On convergence of regularized modiﬁed Newton's method for nonlinear ill-posed problems

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Article in Journal of Inverse and Ill-Posed Problems · May 2010

DOI: 10.1515/JIIP.2010.004

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On convergence of regularized modified Newton’s method for nonlinear ill-posed problems

Santhosh George

Abstract. In this paper we consider regularized modified Newton’s method for approxi- mately solving the nonlinear ill-posed problemF .x/ Dy;where the right hand side is replaced by noisy datay^ı 2 Y withky y^ık ı andF W D.F / X ! Y is a nonlinear operator between Hilbert spacesX andY: Under the assumption that Fréchet derivativeF⁰ofF is Lipschitz continuous, a choice of the regularization parameter and a stopping rule based on a majorizing sequence are presented. We prove that under a general source condition onx0 Ox;the errork Oxx_k;˛^ı kbetween the regularized approximation x_k;˛^ı (x0WDx_0;˛^ı ) and the solutionxOis of optimal order.

Keywords. Tihkonov regularization, regularized Newton’s method, balancing principle.

2000 Mathematics Subject Classification. 65J20, 65J15, 47J06.

1 Introduction

In this paper we consider nonlinear ill-posed problems

F .x/Dy; (1.1)

whereF WD.F / X !Y is a nonlinear operator with non-closed rangeR.F / and X; Y are infinite dimensional real Hilbert spaces with corresponding inner producth:; :iand normk:krespectively. We assume throughout that

Equation (1.1) has a solution xO (not necessarily unique), which in general does not depend continuously on the right hand side datay:

only noisy datay^ı 2Y with

kyy^ık ı (1.2)

are available.

F possesses a locally uniformly bounded Fréchet derivative F⁰.:/in a ball Br.x/O of radiusr aroundxO 2X:

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Since (1.1) is ill-posed, one has to regularize (1.1). Tikhonov regularization (cf. [6–10, 20]) is one of the most widely used regularization method for solving linear and nonlinear ill-posed problems. In this method a regularized approxima- tionx_˛^ıis obtained by solving the minimization problem

x2D.F /min J˛.x/; J˛.x/D kF .x/y^ık²C˛kxx0k² (1.3) with an initial guessx0 2Xand a properly chosen regularization parameter˛ >

0: It is known [21]that if x_˛^ı is an interior point of D.F /; then the regularized approximationx_˛^ısatisfies the Euler equation

F⁰.x/.F .x/y^ı/C˛.xx0/D0 (1.4) of Tikhonov functionalJ˛.x/:Here and belowF⁰.x/is the adjoint of the Fréchet derivativeF⁰.x/:

Convergence of a global minimizer of (1.3) was established in [7, 19, 21] and convergence rates under logarithmic source conditions onx0 Oxhave been established in [15].

Many authors considered iterative methods like Landweber’s method [4, 5, 11], iteratively regularized Gauss–Newtons method [3, 12, 13], etc. for solving (1.1).

In [15] fixed point iteration

x⁰DProj_{D.F /}x0; x^kC1DProj_{D.F /}ˆ.x^k/ (1.5) with ˆ.x/ D x .F⁰.x/F⁰.x/C˛I /¹ŒF⁰.x/.F .x/y^ı/C˛.x x0/ where Proj_{D.F /} is the projection (with respect to the norm on X) on D.F /has been considered and proved that x^k converges to the stationary pointx_˛^ı of the Tikhonov functional J˛.x/: However no error estimate forkx^k Oxkhas been given in [15]. In [16], a general sequence.z^˛_l/converging to the solutionx_˛^ıof the Tikhonov functionalJ˛.x/were considered and obtained estimate forkz_l^˛ Oxk;

under the following assumptions.

Assumption 1.1. There existsv2Xsuch that

x0 OxD'.F⁰.x/O F⁰.x//vO (1.6) where' W Œ0; ! R^C; > kF⁰.x/kO ²;and the function‰./ D ^'./1=2 is non- decreasing. Moreover'satisfies,

æ ˛

'.˛/ inf

˛

'./; 0 < ˛:

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In this paper we consider a modified form called regularized modified Newton’s method defined iteratively by

x_nC1;˛^ı Dx_n;˛^ı .F⁰.x0/F⁰.x0/C˛I /¹ŒF⁰.x0/.F .x_n;˛^ı /y^ı/C˛.x_n;˛^ı x0/;

(1.7) x_0;˛^ı WD x0 for solving (1.1). Note that the methods considered in [14, 19, 21], require the computation of the Fréchet derivative F⁰.:/at global minimizers x_˛^ı of the Tikhonov functionalJ˛.x/and the methods considered in [15] require the computation of the Fréchet derivative F⁰.:/at each iterationx_n;˛^ı converging to the global minimizer of the Tikhonov functionalJ˛.x/:

Observe that the method (1.7) requires the computation of Fréchet derivative F⁰.:/only at one pointx0:This is one of the advantage of the method considered in this paper. Another advantage of our method is that the stopping rule in this paper is based on a majorizing sequence (see [1]) and hence it is not depending on the method.

In Section 2 we provide some preparatory result and derive error bounds for kx^ı_˛ Oxkunder certain general source conditions which include the logarithmic source conditions consider in [15]. In Section 3 we consider the regularized modified Newton’s method defined in (1.7) and prove thatx_n;˛^ı converges to the solution x_˛^ı of (1.2). The analysis of this section is based on a majorizing sequence. Also in this section, using an error estimate forkx_˛^ı Oxk, we obtain an error estimate forkx_n;˛^ı Oxk:In Section 4 we derive error bounds forkx_n;˛^ı Oxkby choosing the regularization parameter˛by an a priori as well as by the balancing principle pro- posed by Pereverzev and Schock [18]. Algorithm for implementing the balancing principle and the stopping rule for the iteration are given in Section 5 and finally the paper ends with some concluding remarks in Section 6.

2 Preparatory results

Throughout this paper we assume that the operatorF satisfies the following assumptions.

Assumption 2.1. There existsr > 0such thatBr.x/O D.F /andF is Fréchet differentiable at allx2Br.x/:O

Assumption 2.2. There exists a constantk0> 0such that for everyx; u2Br.x/O andv2X;there exists an elementˆ.x; u; v/ 2Xsatisfying

ŒF⁰.x/F⁰.u/vDF⁰.u/ˆ.x; u; v/;kˆ.x; u; v/k k0kvkkxuk for allx; u2Br.x/O andv2X:

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The next assumption on source condition is based on a source function'and a property of the source function':We will use this assumption to obtain an error estimate fork Oxx_˛^ık:

Assumption 2.3. There exists a continuous, strictly monotonically increasing function'W.0; a!.0;1/witha kF⁰.x/O F⁰.x/kO satisfying

lim_!0'./D0

for˛1,'.˛/˛

sup₀^˛'./_C˛ c''.˛/; 82.0; a

there existsv2Xsuch that

x0 OxD'.F⁰.x/O F⁰.x//v:O (2.1) Remark 2.4. The above asumption on source function'includes the logarithmic source condition considered in [15].

We will be using the following theorems from [21] for our error analysis.

Theorem 2.5 (cf. [21], Theorem 2.7). Let x^ı_˛ be the solution of the regularized problem (1.4) andx˛ be a solution of (1.4) withy^ıreplaced by the exact datay. Assume Assumption 2.2 with radiusr D ^p^ı_˛ C2kx₀ Oxk:Ifk0kx₀ Oxk < 1;

then

kx_˛^ıx˛k ı p˛p

1k0kx0 Oxk: (2.2) The following theorem is essentially a reformulation of Theorem 2.6 proved in [21]. For the sake of completion, we supply its proof as well.

Theorem 2.6. Letx˛be a solution (1.4) withy^ıreplaced by the exact datay. As- sume Assumption 2.2 and Assumption 2.3 with radiusr D2kx₀ Oxk:If2k0kx₀

O

xk< 1;then

kx˛ Oxk c''.˛/kvk

p12k0kx0 Oxk: (2.3)

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Proof. Note thatF⁰.x˛/.F .x˛/y/C˛.x˛x0/D0;so .F⁰.x/O F⁰.x/O C˛I /.x˛ Ox/

D.F⁰.x/O F⁰.x/O C˛I /.x˛ Ox/F⁰.x˛/.F .x˛/y/˛.x˛ x0/ D˛.x0 Ox/CF⁰.x/O F⁰.x/.xO ˛ Ox/F⁰.x˛/.F .x˛/y/

D˛.x0 Ox/CF⁰.x/O ŒF⁰.x/.xO ˛ Ox/.F .x˛/y/

ŒF⁰.x˛/F⁰.x/O .F .x˛/y/

D˛.x0 Ox/CF⁰.x/O ŒF⁰.x/.xO ˛ Ox/.F .x˛/F .x//O ŒF⁰.x˛/F⁰.x/O .F .x˛/F .x//:O

Thus

x˛ OxDs1Cs2Cs3 (2.4)

where

s1WD˛.F⁰.x/O F⁰.x/O C˛I /¹.x0 Ox/;

s2WD .F⁰.x/O F⁰.x/O C˛I /¹ŒF⁰.x˛/F⁰.x/O .F .x˛/F .x//;O and

s3WD.F⁰.x/O F⁰.x/O C˛I /¹F⁰.x/O ŒF⁰.x/.xO ˛ Ox/.F .x˛/F .x//:O It follows from estimates (2.15) and (2.17) in [21] (also see estimate (A.7) in [14]), that

ks2k k0kx0 Oxkkx˛ Oxk and

ks3k k0kx0 Oxkkx˛ Oxk:

Therefore by (2.4), to complete the proof it is enough to prove that ks₁k c''.˛/kvk:But by Assumption 2.3, we have

ks₁k D k˛.F⁰.x/O F⁰.x/O C˛I /¹'.F⁰.x/O F⁰.x//vkO c''.˛/kvk:

This completes the proof of the theorem.

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3 Regularized modified newton’s method

In this section we prove thatx_n;˛^ı converges tox_˛^ıand provide an error estimate for kx^ı_n;˛x^ı_˛k:Our analysis is based on a majorizing sequence. Recall (see [1], Def- inition 1.3.11) that a nonnegative sequence.tn/is said to be a majorizing sequence of a sequence.xn/inX if

kxnC1xnk tnC1tn; 8n0:

We use the sequence.tn/; n0;given byt0D0; t1D; tnC1DtnC k0

.1 Qr/.tntn1/ (3.1)

whererQ 2Œ0; 1/;as a majorizing sequence, of the sequence.x^ı_n;˛/:The following lemma is essentially a reformulation of a lemma in [8].

Lemma 3.1. Assume there exist nonnegative numbers k0; and rQ 2 Œ0; 1/such

that k0

.1 Qr/ Qr: (3.2)

Then the sequence.tn/defined in (3.1) is increasing, bounded above byt WD

1Qr;and converges to sometsuch that0 < t _1Q_r:Moreover, forn0, 0tnC1tn Qr.tntn1/ Qrⁿ; (3.3) and

ttn rQⁿ

1 Qr: (3.4)

Proof. Since the result holds for D0; k0 D 0orr D 0;we assume thatk0 ¤ 0; ¤0andrQ ¤0:Observe thatt1t0D0;assume thattiC1ti 0;for alli kfor somek:Thent_kC2t_kC1D _.1Q⁰_r/.t_kC1t_k/0;so by induction tnC1tn0for alln0:Now since

k0 .1 Qr/ Qr

the estimate (3.3) follows from (3.1). Further observe that

tkC1tkC Qr.tktk1/ C QrC C Qr^k D 1 Qr^kC1

1 Qr <

1 Qr:

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Hence the sequence.tn/; n 0 is bounded above by _1Q_rInondecreasing, so it converges to somet _1Q_r;and

ttnD lim

i!1tnCitn lim

i!1 i1X

jD0

.tnC1Cj tnCj/ rQⁿ 1 Qr: That completes the proof of the lemma.

To prove the convergence of the sequence.x_n;˛^ı /defined in (1.7) we introduce the following notations: LetR˛.x0/WDF⁰.x0/F⁰.x0/C˛I and

G.x/WDxR˛.x0/¹ŒF⁰.x0/.F .x/y^ı/C˛.xx0/: (3.5) Note that with the above notation,G.x_n;˛^ı /Dx_nC1;˛^ı and

kR˛.x0/¹F⁰.x0/F⁰.x0/k 1: (3.6) The following lemma based on the Assumption 2.2 will be used in due course.

Lemma 3.2. Foru; v2Br.x0/

F .v/F .u/F⁰.x0/.vu/DF⁰.x0/ Z ₁

0 ˆ.uCt.vu/; x0; vu/dt:

Proof. Using the fundamental theorem of integration, foru; v2Br.x0/we have F .v/F .u/D

Z ₁

0 F⁰.uCt.vu//.vu/dt so by Assumption 2.2 we have

F .v/F .u/F⁰.x0/.vu/DF⁰.x0/ Z ₁

0 ˆ.uCt.vu/; x0; vu/dt:

This completes the proof of the lemma.

Hereafter we assume that r max

° ı

p˛ C2kx0 Oxk; t± :

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Theorem 3.3. Let the assumptions in Lemma 3.1 with D ^{kF .x}^p⁰^/y_˛ ^ı^k and As- sumption 2.2 be satisfied. Then the sequence.x_n;˛^ı /defined in (1.7) is well defined andx_n;˛^ı 2Bt.x0/for alln0:Further.x_n;˛^ı /is Cauchy sequence inBt.x0/ and hence converges tox_˛^ı 2Bt.x0/ Bt.x0/andF⁰.x0/.F .x_˛^ı/y^ı/C

˛.x_˛^ıx0/D0:

Moreover, the following estimates hold for alln0;

kx_nC1;˛^ı x_n;˛^ı k tnC1tn; (3.7) and

kx_n;˛^ı x^ı_˛k rQⁿkF .x₀/y^ık

p˛.1 Qr/ : (3.8)

Proof. LetGbe as in (3.5). Then foru; v2Bt.x0/;

G.u/G.v/DuvR˛.x0/¹ŒF⁰.x0/.F .u/y^ı/C˛.ux0/ CR˛.x0/¹ŒF⁰.x0/.F .v/y^ı/C˛.vx0/ DR˛.x0/¹ŒR˛.x0/.uv/F⁰.x0/.F .u/F .v//

C˛R˛.x0/¹.vu/

DR˛.x0/¹F⁰.x0/ŒF⁰.x0/.uv/.F .u/F .v//C˛.uv/

C˛R˛.x0/¹.vu/

DR˛.x0/¹F⁰.x0/ŒF⁰.x0/.uv/.F .u/F .v//

Thus by Lemma 3.2, Assumption 2.2 and (3.6) we have

kG.u/G.v/k k0tkuvk: (3.9) Now we shall prove that the sequence.tn/where.tn/defined in Lemma 3.1 is a majorizing sequence of the sequence.x_n;˛^ı /andx_n;˛^ı 2Bt.x0/;for alln0:

Note thatkx_1;˛^ı x0k D kR˛.x0/¹F⁰.x0/.F .x0/y^ı/k ^{kF .x}^p⁰^/y_˛ ^ı^k D Dt1t0;assume that

kx_iC1;˛^ı x_i;˛^ı k tiC1ti; 8i k (3.10) for somek:Then

kx_kC1;˛^ı x0k kx^ı_kC1;˛x_k;˛^ı k C kx_k;˛^ı x^ı_k1;˛k C C kx_1;˛^ı x0k tkC1tkCtktk1C Ct1t0

DtkC1t:

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Sox_iC1;˛^ı 2Bt.x0/for allik;and hence, by (3.9) and (3.10), kx^ı_kC2;˛x_kC1;˛^ı k k0tkx_kC1;˛^ı x^ı_k;˛k k0

.1 Qr/.tkC1tk/DtkC2tkC1: Thus by inductionkx_nC1;˛^ı x_n;˛^ı k tnC1t_nfor alln0and hence.tn/; n0 is a majorizing sequence of the sequence.x_n;˛^ı /:In particularkx^ı_n;˛x0k tn t;i.e.,x_n;˛^ı 2Bt.x0/;for alln0:So.x_n;˛^ı /; n0is a Cauchy sequence and converges to somex_˛^ı 2Bt.x0/Bt.x0/and

kx_˛^ıx_n;˛^ı k ttn rQⁿ

.1 Qr/ D rQⁿkF .x₀/y^ık p˛.1 Qr/ :

Now by lettingn! 1in (1.7) we obtainF⁰.x0/.F .x_˛^ı/y^ı/C˛.x_˛^ıx0/D0:

4 Error bounds

Combining the estimates in Theorem 2.5, Theorem 2.6 and Theorem 3.3 we obtain the following,

Theorem 4.1. Let x^ı_n;˛ be as in (1.7) and let the assumptions in Theorem 2.5, Theorem 2.6 and Theorem 3.3 be satisfied. Then

kx_n;˛^ı Oxk kF .x0/y^ık .1 Qr/

rQⁿ

p˛C p 1

1k0kx0 Oxk pı

˛ C c'kvk

p12k0kx0 Oxk'.˛/: (4.1) Let

n_ı WDmin¹nW Qrⁿ ıº (4.2)

and let C WDmax

²kF .x0/y^ık

1 Qr Cp 1

1k0kx₀ Oxk

; c'kvk p12k0kx₀ Oxk

³ : (4.3) Theorem 4.2. Letx_n;˛^ı be as in (1.7),nı be as in (4.2) andC be as in (4.3). Let the assumptions in Theorem 4.1 be satisfied. Then

kx_n^ı_ı_;˛ Oxk C ı

p˛C'.˛/

(4.4)

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4.1 A priori choice of the parameter

Note that, if the regularization parameter˛_ı WD˛.ı/satisfies

p˛ı'.˛ı/Dı; (4.5)

then the error p^ı

˛ C'.˛/is of optimal order.

Let ./ WD p

'¹./; 0 < a: Then by (4.5), ı D .'.˛_ı//; so

˛ı D '¹.¹.ı//: Hence for the a priori choice˛ D '¹.¹.ı// we have the following theorem.

Theorem 4.3. Let./ Dp

'¹./for0 < a;and assumptions in Theo- rem 4.2 holds. Forı > 0;let˛D'¹.¹.ı//and letn_ıbe as in (4.2). Then

kx_n^ı_ı_;˛ Oxk DO.¹.ı//:

The disadvantage of the above a priori parameter choice is that, in practice an index function ' describing a solution smoothness is usually unknown. So one has to apply a posteriori rules which choose ˛ from quantities that arise during calculations. The well-known a posteriori rules that have been used for Tikhonov regularization of nonlinear ill-posed problems are the discrepancy principle [2,22], the rules considered in [15, 21] and the balancing principle considered in [18]. In the next subsection we considere the balancing principle considered in [18] for choosing the regularization parameter˛in (1.7).

4.2 An adaptive choice of the parameter

In this subsection, we will present the balancing principle studied in [16–18] for choosing the parameter˛in (1.7).

In the balancing principle, the regularization parameter˛is selected from some finite set

DM.˛/WD ¹˛iDⁱ˛0; iD0; 1; : : : ; Mº (4.6) where > 1andMis such that1˛M:We choose˛0WDı;because we expect to have an accuracy of order at leastO.p

ı/and from Theorem 4.2, it follows that such an accuracy cannot be guaranteed for˛ < ı:

Letxi WDx_n^ı_ı_;˛_i:Then by (3.8) and (4.2) we have kxix_˛^ı_ik kF .x0/y^ık

1 Qr pı

˛i; 8i D0; 1; : : : ; M: (4.7) The parameter choice strategy that we are going to consider in this paper, we selects ˛ D ˛i from DM.˛/ and operates only with corresponding xi; i D0; 1; : : : ; M:

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Theorem 4.4. Assume that there exists i 2 ¹0; 1; 2; : : : ; Mº such that '.˛i/

pı˛i:Let assumptions of Theorem 4.2 be satisfied and let

lWDmax

²

iW'.˛i/ ı p˛i

³

< M;

kWDmax

²

iW kx_ixjk 4C ı

p˛j; j D0; 1; 2; : : : ; i

³

(4.8) whereC is as in (4.3). Thenlkand

k Oxx_kk c ¹.ı/

wherecD6Cp :

Proof. To see thatl k;it is enough to show that, for eachi 2 ¹1; 2; : : : ; Mº;

'.˛i/ ı

p˛i H) kxixjk 4C ı

p˛j; 8j D0; 1; : : : ; i:

Forj i;by (4.4) we have

kx_ixjk kx_ixk C kxxjk C

'.˛i/C ı p˛i

CC

'.˛j/C ı p˛j

4C ı p˛j:

Thus the relationl kis proved. Next we observe that k Oxxkk k Oxxlk C kx_lxkk

C

'.˛_l/C ı p˛_l

C4C ı p˛_l 6C ı

p˛_l: Now since˛ı ˛lC1˛l;it follows that

pı

˛_l p

ı

p˛_ı Dp

'.˛ı/Dp ₁ .ı/:

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5 Implementation of adaptive choice rule

In this section we provide an algorithm for the determination of a parameter ful- filling the balancing principle (4.8) and also provide a starting point for the iteration (3.3) approximating the unique solutionx_˛^ı of (1.3). We choose the starting pointx0such thatx02D.F /andkF .x0/y^ık _4k^p^ı₀:

The choice of the stopping indexn_ıinvolves the following steps:

Choose˛0Dıand > 1:

Chooser > 0Q such thatr <Q ¹₂ 1

r

1 ^4k⁰^{kF .x}^p_ı⁰^/y^ı^k :

Choose the parameter˛M D^M˛0big enough but not too large.

Choosenısuch thatnıDmin¹nW Qrⁿ ıº:

Finally the adaptive algorithm associated with the choice of the parameter speci- fied in Theorem 4.4 involves the following steps:

5.1 Algorithm

Seti 0

solvexi WDx^ı_n_ı_;˛_iby using the iteration (3.3).

Ifkxixjk> 4C

pı

p^j; j i;then takek Di1:

Seti DiC1and return to step 2.

6 Concluding remarks

In this paper we have considered a regularized modified Newton’s method for obtaining approximate solutions for a nonlinear operator equation F .x/ D y;

when the available data isy^ıin place of the exact datay:The procedure involves finding the fixed point of the function

G.x/Dx.F⁰.x0/F⁰.x0/C˛I /¹ŒF⁰.x0/.F .x/y^ı/C˛.xx0/ in an iterative manner.

It should of course be mentioned that the method requires to compute the Fréchet derivativeF⁰.:/only at one pointx0unlike the method considered in [15].

Also it should be noted that the proof for the convergence result and the stopping rule are based on a majorizing sequence, which is independent of the method. For

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choosing the regularization parameter˛we made use of the balancing principle suggested in [18].

In a future work, it is envisaged to investigate the method considered in [15]

to obtain quadratic convergence of the iterate to the solution x_˛^ı of the Tikhonov functionalJ˛.x/:

Acknowledgments.The author thanks National Institute of Technology Karnataka, India, for the financial support under seed money grant No.RGO/O.M /SEED GRANT/106/2009.

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Received December 7, 2009.

Author information

Santhosh George, Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Surathkal, Mangalore 575025, India.

E-mail:[email protected]