Regularization for a Common Solution of a System of Ill-Posed Equations Involving Linear Bounded Mappings 1

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1 Applied Mathematical Sciences, Vol. 5, 2011, no. 76, Regularization for a Common Solution of a System of Ill-Posed Equations Involving Linear Bounded Mappings 1 Nguyen Buong and Nguyen Dinh Dung Vietnamese Academy of Science and Technology Institute of Information Technology 18, Hoang Quoc Viet, Cau Giay, Ha Noi, Vietnam nbuong@ioit.ac.vn Abstract. The purpose of this paper is to give a regularization method for solving a system of ill-posed equations involving linear bounded mappings in real Hilbert spaces and an example of finding a common solution of two systems of linear algebraic equations with singular matrices. Mathematics Subect Classification: 47H17 Keywords: Hilbert spaces, Tikhonov regularization, singular matrix 1. INTRODUCTION Let X and Y be Hilbert spaces with scalar product and norm of X denoted by the symbols.,. X and. X, respectively. Let A, =1,..., N, be N linear bounded mappings from X into Y. Consider the following problem: find an element x X such that where f is given in Y a priori. Set A x = f, =1,..., N, (1.1) S = {x X : A x = f }, =1,..., N, and S = N S. Here, we assume that S. From the properties of A it is easy to see that each set S is a closed convex set in X. Therefore, S is also a closed convex subset in X. 1 This work was supported by the Vietnamese National Foundation of Science and Technology Development.

2 3782 Nguyen Buong and Nguyen Dinh Dung We are specially interested in the situation where the data f is not exactly known, i.e., we have only an approximation f δ of the data f satisfying f f δ Y δ, δ 0,,..., N. (1.2) With the above conditions on A, each -equation in (1.1) is ill-posed. By this we mean that the solution set S does not depend continuously on the data f. Therefore, to find a solution of each -equation in (1.1) one has to use stable methods. One of those methods is the variational variant of Tikhonov s regularization that consists of minimizing the functional A x f δ 2 Y + α x x 2 X, (1.3) where x is some element in X \ S, and chosing a value of the regularization parameter α = α(δ 1, δ N ) > 0. It proved in [1] that each -minimization problem of (1.3) has unique solution x αδ, and if δ 2/α, α 0 then {xαδ } converges to a solution x satisfying x x X = min x S x x X, =1,..., N. Our problem: find x δ α such that x δ α x as δ,α 0, a relation α = α(δ 1, δ N ) such that x δ α(δ 1, δ N ) x as δ 0, and finally estimate the value x δ α(δ 1, δ N ) x where x is a x -minimal norm element in S (x MNS). Burger and Kaltenbacher [2] used the Newton-Kaczmarz method cyclically for regularizing each separate equation in (1.1) under a source condition on each mapping A. The Steepest-Descent-Kaczmarz method is used cyclically by Haltmeier, Kowar, Leitão, and Scherzerfor [3] for regularizing each separate equation in (1.1) under a local tangential cone condition on each mapping A. Note that the system of equations (1.1) can be written in the form Ax = f, (1.4) where A : X Y := Y 1... Y N by Ax := (A 1 x,..., A N x), and f := (f 1,..., f N ). Very recently, Halmeier, Leitão and Scherzer [4] considered the Landweber-Kaczmarz method to solve (1.4) under a local tangential cone condition also on each mapping A. Equation (1.4) can be seen as a special case of (1.1) with N = 1. Howerver, one potential advantage of (1.1) over (1.4) can be that it might better reflect the structure of the underlying information (f 1,..., f N ) leading to the couplet system, than a plain concantenation into one single data element f could. In [5], for finding a common zero for a finite family of potential hemicontinuous monotone mappings from a reflexive Banach space E into E, the adoint of E, the first author proposed a regularization method by solving the

3 Regularization for a common solution 3783 following regularized equation α μ A (x)+αu(x) =θ, =0 μ 0 =0<μ <μ +1 < 1, =1, 2,..., N 1, (1.5) where U is a normalized duality mapping of E and estimated convergence rate of the regularized solution under a smooth condition only for one A 1, i.e., A 1( x) z = U(x 0 ), for some element z E. In this paper, to solve (1.1), we consider a new regularization method based on the following unconstrained optimization problem: min x X A x f δ 2 Y + α x x 2 X. (1.6) We shall show convergence rate of regularized solution in (1.6) under a source condition only on A 1 in the next section. In section 3, we give an example showing that problem (1.4), perhaps, is well-posed, although each equation in (1.1) is ill-posed. So, the cyclical regularization for each separate equation in (1.1) as in [2] and [3] is dispensable. Moreover, it does not exploir the wellposed property of the given system of equations. Meantime, our method (1.6) still uses the property. Above and below, the symbols and denote the weak convergence and convergence in norm, respectively, and a b is meant a = O(b) and b = O(a). 2. MAIN RESULTS Under the assumptions on A it can be easy to show that problem (1.6) admits a unique solution. We shall first address two questions. Is the problem (1.6) stable in the sense of continuous dependence of the solution on the data f δ? Secondly, do the solutions of (1.6) converge toward a solution of (1.1) as α, δ 0. In [6], stability has been proved for the case N = 1. For the convenience of the reader, we provide the whole argument in the case of arbitrary N 1. Theorem 2.1. Let α>0, f δ k f δ with δ 0, ask, and x k be a minimizer of (1.6) with f δ replaced by f δ k. Then the sequence of {x k } converges to a minimizer of (1.6). Proof. Obviously, x δ α is a solution of (1.6) if and only if it is a solution of the following equation Bx + α(x x )= f δ, (2.1) where B = A A and fδ = A f δ,

4 3784 Nguyen Buong and Nguyen Dinh Dung where A denotes the adoint mapping of A. Obviously, if f δ k f k δ = as k. Denote by x δk α A f δ k f δ = A f δ the solution of the following equation Bx + α(x x )= f δ k, k f δ = f δ k then A f δ k. (2.2) From (2.1), (2.2) and the monotone property of the mapping B it implies that x δk α x δ α X f k δ f δ X /α, for each α>0. This together with f δ k f δ follows that x δk α Theorem is proved. xδ α as k. Further, without loss of generality, assume that δ = δ, δ 0. Theorem 2.2. Let α(δ) be such that α(δ) 0,δ/α(δ) 0 as δ 0. Then the sequence {x δ α }, where δ 0,α = α(δ) and xδ α is a solution of (1.6), converges to an x -MNS x of (1.1). Proof. Clearly, we have, for each y S, that By = f, N f = A f. (2.3) So, from (2.1) and (2.3), we obtain that Bx δ α By, xδ α y + α xδ α x,x δ α y = f δ f,x δ α y. This together with the nonegative property of B implies that x δ α y X x y X + A δ (Y X) y S. (2.4) α Since A are the bounded linear mappings and δ/α(δ) 0, the sequence {x δ α(δ) } is bounded. Then, there exists a subsequence {x k := x δ k α(δk )} of the sequence {x δ α(δ) } converging weakly to some element x H as k.now, we shall prove that x S. Indeed, from (1.6) we can obtain the following inequalities A l x k f δ k l 2 Y l A x k f δ k 2 Y A y f δ k Y + α(δ k ) y x 2 X Nδ 2 k + α(δ k ) y x 2 X,

5 Regularization for a common solution 3785 for any y S and l =1,..., N. Since each functional A l x f δ k l 2 is weakly lower semicontinuous in x [7], x k x and δ k,α(δ k ) 0ask, we obtain from the last inequality that A l x = f l,l =1,..., N. So, x S. Now, from (2.4), δ/α(δ) 0, as δ 0, the weakly lower semicontinuity of norm and that any closed convex subset in H has only one x -MNS, it follows that x δ α(δ) x as δ 0. This completes the proof. Theorem 2.3. Assume that there exists ω Y 1 such that x x = A 1ω. Then for the choice α δ p, 0 <p<1, we obtain x δ α(δ) x X = O(δ p/2 ). Proof. Using (1.6) with x = x and δ = δ for all =1,..., N, we obtain So, we have that A x δ α(δ) f δ 2 Y + α(δ) x δ α(δ) x 2 X A x f δ 2 Y + α(δ) x x 2 X. A x δ α(δ)) f δ 2 Y + α(δ) x δ α(δ) x 2 X Nδ 2 + α(δ)( x x 2 X x δ α(δ) x 2 X + x δ α(δ) x 2 X). Since x x 2 X xδ α(δ) x 2 X + xδ α(δ) x 2 X =2 x x, x x δ α(δ) we obtain that A 1 x δ α(δ) f1 δ 2 Y 1 + α(δ) x δ α(δ) x 2 X Nδ 2 +2α(δ) ω, A 1 ( x x δ α(δ) ) Nδ 2 +2α(δ) ω, f 1 f1 δ + f 1 δ A 1x δ α(δ) Nδ 2 +2α(δ) ω Y1 (δ + A 1 x δ α(δ) f 1 δ Y 1 ). Therefore, A 1 x δ α(δ) f 1 δ 2 Y 1 + α(δ) x δ α(δ) x 2 X Nδ2 +2α(δ) ω Y1 δ +2α(δ) ω Y1 A 1 x δ α(δ) f 1 δ Y 1. This together with the implication (a, b, c 0,a 2 ab + c 2 ) a b + c (2.5) (2.6)

6 3786 Nguyen Buong and Nguyen Dinh Dung implies A 1 x δ α(δ) f δ 1 Y 1 [ Nδ 2 +2 ω Y1 α(δ)δ] 1/2 +2 ω Y1 α(δ) and hence if α(δ) δ p, 0 <p<1. x δ α(δ) x X = O(δ p/2 ), 3. NUMERICAL EXAMPLES For illustration, we consider the following problem of finding a common solution of two systems of linear algebraic equations A x = f, =1, 2, (3.1) where A 1 = ,A 2 = 2 1 0,f 1 = 2 3,f 2 = It is easy to verify that system (3.1) possesses a unique common solution x = (1; 1; 1). Since det A 1 = det A 2 = 0, each equation in (3.1) is ill-posed. Based on the results in section 2, the common solution of (3.1) can be found by solving the following optimization problem min A 1x f A 2 x f α x x 2, (3.2) x R 3 where x = x 2 i + x2 2 + x 2 3 for every x =(x 1 ; x 2 ; x 3 ) R 3. It is not difficult to verify that (3.2) is equivalent to the following equation Bx + α(x x )= f, (3.3) where B = A 1 A 1 + A 2 A 2 = , f = A 1 f 1 + A 2 f 2 = 26 20, and x is any vector in R 3. Since det B = 996 and (3.3) can be considered as a regularization equation for the well-posed equation Bx = f, we can use Jacoby or Gauss-Seidel iteration methods for finding a unique solution x α of (3.3). The following table 1 shows the calculation results for the approximation solution x α =(x α 1 ; xα 2 ; xα 3 )at15th iteration with the started point x 0 =(2, 2, 2). Table th approximation solution with started point x 0 = (2; 2; 2).

7 Regularization for a common solution 3787 α x α 1 x α 2 x α 3 x x α Now, we give another example with det B = 0. We consider the case that A 1 = ,A 2 = , and Then, f 1 = ,f 2 = B = , f = Since the rank of the matrix B = 3, it is not difficult to verify that the set of common solutions of (3.1) is a line passed through two points x = (1; 1; 1; 1) and x = (1; 3; 6; 2). So, the solution of minimal norm is the vector x = (1; 7/15; 1/3; 11/15) (1; ; ; ). Table 2 shows our calculation result with δ =10 n. To solve regularized equation (3.3) with f replaced by f δ we use the iterative regularization [8] x (k+1) = x (k) α k (Bx (k) + ε k (x (k) x (0) ) f δ ),x (0) R 4 any vector, with the stopping rule in [9]: Bx (K) f δ 2 τδ < Bx (k) f δ 2,τ>1, for all k =1,..., K 1 and every fixed δ. In our example, we have that L = B = We chose ε 0 =0.1 and ε k+1 = ε k /(1 + ε 3 k ). Then, the sequence {ε k } satisfies the conditions: ε k > 0,ε k 0 and (ε k ε k+1 )/(ε 3 k ε k+1) = 1 and λ =(ε 0 + L 2 )/2 = By taking α k = cε k with c =1/(2λ) = we have that (1 cλ)cε 2 0 = < 1. Clearly, condition (2.6) in [8] is equivalent to τ ( x x (0) [(1 + ε 3 0 )(1 + ε2 0 )+2ε ) 2 0] +1. (1 cλ 2(ε 0 /c))ε 0 Therefore, with x (0) = (0; 0; 0; 0), we obtain that x x (0) = x =1.3663, and hence τ

8 3788 Nguyen Buong and Nguyen Dinh Dung Table 2. Approximation solutions with started point x (0) = (0; 0; 0; 0), τ = 100 and δ =10 n,n=1, 2, n K x (K) 1 x (K) 2 x (K) 3 x (K) 4 B f δ τδ References [1] A.N. Tikhonov and V.Y. Arsenin, Solutions of ill-posed problems, Wiley, N.Y [2] M. Burger and B. Kaltenbacher, Regularization Newton-Kaczmarz methods for nonlinear ill-posed problems, SIAM J. Numer. Analysis, 44 (2006) [3] M. Haltmeier, R. Kowar, A. Leitao, and O. Scherzer, Kaczmarz methods for nonlinear ill-posed equations I: convergence analysis, Inverse problem and Imaging, 1 (2) (2007) , II: Applications, 1 (3) (2007) [4] A.D. Cezaro, M. Haltmeier, A. Leitao, and O. Scherzer, On steepest-descent-kaczmarz method for regularizing systems of nonlinear ill-posed equations, Applied Mathematics and Computations, 202 (2) (2008) [5] Ng. Buong, Regularization for unconstrained vector optimization of convex functionals in Banach spaces, Zh. Vychisl. Mat. i Mat. Fiziki, 46(3) (2006) [6] H.W. Engl, K. Kunisch, and A. Neubauer, Convergence rates for Tikhonov regularization of non-linear ill-posed problems, Inverse Problems 5 (1989) [7] M.M. Vainberg, Variational method and method of monotone mappings, Moscow, Nauka 1972 (in Russian). [8] A.B. Bakushinsky, A. Goncharsky, Ill-posed problems: Theory and Applications, Kluwer Academic [9] A.B. Bakushinsky and A. Smirnova, A posteriori stopping rule for regularized fixed point iterations, Nonl. Anal. 64 (2006) Received: June, 2011