DU Shu-kai

(School of Mathematics and Statistics,Wuhan University,Wuhan 430072,China)

ON CONVERGENCE CONDITIONS OF LEAST-SQUARES PROJECTION METHOD FOR OPERATOR EQUATIONS OF THE SECOND KIND

DU Shu-kai

(School of Mathematics and Statistics,Wuhan University,Wuhan 430072,China)

In this paper,we investigate the convergence conditions of least-squares projection method for compact operator equations of the second kind.By technics in functional analysis and Moore-Penrose inverse,we obtain 4 new mutually equivalent convergence conditions,which build the connections among several types of convergence conditions and provide us with more choices to examine the convergency of the approximation scheme.A simple and important example is also studied as an application of the theorem.

convergence condition;least-squares projection;operator equation

1 Introduction

Operator equation was one ofthe principaltools in a large area ofapplied mathematics, and the literature discussing around this topic is vast.In this paper,we will limit our discussion on the compact operator equations of the second kind,which has the form

where K:X → X is a compact operator and b ∈ X is given.Thanks to the First and the Second Riesz Theorem,we know dim N(T) ＜ ∞ [6,Theorem 3.1,p.28],and R(T) was closed[6,Theorem 3.2,p.29].We aim to obtain the best-approximate solution of(1.1), which is denoted as x†:=T†b,where T†is the Moore-Penrose inverse of T.Note that as R(T)is closed,T†is naturally guaranteed to be bounded.

Due to the complexity of the specific problems that has form(1.1),it is diffi cult for us to find a universal solution to all the problems.A more promising strategy is finding the numericalsolution,which involve approximating the abstract space and operator with finite freedoms.Let{Xn}be a sequence of finite-dimensionalsubspaces of X such thatand for each n ∈ N set Tn:=T Pn,where Pn:=PXnis the orthogonal projection from X onto Xn.Note that(1.2)implies

here we say{(Xn,Tn)}n∈Nis a LSA(least-squares approximation setting)for T,and all of our following discussions will be based on this setting.Our target is to fi nd suitable LSA such that

namely,x†:=T†b can be approximated by x†n:=T†nb.There were many works touching upon this problem such as Du[1,2],Groetsch[4],Groetsch-Neubauer[3],and Seidman[5]. Note that(1.4)does not naturally holds for equation(1.1),as Du’s example[2,Example 2.10] shows.To guarantee the convergency of the approximation scheme{T†n}for T†.Groetsch [4,Proposition 0]and Du[2,Theorem 2.8(d)]provide the following convergence conditions

where(1.5)is the stability condition of LSA{(Xn,Tn)}n∈N,that is,

However,as will show in a simple and important example,a direct examination of(1.5) could be diffi cult,but the examination of some other equivalent condition of(1.5)that we will soon give in our theorem can be very easy.

In this paper,we willgive some equivalent characterizations for(1.4)(or(1.5)).These equivalent characterizations can not only increase our understanding on the convergence of this approximation scheme by offering different perspectives,but also provide us with some simple and ‘easy to check’criteria to examine the convergence.To proceed,we need the following notation

Theorem 1.1For the compact operator equation(1.1)with LSA,the following propositions are equivalent:

(c)There holds

(e)There is a n∗∈ N such that Xn∗⊇ N(T).

In Section 2,we willgive some lemmas and the proofof Theorem 1.1.In Section 3,we willstudy some examples to further explain the theorem.

2 Proofs

To prove Theorem 1.1 we need to prepare several lemmas.

Lemma 2.1Let T ∈ B(X)with dim N(T) ＜ ∞,and have LSA{(Xn,Tn)}.

(a)There hold

(b)There is a n∗∈ N such that

(c)If R(T)is closed and

then(1.5)holds.

Proof(a)It is clear that

and therefore

N(T)∩ Xn⊆ N(T)∩ Xn+1(∀n), dim(N(T)∩ Xn) ≤ dim N(T) ＜ ∞,

and therefore there is a n∗∈ N such that

This implies that

(c)Assume that sup?Tn†?=+∞.Then by the uniform boundedness principle,there is an u ∈ X such thatn

Hence there exists a subsequencesuch thaDue to(b),

there is a n∗∈ N such that

Hence it follows from(2.1)that N(T) ∩ Xn=N(T) ⊆ Xnfor n ≥ n∗,and therefore

kknknk

This with T†∈ B(X)(by R(T)being closed)implies that

that contradicts with ‖vk‖ =1.

Lemma 2.2Let T ∈ B(X)have LSA{(Xn,Tn)}.Then

ProofIt is clear that s-limG(Tn) ⊆ w-l~im G(Tn).Hence,we need only to show that

n→∞n→∞

Let(x,y) ∈ G(T).Then(x,Tnx) ∈ G(Tn),and(x,Tnx) → (x,y)(n → ∞).Therefore (x,y) ∈ s-lim G(Tn).This gives that G(T) ⊆ s-limG(Tn).Let(x,y) ∈ w-l~imG(Tn).

n→∞∞n→∞n→∞

Then there is a sequence{(xn,yn)}such that ∪k=nG(Tk) ∍ (xn,yn) ⇀ (x,y)(n → ∞). Thus there is a sequence{kn}such that

Note that for all v ∈ X there holds

Thus we have 〈T x − y,v〉=0 ∀v ∈ X,that is,(x,y) ∈ G(T).This gives thatG(T).

Lemma 2.3Let H be a Hilbert space and{Hn}a sequence of closed subspaces of H. Then

in the case that{PHn}is strongly convergent,

ProofSee[1,Lemma 2.13].

Next,we prove Theorem 1.1 as follows.

Proof of Theorem 1.1Note that,by Lemma 2.1(a),PN(Tn)=PN(T)∩Xn+I − Pn, and there is a n∗∈ N such that

Therefore it follows that

that is,

Due to this with the uniform boundedness principle,it is clear that

Let(a)be valid,namely,(1.4)and(1.5)hold(by(2.4)).Then for all x ∈ X there hold

Thus we obtain that

Let(y,x) ∈ G （T†）.Then

By Lemma 2.2,s-nl→im∞G(Tn)=G(T),hence there is a squence{xn}such that

This with(2.6)implies that

and therefore

Hence,there is a sequence{kn}such that

This with Tk†

nyn∈ N(Tkn)⊥⊆ Xknand(2.6)implies that

and for any v ∈ X,

Thus we have

Hence to prove(c)we need only to show that

Assume that(2.7)is not valid.Then there is a x0with ‖x0‖ =1. Note that,by(2.2)with Lemma 2.3,

It follows that x0∈ N(T) ∩ (N(T) ∩ Xn∗)⊥,and that x0satisfies

and(noting N(Tn)⊥=(N(T)∩ Xn)⊥∩ Xn⊆ Xn)

So we have

This contradicts with ‖x0‖ =1.

(c)=⇒ (d)Let(c)be valid.For any b,bn∈ X with ‖bn− b‖ → 0(n → ∞)we need only to show that

Due to Lemma 2.3,(c)is equivalent toBy Lemma 2.1(a)and Lemma 2.3,the above equation is equivalent to(2.1).Then by Lemma 2.1(c),we obtain(1.5),that is

Let x ∈ w-l~im T−1（Pb）.Then there is a squence{x}with an integer sequence

n→∞nR(Tn)nn{kn}such that

Note that for all v ∈ X there holds

and by use of(2.9),

That gives x ∈ s-lim T−1（Pb）.So we get

n→∞nR(Tn)n

Thus we get(d).

(d)=⇒ (e)Let(d)be valid.It is clear that

By(2.2)and Lemma 2.3,there is a n∗∈ N such that

That gives(e).

(e)=⇒ (a)Let(e)hold.Then(2.1)is valid.Hence we have(1.5)by Lemma 2.1(c), that is(a)holds.

3 Example

In Theorem 1.1,one thing worth to notice is condition(e),which claims that the strong convergence of the LSA{(Xn,Tn)}n∈Nis equivalent to

Note that the examination of this condition does not involve any computation of operator norm or generalized-inverse,which are unavoidable in the examination of the stability condition(condition(a)in Theorem 1.1).Here we willlook at a simple example to see how the condition(e)can be used in specific integralequation.

Example 1Let X:=L2[−π,π],and we choose the approximation space as

We consider the below integralequation of the second kind:

where f ∈ L2[−π,π]is known,and we want to get ϕ.Let{Xn,Tn}be the LSA of the equation.It is easy to check that N(T)=span{x},and for x,it has the Fourier series As the non-zero coeffi cients in the series has infinite term,so it is obvious that there is no n∗∈Nsuch that N(T) ⊆ Xn∗,namely,the convergence condition(e)in Theorem 1.1 doesrding to the Theorem 1.1,the stability condition fails in this case,and

Here we look again on the stability condition(a),namely,

We notice that to examine this condition,we need to compute generalized-inverse and operator norm,the cost ofwhich is almost equalto computing the minimalspectralof Tn.Thus, it is hard to find a unified method to achieve this task.

Condition(e)in Theorem 1.1 also give the clue to choose convergent approximation scheme.For Example 1,to guarantee the convergence,we choose the approximation space as

The above is the subspace spanned by the first(n+1)terms of the sequence of Legendre polynomial on[−π,π].Now the LSA{Xn,Tn}possesses convergency as a result of X2⊇N(T).

[1]Du N.The basic principles for stable approximations to orthogonal generalized inverses of linear operators in Hilbert spaces[J].Numer.Funct.Anal.Optim.,2005,26(6):675–708.

[2]Du N.Finite-dimensionalapproximation settings for infi nite-dimensional Moore–Penrose inverses[J]. SIAM J.Numer.Anal.,2008,46(3):1454–1482.

[3]Groetsch C W,Neubauer A.Convergence of a general pro jection method for an operator equation of the fi rst kind[J].Houston J.Math.,1988,14(2):201–208.

[4]Groetsch C W.On a regularization-Ritz method for Fredholm equations of the fi rst kind[J].J.Integ. Equ.,1982,4(2):173–182.

[5]Seidman T I.Nonconvergence results for the application of least-squares estimation to ill-posed problems[J].J.Optim.The.Appl.,1980,30(4):535–547.

[6]Kress R.Linear integral equations(2rd ed.)[M].New York:Springer-Verlag,1999.

第二类算子方程最小二乘投影法的收敛性条件

杜书楷
(武汉大学数学与统计学院,湖北 武汉 430072)

本文研究针对第二类紧算子方程的最小二乘投影法的收敛条件.通过泛函分析及广义逆理论,得到了四个新的互相等价的收敛性条件,这些条件建立起了几种不同收敛性之间的联系并为人们检验逼近框架的收敛性提供了更多地选择.文中也给出了对一些简单且重要的例子的研究,以作为主要定理应用的范例.

收敛条件;最小二乘投影法;第二类算子方程

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∗Received date:2014-12-19 Accepted date:2015-05-06

Biography:Du Shukai(1990–),male,born at Wuhan,Hubei,master,major in Numerical Analysis.