZBIÛNOSTI DLQ ODNOHO KLASU METODIV … 847
Poznaçymo Y0 = Rang( )A A∗ . Dlq dovil\noho elementa u Y∈ 0 , u A A= ∗ v,
v ∈X , z uraxuvannqm (15) i (16), a takoΩ umov teoremy, analohiçno [6] (dyv.
teoremuI3.1) oderΩymo
S unα, =
S A Anα,
∗ v ≤
≤
S A A A A S A A A A S A An h h n h h h n h n n h n h nα α α, , , , , , ,( ) ( )∗ ∗ ∗ ∗ ∗− + − +v v v ≤
≤
v c S A A S c nn h n
rn
2 1
2
12α α χ α, ,− + +( )− ≤
≤
v c h c n rn
2 1
2
12+ +( )− χ α → 0. (17)
Oskil\ky mnoΩyna Y0 skriz\ wil\na v Y, to, zhidno z teoremog Banaxa – Ítejn-
hauza, my dovely zbiΩnist\ do nulq druhoho dodanka z (14) dlq dovil\noho
x A† ( )∈ ⊥Ker = Rang( )A A∗ .
Zalyßylos\ doslidyty ostanng skladovu iz (14). Vnaslidok (4) ma[ misce
spivvidnoßennq A Ph n n,
∗
2
= Ah n,
∗ , z qkoho vyplyva[
R A A xn h nα, ,
†( )− : = g A A A A A xh n h n h n h nα( ) ( ), , , ,
†∗ ∗ − =
= g A A A A A P A xh n h n h n h n h n nα( )( ), , , , ,
†∗ ∗ ∗−
2
= R A P A xn h n nα, ,
†( )−
2
.
Z lemyI2 [3] ma[mo
( ),
†A P A xh n hn−
2
≤ c I P xrn
n3 2
2− −( ) † ,
de c3 = c r r3( , )β — vidoma konstanta. Vykorystovugçy navedene vywe spivvid-
noßennq ta (9), znaxodymo
R A A xn h nα, ,
†( )− = R A P A xn h n nα, ,
†( )−
2
≤
≤ R A P A x P A A xn h n h hn nα, ,
† †( ) ( )− + −( )2 2
≤
≤ χ α∗
− − − +( )1 2
3 2
2/ † †( )c I P x h xrn
n .
Z uraxuvannqm umov teoremy oçevydno, wo tretij dodanok takoΩ zbiha[t\sq do
nulq pry δ , h → 0.
Teoremu dovedeno.
4. Obhovorennq rezul\tativ. OtΩe, v teoremiI1 stverdΩu[t\sq, wo pry vy-
konanni umov (13) dlq dovil\noho metodu z A harantovano zbiΩnist\ joho ap-
roksymacij do normal\noho rozv’qzku bud\-qkoho rivnqnnq (1) z A r∈H . Po-
stavymo za metu z’qsuvaty, pry qkyx umovax metody z A zabezpeçugt\ zadanu
ßvydkist\ zbiΩnosti nablyΩen\ do toçnoho rozv’qzku.
Dlq c\oho vvedemo dodatkovi prypuwennq. Nexaj element x†
pry deqkyx
dijsnyx ρ, ν > 0 naleΩyt\ mnoΩyni, wo ma[ vyhlqd
M Aν ρ, ( ) =
x x A: ,= ≤{ }ν ρv v , (18)
de A = ( ) /A A∗ 1 2. Budemo vvaΩaty, wo znaçennq ρ > 0 vidomo, a dlq nevidomo-
ho parametra ν zadano interval joho moΩlyvyx znaçen\ ν ν∈[ , ]1 1 , 1 < ν1 < ∞ .
Vidomo (dyv. [9, c. 15]), wo ßvydkist\ zbiΩnosti do bud\-qkoho elementa z mno-
Ωyny M Aν ρ, ( ) bude optymal\nog za porqdkom, qkwo ocinka poxybky dorivng[
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
848 {. V. LEBED{VA
O h( ) /( )δ ν ν+ +1
.
Utoçnymo umovy (13) na vybir parametriv dyskretyzaci] n ta rehulqryzaci] α .
OtΩe, nexaj u sxemi dyskretyzaci] (4) za parametr n vybyra[t\sq najmenße na-
tural\ne çyslo, qke zadovol\nq[ umovu (dyv. spivvidnoßennq (10) z [5])
2 2− rnn ≤ 2
4
5c
c h( )δ + , (19)
de
c4 = 2
2 1
2
2
2
r
r r−
β ρ , c5 = ρ β
1
2
2 1
2
2+
−
r
r
r .
Vybir parametra rehulqryzaci] α zdijsng[t\sq vidpovidno do uzahal\nenoho
pryncypu vidxylu [10]
b c h1 5( )δ + ≤ P f A xn h n2 δ α− , ≤ b c h2 5( )δ + , 2 < b1 < b2 . (20)
Zaznaçymo, wo klas takyx metodiv bulo pobudovano u roboti [5] u vypadku,
koly h = 0, a potim uzahal\neno u roboti [11] na vypadok dosyt\ malyx h > 0.
Teorema02. Nexaj A, Ah
r∈H , x†
n aleΩyt\ (18). Todi v ramkax bud\-
qkoho metodu z A pry vykonanni umov (19) i (20) harantovano optymal\nu za
porqdkom ßvydkist\ zbiΩnosti do x†
. Pry c\omu potuΩnist\ mnoΩyny ska-
lqrnyx dobutkiv vyhlqdu (5), wo zadiqni pry dyskretyzaci] koefici[ntiv Ah , fδ ,
ma[ porqdok
O h hr r( ) log ( )/ /δ δ+ +( )− + −1 1 1 1 .
Ce tverdΩennq dovodyt\sq analohiçno teoremiII1 [11].
Takym çynom, z teoremI1 i 2 vyplyva[, wo v ramkax zaproponovanoho pidxodu
do rozv’qzuvannq nekorektnyx zadaç u vypadku, koly parametry dyskretyzaci] n
ta rehulqryzaci] α zadovol\nqgt\ lyße umovu (13), harantu[t\sq zbiΩnist\ do
bud\-qkoho normal\noho rozv’qzku rivnqnnq (1). Vodnoças dovil\nyj metod z
klasu A u razi bil\ß strohoho vyboru n ta α (umovy (19), (20)) zabezpeçu[
zbiΩnist\ iz optymal\nog za porqdkom ßvydkistg do vsix normal\nyx rozv’qz-
kiv iz mnoΩyny (18).
Efektyvnist\ zaproponovanyx metodiv proilgstru[mo na prykladi rivnqnnq
˜ ( )Ax t ≡
0
1
∫ k t x d( , ) ( )τ τ τ = f t( ) , (21)
de x t( ) , f t L( ) ( , )∈ 2 0 1 ta
k t( , )τ =
π τ τ
π τ τ
2
2
1 1 0 1
1 1 0 1
sinh sinh( ) sinh , ,
sinh sinh( ) sinh , .
/
/
t t
t t
− ≤ ≤ ≤
− ≤ ≤ ≤
Vidomo (dyv. [12], rozdil 2, § 15), wo funkciq k t( , ) /τ π2
[ funkci[g Hrina
dlq odnoridno] krajovo] zadaçi
f t f tII( )( ) ( )− = 0, f ( )0 = f ( )1 = 0,
ta operator à di[ z L2 0 1( , ) v W2
2 0 1[ , ]. Zvidsy vyplyva[, wo ̃A ∈H 2
ta spiv-
vidnoßennq (2) vykonu[t\sq z konstantog βr = 1.
V qkosti pravo] çastyny viz\memo funkcig
f t( ) = π4
1
1 2 1 1 1
2 1
1
2sinh
sinh ( sinh )( cosh )
sinh
sinh sinht t t− − +
+
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
PRO UMOVY ZBIÛNOSTI DLQ ODNOHO KLASU METODIV … 849
+ t t
t
( cosh )cosh sinh cosh sinh1 1
2
1 1− + −
,
a za rozv’qzok x M A†
, ( )˜∈ 11 — funkcig
x t†( ) : = ˜ ( )A t⋅ 1 = π2 1 1
1
1sinh ( cosh )
sinh
cosht
t
− − +
. (22)
Pravu çastynu rivnqnnq (21) zburg[mo za pravylom f tδ( ) = f t( ) + δ , de za
riven\ poxybky vybrano znaçennq δ = 10 3− , δ = 10 4− . Oskil\ky obçyslennq
toçnyx komponent vektora (5) [ texniçno skladnog zadaçeg, ma[ sens zamist\
funkci] k t( , )τ vzqty sumu rqdu Tejlora k th( , )τ z ocinkog poxybky
k t k th L L( , ) ( , ) ( , ) ( , )τ τ− ×2 20 1 0 1 ≤ h,
de h = 2 ⋅ 10 – 5 , a same
k th( , )τ =
π τ τ τ τ
π τ τ τ τ τ τ τ
2
3 5 3 5
2
3 5 3 5
1
1 1
6
1 1
120
1 1
6
1
120
0 1
1
1
6
1
120
1 1
6
1 1
120
1 0 1
sinh
( ) ( ) , ,
sinh
( ) ( ) , .
t t t t
t
− + − + −
+ +
≤ ≤ ≤
+ +
− + − + −
≤ ≤ ≤
Oskil\ky c4 = 16 15/ ta c5 = 7 3/ , z (19) vyplyva[, wo dlq dosqhnennq opty-
mal\no] za porqdkom ßvydkosti zbiΩnosti parametr dyskretyzaci] n slid vyby-
raty rivnymII3 pry δ = 10 3−
ta 4 pry δ = 10 4− . Dlq dyskretyzaci] koefici-
[ntiv rivnqnnq (21) za bazys E = { }ei i =
∞
1 vybrano ortonormovanu systemu polino-
miv LeΩandra, zmiwenu na vidrizok [0, 1], ta zadiqno sxemu (4), de
( , )x ek =
0
1
∫ e t x t dtk( ) ( ) , ( )˜ ,A e eh j i =
0
1
0
1
∫ ∫ e t k t e dt di h j( ) ( , ) ( )τ τ τ .
Dlq rehulqryzaci] rivnqnnq (21) viz\memo metod Landvebera.
Rozpyßemo dali zaproponovanyj alhorytm rozv’qzannq zadaçi (21).
1. Vyxidni dani: Ãh
r∈H , fδ , h, δ, ρ.
2. Vybir rivnq dyskretyzaci] n = n h( , )δ za pravylom (19).
3. Obçyslennq znaçen\ informacijnyx funkcionaliv
( , )e fi δ , i n∈[ , ]1 2 ,
( ), ˜e A ei h j , ( , )i j n∈Γ .
4. Vybir znaçen\ parametriv: 0 2
2< <µ
˜
,Ah n
, 2 < b1 < b2 .
5. Obçyslennq nablyΩen\ zhidno z pravylom
x0 = 0, xl = x A A x P fl h n h n l n−
∗
−− −1 1 2
µ δ, ,( )˜ , l = 1, 2, … ,
i pryncypom vidxylu (20).
Oskil\ky
˜ ,,Ah n
2
0 90097≤ pry n = 3 i
˜ ,,Ah n
2
0 90098≤ pry n = 4, to
znaçennq parametra µ docil\no vybyraty z intervalu (0, 2].
Çyslovi rozraxunky bulo vykonano pry µ = 1; 1,5; 2 ta b2 = 2,25; 2,5;
2,75, a za b1 vzqto stalu 2,1. Qk vyqvylos\, najkrawa toçnist\ nablyΩennq
do rozv’qzku (22) dosqha[t\sq pry b2 = 2,25 (u sensi znaçennq perßo] vidminno]
vid 0 cyfry). Otrymano nastupni rezul\taty çyslovyx rozraxunkiv: pry δ =
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
850 {. V. LEBED{VA
= 10 3−
dosqhnuto toçnist\ 0,0182 (µ = 1) i 0,0181 (µ = 2), a pry δ = 10 4−
— toçnist\ 0,0050 (µ = 1) i 0,0051 (µ = 2).
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ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
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| id | umjimathkievua-article-3201 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:38:05Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/fd/299135d21c0ce8b6a3b56a269e1f86fd.pdf |
| spelling | umjimathkievua-article-32012020-03-18T19:48:06Z On the conditions of convergence for one class of methods used for the solution of ill-posed problems Про умови збіжності для одного класу методів розв'язання некоректних задач Lebedeva, E. V. Лебедева, Є. В. We propose a new class of projection methods for the solution of ill-posed problems with inaccurately specified coefficients. For methods from this class, we establish the conditions of convergence to the normal solution of an operator equation of the first kind. We also present additional conditions for these methods guaranteeing the convergence with a given rate to normal solutions from a certain set. Предложен новый класс проекционных методов для решения некорректных задач с неточно заданными коэффициентами. Для методов из этого класса установлены условия сходимости к нормальному решению операторного уравнения I рода. Также приведены дополнительные условия на эти методы, при выполнении которых обеспечивается сходимость с заданной скоростью к нормальным решениям, принадлежащим определенному множеству. Institute of Mathematics, NAS of Ukraine 2008-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3201 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 6 (2008); 843–850 Український математичний журнал; Том 60 № 6 (2008); 843–850 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3201/3148 https://umj.imath.kiev.ua/index.php/umj/article/view/3201/3149 Copyright (c) 2008 Lebedeva E. V. |
| spellingShingle | Lebedeva, E. V. Лебедева, Є. В. On the conditions of convergence for one class of methods used for the solution of ill-posed problems |
| title | On the conditions of convergence for one class of methods used for the solution of ill-posed problems |
| title_alt | Про умови збіжності для одного класу методів розв'язання некоректних задач |
| title_full | On the conditions of convergence for one class of methods used for the solution of ill-posed problems |
| title_fullStr | On the conditions of convergence for one class of methods used for the solution of ill-posed problems |
| title_full_unstemmed | On the conditions of convergence for one class of methods used for the solution of ill-posed problems |
| title_short | On the conditions of convergence for one class of methods used for the solution of ill-posed problems |
| title_sort | on the conditions of convergence for one class of methods used for the solution of ill-posed problems |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3201 |
| work_keys_str_mv | AT lebedevaev ontheconditionsofconvergenceforoneclassofmethodsusedforthesolutionofillposedproblems AT lebedevaêv ontheconditionsofconvergenceforoneclassofmethodsusedforthesolutionofillposedproblems AT lebedevaev proumovizbížnostídlâodnogoklasumetodívrozv039âzannânekorektnihzadač AT lebedevaêv proumovizbížnostídlâodnogoklasumetodívrozv039âzannânekorektnihzadač |