A New Regularization Method for a Parameter Identification Problem in a Non-linear Partial Differential Equation

NAIR M.Thamban and ROY Samprita Das

1 Department of Mathematics,BITS Pilani,K K Birla Goa Campus,Zuarinager,Goa 403726,India.

2 Department of Mathematics and Statistics,IISER Kokota,Nadia,West Bengal 741246,India.

Abstract. We consider a parameter identification problem associated with a quasilinear elliptic Neumann boundary value problem involving a parameter function a(·) and the solution u(·),where the problem is to identify a(·) on an interval I:=g(Γ) from the knowledge of the solution u(·) as g on Γ,where Γ is a given curve on the boundary of the domain Ω⊆R3 of the problem and g is a continuous function.The inverse problem is formulated as a problem of solving an operator equation involving a compact operator depending on the data,and for obtaining stable approximate solutions under noisy data,a new regularization method is considered.The derived error estimates are similar to,and in certain cases better than,the classical Tikhonov regularization considered in the literature in recent past.

Key Words: Ill-posed;regularization;parameter identification.

1 Introduction

Let Ω be a bounded domain in R3withC1,1boundary.Consider the problem of finding a weak solutionu∈H1(Ω) of the partial differential equation

with boundary condition

wherea∈H1(R) andj∈L2(∂Ω).One can come across this type of problems in the steady state heat transfer problem withubeing the temperature,athe thermal conductivity which is a function of the temperature,andjthe heat flux applied to the surface.In this regard,the following result is known (see[1-3]):

Theorem 1.1.Let a≥κ0＞0a.e.for some constant κ0andThen there exists u∈H1(Ω)such that(1.1)and(1.2)are satisfied.If,in addition,with p＞3,then u ∈

In view of the above theorem,we assume that,

Supposeγ:[0,1]→∂Ω is aC1-curve on∂Ω andg:Γ→R such thatg◦γ∈C1([0,1]),where Γ is the range ofγ.One of the inverse problems associated with (1.1)-(1.2) is:

Problem (P):To identify an a∈H1(R)on I:=g(Γ)such that the corresponding u satisfies(1.1)-(1.2)along with the requirement

In the following we shall use the same notation fora∈H1(R) and for its restriction onIas a function inH1(I).

We shall see that the Problem (P) is ill-posed,in the sense that the solutiona|Idoes not depend continuously on the datagandj(see Section 2).To obtain a stable approximate solution for the Problem (P),we use a new regularization method which is different from some of the standard ones in the literature.We discuss this method in Section 3.

The existence and uniqueness of solution for the Problem (P) is known under some additional conditions onγandg,as specified in Section 2 (see,e.g.,[3,4]).In [2] and[3] the problem of finding a stable approximate solution of the problem is studied by employing Tikhonov regularization with noisy data.In [2],with the noisy datagδ,in place ofg,satisfying‖g-gδ‖L2(Γ)≤δ,convergence rateis obtained whenevera∈H4(I) and its trace is Lipschitz on∂Ω,whereaδis the approximate solution obtained via Tikhonov regularization.In [3],the rateis obtained without the additional assumption ona,where noise injas well asgis also considered as

It is stated in[3]that“the rateis possible with respect toH1-norm,provided some additional smoothness conditions are satisfied”;however,the details of the analysis is missing.

Under our newly introduced method,we obtain the above type of error estimates using appropriate smoothness assumptions.In particular we prove that,ifg0∈R is such thatI=[g0,g1]and ifa(g0) is known or is approximately known,and the perturbed datajδandgδbelong toW1-1/p,p(∂Ω) forp＞3 andC1(Γ),respectively,satisfying (1.5),then the convergence rate iswith respect toL2-norm.With additional assumption that the exact solution is inH3(I) we obtain a convergence rateO(δ2/3) with respect toL2-norm.Again,in particular,ifg◦γis inH4([0,1]),the rateO(δ2/3) with respect toL2-norm is obtained under a weaker condition on perturbed datagδ,namely,gδ∈L2(Γ) with‖g-gδ‖L2(Γ)≤δ.Also,in the new method we do not need the assumption ongδmade in [3] which isgδ(Γ)⊂g(Γ).Thus some of the estimates obtained in this paper are improvements over the known estimates,and are also better than the expected best possible estimate,namelyO(δ3/5),in the context of Tikhonov regularization,as mentioned in[3].

The paper is organized as follows:In Section 2 we present a theorem which characterize the solution of the inverse Problem (P) in terms of the solution of the Laplace equation with an appropriate Neumann condition.Also,the inverse problem is represented as the problem of solving a linear operator equation,where the operator is written as a composition of three injective bounded operators,one of which is a compact operator,and prove some properties of these operators.The new regularization method is defined in Section 3,and error estimates with noisy as well as exact data are derived.In Section 4 we present error analysis with some relaxed conditions on the perturbed data.In Section 5 a procedure is described to relax a condition on the exact data and corresponding error estimate is derived.In Section 6 we illustrate the procedure of obtaining a stable approximate solution to the Problem (P).

2 Operator theoretic formulation

Throughout the paper we denote the range of the functiong:Γ→R asI:=[g0,g1],that isg0andg1are the left and right end-points of the closed intervalg(γ([0,1])).

The following theorem,proved in[4],helps us to identify the solution of the Problem (P).

Theorem 2.1.The Problem (P) has a unique solution,and it is the unique a∈H1(I)such that

where M is a constant andsatisfies

It is known that ifj ∈W1-1/p,p(∂Ω) forp＞3,thenvsatisfying (2.2)-(2.3) belongs toW2,p(Ω),and

for some constantC＞0(see Theorems 2.3.3.2 and 2.4.2.7 in[5]).

In view of Theorem 2.1,the inverse Problem (P) can be restated as follows:Givenjandgas in the Problem (P),letsatisfy (2.2) and (2.3) along with the condition

Then,a∈H1(I) is the solution of the Problem (P) if and only if

The above equation can be represented as an operator equation

wherevjis the solution of (2.2)-(2.5) and the operatorT:L2(I)→L2[0,1]is defined by

Theorem 2.2.The operator T:L2(I)→L2[0,1]defined in(2.7)is an injective compact operator of infinite rank.

Proof.Note that for everyw∈L2(I) and for everys,τ ∈[0,1],we have

Sinceg◦γis continuous,the set{Tw:‖w‖L2(I)≤1}is equicontinuous and uniformly bounded inC[0,1].Hence,Tis a compact operator fromL2(I) toC[0,1].Since,the inclusionC[0,1]⊆L2[0,1]is continuous,it follows thatT:L2(I)→L2[0,1]is also a compact operator.We note thatTis injective.Hence,Tis of infinite rank.

It is to be observed that the compact operatorTdefined in (2.7) depends on theg.Thus,problem of solving the operator equation (2.6) based on the data (g,j) is non-linear as well as ill-posed.In order to propose a new regularization method for obtaining stable approximate solutions,we represent the operatorTas a composition of three operators,that is,

where,forr∈{0,1},

are defined as follows:

Clearly,T1,T2,T3are linear operators and

Here,we used the convention thatH0(I):=L2(I).

By the above representation ofT,the operator equation (2.6) can be split into three equations:

To prove some properties of the operatorsT1,T2,T3,we specify the requirements onj,gandγ,namely the following.

Assumption 2.1.Let j∈W(1-1/p),p(∂Ω)with p＞3and=0.Let γ:[0,1]→∂Ωbe a C1-curve on ∂Ωand g:Γ→Rbe such that g∈C1(Γ),

for some positive constants Cγ,,Cg and.

Next we state a result from analysis which will be used in the next result and also in many other results that follow.

Lemma 2.1.Let h1and h2be two continuous functions on intervals J1and J2respectively,such that h2(J2)=J1.Also,letbe continuous with.Then,

We shall also make use of the following proposition.

Proposition 2.1.Let Cg,Cγ,be as in Assumption2.1.Then for any w∈L2(I),

Proof.By Lemma 2.1 and the inequalities (2.14) and (2.15) in Assumption 2.1,we have

From the above,we obtain the required inequalities in (2.16).

Theorem 2.3.Let r∈{0,1},and let

be defined as in(2.8),(2.9)and(2.10),respectively.Then,T2is a compact operator,and for every w∈L2(I),

In particular,T1and T3are bounded operators with bounded inverse from their ranges.

Proof.SinceH1(I) andH2(I) are compactly embedded inL2(I) (see,e.g.,[6]),T2is a compact operator of infinite rank.Now,letw∈H1(I) andτ ∈I.Then

Hence,using the fact that (T1(w))′=wand (T1(w))′′=w′,we have

Thus,(2.17) is proved.By the inequalities in (2.16) we obtain

for everyw ∈L2(I).The inequalities in (2.17) and (2.19) also show thatT1andT3are bounded operator with bounded inverse from their ranges.

3 The new regularization

We know that the Problem (P) is ill-posed.We may also recall that the operator equation (2.6) is equivalent to the system of operator equations (2.11)-(2.13),wherein Eq.(2.12) is ill-posed,sinceT2is a compact operator of infinite rank.Thus,in order to regularize (2.6),we shall replace Eq.(2.12) by a regularized form of it using a family of bounded operators,α＞0.

Note thatT2:H2(I)→L2(I) is defined by

for eachα＞0.

Theorem 3.1.For α＞0,let:H2(I)→L2(I)be defined as in(3.1)Then,

In particular,is a bounded operator with.Further,

Proof.We observe that,for anyw∈H2(I),

In order to define a regularization family forT2,we introduce the space

Note that,forw∈H2(I),w∈Wif and only if

for someξ ∈H1(I) satisfyingξ(g1)=0.

Now,we prove some results associated withW.

Proposition 3.1.The space W defined in(3.2)is a closed subspace of H2(I)and

where Q:H2(I)→H2(I)is the orthogonal projection onto W.

Proof.Let (wn) inWbe such thatwn →w0inH2(I) for somew0∈H2(I).By a Sobolev imbedding Theorem [6],H2(I) is continuously imbedded in the spaceC1(I) withC1-norm.Therefore,w0∈C1(I),and

Thus,sincewn ∈W,in particular

Hencew0∈W.ThusWis closed.Now,letQ:H2(I)→H2(I) be the orthogonal projection ontoW.Then,fory∈L2(I) andw∈Wwe have,

Proposition 3.2.Let α＞0.Let L:H2(I)→H2(I)be defined by

for every x∈H2(I),t∈I.Then we have the following.

(i) For anyx∈H2(I),Lx∈C∞(I)⊂H2(I),α(Lx)′′=Lxand.

(ii)Lis a bounded linear operator.

(iii) The mapid-Lis a projection ontoW,whereidis the identity map onH2(I).

Proof.Clearly,Lis a linear operator,and for anyx ∈H2(I),we haveLx ∈C∞(I)⊂H2(I) andα(Lx)′′=Lx.To show thatLis continuous,let (xn) be a sequence inH2(I) such that‖xn-x‖H2(I)→0 for somex∈H2(I).By a Sobolev imbedding Theorem[6],H2(I) is continuously imbedded in the spaceC1(I) withC1-norm,and so we have|xn(g0)-x(g0)|→0 and|x′n(g1)-x′(g1)|→0 asn→∞.Using this,it can be shown thatLis continuous.Now again by definition ofL,for anyx∈H2(I) we have

so that (id-L)(x-Lx)=x-Lx-L(x-Lx)=x-Lx.Hence,using the definition of the spaceW,we haveid-Lis a projection ontoW.

We shall use the notation

whereLis the bounded operator as in Proposition 3.2.

Theorem 3.2.Let0＜α＜1.Then,for every w∈W,

Proof.First we observe,by integration by parts,that forw1,w2∈W,Hence,for everyw∈W,

Since 0＜α＜1,for everyw∈W,

This completes the proof.

At this point let us note that,by (3.4),is bounded below onW.Henceforth,we shall use the same notation forand its restriction toW,that is,

and the adjoint of this operator will be denoted.The following lemma is used to prove some important properties of,which plays an important role in formulating the new regularization method.Its proof follows from properties of closed range operators,using some standard tools of functional analysis (e.g.,for (3.7) below,see Theorem 11.1.10 in[7]).

Lemma 3.1.Let H1and H2be Hilbert spaces and let S:H1→H2be a bounded linear operator with closed range.Then,

Suppose,in addition,that there exist c＞0such that‖Sx‖≥c‖x‖for all x∈H1.Then

Further,if‖·‖0is any norm on H1and if c0＞0is such that‖Sx‖≥c0‖x‖0for all x∈H1,then

where S†:=(S*S)-1S*,the generalized inverse of S.Here,R(S)and N(S)respectively,denote the range and null space of the operator S.

Corollary 3.1.Let0＜α＜1andbe as in(3.6).Then for every y∈L2(I),

Proof.TakingH1=WandH2=L2(I) in Lemma 3.1,the inequalities in (3.10) and (3.11) follow from (3.9) by taking the norm‖·‖0as‖·‖H2(I)and‖·‖H1(I)respectively,onWand by using (3.4) and (3.5),respectively.

LetRα:L2(I)→Wforα＞0 be defined by

We note that,by Corollary 3.1,Rαis a bounded operator fromL2(I) toW(with respect to the norm‖·‖H2(I)),for eachα＞0.Since,we have

Next,we prove that{Rα}α＞0,defined as in (3.12),is a regularization family forT2:W →L2(I).Towards this aim,we first prove the following theorem.

Theorem 3.3.For α＞0,let Rα be as in(3.12),and let CL be as in(3.3).Then the following results hold.

Proof.(i) Letw∈W.By (3.13),we have

Hence,using (3.10),

Thus,‖RαT2w‖H2(I)≤2‖w‖H2(I)for everyw∈W.

(ii) Letw∈W∩H4(I).Let us note thatw′′is in the domain ofT2and hence is inH2(I)(may not be inW).By Proposition 3.2,w′′-Lw′′∈Wand.Thus,using the above fact,along with the fact thatis in the domain ofT2,by (3.13) and (i) above,we have

we obtain the required inequality.

(iii) Forw∈W,using (3.11),we have.Thus,the proof is complete.

Lemma 3.2.The space W ∩H4(I)is dense in W.

Proof.Letw ∈W.SinceH4(I) is dense inH2(I) as a subspace ofH2(I) (see,e.g.,[6]),there exists a sequence (wn) inH4(I) such that

Now,defineP:H2(I)→Wby

SinceH2(I) is continuously imbedded inC1(I)[6],(3.14) implies that|wn(g0)-w(g0)|→0 andasn→0.Thus,asIis bounded we have

Again by definition ofPandWwe havePwn∈W∩H4(I) andPw=w.Hence from (3.14) and (3.15) we have the proof.

Theorem 3.4.Let w∈W,and let{Rα}α＞0be as in(3.12).Then

In particular,{Rα}α＞0is a regularization family for T2.

Proof.By Theorem 3.3,(RαT2) is a uniformly bounded family of operators fromWtoWand‖RαT2w-w‖H2(I)→0 asα→∞for everyx ∈W ∩H4(I).SinceW ∩H4(I) is dense inW(see Lemma 3.2),by a result in functional analysis (see Theorem 3.11 in[7]),we obtain‖RαT2w-w‖H2(I)→0 asα→∞for everyw∈W.Thus{Rα}α＞0is a regularization family forT2.

Throughout,we assume thata0∈H1(I) is the unique solution of the Problem (P).Thus,Eqs.(2.11)-(2.13) have solutions namely,ζ0,b0anda0,respectively.That is,

Having obtained the regularization family{Rα}α＞0forT2as in (3.12),we may replace the solutionb0of Eq.(2.12) by

The regularized solutionaαfor the Problem (P) is defined along the following lines:

Sincebα ∈W ⊂R(T1),each of the above equations has unique solution.In fact,ζ0=T2b0withb0=T1a0,wherea0is the unique solution of (2.6).Note that,the operator equation (3.20) has a unique solution,becauseis bounded below,and (3.21) has a unique solution asT1is injective with rangeW,andbα ∈W.Hence we have,aα(g1)=0.Thus to obtain convergence of{aα}toa0asα→0,it is necessary thata0(g1)=0.Therefore,in this section,we assume that,

We shall relax this condition in Section 5,by appropriately redefining regularized solutions.

3.1 Error estimates under exact data

Forα＞0,letaαbe defined via Eqs.(3.19)-(3.21).Also,Leta0be the unique solution to the Problem (P) satisfying (3.22).Then,we look at the estimates for the error term (a0-aα) in bothL2(I) andH1(I) norms in the following theorem.

Theorem 3.5.The following results hold.

3.If a0∈H3(I),then with CL is as in(3.3),

Proof.By our assumption,a0(g1)=0.Therefore,by definition ofT1and the spaceW,we haveb0=T1(a0)∈W.Now let us first observe that,by the definition ofbα

Hence,by the inequality (2.17),forr∈{0,1},we have,

and hence,by Theorem 3.4,‖a0-aα‖H1(I)→0 asα→0.Thus we have proved (1).

Also,sinceb0∈W,from (3.23) and Theorem 3.3(iii),we have

which proves (2).Now,leta0∈H3(I).Thenb0∈H4(I).Sinceb0∈W,we haveb0∈W ∩H4(I).Hence proof of (3) follows from (3.23) and Theorem 3.3(ii).

3.2 Error estimates under noisy data

In practical situations the observations of the datajandgmay not be known accurately and we may have some noisy data instead.In this section we assume that the noisy datagεandjδare such that

for some known noise levelεandδ,respectively.At this point,let us note that a weaker condition on perturbed datajδ,for examplejδ ∈L2(∂Ω),is not very feasible to work with.This is because,in that case the corresponding solutionvjδof (2.3)-(2.5) withjδin place ofj,is not continuous and hence its restriction on Γ does not make sense.In practical situations,if such a perturbed data arise,one may work with an appropriate approximation which is inW1-1/p,p(∂Ω) withp＞3.For the perturbed datagε,in the next section we consider the case when it is in a more general space which isL2(Γ).

Corresponding to the dataj,jδas above,we denote

Lemma 3.3.Let γ0be a C1curve onR2and letΓ0={(x,γ0(x))∈R2:d0≤x ≤d1} for some d0,d1inRwith d0＜d1.Then

Proof.Let.Then,using H¨older’s inequality we have

Lemma 3.4.Let w∈H1(∂Ω)and γ be a curve on ∂Ωsuch that|γ′(t)|is bounded away from0as in(2.14).Then there exists C0＞0such that

Proof.Letw∈H1(∂Ω).Since Ω is withC1boundary,

for some elementsω1,···,ωm ∈H1(R2)(see,e.g.,[5,6]).Also,there exists a set{σ1,···,σm}of diffeomorphisms from some neighbourhoods in∂Ω to R2,which satisfies

For anyi ∈{1,···,m},sinceσiis a diffeomorphismσi◦γis a curve in R2.Asis compact andσiis one-one there exists constantCσ＞0 such thatfor allx ∈γ([0,1]) and 1≤i ≤m.Hence,by Lemma 2.1,(3.30) and property ofγalong with (2.14),we obtain

Hence,using (3.28) and (3.29),we get

This completes the proof.

Proposition 3.3.Let.Let(Ω)be the solution of(2.3)-(2.5)within place of j,such that it satisfies(2.1).Then there existssuch that

Proof.Sinceis inW1-1/p,p(∂Ω),we know thatand

for some constantC5＞0 (see inequality (2.4)).By trace theorem for Sobolev spaces[5],and by continuous imbedding ofW(2-1/p),p(∂Ω) intoW1,p(∂Ω),we haveW2-1/p,p(∂Ω)⊆W1,p(∂Ω) and

for some constantsC6,C7＞0.

Thus,using (3.31),(3.32) and withvin place ofwin Lemma 3.4,we have,

Corollary 3.2.Let j be as in Assumption2.1and jδ satisfy(3.24)and(3.26).Let f and fδ be as in(3.27).Then

whereis as in Proposition3.3.

Proof.By Proposition 3.3 we have

Lemma 3.5.For ε＞0,

where Cg andare as in(2.15).In particular,if0＜ε≤Cg/2then

Proof.For anysin[0,1],we have

by (2.15),we obtain (3.34).The relations in (3.35) are obvious by the assumption onε.

Remark 3.1.Since,γ′satisfies (2.14),and,(gε)′satisfies (3.35) forε＜Cg/2,it follows thatgε(Γ) is a non-degenerate closed interval,that is,Iε:=gε(Γ)=for somewith.

The following lemma will help us in showing thatI∩Iεis a closed and bounded (nondegenerate) interval.

Lemma 3.6.Let ϕ1,ϕ2be in C([ξ1,ξ2])for some ξ1and ξ2inR,and let η＞0be such that

Let I1:=ϕ1([ξ1,ξ2])=[a1,b1]and I2:=ϕ2([ξ1,ξ2])=[a2,b2]for some a1,b1,a2and b2inR.If a1＜b1and a2＜b2and η＞0is such that

and I1∩I2=[a,b]is a non-degenerate interval,that is,a＜b.

Proof.Supposea1＜b1anda2＜b2.Since,for some,and since,we obtain

Thus,(3.38) is proved.

To prove the remaining,let us first consider the casea1≤a2.Then,,where=min{b2,b1}.Note that,by (3.37) and (3.39),we have

Thus,b1＞a2,and also,asb2＞a2we have,

Next,leta1＞a2.In this case,,where.Note,again by (3.37) and (3.39),that

Thus,b2＞a1,and also,asb1＞a1we have,

Hence,combining both the cases,we have the proof.

Remark 3.2.Lets1ands0in [0,1] be such thatg0=g(γ(s0)) andg1=g(γ(s1)).Let us recall thatI:=[g0,g1]andIε:=.Sincegandgεare inC1(Γ),we haveg◦γandgε◦γare inC1([0,1]).Also,

Thus,by Lemma 3.6,we have

Hence,takingε＜(g1-g0)/4,we have

and thus,2ε＜min{(g1-g0),.Hence by Lemma 3.6,I∩Iεis a closed and bounded non-degenerate interval.Let us denote this interval by.Thus,

Next,we shall make use of the following lemma which can be proved using the Sobolev imbeding theorem[6].

Lemma 3.7.There exists a constant C＞0such that for any closed interval J,

where CJ:=Cmax{4,(2|J|+1)}.In particular,for any interval J0such that J0⊆J,

Ify∈W1,∞(J1) then using (3.42) we obtain

and additionally ify′′∈L∞(J1),then

Lemma 3.8.Let J1and J2be closed intervals such that J2⊆J1and let CJ1be as in Lemma3.7.Let y∈H2(J1),then we have the following.

Proof.LetJ1=[a,b]andJ2=[c,d]for somea≤bandc≤d.IfJ1=J2thenJ1J2=∅,and in that case the result holds trivially.So let us consider the cases when eithera＜cord＜b,or both holds.Without loss of generality let us assume thata＜candd＜b.Lety∈H2(J1).Then by (3.42)yandy′are inL∞(J1).Thus takingJ0=[a,c]in (3.43) we have

and takingJ0=[d,b]in (3.43) we have

Hence we have (i).Next,additionally if,y′′∈L∞(J1),havingJ0=[a,c]in (3.44) we obtain

and havingJ0=[d,b]in (3.44) we obtain

Hence we have (ii).

Lemma 3.9.Let ϕ1,ϕ2,I1,I2and η be as in Lemma3.6satisfying all the assumptions there.Then,for any interval I3⊂I1∩I2and y∈C1(I1)

Assume,further,that ϕ1,ϕ2∈C1([ξ1,ξ2])satisfyingfor some constants Cϕ1,Cϕ2＞0.Then,for y∈H2(I1)

withand CI is as in Lemma3.7.

Proof.By Lemma 3.6,we haveI1∩I2to be a closed non-degenerate interval.LetI3be an interval inI1∩I2.Then fory∈C1(I1) using fundamental theorem of calculus and H¨older’s inequality we have

Hence,using (3.42) we have (3.46).

Now,additionally letε ≤Cg/2.Then,by (2.14) and (3.35)gεandγare bijective,and so (gε◦γ)-1is continuous.Thusis a closed non-degenerate interval.In other words

Theorem 3.6.Letbe as defined in(3.52).Then,for ζ ∈W,

Proof.Letζ ∈W.For anys∈[0,1],by (2.14) and (2.15),we have

By (3.50) and (3.51),we have

respectively.Nowζ ∈W ⊂H2(I).Then,by definition ofT3and,we have

Hence,takingϕ1asg◦γandϕ2asgε◦γin Lemma 3.9,we have

This completes the proof.

Theorem 3.7.The map,defined as in(3.52),is bounded linear and bounded below.In fact,for every,

where Cγ,and Cg,are as in(2.14)and(2.15),respectively.

Proof.Clearly,is a linear map.Since (2.14) and (3.35) hold,using Lemma 2.1,and (3.52) we obtain

Hence we have the proof.

Now,by Theorem 3.7,we know thatis a bounded linear operator which is bounded below.Thus using Lemma 3.1,the operator

is a bounded linear operator and is the generalized inverse of.The following theorem,which also follows from Lemma 3.1,shows that the family

is in fact uniformly bounded.

Theorem 3.8.For every ζ ∈L2([0,1]),

whereare as in(2.14)and(2.15).

In order to obtain an approximate solution of (2.6) under the nosy data (jδ,gε) satisfying (3.25) and (3.26),we adopt the following operator procedure:First we consider the following operator equation

belongs toL2(I).Next,we consider the operator equation

Letbα,ε,δbe the unique solution of Eq.(3.56).Thus by solving the operator equations (3.55) and (3.56) we obtainbα,ε,δ.Sincebα,ε,δ∈W ⊂R(T1),is the solution of the equation

We show thataα,ε,δis a candidate for an approximate solution to the Problem (P).

Lemma 3.10.Under the assumptions in Assumption2.1on(j,g),let a0∈H1(I)be the solution of T(a)=fj.Assume further that a0(g1)=0.For ζ ∈L2(I),let bα,ζ ∈H2(I)be such that

and let.Then

where Cα＞0is such that Cα →0as α→0.In addition,if a0∈H3(I),then

Here CL is as(3.3).

Proof.Letb0=T1(a0).Then,asa0(g1)=0,we haveb0∈W.Now,by definition ofaα,ζand,H1(I) andH2(I) norms,forr∈{0,1}

Hence,forr∈{0,1},

By Theorem 3.4 we have

Also,by Theorem 3.3-(iii) we have

Again,using (3.10) and (3.11),we have

Thus combining (3.61),(3.62) and (3.64) we have (3.57) with

and combining (3.61),(3.63) and (3.65) we have (3.58).

Next,leta0∈H3(I),b0=T1(a0)∈W ∩H4(I).Then,using theorem 3.3-(ii) we have,forr∈{0,1},

Thus combining (3.61),(3.64) and (3.66) we have (3.59),and combining (3.61),(3.65) and (3.66) we have (3.60).

Now,we prove one of the main theorems of this paper.

Theorem 3.9.Let ε ＜min{(g1-g0)/4,Cg/2}.Let a0,g and j be as in Lemma3.10.Let gε ∈C1(Γ),jδ ∈W1-1/p,p(∂Ω)with p＞3,ζε,δ be the solution of(3.55),andwhere bα,ε,δ is the solution of(3.56).Also,let gε and jδ satisfy(3.25)and(3.26),respectively.Then

where Cα＞0is such that Cα →0as α→0.

In addition if a0∈H3(I),then

Now by definition,bα,ε,δis the unique solution of Eq.(3.56).Thus,withζε,δin place ofζin Lemma 3.10,we have the proof.

Remark 3.3.Leta0andaα,ε,δbe as defined in Theorem 3.9.Then (3.67) and (3.68) take the forms

respectively,whereCα＞0 is such thatCα→0 asα→0,and if,in addition,a0∈H3(I),then (3.69) and (3.70) take the forms

respectively,whereK1,K2,K3,K4are positive constants independent ofα,ε,δandCL ≥‖id-L‖,whereLis the bounded operator as in Proposition 3.2.Then,choosingandε=δin (3.67) we have

Thus using the new regularization method we obtain a result better than the orderO(1) in[3]obtained using Tikhonov regularization.On choosingα=δ=εin (3.68) we have

which is same as the estimate obtained in[3].Next,under the source conditiona0∈H3(I) and forandε=δ,(3.69) gives the order as

This estimate is similar to a result obtained in [2] with source conditiona0∈H4(I) and trace ofa0being Lipschitz which is stronger than the source condition needed in our result,whereas under the same source conditiona0∈H3(I),the choice ofα=δ2/3andε=δin (3.70) gives the rate as

This is better than the rateO(δ3/5) mentioned in[3]as the best possible estimate underL2(I) norm (under realistic boundary condition) using Tikhonov regularization.

4 Relaxation of assumption on perturbed data

In the previous section we have carried out our analysis assuming that the perturbed datagεis inC1(Γ),along with (3.25).This assumption can turn out to be too strong for implementation in practical problems.Hence,here we consider a weaker and practically relevant assumption on our perturbed datagε,namelygε ∈L2(Γ) with

What we essentially used in our analysis in Section 3 to derive the error estimates is thatgε◦γis close tog◦γin appropriate norms.Here,we considerin place ofgε◦γ,where Πh:L2([0,1])→L2([0,1]) is the orthogonal projection onto a subspace ofW1,∞([0,1]),and we show thatis close tog◦γin appropriate norms,and then obtain associated error estimates.For this purpose,we shall also assume more regularity ong◦γ,namely,g◦γ∈H4([0,1]).

Let Πh:L2([0,1])→L2([0,1]) be the orthogonal projection onto the spaceLhwhich is the space of all continuous real valued piecewise linear functionswon[0,1]defined on a uniform partition 0=t0＜t1＜···tN=1 of mesh sizeh,that is,ti:=(i-1)hfori=1,···(N+1) andh=1/N.Thus,w ∈Lhif and only ifw ∈C[0,1] such thatw|[ti-1,ti]is a polynomial of degree at most 1.Let.

In the following,forw ∈L2([0,1]) andτh ∈Th,we use the notationandwheneverw|τhbelong toHm(τh) andWm,∞(τh),respectively.As a particular case of inverse inequality stated in Lemma 4.5.3 in[8],form∈{0,1},we have

whereis a positive constant.

Proposition 4.1.Let w ∈L2([0,1]),m ∈N∪{0} and τh ∈Th.Then the following inequalities hold.

where C0:=2C[0,1]with C[0,1]as in(3.42)andis as in(4.2).

Proof.Iffor somej∈N∪{0},then using (3.42) and the fact thatτhis of lengthh,we obtain

whereI0:=[0,1].Hence,we have

Thus,takingC0=2CI0,we have (4.3).

By repeatedly using (3.42) and then by (4.3),we obtain

As we have takenC0=2CI0,we have the proof of (4.4).

Since Πhis an orthogonal projection,from (4.2) we obtain,

and,by repeatedly using (4.3) we have

Hence we have the proof of (4.5).

For simplifying the notation,we shall denote

Theorem 4.1.Let τh ∈Th and(4.6)be satisfied.Then,the following inequalities hold.

Proof.Using triangle inequality we have

Assumption (2.14),Lemma 2.1 and (4.1) imply

so that,using (4.2) and the fact that Πhis an orthogonal projection,we have

By (4.4) and (4.5),

Thus,using (4.7),(4.10) and (4.12),and taking,we have (i).By (4.4) and (4.5),

Hence,using (4.8) and (4.11),and takingwe have (ii).

To prove (iii) and (iv),lets∈[0,1].Note that

Using (2.14) and (2.15) the above implies

Hence using (ii) we have (iii) and (iv).

From (iii) and (iv) in Theorem 4.1 we obtain the following corollary.

Corollary 4.1.Let h be such that

Hence,combining (4.26) and (4.27) we have (4.21),and combining (4.26) and (4.28) we have (4.22).Hence,is bounded linear and bounded below.Since,satisfies (4.21) and (4.22),from Lemma 3.1,we obtain (4.23).

Using the fact that Πhis a projection,and Lemma 2.1 and (2.14),we obtain,

and,using the fact that Πhis an orthogonal projection,and (4.5),

Now,ζ ∈Wimplies.Hence,takingϕ1andϕ2asandrespectively,in the first part of Lemma 3.9,(3.42) and (4.31),we have,

Now,by (3.42),ζ ∈Wimpliesζ ∈W1,∞(I).Hence,as (4.33) and (4.34) hold,by Lemma 3.8-(i) and then by (3.42),we have

Thus,from (4.35) we have (4.24).

Ifζ ∈H3(I),then,since (4.33) and (4.34) hold,by Lemma 3.8-(ii) and then by (3.42),

Thus,from (4.35) we have (4.25).

Proposition 4.3.Let a0and g be as defined in Lemma3.10.Let h and ε satisfy the relations in(4.13)and(4.16).Let gε ∈L2(I)be such that(4.1)is satisfied.Then,b0=T1(a0)satisfies,

and,in addition,if a0∈H2(I),then,

Proof.Since,handεsatisfy (4.13),for anyτh ∈Th,as (4.17) holds,by Lemma 3.8-(i) and then by (3.42),we have

and,ifa0∈H2(I),b0∈H3(I) and so,by Lemma 3.8-(ii) and then by (3.42),

Theorem 4.3.Let a0,g and j be as in Lemma3.10.Let gε∈L2(I),jδ∈W1-1/p,p(∂Ω)with p＞3.Also,let gε and jδ satisfy(3.26)and(4.1),respectively,and h and ε satisfy the relations in(4.13)and(4.16),and.Then the following results hold.

In the above Cα＞0is such that Cα →0as α→0,b0=T1(a0),

and C0,CL,Cγ are constants as defined in(2.14), (2.15), (3.42), (4.3),Proposition3.2,Theorem4.1-(ii) respectively.

Proof.By definition ofζε,δ,h,

Hence,from (4.46) and (4.47) we have

Thus,from (4.38),(4.45) and (4.48) we have

Ifa0∈H2(I) thenb0∈H3(I),and thus from (4.39),(4.45) and (4.48) we have,

Our aim is to find an estimate for the error term (a0-aα,ε,δ,h) inL2(I) andH1(I) norms.Nowbα,ε,δ,his the unique solution of equation (4.37).Thus,by Lemma 3.10 we need an estimate of‖ζε,δ,h-b0‖L2(I)in order to find our required estimates.Inequalities (4.49) and (5.19) give us estimates of‖ζε,δ,h-b0‖L2(I)under different conditions onb0.Hence,takingζε,δ,hin place ofζin Lemma 3.10 we have the proof.

Remark 4.1.Suppose

Then,forε=δandh=δ1/2,(4.13) and (4.16) are satisfied.Hence,by Theorem 4.3,we have the following:

2.Ifa0∈H3(I) andα=δ2/3,then

3.Choosingα=δ,we have

4.Ifa0∈H2(I),then

Resultsin (1) and (2) above are analogous to the corresponding results fora0-aα,ε,δin Remark 3.3.The estimate in (4) is same as the corresponding estimate in Remark 3.3,except for the fact that here we need an additional condition thata0∈H2(I).

5 With exact solution having non-zero value at g1

In the previous two sections we have considered the exact solution with assumption thata0(g1)=0.Here we consider the case whenbut is assumed to be known.Leta0(g1)=c.Sincea0is the solution to the Problem (P),by (2.6) we havefj=T(a0) which implies

Now by definition ofTwe have

Thus,combining (5.1) and (5.2) we have

Hencea0-cis the solution of the following operator equation,

where clearlyfj-c(gγ-g0)∈L2([0,1]).Also,(a0-c)(g1)=0.Now,let us define

Thenb0,c ∈W.Thus,the analysis of the previous two sections can be applied here to obtain a stable approximate solution of Eq.(5.4).Let,wherebc,αis the solution to the following equation.

whereζcis the solution of the equation

Now,letgεandjδbe the perturbed data as defined in Theorem 4.3.Also,letgbe such thatg◦γ∈H4([0,1]).Letbe the solution of the following equation

Then we have the following theorem.

Theorem 5.1.Let a0,c and b0,c be as defined in the beginning of the section.Let g and j be as defined in Lemma3.10,and gγ∈H4([0,1]).Let h and ε satisfy(4.13)and(4.16),respectively.Also,let gε ∈L2(Γ),jδ ∈W1-1/p,p(∂Ω)with p＞3,and gε and jδ satisfy(3.26)and(4.1)respectively.Let,and let

where Cα＞0is such that Cα →0as α→0.Further,we have the following.

Ifa0,c ∈H2(I),from (4.39),(5.14) and (5.17) we have,

By definition,bc,α,ε,δ,his the unique solution of Eq.(5.8).Also,a0,c ∈H2(I)∩Wimpliesb0,c ∈H3(I)∩W.Thus,puttingζc,ε,δ,hin place ofζin Lemma 3.10,we have the proof using (4.49) and (5.19).

From Theorem 5.1,we see thatc+ac,α,ε,δ,his a stable approximate solution of the Problem (P),with error estimates obtained from Theorem 5.1.

Remark 5.1.Let us relax the assumption on the exact solutiona0even more.Let us assume thata0(g1) is not equal to the known numbercbut is known to be“close”to it,i.e,

Thus,using similar arguments as in the proof of Theorem 5.1,we obtain estimates for

Using the fact that

we obtain (ac,α,ε,δ,h+c) as a stable approximate solution to the Problem (P),and obtain the corresponding error estimates.

6 Illustration of the procedure

In order to find a stable approximate solution of the Problem (P) using the new regularization method we have to undertake the following.

Letjδ ∈W1-1/p,p(∂Ω) withp＞3,gε ∈L2(∂Ω) be the perturbed data satisfying (3.26) and (4.1) respectively,and letAlso let us assumeg◦γ ∈H4([0,1]).Then,by the following steps we obtain the regularized solutionaα,ε,δ.

Acknowledgement

The work on this paper was completed while the authors were at Department of Mathematics,I.I.T.Madras.The authors thank the referee (s) for positive comments and for many useful suggestions which helped to improve the presentation of the first draft of the paper.