Dynamical systems method for solving linear ill-posed problems

Transcription

1 ANNALES POLONICI MATHEMATICI * (2*) Dynamical sysems mehod for solving linear ill-posed problems by A. G. Ramm (Manhaan, KS) Absrac. Various versions of he Dynamical Sysems Mehod (DSM) are proposed for solving linear ill-posed problems wih bounded and unbounded operaors. Convergence of he proposed mehods is proved. Some new resuls concerning he discrepancy principle for choosing he regularizaion parameer are obained. 1. Inroducion. In his paper we presen par of he resuls from he auhor s invied plenary alk a he inernaional conference on mahemaical analysis and applicaions ICMAAS6, held in Egyp. This par deals wih linear ill-posed problems. Some of he ideas and resuls in his paper are aken from he papers of he auhor, cied in he bibliography, bu many resuls are new, including Theorems 2 8 and 1. The Dynamical Sysems Mehod (DSM) is developed in [7], [8], [9] [33]. The discrepancy principle was discussed earlier in [5]. Is analogs and new versions have been sudied recenly in [9] [11] and in [7], [8] for DSM. Consider an equaion (1.1) Au f =, where A is an operaor in a Banach space X. If A is a homeomorphism of X ono X (i.e., a coninuous injecive and surjecive map in X which has a coninuous inverse) hen problem (1.1) is called well-posed in he Hadamard sense. Oherwise i is called ill-posed. The DSM for solving equaion (1.1) consiss of solving he Cauchy problem (1.2) u = Φ(, u), u() = u, u = du d, 2 Mahemaics Subjec Classificaion: 47A52, 47B25, 65R3. Key words and phrases: ill-posed problems, dynamical sysems mehod (DSM), regularizaion parameer, discrepancy principle, unbounded operaors, linear operaor equaions. [1]

2 2 A. G. Ramm which we call a dynamical sysem, where Φ is chosen so ha he problem (1.2) has a unique soluion u(), defined for all, and such he limi u( ) := lim u() exiss and saisfies (1.3) A(u( )) = f. We do no assume ha he soluion o (1.1) is unique, bu we do assume ha i exiss. If (1.3) holds, hen we say ha DSM is jusified for solving equaion (1.1). There is a large body of lieraure on solving ill-posed problems (see, e.g., [2], [7], [39] and references herein). Variaional regularizaion, ieraive regularizaion, quasisoluions and quasiinversion are some of he mehods for sable soluion of ill-posed problems discussed in he lieraure. In his paper several new mehods for sable soluion of linear ill-posed problems are discussed. They are based on he Dynamical Sysems Mehod. This mehod has been developed fairly recenly for solving a wide variey of linear and nonlinear ill-posed problems [9] [38] alhough i was proposed already in [1] for solving well-posed problems. Because of space limiaions we will no discuss solving nonlinear illposed problems by he DSM, and refer he reader o [7], [8], [17], [33] [38]. Here we describe a new version of DSM for solving linear ill-posed problems. There are many pracical problems of his ype. We only menion solving ill-condiioned linear algebraic sysems and Fredholm equaions of he firs kind ([32], [33]). The novel poins in our resuls are no only he mehod of solving hese problems by DSM bu also he applicabiliy of he mehod o unbounded operaors ([2], [22] [23], [25]). In he lieraure, a widelydiscussed mehod for solving ill-posed problems (1.1) is he mehod of variaional regularizaion inroduced by Phillips [6] and sudied by Tikhonov [39], Morozov [5], Ivanov [2], Ramm ([7], [8]) and many oher auhors under he assumpion ha he operaor A in (1.1) is a linear bounded operaor. There are also some resuls on regularizaion of unbounded operaors (e.g. [4], [28] [2] [22], [25]). The variaional regularizaion mehod for sable soluion of (1.1) consiss in solving he problem (1.4) F (u) := Au f δ 2 + a u 2 = min, where a > is a consan, called a regularizaion parameer, f δ is he noisy daa, i.e., an elemen which saisfies he inequaliy f δ f δ and which is given ogeher wih he noise level δ >, while he exac daa f is no known. A sable soluion o (1.1) is an elemen u δ such ha lim δ u δ y =, where Ay = f and y is he unique minimal-norm soluion of he linear equaion (1.1). If X is a Hilber space H, which we assume below, hen he minimal-norm soluion is he soluion which is orhogonal o he null space N of A, N = N(A) = {u : Au = }. If he linear operaor A in (1.1) is

3 Dynamical sysems mehod 3 unbounded, hen we assume ha i is closed and densely defined, so ha is adjoin A is densely defined and closed (see, e.g., [3]). The DSM for such operaors is developed in [2] [22], [25], [28], [33], and [8]. If A is bounded, hen a necessary and sufficien condiion for u o be he minimizer of he quadraic funcional (1.4) is he Euler equaion (1.5) T a u = A f δ, T a := T + ai, T := A A, where I is he ideniy operaor and T is a selfadjoin operaor. Equaion (1.5) has a unique soluion u a,δ = Ta 1 A f δ. One can choose a = a(δ) so ha lim δ a(δ) = and u δ := u a(δ),δ is a sable soluion o (1.1): (1.6) lim δ u δ y =, where Ay = f and y N. There are a priori choices of a(δ) and a poseriori ones. An a priori choice is based on he esimae (1.7) where T 1 a A f δ y Ta 1 A (f δ f) + Ta 1 A f y δ 2 a + η(a), (1.8) η 2 (a) = T 1 a T y y 2 = a 2 d(e s y, y) (s + a) 2 P Ny 2 as a, and P N is he orhoprojecor ono N. Since we assume ha y N, equaion (1.8) implies (1.9) lim a η(a) =. The erm δ/(2 a) in (1.7) appears due o he esimae (1.1) T 1 a A = A Q 1 a = UQ 1/2 Q 1 Q 1/2 Q 1 a = sup s a s s + a = 1 2 a. Here Q := AA, U is a parial isomery, Q a := Q + ai, and we have used he formula (1.11) Ta 1 A = A Q 1 a, he polar decomposiion A = UQ 1/2, and he specral heorem for he selfadjoin operaor Q, namely g(q) = sup s g(s). Formula (1.11) is obvious if A is bounded: muliply (1.11) by T a on he lef and hen by Q a on he righ, and ge A (AA + ai) = (A A + ai)a, which is an obvious ideniy. Since he operaors Q a and T a are boundedly inverible, one may reverse seps and ge (1.11). Thus a priori choices of a(δ), which imply (1.6),

4 4 A. G. Ramm are δ (1.12) lim =, δ a(δ) lim a(δ) =. δ There are many funcions a(δ) saisfying (1.12). One can find an opimal value a(δ) by minimizing he righ-hand side of (1.7) wih respec o a. Alernaively, one may calculae a(δ) by solving he equaion δ = 2 a(δ) η(a) for a for a fixed small δ >. If A is closed, densely defined in H, unbounded, and a = cons >, hen he auhor has proved in [22] ha he operaor Ta 1 A, wih he domain D(A ), is closable, is closure, denoed again Ta 1 A, is a bounded operaor defined on all of H, Ta 1 A 1/(2 a), and (1.1) holds. For convenience of he reader le us skech he proof of hese claims. To check ha Ta 1 A is closable, one akes h n D(Ta 1 A ) = D(A ) such ha h n and Ta 1 A h n g as n, and checks ha g =. Indeed, le u H be arbirary. Then (1.13) (g, u) = lim n 1 (Ta A h n, u) = lim n (h n, AT 1 a u) =. Since u is arbirary, his implies g =, as claimed. Noe ha Ta 1 u D(A), so ha he above calculaion is jusified. If one drops he index n and he lim in (1.13), hen one can see ha he adjoin o he closure of Ta 1 A is he operaor ATa 1, defined on all of H and bounded: ATa 1 = UT 1/2 Ta a. Since A = A, one ges T 1 a A 1/(2 a). Finally, formula (1.11) can be proved for an unbounded closed, densely defined operaor A as above, if one checks ha he operaor A AA is densely defined. This is indeed he case, because he operaor A AA A = T 2 is densely defined if T is, and D(T 2 ) D(A AA ). Le us now describe an a poseriori choice of a(δ) which implies (1.6) and which is called he discrepancy principle. This principle was discussed in [5], [7]. I consiss in finding a(δ) from he equaion (1.14) Au a,δ f δ = Cδ, 1 < C < 2, where C = cons, f δ > Cδ and u a,δ = Ta 1 A f δ. One can prove (see e.g. [7]) ha equaion (1.14) for a small fixed δ > has a unique soluion a(δ) wih lim δ a(δ) = and u δ = u a(δ),δ saisfies (1.6), i.e. u δ is a sable soluion o (1.1). To prove hese claims one denoes by P N he orhogonal

5 Dynamical sysems mehod 5 projecion ono a subspace N, and wries (1.14) as (1.15) C 2 δ 2 = [ATa 1 A I]f δ 2 = [QQ 1 a I]f δ 2 a 2 d(e δ f δ, f δ ) = (s + a) 2 =: h(a, δ), and akes ino accoun ha h(a, δ) is, for a fixed δ >, a coninuous monoone funcion of a, wih h(, δ) = f δ 2 > C 2 δ 2 and h(+, δ) = P N f δ δ 2, so ha here exiss a unique a = a(δ) such ha h(a(δ), δ) = C 2 δ 2. Here we have se N := N(A ) and used he obvious relaion N(Q) = N(A ), he inequaliy P N f δ P N (f δ f) + P N f f δ f = δ, and he relaion P N f =. This las relaion follows from he relaions f R(A) and R(A) N. Le us now check ha if a(δ) solves (1.5) hen u δ = u a(δ),δ saisfies (1.6). One has F (u δ ) F (y), so (1.16) Au δ f δ 2 + a(δ) u δ 2 δ 2 + a(δ) y 2. Since Au δ f δ 2 = C 2 δ 2 > δ 2 and a(δ) >, one ges (1.17) u δ y. Therefore one can selec a weakly convergen sequence u n = u δn u as n. Le us prove ha u = y and lim n u n y =. Since his holds for any subsequence, i will hen follow ha (1.6) holds. To prove ha u = y noe ha (1.17) implies u y, and ha u solves (1.1). Since he minimal-norm soluion o (1.1) is unique, i follows ha u = y. To check ha u solves (1.1) we noe ha lim δ Au δ f =, as follows from (1.16) because lim δ a(δ) =. The relaions u δ u and Au δ f as δ imply Au = f and lim δ u δ u =. Indeed, le us firs check ha Au = f. One has (f, g) = lim δ (Au δ, g) = (u, A g) g D(A ). Thus u D(A) and Au = f, as claimed. As proved above, his implies ha u=y. Therefore u δ y and u δ y. This implies ha lim δ u δ y =. Indeed, (1.18) u δ y 2 = u δ 2 + u 2 2 Re (u δ, y) 2 y 2 2 Re (u δ, y) as δ. Thus, he relaion (1.6) is proved for he choice of a(δ) by he discrepancy principle. The drawback of he a priori choice of a(δ) is ha i is nonunique and alhough i guaranees convergence (1.16), he error of he mehod can be large if δ > is fixed. The drawback of he discrepancy principle is he

6 6 A. G. Ramm necessiy of solving he nonlinear equaion (1.14) and also a possible large error for a fixed δ. In Secion 2 we discuss he DSM for solving linear equaions (1.1). 2. DSM for solving linear problems. We assume firs ha he linear closed densely defined in H operaor in (1.1) is selfadjoin, A = A. This is no an essenial resricion: every solvable linear equaion (1.1) is equivalen o he equaion T u = A f, where T = T = A A. Indeed, if Au = f, hen applying A and assuming f D(A ), one ges T u = A f. Conversely, if T u = A f and f = Ay, hen T u = T y. Muliply he equaion = T (u y) by u y o ge = (A A(u y), u y) = Au Ay 2. Thus Au = Ay = f. If A is bounded, hen f D(A ) for any f H. If A is unbounded, hen D(A ) is a linear dense subse of H. In his case, if f D(A ), hen we define a soluion of he equaion T u = A f by he formula u = lim a Ta 1 A f. As we have proved in Secion 1, for any f R(A) his limi exiss and equals he minimal-norm soluion y: lim a Ta 1 A Au = y if Au = f. This is rue because lim a Ta 1 T u = u P N u = y. The DSM for solving equaion (1.1) wih a linear selfadjoin operaor can be consruced as follows. Consider he problem (2.1) u a = i(a + ia)u a if, u() = ; u = du d, where a = cons >. Our firs resul is formulaed as Theorem 1. Theorem 1. If Ay = f and y N, hen (2.2) lim a lim u a () = y. Our second resul shows ha he mehod, based on Theorem 1, gives a sable soluion of he equaion Au = f. Assume ha f δ f δ, and le u a,δ () be he soluion o (2.1) wih f δ in place of f. Theorem 2. There exis = δ wih lim δ δ =, and a = a(δ) wih lim δ a(δ) =, such ha u δ := u a(δ),δ ( δ ) saisfies (1.6). We will discuss he ways o choose a(δ) and δ afer he proofs of hese heorems are given. From he numerical poin of view, if one inegraes problem (2.1) wih he exac daa f on he inerval T, and T is fixed, hen one is ineresed in choosing a = a(t ) such ha lim T u a(t ) (T ) y =. We will give such a choice of a(t ). Before we sar proving hese wo heorems, le us explain he ideas of he proof. Suppose B is a linear operaor and is inverse B 1 exiss and is bounded.

7 Finally, assume ha (2.3) lim e B =. Dynamical sysems mehod 7 This will happen, for example, if Re B c, where c > is a consan. Under hese assumpions one has (2.4) e Bs ds = B 1 (e B I), lim e Bs ds = B 1. The operaor ebs ds solves he problem (2.5) Ẇ = BW + I, W () =, where I is he ideniy operaor. If lim e B =, hen (2.6) lim W () = B 1. The basic idea of he DSM is he represenaion of he inverse operaor as he limi as of he soluion o he Cauchy problem (2.5). Proof of Theorem 1. The soluion o (2.1) is u a () = e i(a+ia)( s) (if) ds = [i(a + ia)] 1 (e i(a+ia) I)( if). Since e i(a+ia) = e a as, one ges (2.7) lim u a () = (A + ia) 1 f. Since f = Ay, one has (2.8) η(a) := (A + ia) 1 Ay y = a (A + ia) 1 y as a. The las relaion follows from he assumpion y N. Indeed, by he specral heorem one has lim a η2 (a) = lim a 2 (A + ia) 1 y 2 a 2 = lim a a s 2 + a 2 d(e sy, y) Theorem 1 is proved. = (E E )y 2 = P N y 2 =. Remark 1. In a numerical implemenaion of Theorem 1 one chooses τ and a = a(τ), and inegraes (2.1) on he inerval [, τ]. One chooses τ so ha (2.9) lim τ u a(τ) y =.

8 8 A. G. Ramm In order o choose a(τ), noe ha u a () u a ( ) e a /a, as follows from he derivaion of (2.7). Therefore, by Theorem 1, he relaion (2.9) holds if e a(τ)τ (2.1) lim τ a(τ) =, lim a(τ) =. τ For example, one may ake a(τ) = τ γ, where < γ < 1 is a consan. Proof of Theorem 2. Le us sar wih he formula (2.11) u a,δ () = e i(a+ia)( s) ( if δ ) ds = [i(a + ia)] 1 (e i(a+ia) I)(if δ ). Thus (2.12) E := u a,δ () y u a,δ () u a () + u a () y. One has (2.13) u a,δ () u a () f δ f a and I e ia a 2δ a, (2.14) u a () y e a f + η(a), a where η(a) is defined in (2.8), lim a η(a) =. Since f f δ + δ c, one obains from (2.12) (2.14): (2.15) lim δ E = provided ha = δ, a = a(δ) and (2.16) lim δ δ =, Theorem 2 is proved. lim a(δ) =, δ e a(δ)δ lim δ a(δ) δ =, lim δ a(δ) =. Remark 2. There are many choices of δ and a(δ) saisfying relaions (2.16). If one has an esimae of he rae of decay of η(a) as a, hen one may obain some rae of convergence of E o zero as δ. However, i is impossible, in general, o ge a rae of decay of η(a) as a wihou addiional assumpions on he daa f or on he soluion y. A ypical assumpion is y = Az, ha is, y R(A). If fracional powers of A are defined (which is he case when A, for example) hen one may assume y = A γ z, γ >. Le us show how o ge he rae of decay of η(a) under such assumpions. Assume, for example, ha y = Az. Then (2.17) η 2 (a) = a 2 s 2 a 2 + s 2 d(e sz, z) a 2 z 2,

9 Dynamical sysems mehod 9 and he error bound is (2.18) E 2δ a + c e a 1 a + c 2a, c 2 = z, c 1 = f δ + δ. Choose, for example, a = δ γ, < γ < 1, and δ = δ µ, µ > γ. Then lim δ e a(δ) δ/a(δ) = and (2.15) holds wih he rae δ ν, ν = min(1 γ, γ). If γ = 1/2 hen max <γ<1 min(1 γ, γ) is equal o 1/2, and for γ = 1/2 one ges E = O(δ 1/2 ) if δ = δ µ wih µ > 1/2. 3. Second version of he DSM. Consider problem (2.1) wih a = a(). Le us assume ha (3.1) < a(), a + a 2 L 1 (, ), a(s) ds =. The soluion o his problem is (3.2) u() = e ia( s) s a(p) dp ds ( if). Theorem 3. Under he above assumpions one has (3.3) lim u() y =. Proof. Since f = Ay, inegraing by pars one ges u() = e ia e ias s a dp y e ia( s) a(s)e s a(p) dp ds y. Thus (3.4) u() = y e ia a dp y e ia( s) a(s)e s a dp dsy, and (3.5) u() y e a dp y + e ia( s) a(s)e s a dp ds y =: J 1 + J 2. By he las assumpion of (3.1) one ges (3.6) lim J 1 =. We now prove ha (3.7) lim J 2 =. Using he specral heorem, one ges (3.8) J2 2 = d(e λ y, y) e iλ( s) a(s)e s a dp ds 2.

10 1 A. G. Ramm Le us prove ha (3.9) lim e iλ( s) a(s)e s a dp ds = λ. From (3.8), (3.9) and he assumpion y N, he conclusion (3.7) follows. To verify (3.9), inegraing by pars one ges (3.1) J 3 := e iλ( s) a(s)e s a dp ds Thus = eiλ( s) iλ a(s)e s a dp + 1 iλ e iλ( s) [a (s) + a 2 (s)]e s a(p) dp ds. (3.11) J 3 = a() iλ + eiλ a dp a() + J 4, λ, iλ where J 4 denoes he las inegral in (3.1). The firs wo erms in (3.11) end o zero as because of he assumpions abou a(). Assumpions (3.1) imply ha (3.12) lim J 4 =. Thus, Theorem 3 is proved. Remark 3. For example, he funcion a()=c /(c 1 +) b, where c, c 1 > are arbirary consans and b (1/2, 1) is a consan, saisfies assumpions (3.1). Le us prove ha Theorem 3 yields a sable soluion o equaion (1.1). Theorem 4. There exiss a sopping ime δ, wih lim δ δ =, such ha (1.6) holds wih u δ = u δ ( δ ), where u δ () is he soluion o problem (2.1) wih a = a() and f δ in place of f, wih f δ f δ. Proof. One has (3.13) u δ () y u δ () u() + u() y, where u() solves problem (2.1) wih a = a() and exac daa. We have proved in Theorem 3 ha (3.14) lim u() y =. We have (3.15) u δ () u() e s a(p) dp ds f δ f δ a().

11 Dynamical sysems mehod 11 Here he esimae e s a dp ds 1/a() was used, which is derived easily: e s a dp ds 1 a(s)e s a dp dp = 1 a() a() e s a dp Choose δ so ha (3.16) lim δ δ =, = 1 a() (1 e a dp ) 1 a(). lim δ δ a( δ ) =. This is obviously possible. Then (3.13) (3.15) imply (3.17) lim δ u δ ( δ ) y =. Theorem 4 is proved. 4. Third version of DSM for solving equaion (1.1) (4.1) u = u + T 1 a() A f, u() =, where f = Ay, y N, and a() > is a monoonically decaying coninuous funcion such ha lim a() = and a() d =. We could ake he iniial condiion u() = u arbirary. The conribuion o he soluion of problem (4.1) which comes from he iniial condiion u is he erm u e. I decays exponenially fas and our argumens do no depend on his erm essenially. To simplify and shoren our argumen we ake u =. Theorem 5. Under he assumpions of Theorem 3 he soluion u() o problem (4.1) exiss, is unique, is defined for all, and lim u() = y, where y is he minimal-norm soluion o (1.1). Proof. One has u() = e ( s) T 1 a(s) A Ay ds = e ( s) y ds Thus (4.2) u() y e y + e ( s) a(s) T 1 a(s) y ds. e ( s) a(s)t 1 a(s) y ds. One can easily check ha if b(s) is a coninuous funcion on [, ) and b( ) = lim s b(s) exiss, hen (4.3) lim e ( s) b(s) ds = b( ).

12 12 A. G. Ramm Thus, Theorem 5 will be proved if one checks ha (4.4) lim a Ta 1 y = y H. a To prove (4.4) one wries, using he specral heorem, (4.5) a 2 T 1 a y 2 = Theorem 5 is proved. a 2 (s + a) 2 d(e sy, y) = P N y 2 =. Le us prove ha he DSM mehod (4.1) yields a sable soluion o problem (1.1). Assume ha f is replaced by f δ, wih f δ f δ, in equaion (4.1), and denoe by u δ () he corresponding soluion. Then, using he esimae (3.13), one ges (4.6) lim u δ ( δ ) y = δ provided ha (4.7) lim δ δ =, lim δ δ a(δ ) =. To check he sufficiency of he second condiion of (4.7) for (4.6) o hold, one proceeds as follows: u δ () u() = e ( s) T 1 (4.8) a(s) A (f δ f) ds δ e ( s) 1 2 a(s) ds δ 2 a(). Here we have used he monooniciy of a(), which implies a() a(s) if s, and he esimae Ta 1 A 1/(2 a), which was proved earlier. Le us sae he resul we have proved. Theorem 6. If δ is chosen so ha (4.7) holds, hen he soluion u δ () o problem (4.1) wih noisy daa f δ in place of f saisfies (4.6). 5. A new discrepancy principle. The usual discrepancy principle is described in he inroducion. I requires solving nonlinear equaion Au a,δ f = Cδ wih C = cons, 1 < C < 2, where u a,δ = Ta 1 A f δ. Thus one has o know he exac minimizer u a,δ of he funcional F (u) = Au f δ 2 + a u 2, or he exac soluion of he equaion T a u = A f δ. In his secion we discuss he following quesion: How does one formulae he discrepancy principle in he case when u a,δ is no he exac soluion of he minimizaion problem F (u) = min, bu an approximae soluion?

13 Dynamical sysems mehod 13 Le us sae he resul. Theorem 7. Assume ha A is a bounded linear operaor in a Hilber space H, ha f = Ay, y N, f δ f δ, f δ > Cδ, C = cons, C (1, 2), and u a,δ is any elemen which saisfies he inequaliy (5.1) F (u a,δ ) m + (C 2 1 b)δ 2, where (5.2) F (u) = Au f δ 2 + a u 2, m = inf u H F (u), b = cons > and C 2 > 1 + b. Then he equaion (5.3) Au a,δ f δ = Cδ has a soluion for any fixed δ >, wih lim δ a(δ) =, and (5.4) lim δ u δ y =, where u δ = u a(δ),δ and a(δ) solves (5.3). Proof. To prove he exisence of a soluion o (5.3), we denoe Au a,δ f δ by h(δ, a), check ha h(δ, +) < Cδ, h(δ, ) > Cδ, and noe ha h(δ, a) is a coninuous funcion of a on he inerval (, ). This implies he exisence of a soluion a = a(δ) o equaion (5.3). As a, one has a u a,δ 2 F (u a,δ ) m + (C 2 1 b)δ 2 F () + (C 2 1 b)δ 2, so, wih c := F () + (C 2 1 b)δ 2, one obains Thus u a,δ c/ a as a. (5.5) h(δ, ) = A f δ = f δ > Cδ. As a, one has Since one has and h 2 (δ, a) F (u a,δ ) m + (C 2 1 b)δ 2 F (y) + (C 2 1 b)δ 2. F (y) = δ 2 + a y 2, h 2 (δ, a) (C 2 b)δ 2 + a y 2, (5.6) h(δ, +) (C 2 b) 1/2 δ < Cδ. Finally, he coninuiy of h(δ, a) wih respec o a (, ) for any fixed δ > follows from he coninuiy of he bounded operaor A and he coninuiy of u a,δ wih respec o a (, ). Thus, he exisence of a soluion a = a(δ) > of equaion (5.3) is proved. One akes a soluion for which lim δ a(δ) =.

14 14 A. G. Ramm Such a soluion exiss because h(δ, +) and m = m(δ, a) end o zero as δ and a. Le us prove (5.4). One has (5.7) F (u δ ) = Au δ f δ 2 + a(δ) u δ 2 Ay f δ 2 + a(δ) y 2 δ 2 + a(δ) y 2. Since Au δ f δ = Cδ, C > 1 i follows from (5.7) ha (5.8) u δ y. Thus one may assume ha u δ u as δ. Le us prove ha Au = f. Firs, we observe ha so Au δ f Au δ f δ + f δ f Cδ + δ, (5.9) lim δ Au δ f =. Secondly, for any v H we have (5.1) (f Au, v) = lim (Au δ Au, v) = lim (u δ u, A v) =, δ δ because u δ u. Since v is arbirary, one concludes from (5.1) ha Au = f. From (5.8) i follows ha u y. As he minimal-norm soluion o he equaion Au = f is unique, one obains u = y. Thus, u δ y and u δ y. This implies (5.4), as follows from (1.18). Theorem 7 is proved. 6. Discrepancy principle does no yield uniform convergence wih respec o he daa. In his secion we make he following assumpion. Assumpion A. A is a linear bounded operaor in a Hilber space H, N := N(A) = N(A ) := N = {}, A 1 is unbounded, Ay = f, f δ f δ, f δ > δ. Le a = a(δ) be chosen by he discrepancy principle, (6.1) Au a,δ f δ = Cδ, u δ = u a(δ),δ = T 1 a(δ) A f δ. Consider he se S δ := {v : Av f δ δ}. We are ineresed in he following quesion: given {f δ } δ (,δ ), where δ > is a small number, and assuming ha a(δ) is he soluion o (6.1), can one guaranee uniform convergence wih respec o he daa f? In oher words, is i rue ha (6.2) lim δ sup v S δ u δ v =? The answer is no.

15 Theorem 8. There exis f δ such ha (6.3) lim δ sup v S δ u δ v c >, Dynamical sysems mehod 15 c = cons. Proof. Se Ta 1 A =: G, u δ = Gf δ, G = 1/(2 a). We have proved in (1.1) ha G 1/2 a, bu in fac equaliy holds because in (1.1), U is uniary under Assumpion A. Thus, one can find an elemen p = p a wih p = δ/2 such ha (6.4) Gp 1 δ G p = 2 8 a. Assumpion A implies ha he ranges R(A) and R(T ) are dense in H. Thus one can find an elemen z = z a,δ such ha (6.5) f δ AT b z p δ/8, b (, 1), b = cons. For any v one has (6.6) Gf δ v Gf δ GAv + GAv v. Take v = T b z and le M > be an arbirarily large fixed consan. Then (6.7) lim sup δ v S δ, v=b b z, z M GAv v =, because (6.8) GAv v = Ta 1 T v v = at 1 T b s b z = a sup s s + a = cab, where c = b b (1 b) 1 b. From (6.7) and (6.6) one sees ha (6.9) lim if and only if (6.1) lim sup δ v S δ, v=t b z, z M sup δ v S δ, v=t b z, z M Take z = z a, v = T b z. Then Gf δ v = Gf δ GAv =. (6.11) f δ Av p + f δ AB b z p δ/2 + δ/8 = 5δ/8 δ, and, using (6.5), one ges (6.12) Gf δ GAv = G(f δ Av p) + Gp Gp G δ 8 δ ( 1 a 8 1 ) = δ a. If δ/ a c >, hen, according o (6.12), (6.1) fails. Le us find f δ such ha for a = a(δ), defined by he discrepancy principle, one has δ/ a c >. This will complee he proof of Theorem 8. Le us

16 16 A. G. Ramm assume for simpliciy ha A = A > is compac. Then T = A A = A 2, and equaion (6.1) becomes (6.13) C 2 δ 2 = [A(A 2 + a) 1 A I]f δ 2 [ λ 2 ] 2 j = λ 2 j + a 1 f δj 2 = j=1 j=1 a 2 f δj 2 (λ j + a) 2. Here λ j are he eigenvalues of A 2, f δj = (f δ, ϕ j ), A 2 ϕ j = λ j ϕ j, ϕ j = 1. Assume, for example, ha λ j = 1/j and f δj 2 = 1/j 2. Then (6.13) becomes (6.14) C 2 δ2 a 2 = j 2 (j 1 =: I(a). + a) 2 Noe ha where One has I 1 (a) = 1 j=1 I(a) I 1 (a) as a, I 1 (a) := x 2 1 (x 1 + a) 2 dx = = a 1 [ 1 a a x 2 (x 1 + a) 2 dx. ds (s + a) 2 = (s + a) 1 1 ] = 1 [1 + O(a)], a. a This and (6.14) imply δ/ a c > as δ. Theorem 8 is proved. 7. Ieraive processes for solving equaion (1.1). In his secion convergen ieraive processes for solving equaion (1.1) are consruced in he case when A is a closed, densely defined, unbounded operaor in H. Consider he process (7.1) u n+1 = Bu n + T 1 a A f, u 1 N, B := at 1 a, where a = cons > and he iniial elemen u 1 is arbirary in he subspace N, where N := N(A) = N(T ), T = A A, T a = T + ai. Noe ha B and B 1. Theorem 9. Under he above assumpions one has (7.2) lim n u n y =. Proof. Le w n = u n y. Then (7.3) w n+1 = Bw n = B n w, w := u 1 y, w N.

17 Le us prove ha Dynamical sysems mehod 17 (7.4) lim n Bn w =. If (7.4) is verified, hen Theorem 9 is proved. We have (7.5) B n w 2 = a 2n (a + s) 2n d(e sw, w) = s>b + s b =: J 1 + J 2, where E s is he resoluion of he ideniy corresponding o he operaor T, and b > is a small number which will be chosen laer. For any fixed b > one has lim n J 1 = because a/(a + s) a/(a + b) < 1 if s b. b d(e sw, w) = because On he oher hand, J 2 b d(e sw, w), and lim b w N = E H. Therefore, given an arbirary small number η > one can choose b > such ha J 2 η/2. Fix such a b and choose n sufficienly large so ha J 1 η/2. Then B n w 2 η. Since η is arbirarily small, we have proved (7.4). Theorem 9 is proved. Remark 4. The ieraive process (7.1) yields a sable soluion of equaion (1.1). Indeed, le f δ be given wih f δ f δ, and le u n,δ be defined by (7.1) wih f δ in place of f. Le w n,δ = u n,δ y. Then so (7.6) w n+1,δ = w n+1,δ = Bw n,δ + Ta 1 A (f δ f), n B j T 1 A (f δ f) + B n (u 1 y), (u 1 y) N. j=1 We have proved above ha (7.7) B n (u 1 y) =: E(n) as n. One has (7.8) n B j Ta 1 A (f δ f) j= (n + 1)δ 2 a, because B 1 and Ta 1 A 1/(2 a). From (7.7) and (7.8) one finds he sopping rule, i.e., he number n(δ) such ha lim δ w n(δ),δ =. This n(δ) is found for any fixed small δ as he minimizer for he problem (7.9) (n + 1)δ 2 a + E(n) = min. Alernaively, one can find n 1 (δ) from he equaion (7.1) E(n) = (n + 1)δ 2 a. Clearly n(δ) and n 1 (δ) end o as δ.

18 18 A. G. Ramm 8. Discrepancy principle for DSM. In his secion we formulae and jusify a discrepancy principle for DSM. Le us sar wih he version (4.1). We assume ha a() > is a monoonically decaying wice coninuously differeniable funcion, lim [a() + ȧ + ä] =, ä >, and lim ȧ()/a() =, for example, a() = c 1 /(c + ) b, where c 1, c and b are posiive consans, wih b (.5, 1). For his a() all he assumpions (3.1) hold. Theorem 1. The equaion (8.1) AT 1 a() A f δ f δ = Cδ, C = cons, 1 < C < 2, has a soluion = δ, wih lim δ δ =, such ha (4.6) holds, where u δ () is he soluion o (4.1) wih f δ in place of f, and f δ > Cδ. Proof. We have proved earlier ha equaion (8.1) has a unique soluion a = a δ and lim δ a δ =. If a() is a monoonically decaying funcion such ha lim a() =, hen he equaion a δ = a() uniquely defines = δ such ha a( δ ) = a δ, and lim δ δ =. Le us skech he proof of (4.6), where u δ ( δ ) = δ e ( δ s) T 1 a(s) A f δ ds and δ as δ. We have proved (cf. (1.17)) ha T 1 a( δ ) A f δ y, and (8.2) lim T 1 δ a( δ ) A f δ y =. (cf (1.18)). I is clear ha lim e ( s) g(s) ds = g( ) provided ha g is a coninuous funcion and g( ) := lim g() exiss. Noe ha lim s δ T 1 a( δ ) A f δ T 1 a(s) A f δ =. We have (8.3) δ e ( δ s) T 1 a(s) A f δ ds = y + o(1) as δ. To check his we use equaion (8.2), he formula T 1 a( δ ) T 1 a(s) = T 1 a(s) [a( δ) a(s)]t 1 a( δ ), he esimaes T 1 a( δ ) A f δ y and T 1 a(s) 1/a(s), and he relaion ( s) a(s) a() lim e ds =, a(s) which holds due o our assumpions on a(). Theorem 1 is proved. References [1] M. Gavurin, Nonlinear funcional equaions and coninuous analogs of ieraive mehods, Izv. Vyssh. Uchebn. Zaved. Ma. 5 (1958), (in Russian).

19 Dynamical sysems mehod 19 [2] V. Ivanov, V. Vasin and V. Tanana, Theory of Linear Ill-Posed Problems, Nauka, Moscow, 1978 (in Russian). [3] T. Kao, Perurbaion Theory for Linear Operaors, Springer, New York, [4] O. Liskoves, Regularizaion of equaions wih a closed linear operaor, Differenial Equaions 6 (197), [5] V. Morozov, Mehods of Solving Incorrecly Posed Problems, Springer, New York, [6] D. Phillips, A echnique for numerical soluion of cerain inegral equaions of he firs kind, J. Assoc. Compu. Mach. 9 (1962), [7] A. G. Ramm, Inverse Problems, Springer, New York, 25. [8], Dynamical Sysems Mehod for Solving Operaor Equaions, Elsevier, Amserdam, 27 [9], On he discrepancy principle, Nonlinear Func. Anal. Appl. 8 (23), [1], Discrepancy principle for he dynamical sysems mehod, Comm. Nonlinear Sci. Numer. Simulaion 1 (25), [11], A new discrepancy principle, J. Mah. Anal. Appl. 31 (25), [12], Dynamical sysems mehod for solving nonlinear operaor equaions, In. J. Appl. Mah. Sci. 1 (24), [13], Dynamical sysems mehod for solving operaor equaions, Comm. Nonlinear Sci. Numer. Simulaion 9 (24), [14], Inequaliies for soluions o some nonlinear equaions, Nonlinear Func. Anal. Appl. 9 (24), [15], Dynamical sysems mehod and surjeciviy of nonlinear maps, Comm. Nonlinear Sci. Numer. Simulaion 1 (25), [16], DSM for ill-posed equaions wih monoone operaors, ibid. 1 (25), [17], Dynamical sysems mehod (DSM) and nonlinear problems, in: Specral Theory and Nonlinear Analysis, J. Lopez-Gomez (ed.), World Sci., Singapore, 25, [18], Dynamical sysems mehod for nonlinear equaions in Banach spaces, Comm. Nonlinear Sci. Numer. Simulaion 11 (26), [19], Dynamical sysems mehod and a homeomorphism heorem, Amer. Mah. Monhly 113 (26), [2], Dynamical sysems mehod (DSM) for unbounded operaors, Proc. Amer. Mah. Soc. 134 (26), [21], Dynamical sysems mehod (DSM) for selfadjoin operaors, J. Mah. Anal. Appl. 328 (27), [22], On unbounded operaors and applicaions, Appl. Mah. Le. 21 (28), [23], Ill-posed problems wih unbounded operaors, J. Mah. Anal. Appl. 325 (27), [24], Two resuls on ill-posed problems, In. J. Appl. Mah. Sais. 11 (27), [25], Ieraive soluion of linear equaions wih unbounded operaors, J. Mah. Anal. Appl. 33 (27), [26], On a new noion of regularizer, J. Phys. A 36 (23), [27], Linear ill-posed problems and dynamical sysems, J. Mah. Anal. Appl. 258 (21), [28], Regularizaion of ill-posed problems wih unbounded operaors, ibid. 271 (22), [29], A DSM proof of surjeciviy of monoone nonlinear mappings, Ann. Polon. Mah. 95 (28), [3], DSM for general nonlinear equaions, Nonlinear Anal. 69 (28),

20 2 A. G. Ramm [31], Discrepancy principle for DSM II, Comm. Nonlinear Sci. Numer. Simulaion 13 (28), [32] A. F. Ramm and N. S. Hoang, Solving ill-condiioned linear algebraic sysems by he dynamical sysems mehod, Inverse Problems Sci. Engrg. 16 (28), [33],, Dynamical sysems mehod for solving linear finie-rank operaor equaions, Ann. Polon. Mah. 95 (29), [34],, A new version of he Dynamical Sysems Mehod (DSM) for solving nonlinear equaions wih monoone operaors, Differenial Equaions Appl. 1 (29), [35],, A nonlinear inequaliy, J. Mah. Inequal. 2 (28), [36],, Dynamical Sysems Gradien mehod for solving nonlinear equaions wih monoone operaors, Aca Appl. Mah., o appear. [37],, An ieraive scheme for solving nonlinear equaions wih monoone operaors, BIT 48 (28), [38],, A discrepancy principle for equaions wih monoone coninuous operaors, Nonlinear Anal., o appear. [39] A. Tikhonov, A. Leonov and A. Yagola, Nonlinear Ill-Posed Problems, Chapman and Hall, London, Mahemaics Deparmen Kansas Sae Universiy Manhaan, KS , U.S.A. ramm@mah.ksu.edu Received and in final form (191)