Ann. Polon. Math., 95, N1,(2009),

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1 Ann. Polon. Math., 95, N1,(29), Corresponding author. 1

2 Dynamical systems method for solving linear finite-rank operator equations N. S. Hoang A. G. Ramm Mathematics Department, Kansas State University, Manhattan, KS , USA Abstract A version of the Dynamical Systems Method (DSM) for solving ill-conditioned linear algebraic systems is studied in this paper. An a priori and a posteriori stopping rules are justified. An iterative scheme is constructed for solving ill-conditioned linear algebraic systems. Keywords. Ill-posed problems, Dynamical Systems Method, Variational Regularization 1 Introduction We want to solve stably the equation Au = f, (1) where A is a linear bounded operator in a real Hilbert space H. We assume that (1) has a solution, possibly nonunique, and denote by y the unique minimal-norm solution to (1), y N := N(A) := {u : Au = }, Ay = f. We assume that the range of A, R(A), is not closed, so problem (1) is ill-posed. Let f δ, f f δ δ, be the noisy data. We want to construct a stable approximation of y, given {δ, f δ, A}. There are many methods for doing this, see, e.g., [4] [6], [7], [14], [15], to mention some (of the many) books, where variational regularization, quasisolutions, quasiinversion, and iterative regularization are studied, and [7]-[12], where the Dynamical Systems Method (DSM) is studied systematically (see also [1], [14], [13], and references therein for related results). The basic new results of this paper are: 1) a new version of the DSM for solving equation (1) is justified; 2) a stable method for solving equation (1) with noisy data by the DSM is given; a priori and a posteriori stopping rules are proposed and justified; 3) an iterative method for solving linear illconditioned algebraic systems, based on the proposed version of DSM, is formulated; its convergence is proved; 4) numerical results are given; these results show that the proposed method yields a good alternative to some of the standard methods (e.g., to variational regularization, Landweber iterations, and some other methods). 2

3 The DSM version we study in this paper consists of solving the Cauchy problem u(t) = P(Au(t) f), u() = u, u N, u := du dt, (2) and proving the existence of the limit lim t u(t) = u( ), and the relation u( ) = y, i.e., lim u(t) y =. (3) t Here P is a bounded operator such that T := PA is selfadjoint, N(T) = N(A). For any linear (not necessarily bounded) operator A there exists a bounded operator P such that T = PA. For example, if A = U A is the polar decomposition of A, then A := (A A) 1 2 is a selfadjoint operator, T := A, U is a partial isometry, U = 1, and if P := U, then P = 1 and PA = T. Another choice of P, namely, P = (A A + ai) 1 A, a = const >, i s used in Section 3. For this choice Q := AP. If the noisy data f δ are given, f δ f δ, then we solve the problem u δ (t) = P(Au δ (t) f δ ), u δ () = u, (4) and prove that, for a suitable stopping time t δ, and u δ := u δ (t δ ), one has lim u δ y =. (5) δ An a priori and an a posteriori methods for choosing t δ are given. In Section 2 these results are formulated and recipes for choosing t δ are proposed. In Section 3 a numerical example is presented. 2 Formulation and results Suppose A : H H is a linear bounded operator in a real Hilbert space H. Assume that equation (1) has a solution not necessarily unique. Denote by y the unique minimal-norm solution i.e., y N := N(A). Consider the DSM (2) where u N is arbitrary. Denote The unique solution to (2) is T := PA, Q := AP. (6) t u(t) = e tt u + e tt e st dspf. (7) Let us first show that any ill-posed linear equation (1) with exact data can be solved by the DSM. We assume below that P = (A A + ai) 1 A, where a = const >. With this choice of P one has N(T) = N(A), T 1. 3

4 2.1 Exact data The following result is known (see [7]) but a short proof is included for completeness. Theorem 1 Suppose u N and T = T. Then problem (2) has a unique solution defined on [, ), and u( ) = y, where u( ) = lim t u(t). Proof. Denote w := u(t) y, w := w() = u y. Note that w N. One has The unique solution to (8) is w = e tt w. Thus, ẇ = Tw, T := PA, w() = u y. (8) w 2 = e 2t d E w, w. where u, v is the inner product in H, and E is the resolution of the identity of T. Thus, w( ) 2 = lim t e 2t d E w, w = P N w 2 =, where P N = E E is the orthogonal projector onto N. Theorem 1 is proved. 2.2 Noisy data f δ Let us solve stably equation (1) assuming that f is not known, but f δ, the noisy data, are known, where f δ f δ. Consider the following DSM Denote u δ = P(Au δ f δ ), u δ () = u. (9) w δ := u δ y, T := PA, w δ () = w := u y N. Let us prove the following result: Theorem 2 If T = T, lim δ t δ =, lim δ t δ δ =, and w N, then Proof. One has lim w δ(t δ ) =. δ ẇ δ = Tw δ + ζ δ, ζ δ = P(f δ f), ζ δ P δ. (1) The unique solution of equation (1) is w δ (t) = e tt w δ () + t e (t s)t ζ δ ds. 4

5 Let us show that lim δ w δ (t δ ) =. One has lim w δ(t) lim e tt w δ () + lim t t t t e (t s)t ζ δ ds. (11) Let E be the resolution of identity corresponding to T. One uses the spectral theorem and gets: Note that t e (t s)t dsζ δ = = t 1 e t de ζ δ e (t s) ds e tet 1 T de ζ δ = since 1 x e x for x. From (12) and (13), one obtains t e (t s)t dsζ δ 2 = 1 e t de ζ δ. (12) t, >, t, (13) 1 e t 2 d E ζ δ, ζ δ t 2 d E ζ δ, ζ δ = t 2 ζ δ 2. This estimate follows also from the inequality: e (t s)t 1, which holds for T = T and t s. Indeed, one has t e (t s)t ds t, and estimate (14) follows. Since ζ δ P δ, from (11) and (14), one gets ( ) lim w δ(t δ ) lim e tδt w δ () + t δ δ P =. δ δ Here we have used the relation: lim δ e tδt w δ () = P N w =, (14) and the last equality holds because w N. Theorem 2 is proved. From Theorem 2, it follows that the relation t δ = C, γ = const, γ (, 1) δγ where C > is a constant, can be used as an a priori stopping rule, i.e., for such t δ one has lim u δ(t δ ) y =. (15) δ 5

6 2.3 Discrepancy principle In this section we assume that A is a linear finite-rank operator. Thus, it is a linear bounded operator. Let us consider equation (1) with noisy data f δ, and a DSM of the form u δ = PAu δ + Pf δ, u δ () = u, (16) for solving this equation. Equation (16) has been used in Section 2.2. Recall that y denotes the minimal-norm solution of equation (1), and that N(T) = N(A) with our choice of P. Theorem 3 Let T := PA, Q := AP. Assume that Au f δ > Cδ, Q = Q, T = T, and T is a finite-rank operator. Then the solution t δ to the equation h(t) := Au δ (t) f δ = Cδ, C = const, C (1, 2), (17) does exist, is unique, lim δ t δ =, and where y is the unique minimal-norm solution to (1). Proof. Denote One has lim u δ(t δ ) y =, (18) δ v δ (t) := Au δ (t) f δ, w(t) := u(t) y, w := u y. d dt v δ(t) 2 = 2 A u δ (t), Au δ (t) f δ = 2 A[ P(Au δ (t) f δ )], Au δ (t) f δ = 2 AP(Au δ f δ ), Au δ f δ, where the last inequality holds because AP = Q. Thus, v δ (t) is a nonincreasing function. Let us prove that equation (17) has a solution for C (1, 2). One has the following commutation formulas: Using these formulas and the representation one gets: e st P = Pe sq, Ae st = e sq A. u δ (t) = e tt u + t e (t s)t Pf δ ds, (19) v δ (t) = Au δ (t) f δ = Ae tt u + A t e (t s)t Pf δ ds f δ t = e tq Au + e tq e sq dsqf δ f δ = e tq A(u y) + e tq f + e tq (e tq I)f δ f δ = e tq Aw e tq f δ + e tq f = e tq Au e tq f δ. (2) 6

7 Note that lim t e tq Aw = lim Ae tt w = AP N w =. t Here the continuity of A and the following relation were used. Therefore, lim t e tt w = lim t e st de s w = (E E )w = P N w, lim v δ(t) = lim e tq (f f δ ) f f δ δ, (21) t t where e tq 1 because Q. The function h(t) is continuous on [, ), h() = Au f δ > Cδ, h( ) δ. Thus, equation (17) must have a solution t δ. Let us prove the uniqueness of t δ. If t δ is non-unique, then without loss of generality we can assume that there exists t 1 > t δ such that Au δ (t 1 ) f δ = Cδ. Since v δ (t) is nonincreasing and v δ (t δ ) = v δ (t 1 ), one has Thus, Using (19) and (22) one obtains v δ (t) = v δ (t δ ), t [t δ, t 1 ]. d dt v δ(t) 2 =, t (t δ, t 1 ). (22) AP(Au δ (t) f δ ) 2 = AP(Au δ (t) f δ ), Au δ (t) f δ =, t [t δ, t 1 ], where AP = Q 1 2 is well defined since Q = Q. This implies Q 1 2(Au δ f δ ) =. Thus From (2) one gets: Q(Au δ (t) f δ ) =, t [t δ, t 1 ]. (23) v δ (t) = Au δ (t) f δ = e tq Au e tq f δ. (24) Since Qe tq = e tq Q and e tq is an isomorphism, equalities (23) and (24) imply This and (24) imply Q(Au f δ ) =. AP(Au δ (t) f δ ) = e tq (QAu Qf δ ) =, t. This and (19) imply d dt v δ 2 =, t. (25) 7

8 Consequently, Cδ < Au δ () f δ = v δ () = v δ (t δ ) = Au δ (t δ ) f δ = Cδ. This is a contradiction which proves the uniqueness of t δ. Let us prove (18). First, we have the following estimate: Au(t δ ) f Au(t δ ) Au δ (t δ ) + Au δ (t δ ) f δ + f δ f tδ e t δq e sq Qds f δ f + Cδ + δ, where u(t) solves (2) and u δ (t) solves (9). One uses the inequality: e t δ Q and concludes from (26), that tδ e sq Qds = I e t δ Q 2, (26) Secondly, we claim that lim Au(t δ) f =. (27) δ lim t δ =. δ Assume the contrary. Then there exist t > and a sequence (t δn ) n=1, t δ n < t, lim n δ n =, such that lim Au(t δ n ) f =. (28) n Analogously to (19), one proves that d dt v 2, where v(t) := Au(t) f. Thus, v(t) is nonincreasing. This and (28) imply the relation v(t ) = Au(t ) f =. Thus, = v(t ) = e t Q A(u y). This implies A(u y) = e t Q e t Q A(u y) =, so u y N. Since u y N, it follows that u = y. This is a contradiction because Thus, Cδ Au f δ = f f δ δ, 1 < C < 2. lim t δ =. (29) δ Let us continue the proof of (18). From (2) and the relation Au δ (t δ ) f δ = Cδ, one has Cδt δ = t δ e t δq Aw t δ e t δq (f δ f) t δ e t δq Aw + t δ e t δq (f δ f) t δ e t δq Aw + t δ δ. 8 (3)

9 We claim that lim t δe tδq Aw = lim t δ Ae tδt w =. (31) δ δ Note that (31) holds if T has finite rank, and w N. It also holds if T is compact and the Fourier coefficients w j := w, φ j, Tφ j = j φ j, decay sufficiently fast. In this case Ae tt w 2 T 1 2 e tt w 2 = j e 2jt w j 2 := S = o( 1 t2), t, j=1 provided that j=1 w j 2 j <. Indeed, S = + j 1 t 2 3 j > 1 t 2 3 := S 1 + S 2. One has S 1 1 t 2 j t 2 3 w j 2 2 j where c > is a constant. From (31) and (3), one gets = o( 1 t 2), S 2 ce 2t 13 = o( 1 t2), t, Thus, lim δ (C 1)δt δ lim δ t δ e t δq Aw =. lim δt δ = (32) δ Now, the desired conclusion (18) follows from (29), (32) and Theorem 2. Theorem 3 is proved. 2.4 An iterative scheme Let us solve stably equation (1) assuming that f is not known, but f δ, the noisy data, are known, where f δ f δ. Consider the following discrete version of the DSM: Let us denote u n := u n,δ when δ, and set u n+1,δ = u n,δ hp(au n,δ f δ ), u δ, = u. (33) w n := u n y, T := PA, w := u y N. Let n = n δ be the stopping rule for iterations (33). Let us prove the following result: Theorem 4 Assume that T = T, h T < 2, lim δ n δ h =, lim δ n δ hδ =, and w N. Then lim w n δ = lim u nδ y =. (34) δ δ 9

10 Proof. One has w n+1 = w n htw n + hζ δ, ζ δ = P(f δ f), ζ δ P δ, w = u y. (35) The unique solution of equation (35) is w n+1 = (I ht) n+1 w + h Let us show that lim δ w nδ =. One has n (I ht) i ζ δ. i= n 1 w n (I ht) n w + h (I ht) i ζ δ. (36) Let E be the resolution of the identity corresponding to T. One uses the spectral theorem and gets: n 1 (I ht) i = h n 1 h i= = h i= (1 h) i de i= 1 (1 h) n 1 (1 h) de = 1 (1 h) n de. Note that 1 (1 h)n hn, >, t, (38) since 1 (1 α) n αn for all α [, 2]. From (37) and (38), one obtains n 1 h 2 (I ht) i ζ δ = i= 1 (1 h)n 2 d E ζ δ, ζ δ (hn) 2 d E ζ δ, ζ δ = (nh) 2 ζ δ 2. Alternatively, this estimate follows from the inequality (I ht) i 1, provided that ht < 2. Indeed, in this case one has n 1 i= (I ht)i n, and this implies estimate (39). Since ζ δ P δ, from (36) and (39), one gets ( ) lim w n δ lim (I ht) n δ w δ () + hn δ δ P =. δ δ Here we have used the relation: lim (I δ ht)n δ w δ () = P N w =, (37) (39) and the last equality holds because w N. Theorem 4 is proved. 1

11 From Theorem 4, it follows that the relation n δ = C, γ = const, γ (, 1) hδγ where C > is a constant, can be used as an a priori stopping rule, i.e., for such n δ one has lim u n δ y =. (4) δ 2.5 An iterative scheme with a stopping rule based on a discrepancy principle In this section we assume that A is a linear finite-rank operator. Thus, it is a linear bounded operator. Let us consider equation (1) with noisy data f δ, and a DSM of the form u n+1 = u n hp(au n f δ ), u = u, (41) for solving this equation. here u is an arbitrary initial approximation. Equation (41) has been used in Section 2.4. Recall that y denotes the minimal-norm solution of equation (1). Example of a choice of P is given in Section 3. Note that N := N(T) = N(A). Theorem 5 Let T := PA, Q := AP. Assume that Au f δ > Cδ, Q = Q, T = T, h T < 2, h Q < 2, and T is a finite-rank operator. Then there exists a unique n δ such that Au nδ f δ Cδ < Au nδ 1 f δ, C = const, C (1, 2). (42) For this n δ one has: lim u n δ y =. (43) δ Proof. Denote v n := Au n f δ, w n := u n y, w := u y. From (41), one gets This implies v n+1 = Au n+1 f δ = Au n f δ hap(au n f δ ) = v n hqv n. v n+1 2 v n 2 = v n+1 v n, v n+1 + v n = hqv n, v n hqv n + v n = v n, hq(2 hq)v n (44) where the last inequality holds because AP = Q and hq < 2. Thus, ( v n ) n=1 is a nonincreasing sequence. 11

12 Let us prove that equation (42) has a solution for C (1, 2). One has the following commutation formulas: (I ht) n P = P(I hq) n, A(I ht) n = (I hq) n A. Using these formulas, the representation n 1 u n = (I ht) n u + h (I ht) i Pf δ, and the identity (I B) n 1 i= Bi = I B n, with B = I hq, I B = hq, one gets: i= v n = Au n f δ n 1 = A(I ht) n u + Ah (I ht) i Pf δ f δ i= n 1 = (I hq) n Au + (I hq) i hqf δ f δ i= (45) = (I hq) n Au (I (I hq) n )f δ f δ = (I hq) n (Au f) + (I hq) n (f f δ ) = (I hq) n Aw + (I hq) n (f f δ ). If hq := V = V is an operator with V 2, then I hq = I V = sup s 2 1 s 1. Note that lim (I n hq)n Aw = lim A(I n ht)n w = AP N w =, where P N is the orthoprojection onto the null-space N of the operator T, and the continuity of A and the following relation lim n (I ht)n w = lim n were used. Therefore, (1 sh) n de s w = (E E )w = P N w, sh < 2, lim v δ(t) = lim (I n n hq)n (f f δ ) f f δ δ, (46) where I hq 1 because Q and hq < 2. The sequence { v n } n=1 is nonincreasing with v > Cδ and lim n v n δ. Thus, there exists n δ > such that (42) holds. Let us prove (43). Let u n, be the sequence defined by the relations: u n+1, = u n, hp(au n, f), u, = u. 12

13 First, we have the following estimate: Au nδ, f Au nδ Au nδ, + Au nδ f δ + f δ f n δ 1 (I hq) i hq f δ f + Cδ + δ. i= Since hq < 2, one has I hq 1. This implies the following inequality: n δ 1 (I hq) i hq = I (I hq)n δ 2, i= and one concludes from (47) that (47) Secondly, we claim that lim Au n δ, f =. (48) δ lim hn δ =. δ Assume the contrary. Then there exist n > and a sequence (n δn ) n=1, n δ n < n, such that lim Au n n δ, f =. (49) Analogously to (44), one proves that v n, v n 1,, where v n, = Au n, f. Thus, the sequence v n, is nonincreasing. This and (49) imply the relation v n, = Au n, f =. Thus, = v n, = (I hq) n A(u y). This implies A(u y) = (I hq) n (I hq) n A(u y) =, so u y N. Since, by the assumption, u y N, it follows that u = y. This is a contradiction because Thus, Cδ Au f δ = f f δ δ, 1 < C < 2. lim hn δ =. (5) δ Let us continue the proof of (43). From (45) and Au nδ f δ = Cδ, one has Cδn δ h = n δ h(i hq) n δ Aw n δ h(i hq) n δ (f δ f) n δ h(i hq) n δ Aw + n δ h(i hq) n δ (f δ f) n δ h(i hq) n δ Aw + n δ hδ. (51) We claim that if w N, ht < 2, and T is a finite-rank operator, then lim n δh(i hq) n δ Aw = lim n δ ha(i ht) n δ w =. (52) δ δ 13

14 From (51) and (52) one gets Thus, lim δ (C 1)δhn δ lim δ n δ h(i hq) n δ Aw =. lim δn δh = (53) δ Now (43) follows from (5), (53) and Theorem 4. Theorem 5 is proved. 3 Numerical experiments 3.1 Computing u δ (t δ ) In [3] a DSM (9) was investigated with P = A and the singular value decomposition (SVD) of A was assumed known. In general, it is computationally expensive to get the SVD of large scale matrices. In this paper, we have derived an iterative scheme for solving ill-conditioned linear algebraic systems Au = f δ without using SVD of A. Choose P = (A A + a) 1 A where a is a fixed positive constant. This choice of P satisfies all the conditions in Theorem 3. In particular, Q = AP = A(A A + ai) 1 A = AA (AA + ai) 1 is a selfadjoint operator, and T = PA = (A A + ai) 1 A A is a selfadjoint operator. Since T = A A + a de = sup A A + a < 1, where E is the resolution of the identity of A A, the condition h T < 2 in Theorem 5 is satisfied for all < h 1. Set h = 1 and P = (A A + a) 1 A in (41). Then one gets the following iterative scheme: u n+1 = u n (A A + ai) 1 (A Au n A f δ ), u =. (54) For simplicity we have chosen u =. However, one may choose u = v if v is known to be a better approximation to y than and v N. In iterations (54) we use a stopping rule of discrepancy type. Indeed, we stop iterations if u n satisfies the following condition Au n f δ 1.1δ. (55) The choice of a affects both the accuracy and the computation time of the method. If a is too large, one needs more iterations to approach the desired accuracy, so the computation time will be large. If a is too small, then the results become less accurate because for too small a the inversion of the operator A A + ai is an ill-posed problem since the operator A A is not boundedly invertible. Using the idea of the choice of the initial guess of the regularization parameter in [2], we choose a to satisfy the following condition: δ φ(a) := A(A A + a) 1 A f δ f δ 2δ. (56) This can be done by using the following strategy: 14

15 1. Choose a := δ A 2 3 f δ as an initial guess for a. 2. Compute φ(a). If a satisfies (56), then we are done. Otherwise, we go to step If c = φ(a) a δ > 3, we replace a by and go back to step 2. If 2 < c 3, then we 2(c 1) replace a by and go back to step 2. Otherwise, we go to step 4. a 2(c 1) 4. If c = φ(a) δ < 1, we replace a by 3a. If the inequality c < 1 has occured in an earlier iteration, we stop the iterations and use 3a as our choice for a in iterations (54). Otherwise we go back to step 2. In our experiments, we denote by DSM the iterative scheme (54), by VR i a Variational Regularization method (VR) with a as the regularization parameter, and by VR n the VR in which Newton s method is used for finding the regularization parameter from a discrepancy principle. We compare these methods in terms of relative error and number of iterations, denoted by n iter. All the experiments were carried in double arithmetics precision environment using MATLAB. 3.2 A linear algebraic system related to an inverse problem for the heat equation In this section, we apply the DSM and the VR to solve a linear algebraic system used in [2]. This linear algebraic system is a part of numerical solutions to an inverse problem for the heat equation. This problem is reduced to a Volterra integral equation of the first kind with [, 1] as the integration interval. The kernel is K(s, t) = k(s t) with k(t) = t 3/2 2κ π exp( 1 4κ 2 t ). Here, we use the value κ = 1. In this test in [2] the integral equation was discretized by means of simple collocation and the midpoint rule with n points. The unique exact solution u n is constructed, and then the right-hand side b n is produced as b n = A n u n (see [2]). In our test, we use n = 1, 2,...,1 and b n,δ = b n + e n, where e n is a vector containing random entries, normally distributed with mean, variance 1, and scaled so that e n = δ rel b n. This linear system is ill-posed: the condition number of A 1 obtained by using the function cond provided in MATLAB is This number shows that the corresponding linear algebraic system is severely ill-conditioned. Table 1 shows that the results obtained by the DSM are comparable to those by the VR n in terms of accuracy. The time of computation of the DSM is comparable to that of the VR n. In some situations, the results by VR n and the DSM are the same although the VR n uses 3 more iterations than does the DSM. The conclusion from this Table is that DSM competes favorably with the VR n in both accuracy and time of computation. Figure 1 plots numerical solutions to the inverse heat equation for δ rel =.5 and δ rel =.1 when n = 1. From the figure one can see that the numerical solutions 15

16 Table 1: Numerical results for the inverse heat equation with δ rel =.5, n = 1i, i = 1, 1. DSM VR i VR n u n n δ y 2 u iter y 2 n δ y 2 u iter y 2 n δ y 2 iter y δ rel =.5 u VR 1.2 δ rel =.1 u VR 1 u DSM 1 u DSM u VRi u VRi.8 u exact.8 u exact Figure 1: Plots of solutions obtained by DSM, VR for the inverse heat equation when n = 1, δ rel =.5 (left) and δ rel =.1 (right). obtained by the DSM are about the same those by the VR n. In these examples, the time of computation of the DSM is about the same as that of the VR n. The conclusion is that the DSM competes favorably with the VR n in this experiment. 4 Concluding remarks Iterative scheme (54) can be considered as a modification the Landweber iterations. The difference between the two methods is the multiplication by P = (A A+aI) 1. Our iterative method is much faster than the conventional Landweber iterations. Iterative method (54) is an analog of the Gauss-Newton method. It can be considered as a regularized 16

17 Gauss-Newton method for solving ill-conditioned linear algebraic systems. The advantage of using (54) instead of using (4.1.3) in [2] is that one only has to compute the lower upper (LU) decomposition of A A + ai once while the algorithm in [2] requires computing LU at every step. Note that computing the LU is the main cost for solving a linear system. Numerical experiments show that the new method competes favorably with the VR in our experiments. References [1] Airapetyan, R., Ramm, A. G., Dynamical systems and discrete methods for solving nonlinear ill-posed problems, Appl.Math.Reviews, vol. 1, Ed. G. Anastassiou, World Sci. Publishers, 2, pp [2] Hoang, N. S. and Ramm, A. G., Solving ill-conditioned linear algebraic systems by the dynamical systems method (DSM), Inverse Problems in Sci. and Engineering, (to appear). [3] Hoang, N. S. and Ramm, A. G., Dynamical systems gradient method for solving illconditioned linear algebraic systems, (submited). [4] Ivanov, V., Tanana, V., Vasin, V., Theory of ill-posed problems, VSP, Utrecht, 22. [5] Lattes, J., Lions, J., Mèthode de quasi-réversibilité et applications, Dunod, Paris, [6] Morozov, V.A., Methods for solving incorrectly posed problems, Springer Verlag, New York, [7] Ramm, A. G., Dynamical systems method for solving operator equations, Elsevier, Amsterdam, 27. [8] Ramm, A. G., Dynamical systems method for solving nonlinear operator equations, International Jour. of Applied Math. Sci., 1, N1, (24), [9] Ramm, A. G., Dynamical systems method for solving operator equations, Communic. in Nonlinear Sci. and Numer. Simulation, 9, N2, (24), [1] Ramm, A. G., Discrepancy principle for the dynamical systems method I, II, Communic. in Nonlinear Sci. and Numer. Simulation, 1, N1, (25), 95-11; 13, (28), [11] Ramm, A. G., Dynamical systems method (DSM) and nonlinear problems, in the book: Spectral Theory and Nonlinear Analysis, World Scientific Publishers, Singapore, 25, (ed J. Lopez-Gomez). [12] Ramm, A. G., Dynamical systems method (DSM) for unbounded operators, Proc.Amer. Math. Soc., 134, N4, (26), [13] Tautenhahn, U., On the asymptotical regularization of nonlinear ill-posed problems, Inverse Problems, 1 (1994) [14] Vainikko, G., Veretennikov, A., Iterative processes in ill-posed problems, Nauka, Moscow, [15] Vasin, V., Ageev, A., Ill-posed problems with a priori information, Nauka, Ekaterinburg,

Dynamical Systems Gradient Method for Solving Ill-Conditioned Linear Algebraic Systems

Acta Appl Math (21) 111: 189 24 DOI 1.17/s144-9-954-3 Dynamical Systems Gradient Method for Solving Ill-Conditioned Linear Algebraic Systems N.S. Hoang A.G. Ramm Received: 28 September 28 / Accepted: 29