Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://phys.msu.su/upload/iblock/824/2006-00-00-novitsky.pdf Äàòà èçìåíåíèÿ: Wed Sep 17 23:26:30 2008 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 21:05:40 2012 Êîäèðîâêà:

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. , , . (), , , . . , , , . , , (). ­ , , - (), , [, 2004]1 . , , , - , . [, 1989]2 , [, 1990]3 , , - , , , . , , . , , . , , .
­ .: 2004. 2 351 c. 3
1

.. - . ­ 400 . .. . ­ .: . ., 1989. ­ .. , ­ .: - , 1990. ­ 288 c.


4 , , [, 1989], [, 1990]. , , ­ , . , [, 2004]. , , , ( ) , , . . , []4 ( ) , ; [, 1991]5 . [, 2000]6 [, 2004]. [, 1993]7 . [, 1987]8 , [, 1991]9 , . , [, 1990]. , .. , (.. ).
.., .., .. . . // Pattern Recognition and Image Analysis, . 5 .. , .. . - . // . ­ 1991. ­ . 3, 8. ­ . 53­62. 6 .., ... - . // . ­ 2000. ­ 6 (15). ­ . 17­23. 7 .., .., .. . // . ­ 1993. ­ . 3, 9. ­ . 43­54. 8 .. . . ­ . ­ 1987. ­ . 27, 6. ­ . 985 ­ 989. 9 .., .., .. . // . ­ 1991. ­ . 3, 7. ­ . 57­70.
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5 ; . , , . , , () , , () , : 1. ; 2. , , , , , ; , , . [, 1989], [, 2000]. - : 1. , - ; 2. ; , ; 3. , , , (.. ); ; , , . , , - , [, 2004]. , - , .


6 , , , . , , , , , . . : 1. ; 2. ; , ; 3. .

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, , ; , , - ; , . . : = Af + , (1) ~ ­ R , ~ R f R, ~ , A : R R ­ , , R ~ R ­ . (1) , . U f (·) U , U : R U ­ , "" , , A, , . () R , U , .. U f . . (1) A, , , , , U , "" , , [A, ] (1) [A, , U ] (1). [A, , U ] (..) R U f : h(R, U ) = sup E R - U f 2 min . (2) U min R : R U; R ­ (2), h(R , U ) .. R U f "" U , .. . A, U , E = 0 (1) , f ­ c Ef = 0 F , f , , [A, F, ] (1). [f R R


10 , 2004] (1) = U f + V , V ­ , : h(R, U, V ) = min E R - 2 .
R

R . . U f [A, If , I , U ] (1). f If Rm , I Rn , If = {f Rm , f j fj f j , j = 1, . . . , m}, I = { Rn , i i i , i = 1, . . . , n}, Ij , fj , . c = (c1 , . . . , cm )T ­ , l = (l1 , . . . , lm )T ­ (.. Ij {cj , lj }, j = 1, . . . , m), (c, l) D(A, I f , I | ), D(A, If , I | ) ­ R2m , A, If , I : D(A, If , I | ) =
m m m m

{c, l : i - i

j =1

aij cj -

j =1

|aij |lj

aij cj +
j =1 j =1

|aij |lj i - i ;

aij ­ A. f Rm I1 , . . . , Im , (3) , [, 2006]10 , 1 , . . . , n . : q (l ) = max q (l), j = 1, . . . , m, (4) (c,l)D(A,If ,I | ) q (l) . , k - f q (l) = lk , f " " q (l) =
m

f j cj - lj cj + lj f j , 0 lj < , i = 1, . . . , n, j = 1, . . . , m}. (3)

.

j =1

lj . q (·) (4)

M(A, If , I | ) = {f Rm , f j fj f j , j = 1, . . . , m, i - i
m



j =1

aij fj i - i , i = 1, . . . , n}.

f
10 .. . , . ­ .: , 2006.


11 [, 2006] m lj min, j = 1, . . . , m, M(A, If , I | ) [c1 - l1 , c1 + l1 ] â . . . â [cm - lm , cm + lm ], , M(A, If , I | ). cj (x), lj (x) Ij (x) {cj (x), lj (x)} fj , , cj (x) Ij (x) fj c lj (x), , . - . [, 2000]11 : , , , , µ
,,

|fj - cj (x)| lj (x), j = 1, . . . , m.

d(·) : R U , = x = u = d(x) ; l(x, y ) ­ , y Y , x X . . . [, 1967]12 . x > l, , 0 < x < l a2 , x = 0 . ()
11 12

, , , = u, = x. , , , P (d(·)) = sup min(µ, (x, u), l(u, d(x))) min .
uU d(·) : RU

µ,, (x, f , y ) = x, = f , = y . µ, (x, u) = sup µ,, (x, f , u), (x, u) R â U,
f F

(x, f , y ), (x, f , y ) R â F â U.

.. . . ­ .: , 2000. ­ 190 . .., .. . ­ .: , 1967. ­ 300

.


12 -q (t) = [f (t) - u(l, t)], q (t) x = l, , f (t) , , u(l, t) . , ( ) , . , . . , , A(, ), ­ Ac . , x t :
t

u(x, t) =
0

n=1

1 + 2 2(n + 2 )

-1

sin

n xâ â a2
n n 2

n n- tg = - . Ac A(, ), . ~~ (6) , min Ac - A(, ) 2 , , ~~

n +

e

-a2 n (t- )

f ( )d Ac f , (5)

­ (5): = 1, = a2 1 . (7) ("") . . 1 "" (7) ( ) (6) ( ) . 1 , 2 ­ . , "" () (7) , (6). , , , , . , (7), , . , (. . 1), (6) . , ,


13
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e

h

e

x Þ À

x

. 1.

. [, 2004]. [, 2000], [, 2004]. "" f " " . "" q (l) = lj , (4) lj = max lj , j = 1, . . . , m, (8) (c,l)D(A,If ,I | ) D(A, If , I | ) (3). c ( ), lj ( ) j Ij ( ) {c ( ), lj ( )}, j = 1, . . . , m, j fj Ij ( ), j = 1, . . . , m, f . fj cj ( ) Ij ( ), lj ( ) , |fj - c ( )| lj ( ), j cj ( ) fj , j = 1, . . . , m. "" (8), (3) , , . . - (4), (3) A f ( [A, I ]). , "" : 2. [A, I ] A­ m â m-.
m

c k ( ) =
i=1

a-i (i - ( i + i )/2), k = 1, . . . , m, k (9)

q (l ) = lk [A, I ] i - i 1 min , lk = i=1,...,m |aik | 2 - aik , aik , i, k = 1, . . . , m, ­ A A-1 .


14 f " " , q (l) = min lj , :
j min lj = j (c,l)D(A,If ,I | )

max

min lj .
j

(10)

Ij {c , lj }, j = 1, . . . , m, c f j , min |fj - c ( )| min lj . j j j

, (10), (3): 3. [A, I ] A­ m â m-. (4) q (l) = min lj
j m

c ( ) = k
i=1

a-i (i - ( i + i )/2), k = 1, . . . , m, k i - i 1 min m , 2 i=1,...,m |aik |
k=1

q = l1 = . . . = l m =

i = , i = , i = 1, . . . , m; ­ , A. (10) : - q (1 - e- / ). 2 | |
Þ À

- aik , aik , i, k = 1, . . . , m, ­ A A-1 . . , (8) -2 2 - , > 0 - 1/|ak-1 k | = || 1 - e (11) lk = -1 2 1/|a | = 2 e (1-k) e- + e - 2 , < 0. mk ||

b

a
. 2.

b

a

. 2, ( (8)) ( -


15 (10)). , q (l) =
m

li , .

, . , , . , , , . , , , , , . , . . l. , , x. f (x, t), 0 x l, 0 t T , , u = u(x, t) x [0, L] t [0, T ] x = 0 x = l : a1 ut (0, t) - b1 u(0, t) = 0, |a1 | + |b1 | = 0; a2 ut (l, t) + b2 u(l, t) = 0, |a2 | + |b2 | = 0, a1 , a2 , b1 , b2 ­ . t = 0 , u = u(x, t) 0 x l: ut = a2 uxx + f (x, t), 0 < x < l, 0 < t < T , a1 ut (0, t) - b1 u(0, t) = 0, |a1 | + |b1 | = 0, (12) a2 ut (l, t) + b2 u(l, t) = 0, |a2 | + |b2 | = 0, u(x, 0) = 0. u x t, a . u x t. u = Af + , A ­ , ­ . f (x, t) = g (x)h(t), x [0, L], t [0, T ], g (·) h(·), ´ h(·) g (·). . h(·) : [0, T ] R1 . , Ii (t) =
t 0

i=1

exp(-a2 i (t - ))h( )d , t [0, T ], ,


16 t
l

u(x, t) =
0

i=1

Ii (t)(2/l) sin( i x) sin( i y )g (y )dy , x [0, L], t [0, T ].

, 0 , ..
l

u(x, t) =
0

i=1

~ (2/l) sin( i x) sin( i y ) exp(-a2 i (t - 0 ))g (y )dy As g (·),

0 x L, 0 t T , (13) gi , i = 1, . . . , n, g (x) n [0; l/n], [l/n; 2l/n], . . ., [(n - 1)l/n; l] . t = t0 > 0 . t0 ­ 1 , 2 , . . . , n [0; l/n], [l/n; 2l/n], . . ., [(n - 1)l/n; l], . i = ui (t0 ) + i , i = 1, . . . , n, ui (t0 ) , {1 , . . . , n } . , u(t0 ) = As g ; u(t0 ) {u1 (t0 ), . . . , un (t0 )}, g {g1 , . . . , gn }, As ~ As (13) L2 [0, l]: 1 1, x [(i - 1)x , ix ] , i = 0, . . . , n, x = l/n. ei = 0, x [(i - 1)x , ix ] / x g (·) , h(·) ­ . Ii (x) = sin( i x)
0 T 0 i=1



l

sin( i y )g (y )dy , x [0, L],

x0 u(x, t) = Ii (x)(2/l) exp(-a2 i (t - ))h( )d , x [0, L], t [0, T ].

, y0 , Ii (x) = sin( i x) sin( i y0 ) u(x, t) =
0 T i=1

~ (2/l) sin( i x) sin( i y0 ) exp(-a2 i (t - ))h( )d At h(·),

0 x L, 0 t T . (14) hj , j = 1, . . . , m, h(t) [0; T /m], [T /m; 2T /m], . . ., [(m - 1)T /m; T ]. x0 .


17 , ­ 1 , 2 , . . . , m [0; T /m], [T /m; 2T /m], . . ., [(m - 1)T /m; T ], . j = uj (x0 ) + j , j = 1, . . . , m, uj (x0 ) ­ x0 , {1 , . . . , m } . , u(x0 ) = At h; u(x0 ) {u1 (x0 ), . . . , um (x0 )}, h {h1 , . . . , hm }, At ~ At (14) L2 [0, l]: 1 1, t [(j - 1)t , j t ] c j (t ) = , j = 0, . . . , m, t = T /n. 0, t [(j - 1)t , j t ] / t ­ ( ´ ) ( ). (10) . 3, , . x0 ­ , t0 ­ . : , .
Þ À

. 3.

, , .. [, 1974]13 , [Dorofeev, 2002]14 , . . , , . 4 . z . ; "" , ; "*" .. , .. . . ­ .: , 1974. ­ 224 . K.Y. Dorofeev, N.N. Nikolaeva, V.N. Titarenko, A.G. Yagola. New approaches to error estimation to ill-posed problems with applications to inverse problems of heat conductivity. // Inverse and Ill-posed Problems. ­ 2002. ­ Volume 10, No. 2. ­ pp. 107­212.
14 13


18 , . , , , .

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, . (. . 5) . . 5 () () ( ­ lj , ­ min lj ), () j j

, () ­ - ( ); ( ), . , ­ , , . , D , , , . D (. 5 ); , . D . -


19 , D . , D () . () (), . , D . , : 1. - ; 2. , ´ , ; 3. , - ; 4. ; 5. ; , .


1. .., .. , - . // . ­ 2004. ­ 3. ­ . 24­27. 2. .., .., .. - . // . ­ 2006. ­ .18, 6. ­ . 15-28. 3. .., .. - . // 8- - -


20 " ". . . ­ .: 2002. ­ . 113-114. 4. .., .. . // 7- - "- ". . . ­ .: 2003. ­ . 106-107.


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