On a new notion of the solution to an ill-posed problem ∗† 0 1 A.G. Ramm 0 2 Mathematics Department, Kansas State University, n Manhattan, KS 66506-2602, USA a J [email protected] 3 ] A N Abstract . h A new understanding of the notion of the stable solution to ill-posed problems t a is proposed. The new notion is more realistic than the old one and better fits m the practical computational needs. A method for constructing stable solutions in [ the new sense is proposed and justified. The basic point is: in the traditional 1 definition of the stable solution to an ill-posed problem Au = f, where A is a linear v 6 ornonlinearoperatorinaHilbertspaceH, itisassumedthatthenoisydata fδ,δ { } 6 are given, f f δ, and a stable solution u := R f is defined by the relation δ δ δ δ 3 || − || ≤ lim R f y = 0, where y solves the equation Au = f, i.e., Ay = f. In this 0 δ→0|| δ δ − || . definition y and f are unknown. Any f B(fδ,δ) can be the exact data, where 1 ∈ B(f ,δ) := f : f f δ . 0 δ δ { || − || ≤ } 0 The new notion of the stable solution excludes the unknown y and f from the 1 definition of the solution. : v i X 1 Introduction r a Let Au = f, (1.1) where A : H H is a linear closed operator, densely defined in a Hilbert space H. → Problem (1.1) is called ill-posed if A is not a homeomorphism of H onto H, that is, either equation (1.1) does not have a solution, or the solution is non-unique, or the solution does not depend on f continuously. Let us assume that (1.1) has a solution, possibly non-unique. Let N(A) be the null space of A, and y be the unique normal solution to ∗key words: ill-posed problems, regularizer, stable solution of ill-posed problems †AMS2010 subject classification: 47A52, 65F22, 65J20 (1.1), i.e., y N(A). Given noisy data f , f f δ, one wants to construct a stable δ δ ⊥ k − k ≤ approximation u := R f of the solution y, u y 0 as δ 0. δ δ δ δ k − k → → Traditionally(see, e.g., [2])onecallsafamilyofoperatorsR aregularizerforproblem h (1.1) (with not necessarily linear operator A) if a) R A(u) u as h 0 for any u D(A), h → → ∈ b) R f is defined for any f H and there exists h = h(δ) 0 as δ 0 such that h δ δ ∈ → → R f y 0 as δ 0. ( ) h(δ) δ k − k → → ∗ Inthisdefinitiony isfixedand( )mustholdforanyf B(f,δ) := f : f f δ . δ δ δ ∗ ∈ { k − k ≤ } In practice one does not know the solution y and the exact data f. The only available information is a family f and some a priori information about f or about the solution δ y. This a priori information often consists of the knowledge that y , where is a ∈ K K compactum in H. Thus y S := v : A(v) f δ, v . δ δ ∈ { k − k ≤ ∈ K} WeassumethattheoperatorAisknownexactly, andwealwaysassumethatf B(f,δ), δ ∈ where f = A(y). Definition: We call a family of operators R(δ) a regularizer if sup R(δ)f v η(δ) 0 as δ 0. (1.2) δ k − k ≤ → → v S ∈ δ There is a crucial difference between our new Definition (1.2) and the standard definition ( ): ∗ In ( ) u is fixed, while in (1.2) v is an arbitrary element of S and the supremum of δ ∗ the norm in (1.2) over all such v must tend to zero as δ 0. → The new definition is more realistic and better fits computational needs because not only the solution y to (1.1) satisfies the inequality Ay f δ, but any v S satisfies δ δ k − k ≤ ∈ this inequality Av f δ, v . The data f may correspond to any f = Av, where δ δ k − k ≤ ∈ K v S , and not only to f = Ay, where y is a solution of equation (1.1). Therefore it is δ ∈ more natural to use definition (1.2) than ( ). ∗ Our goal is to illustrate the practical difference between these two definitions, and to constructregularizerinthesense(1.2)forproblem(1.1)withanarbitrary, notnecessarily bounded, linear operator A, which is closed and densely defined in H. This is done in Section 2. In Section 1 this is done for a class of equations (1.1) with nonlinear operators A : X Y, where X and Y are Banach spaces. In this case we assume that → A1) A : X Y is a closed, nonlinear, injective map, f (A), (A) it is the range of → ∈ R R A, and A2) φ : D(φ) [0, ), φ(u) > 0 if u = 0, D(φ) D(A), the sets = := v : φ(v) c → ∞ 6 ⊆ K K { ≤ c are compact inX for everyc = const > 0,and ifv v,thenφ(v) liminf φ(v ). n n n } → ≤ →∞ The last inequality holds if φ is lower semicontinuous. In Hilbert spaces and in reflexive Banach spaces norms are lower semicontinuous. 2 Let us give some examples of equations for which assumptions A1) and A2) are satisfied. Example 1. A is a linear injective compact operator, f (A), φ(v) is a norm on ∈ R X X, where X is densely imbedded in X, the embedding i : X X is compact, 1 1 1 ⊂ → and φ(v) is lower semicontinuous. Example 2. A is a nonlinear injective continuous operator f (A), A 1 is not − ∈ R continuous, φ is as in Example 1. Example 3. A is linear, injective, densely defined, closed operator, f (A), A 1 is − ∈ R unbounded, φ is as in Example 1, X D(A). 1 ⊆ Let us demonstrate by Example A that a regularizer in the sense ( ) may be not a ∗ regularizer in the sense (1.2). In Example B a theoretical construction of a regularizer in the sense (1.2) is given for some equations (1.1) with nonlinear operators. In Section 2 a novel theoretical construction of a regularizer in the sense (1.2) is given for a very wide class of equations (1.1) with linear operators A. Example A: Stable numerical differentiation. In this Example the results from [3] - [11] are used. This Example is borrowed from [10]. Consider stable numerical differentiation of noisy data. The problem is: x Au := u(s)ds = f(x), f(0) = 0, 0 x 1. (1.3) Z ≤ ≤ 0 The data are: f and a constant M , which defines a compact , where f f δ, the δ a δ K k − k ≤ norm is L (0,1) norm, and consists of the L functions which satisfy the inequality ∞ ∞ K u M , a 0. The norm a a k k ≤ ≥ u(x) u(y) u := sup | − | + sup u(x) if 0 a 1, k ka x y a | | ≤ ≤ x,y [0,1] 0 x 1 x∈=y | − | ≤ ≤ 6 u(x) u(y) ′ ′ u := sup ( u(x) + u(x) )+ sup | − |, 1 < a 2. k ka | | | ′ | x y a 1 ≤ 0 x 1 x,y [0,1] − ≤ ≤ x∈=y | − | 6 If a > 1, then we define fδ(x+h(δ))−fδ(x−h(δ)), h(δ) x 1 h(δ), 2h(δ) ≤ ≤ − R(δ)fδ := fδ(x+hh((δδ)))−fδ(x), 0 ≤ x < h(δ), (1.4) fδ(x)−fδ(x−h(δ)), 1 h(δ) < x 1, h(δ) − ≤   where 1 h(δ) = caδa, (1.5) and c is a constant given explicitly (cf [4]). a 3 We prove that (1.4) is a regularizer for (1.3) in the sense (1.2), and := v : v a K { k k ≤ M , a > 1 . In this example we do not use lower semicontinuity of the norm φ(v) and a } do not define φ. Let S := v : Av f δ, v M . To prove that (1.4)-(1.5) is a regularizer δ,a δ a a { k − k ≤ k k ≤ } in the sense (1.2) we use the estimate δ sup R(δ)f v sup R(δ)(f Av) + R(δ)Av v +M ha 1(δ) δ δ a − k − k ≤ {k − k k − k} ≤ h(δ) ≤ v S v S ∈ δ,a ∈ δ,a caδ1−a1 := η(δ) 0 as δ 0. ≤ → → (1.6) Thus we have proved that (1.4)-(1.5) is a regularizer in the sense (1.2). If a = 1, and M < , then one can prove the following result: 1 ∞ Claim: There is no regularizer for problem (1.3) in the sense (1.2) even if the regu- larizer is sought in the set of all operators, including nonlinear ones. More precisely, it is proved in [5], p.345, (see also [8], pp 197-235, where the stable numerical differentiation problem is discussed in detail) that inf sup R(δ)f v c > 0, δ R(δ) v∈Sδ,1 k − k ≥ where c > 0 is a constant independent of δ and the infimum is taken over all operators R(δ) acting from L (0,1) into L (0,1), including nonlinear ones. ∞ ∞ On the other hand, if a = 1 and M < , then a regularizer in the sense ( ) does 1 ∞ ∗ exist, but the rate of convergence in (*) may be as slow as one wishes, if u(x) is chosen suitably (see [4], [8]). Example B: Construction of a regularizer in the sense (1.2) for some nonlinear equations. Assuming A1) and A2), let us construct a regularizer for (1.1) in the sense (1.2). We use the ideas from [10] and [11]. Define F (v) := Av f +δφ(v) and consider the minimization problem of finding δ δ k − k the infimum m(δ) of the functional F (v) on a set S : δ δ m(δ) := inf F (v), S := v : Av f δ, φ(v) c . (1.7) δ δ δ v S { k − k ≤ ≤ } ∈ δ Here = := v : φ(v) c . c K K { ≤ } The constant c > 0 can be chosen arbitrary large and fixed at the beginning of the argument, and then one can choose a smaller constant c , specified below. Since F (u) = 1 δ δ +δφ(u) := c δ, c := 1+φ(u), where u solves (1.1), one concludes that 1 1 m(δ) c δ. (1.8) 1 ≤ Let v be a minimizing sequence and F (v ) 2m(δ). Then φ(v ) 2c . By assumption j δ j j 1 ≤ ≤ A2), as j , one has: → ∞ v v , φ(v ) 2c . (1.9) j δ δ 1 → ≤ 4 Take δ = δ 0 and denote v := w . Then (1.9) and Assumption A2) imply the m → δm m existence of a subsequence, denoted again w , such that: m w w, A(w ) A(w), A(w) g = 0. (1.10) m m → → k − k Thus A(w) = g and, since A is injective by Assumption A1), it follows that w = u, where u is the unique solution to (1.1). Define now R(δ)f by the formula R(δ)f := v , where v is defined in (1.9). δ δ δ δ Theorem 1.1. R(δ) is a regularizer for problem (1.1) in the sense (1.2). Proof. Assume the contrary: sup R(δ)f v = sup v v γ > 0, (1.11) δ δ k − k k − k ≥ v S v S ∈ δ ∈ δ where γ > 0 is a constant independent of δ. Since φ(v ) 2c by (1.9), and φ(v) c, one δ 1 ≤ ≤ can choose convergent in X sequences w := v w˜, δ 0, and v v˜, such that m δm → m → m → w v γ, w˜ v˜ γ, and A(w˜) = g, A(v˜) = g. By the injectivity of A it follows k m− mk ≥ 2 k − k ≥ 2 that w˜ = v˜ = u. This contradicts the inequality w˜ v˜ γ > 0. This contradiction k − k ≥ 2 proves the theorem. The conclusions A(w˜) = g and A(v˜) = g, that we have used above, follow from the inequalities A(v ) f δ and A(v) f δ after passing to the limit δ 0, using δ δ δ k − k ≤ k − k ≤ → 2 assumption A2). 2 Construction of a regularizer in the sense (1.2) for linear equations If A is a linear closed densely defined in H operator, then T = A A is a densely defined ∗ selfadjoint operator. Let T := T + aI, where a = const > 0. The operator T 1A is a a− ∗ densely defined and closable. Its closure is a bounded operator, defined on all of H, and T 1A 1 . See [12]-[15] for details and other results. Let E be the resolution of the || a− ∗|| ≤ 2√a s identity of the selfadjoint operator T, dρ := d(Esy,y), and K := {u : 0∞s−2pdρ ≤ kp2}, where p (0,1) and k > 0 are constants. R p ∈ Our basic result is: Theorem 2.1. The operator R = T 1 A is a regularizer for problem (1.1) in the δ a−(δ) ∗ 2 sense (1.2) if limδ→0 a(δδ)1/2 = 0 and limδ→0a(δ) = 0. Moreover, if a(δ) = bpδ2p+1, then 2p sup R(δ)fδ y Cpδ2p+1, (2.1) k − k ≤ y , Ay f δ ∈K|| − δ||≤ where 1 Cp = 2 b +cpkpbpp, cp = pp(1−p)1−p, bp := (4pcpkp)−2p2+1. p p 5 The above choice of a(δ) is optimal in the sense that the right-hand side of (2.2) (see below) is minimal for this choice of a(δ). Proof. Let ǫ := sup T 1A f y := sup T 1A f y . || a− ∗ δ − || || a− ∗ δ − || y , Ay f δ ∈K|| − δ||≤ Then, with Ay = f, one has ǫ sup T 1A (f f) +sup T 1A Ay y := J +J , ≤ || a− ∗ δ − || || a− ∗ − || 1 2 where δ J , 1 ≤ 2√a and a2 J2 sup a2 T 1y 2 sup ∞ d(E y,y). 2 ≤ { || a− || } ≤ Z (s+a)2 s 0 Thus, asp J2 max 2k2 = c2k2a2p, 2 ≤ s 0 a+s p p p (cid:0) ≥ (cid:1) because max asp is attained at s = pa and is equal to c ap, where s 0 a+s 1 p p ≥ − c := pp(1 p)1 p, k2 := sup ∞s 2pd(E y,y). p − − p Z − s y 0 ∈K Consequently, J c k ap, 2 p p ≤ and δ ǫ +c k ap. (2.2) p p ≤ 2√a Minimizing the right-hand side of (2.2) with respect to a > 0, one obtains inequality (2.1). The minimizer of the right-hand side of (2.2) is 2 2 a = a(δ) = bpδ2p+1, bp := (4pcpkp)−2p+1, 2p and the minimum of the right-hand side of (2.2) is Cpδ2p+1, where 1 C := +c k bp. (2.3) p p p p 2 b p p 2 Theorem 2.1 is proved. 6 References [1] Dunford, N., Schwartz, J., Linear Operators, Interscience, New York, 1958. [2] V. Morozov, Methods of solving incorrectly posed problems, Springer Verlag, New York, 1984. [3] Ramm, A.G., On numerical differentiation, Mathematics, Izvestija vuzov, 11, (1968), 131-135. (In Russian). [4] Ramm, A.G., Stable solutions of some ill-posed problems, Math. Meth. in the appl. Sci. 3, (1981), 336-363. [5] Ramm, A.G., Scattering by obstacles, D.Reidel, Dordrecht, 1986, pp.1-442. [6] Ramm, A.G., Random fields estimation theory, Longman Scientific and Wi- ley, New York, 1990. [7] Ramm,A.G.,Inequalitiesforthederivatives,Math.Ineq.andAppl.,3,N1,(2000), 129-132. [8] Ramm, A.G., Dynamical systems method for solving operator equations, Elsevier, Amsterdam, 2007. [9] Ramm, A.G., Dynamical systems method for solving linear ill-posed problems, Ann. Polon. Math., 95, N3, (2009), 253-272. [10] Ramm, A.G., On a new notion of regularizer, J.Phys A, 36 (2003), 2191-2195. [11] Ramm, A.G., Regularization of ill-posed problems with unbounded operators, J. Math. Anal. Appl., 271, (2002), 447-450. [12] Ramm, A.G., Ill-posedproblemswithunboundedoperators, J.Math.Anal.Appl., 325, (2007), 490-495. [13] Ramm, A.G., Dynamical systems method (DSM) for selfadjoint operators, J. Math. Anal. Appl., 328, (2007), 1290-1296. [14] Ramm, A.G., Iterative solution of linear equations with unbounded operators, J. Math. Anal. Appl., 330, N2, (2007), 1338-1346. [15] Ramm, A.G., On unbounded operators and applications, Appl. Math. Lett., 21, (2008), 377-382. 7