几类反问题的偏差原则及收敛阶
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摘要
反问题的一个特别重要的属性就是它通常是“不适定”的数学问题，使得它无论在进行理论分析还是在进行数值计算时都有特定的困难。其算子形式为：求解这类问题的普遍方法是正则化方法：用一族与原不适定问题相“邻近”的适定问题的解去逼近原问题的解，如何建立有效的正则化方法是反问题领域中不适定问题研究的重要内容.其中Tikllonov提出求解不适定问题的变分正则化方法包括解如下问题F(u)：=||Au-f_δ||~2+a||u||~2=min，其中a是大于0的常数，称为正则化参数．
如何有效地选取正则化参数是此方法的关键，偏差原则是选取正则化参数的一类重要方法．
首先，本文讨论了基于一类动力系统方法的偏差原则及收敛阶．
假设(1)是Hilbert空间中的可解方程，||A||<∞并且R(A)是非闭的，所以问题(1)是不适定的．可解方程(1)等价于Bu=A~*f，B：=A~*A令(?)(t)是一个单调递减函数，满足(?)(t)>0，lim_(t→0)(?)(t)=0，lim_(t→∞)sup_(t/2)≤s≤t|(?)|(?)~(-2)(t)=0．
解决(1)的动力系统方法，包含解下面的Cauchy问题：此处u_0是任意的．而且可以证明对任意的u_0：上述问题对所有大于零的t都有唯一解．存在y：=u(∞)：=lim_(t→∞)u(t)，Ay=f.并且y是(1)的唯一极小范数解．这些结论在[3]中被证明．如果给定f_δ，使得||f-f_δ||≤δ，那么在f被f_δ代替情况下，定义u_δ(t)为上述问题的解．定义t_δ为停止时刻，t_δ满足lim_(δ→0)||u_δ(t_δ)-y||=0且lim_(δ→0)t_δ=∞．假设f_δ⊥N(A~*)，基于此类动力系统方法的偏差原则如下：
定理1如果A是Hilbert空间的有界线性算子，方程(1)可解，||f_δ||>δ，f_δ⊥N(A~*)，并且c(t)是一个单调递减函数，满足那么方程有唯一的解t_δ，并且成立，此处y是(1)的唯一极小范数解．
进一步地，对由该偏差原则得到的解收敛性作出如下估计：
定理2如果A是Hilbert空间的有界线性算子，方程(1)可解，y是(1)的唯一极小范数解，u_(δ，(?))(t_σ)由动力系统方程解得．那么
令y=A~*z，||z||≤E，可以选取(?)(t_δ)，使得
本文第二部分讨论了求解不适定问题的几类动力系统方法及第二种偏差原则：
(1)自伴算子的动力系统方法
关于自伴算子的DSM方法可如下构造．考虑方程其中，a是大于0的常数．第一个结果由定理3给出：
定理3假设(1)中的A是自伴算子，即A=A~*．如果Ay=f且y⊥V，那么第二个结果基于定理3，假设||f_δ-f||≤δ，则可以得到方程(1)的稳定解u_(a，δ)(t)
定理4存在t=t_δ，lim_(δ→0)t_δ=∞，且a=a(δ)，lim_(δ→0)a(δ)=0，使得u_δ：=u_a(δ)．δ(t_δ)满足
在定理3和定理4证明之后，我们给出a(δ)，t_δ的选取方法．
(2)改进的自伴算子的动力系统方法
我们重新考察一下将其中的a换为a(t)．
定理5假设a(t)>0是单调递减的连续函数，且满足此问题的解为那么
进一步地，定理5可得到厅程(1)的稳定解．
定理6存在一个停止时刻t_δ，lim_(δ→0)t_δ=∞，使得其中，u_δ=u_δ(t_δ)是的解，其中||f_δ-f||≤δ．
(3)动力系统方法的第二种偏差原则
在前面使用的动力系统方法的偏差原则中，我们假设f_δ⊥N(A~*)．如果此假设不成立，我们可使用如下的偏差原则：其中，1 定理7如果其中f=Ay，y⊥N，a(t)>0是单调递减连续函数，使得那么
DSM方法u=-u+T_(a(t)~(-1)A~*f，u(0)=0，可得到方程(1)的稳定解，陈述如下：
定理8如果选取t_δ使得成立，那么的解满足
定理9假设a(t)>0是单调递减二阶可导函数，且满足：方程其中，1Cδ．
本文第三部分计论了一种新的偏差原则及收敛阶：
定理10假设A是Hilbcrt空间H中线性有界算子，f=Ay，y⊥N，||f_δ-f||≤δ，||f_δ||≥Cδ，C是常数，C∈(1，2)，且u_(a，δ)满足不等式：其中b是大于0的常数，且C~2>1+b．那么对任意给定的δ>0，方程有解u_(a，δ)满足以及
定理10可叙述为：
定理10’假定
i)A足Hilbcrt空间的有界线性算子，
ii)方程Au=f可解，y是极小范数解，
iii)||f_δ-f||≤δ，||f_δ||≥Cδ，此处C为大于1的常数．
那么
a)方程对任意δ>0可解，其中u_δ．(?)满足不等式F(u_δ．(?))≤m+(C~2-1-b)δ~2，F(u)：=||A(u)-f_δ||_2+(?)||u||_2，m=m(δ．(?))：=inf_uF(u)，b为大于0的常数，C~2>1+b，
b)如果(?)=c(δ)是(3)的解，记u_δ：=u_δ．c(δ)，那么lim_(δ→0)||u_δ-y||=0．
在此基础上，对由该偏差原则得到的解收敛性作出估计．进一步得到；
定理11假定
i)A是Hilbert空间的有界线性算子，
ii)方程Au=f可解，y是极小范数解，
iii)||f_δ-f||≤δ，||f_δ||≥Cδ．此处C为大于1的常数．那么
c)如果c=c(δ)是(3)的解，记u_δ：=u_δ．c(δ)，那么||u_δ-y||=O(δ~(1/2))．
d)若||u_δ-y||=o(δ~(1/2))，则R(A)为有限维，即收敛阶O(δ~(1/2))是最优的．
A character of Inverse Problem is that it is an ill-posed problem usually. Operator formula:Method used usually of solving ill-posed problems is regularization strategy:a family of well-posed problem converge ill-posed problem. Tikhonov Variational regularization method for stable solution of (1) consists of solving the problem F(u) := ||Au-f_δ||~2+ a||u||~2=min,where a is regularization parameters > 0,α= const. An important method for choosing regularization parameter is discrepancy principle.
Firstly.this paper study a discrepancy principle and orders of convergence based Dynamical Systems Method (DSM).
Assume (l)is a solvable linear equation in a Hilbert space. ||A||<∞, and R(A) is not closed, so problem (1) is ill-posed. A solvable quation (1) is equivalent toBu = A~*f,B :=A~*A. Let (?)(t) be a monotone, decreasing function,(?)(t)>0,lim_(t→0)(?)(t)= 0,lim_(t→∞)sup_(1/2)≤s≤t|(?)|(?)~(-2)(t) = 0.
A DSM (dynamical systems method) for solving (1), consists of solving the followingCauchy problem:where u_0 is arbitrary, and proving that, for any u_0.Cauchy problem has a unique solution for all t>0, there exists y:=u(∞) := lirn_(t→∞)u(t), Ay =f, and y is the unique minimal-norm solution to (1). These results are proved in [3]. If f_δis given. such that ||f-f_δ||≤δ, then u_δ(t) is denned as the solution to Cauchy problem with f replaced by fg. The stopping time is defined as a number t_δ为such that lim_(δ→0)||u_δ(t_δ)-y||=0,and lim_(δ→0)t_δ=∞.Let us assume f_δ⊥N(A~*). Then this discrepancy principle:
theorem 1 If A is a bounded linear operator in a Hilbert space H, equation (1) is solvable,||f_δ||>δ,fδ⊥N(A~*) , and (?)(t) satisfies the assumptions stated above, then equationhas a unique solution t_δ, andholds, where y is the unique minimalnorm solution to (1).
Furthermore,we estimate the convergence of solution for the discrepancy principle: theorem 2 If A is a bounded linear operator in a Hilbert space H, equation (1)is solvable, u_(δ,∈)(t_δl) from DSMobtained.then
Lety=A~*z,||z||≤E , can choose t(t_δ),such that
The second part of this paper study some DSM solving ill-posed problem and another discrepancy principle:
(1)DSM of selfadjoint operator
The DSM for solving equation (1) with a linear selfadjoint operator can be constructedas follows. Consider the problemwhere a=const > 0. Our first result is formulated as Theorem 3.
theorem 3 If Ay = f and y⊥N , then
Our second result shows that the method, based on Theorem 1, gives a stable solution of the equation Au = f. Assume that ||f_δ-f||≤δ, and let u_(a,δ)(t)be the solution towith fs in place of f.
theorem 4 There exist t = t_δ, lim_(δ→0)t_δ=∞:and a=a(δ),lim_(δ→0)a(δ)=0 , such that u_δ:= u_a(δ),δ(t_δ) satisfies
We will discuss the ways to choose a(δ) and t_δafter the proofs of these theorems are given.
(2)Improved DSM of selfadjoint operator
Consider problem u=i(A + ia)u_α-if,u(0)=0;u =((du)/(dt)), with a=a(t).
theorem 5 Assume that a(t) > 0 is a continuous function monotonically decayingto zero as t→∞andThe solution to this problem isthen
Furtlicrmorc,Thoorem 5 yields a stable solution to equation (1).
theorem 6 There exists a stopping time t_δ,lim_(δ→0)t_δ=∞, such thathold , where u_δ=u_δ(t_δ) solving||f_δ-f||≤δ.
(3) Another discrepancy principle based DSM
Suppose that the assumption f_δ⊥N(A~*) does not hold. Then one can use the discrepancy principle of the form:
theorem 7 Ifwhere f = Ay,y⊥N ,that a(t) > 0 is a continuous function monotonically decaying to zero as t→∞andthen
DSM method u = -u + T_(a(t))~(-1)A~*f, u(0) = 0, yields a stable solution to problem (1).
theorem 8 If t_δis chosen so thatholds,then the solution ofsatisfies
theorem 9 Assume that a(t)>0 is a monotonically decaying twice continuously differentiate function.The equationhas a solution t = t_δ,lim_(δ→0)t_δ=∞,such that holds,where u_δis the solution to||f_δll>Cδ.
The third part of this paper study a new discrepancy principle and orders of convergence :
theorem 10 Assume that A is a bounded linear operator in a Hilbert space H, that f = Ay,y⊥N,||f_δ-f||≤6,||f_δ||≥Cδ,C=const,C∈(1.2),and u_(α,β)is any element which satisfies the inequality:whereb=const > 0 , and C~2>1+b. Then equationhas a solution for any fixedδ>0 , lim_(δ→0)a(δ)=0, and
Another version of theorem 10 :
theorem 10 ' Assume that
i)A is a bounded linear operator in a Hilbert spaceⅡ,
ii)Thc equation Au = f solvable , y is the unique minimalnorm solution ,
iii) ||f_δ-f||≤δ,||f_δ||≥Cδ.C = const>1.
then
a)The equationhas a solution tor any fixedδ>0, where u_(δ,(?)) satisfied F(u_(δ,(?)))≤m + (C~2-1-b)δ~2,F(u) := ||A(u)-f_δ||_2+∈||u||_2,m=m(δ.∈) := inf_uF(u), b=canst>0,C~2>1+b,
b) If (?) = (?)(δ) solved ||Au_(δ,(?))-f_δ||=Cδ,u_δ:=u_δ.(?)(δ),then lim_(δ→0)||u_δ-y||=0.
Furthermore,we estimate the convergence of solution for the discrepancy principle:
theorem 11 Assume that
i)A is a bounded linear operator in a Hilbert space H,
ii)The equation Au = f solvable , y is the unique minimalnorm solution ,
iii) ||(?)_δ-(?)||≤δ,||(?)_δ||≥Cδ,C = const>1.
then
c)If t = (?)(δ) solved ||Au_(δ,(?))-f_δ||= Cδ,u_δ:= u_(δ,(?))(δ),then ||u_δ-y||=O(δ~(1/2)).
d)If ||u_δ-y||=o(δ~(1/2)), then R(A) has to be finite dimension, that is, orders of convergence O(δ~(1/2))is optimal.
如何有效地选取正则化参数是此方法的关键，偏差原则是选取正则化参数的一类重要方法．
首先，本文讨论了基于一类动力系统方法的偏差原则及收敛阶．
假设(1)是Hilbert空间中的可解方程，||A||<∞并且R(A)是非闭的，所以问题(1)是不适定的．可解方程(1)等价于Bu=A~*f，B：=A~*A令(?)(t)是一个单调递减函数，满足(?)(t)>0，lim_(t→0)(?)(t)=0，lim_(t→∞)sup_(t/2)≤s≤t|(?)|(?)~(-2)(t)=0．
解决(1)的动力系统方法，包含解下面的Cauchy问题：此处u_0是任意的．而且可以证明对任意的u_0：上述问题对所有大于零的t都有唯一解．存在y：=u(∞)：=lim_(t→∞)u(t)，Ay=f.并且y是(1)的唯一极小范数解．这些结论在[3]中被证明．如果给定f_δ，使得||f-f_δ||≤δ，那么在f被f_δ代替情况下，定义u_δ(t)为上述问题的解．定义t_δ为停止时刻，t_δ满足lim_(δ→0)||u_δ(t_δ)-y||=0且lim_(δ→0)t_δ=∞．假设f_δ⊥N(A~*)，基于此类动力系统方法的偏差原则如下：
定理1如果A是Hilbert空间的有界线性算子，方程(1)可解，||f_δ||>δ，f_δ⊥N(A~*)，并且c(t)是一个单调递减函数，满足那么方程有唯一的解t_δ，并且成立，此处y是(1)的唯一极小范数解．
进一步地，对由该偏差原则得到的解收敛性作出如下估计：
定理2如果A是Hilbert空间的有界线性算子，方程(1)可解，y是(1)的唯一极小范数解，u_(δ，(?))(t_σ)由动力系统方程解得．那么
令y=A~*z，||z||≤E，可以选取(?)(t_δ)，使得
本文第二部分讨论了求解不适定问题的几类动力系统方法及第二种偏差原则：
(1)自伴算子的动力系统方法
关于自伴算子的DSM方法可如下构造．考虑方程其中，a是大于0的常数．第一个结果由定理3给出：
定理3假设(1)中的A是自伴算子，即A=A~*．如果Ay=f且y⊥V，那么第二个结果基于定理3，假设||f_δ-f||≤δ，则可以得到方程(1)的稳定解u_(a，δ)(t)
定理4存在t=t_δ，lim_(δ→0)t_δ=∞，且a=a(δ)，lim_(δ→0)a(δ)=0，使得u_δ：=u_a(δ)．δ(t_δ)满足
在定理3和定理4证明之后，我们给出a(δ)，t_δ的选取方法．
(2)改进的自伴算子的动力系统方法
我们重新考察一下将其中的a换为a(t)．
定理5假设a(t)>0是单调递减的连续函数，且满足此问题的解为那么
进一步地，定理5可得到厅程(1)的稳定解．
定理6存在一个停止时刻t_δ，lim_(δ→0)t_δ=∞，使得其中，u_δ=u_δ(t_δ)是的解，其中||f_δ-f||≤δ．
(3)动力系统方法的第二种偏差原则
在前面使用的动力系统方法的偏差原则中，我们假设f_δ⊥N(A~*)．如果此假设不成立，我们可使用如下的偏差原则：其中，1
DSM方法u=-u+T_(a(t)~(-1)A~*f，u(0)=0，可得到方程(1)的稳定解，陈述如下：
定理8如果选取t_δ使得成立，那么的解满足
定理9假设a(t)>0是单调递减二阶可导函数，且满足：方程其中，1
本文第三部分计论了一种新的偏差原则及收敛阶：
定理10假设A是Hilbcrt空间H中线性有界算子，f=Ay，y⊥N，||f_δ-f||≤δ，||f_δ||≥Cδ，C是常数，C∈(1，2)，且u_(a，δ)满足不等式：其中b是大于0的常数，且C~2>1+b．那么对任意给定的δ>0，方程有解u_(a，δ)满足以及
定理10可叙述为：
定理10’假定
i)A足Hilbcrt空间的有界线性算子，
ii)方程Au=f可解，y是极小范数解，
iii)||f_δ-f||≤δ，||f_δ||≥Cδ，此处C为大于1的常数．
那么
a)方程对任意δ>0可解，其中u_δ．(?)满足不等式F(u_δ．(?))≤m+(C~2-1-b)δ~2，F(u)：=||A(u)-f_δ||_2+(?)||u||_2，m=m(δ．(?))：=inf_uF(u)，b为大于0的常数，C~2>1+b，
b)如果(?)=c(δ)是(3)的解，记u_δ：=u_δ．c(δ)，那么lim_(δ→0)||u_δ-y||=0．
在此基础上，对由该偏差原则得到的解收敛性作出估计．进一步得到；
定理11假定
i)A是Hilbert空间的有界线性算子，
ii)方程Au=f可解，y是极小范数解，
iii)||f_δ-f||≤δ，||f_δ||≥Cδ．此处C为大于1的常数．那么
c)如果c=c(δ)是(3)的解，记u_δ：=u_δ．c(δ)，那么||u_δ-y||=O(δ~(1/2))．
d)若||u_δ-y||=o(δ~(1/2))，则R(A)为有限维，即收敛阶O(δ~(1/2))是最优的．
A character of Inverse Problem is that it is an ill-posed problem usually. Operator formula:Method used usually of solving ill-posed problems is regularization strategy:a family of well-posed problem converge ill-posed problem. Tikhonov Variational regularization method for stable solution of (1) consists of solving the problem F(u) := ||Au-f_δ||~2+ a||u||~2=min,where a is regularization parameters > 0,α= const. An important method for choosing regularization parameter is discrepancy principle.
Firstly.this paper study a discrepancy principle and orders of convergence based Dynamical Systems Method (DSM).
Assume (l)is a solvable linear equation in a Hilbert space. ||A||<∞, and R(A) is not closed, so problem (1) is ill-posed. A solvable quation (1) is equivalent toBu = A~*f,B :=A~*A. Let (?)(t) be a monotone, decreasing function,(?)(t)>0,lim_(t→0)(?)(t)= 0,lim_(t→∞)sup_(1/2)≤s≤t|(?)|(?)~(-2)(t) = 0.
A DSM (dynamical systems method) for solving (1), consists of solving the followingCauchy problem:where u_0 is arbitrary, and proving that, for any u_0.Cauchy problem has a unique solution for all t>0, there exists y:=u(∞) := lirn_(t→∞)u(t), Ay =f, and y is the unique minimal-norm solution to (1). These results are proved in [3]. If f_δis given. such that ||f-f_δ||≤δ, then u_δ(t) is denned as the solution to Cauchy problem with f replaced by fg. The stopping time is defined as a number t_δ为such that lim_(δ→0)||u_δ(t_δ)-y||=0,and lim_(δ→0)t_δ=∞.Let us assume f_δ⊥N(A~*). Then this discrepancy principle:
theorem 1 If A is a bounded linear operator in a Hilbert space H, equation (1) is solvable,||f_δ||>δ,fδ⊥N(A~*) , and (?)(t) satisfies the assumptions stated above, then equationhas a unique solution t_δ, andholds, where y is the unique minimalnorm solution to (1).
Furthermore,we estimate the convergence of solution for the discrepancy principle: theorem 2 If A is a bounded linear operator in a Hilbert space H, equation (1)is solvable, u_(δ,∈)(t_δl) from DSMobtained.then
Lety=A~*z,||z||≤E , can choose t(t_δ),such that
The second part of this paper study some DSM solving ill-posed problem and another discrepancy principle:
(1)DSM of selfadjoint operator
The DSM for solving equation (1) with a linear selfadjoint operator can be constructedas follows. Consider the problemwhere a=const > 0. Our first result is formulated as Theorem 3.
theorem 3 If Ay = f and y⊥N , then
Our second result shows that the method, based on Theorem 1, gives a stable solution of the equation Au = f. Assume that ||f_δ-f||≤δ, and let u_(a,δ)(t)be the solution towith fs in place of f.
theorem 4 There exist t = t_δ, lim_(δ→0)t_δ=∞:and a=a(δ),lim_(δ→0)a(δ)=0 , such that u_δ:= u_a(δ),δ(t_δ) satisfies
We will discuss the ways to choose a(δ) and t_δafter the proofs of these theorems are given.
(2)Improved DSM of selfadjoint operator
Consider problem u=i(A + ia)u_α-if,u(0)=0;u =((du)/(dt)), with a=a(t).
theorem 5 Assume that a(t) > 0 is a continuous function monotonically decayingto zero as t→∞andThe solution to this problem isthen
Furtlicrmorc,Thoorem 5 yields a stable solution to equation (1).
theorem 6 There exists a stopping time t_δ,lim_(δ→0)t_δ=∞, such thathold , where u_δ=u_δ(t_δ) solving||f_δ-f||≤δ.
(3) Another discrepancy principle based DSM
Suppose that the assumption f_δ⊥N(A~*) does not hold. Then one can use the discrepancy principle of the form:
theorem 7 Ifwhere f = Ay,y⊥N ,that a(t) > 0 is a continuous function monotonically decaying to zero as t→∞andthen
DSM method u = -u + T_(a(t))~(-1)A~*f, u(0) = 0, yields a stable solution to problem (1).
theorem 8 If t_δis chosen so thatholds,then the solution ofsatisfies
theorem 9 Assume that a(t)>0 is a monotonically decaying twice continuously differentiate function.The equationhas a solution t = t_δ,lim_(δ→0)t_δ=∞,such that holds,where u_δis the solution to||f_δll>Cδ.
The third part of this paper study a new discrepancy principle and orders of convergence :
theorem 10 Assume that A is a bounded linear operator in a Hilbert space H, that f = Ay,y⊥N,||f_δ-f||≤6,||f_δ||≥Cδ,C=const,C∈(1.2),and u_(α,β)is any element which satisfies the inequality:whereb=const > 0 , and C~2>1+b. Then equationhas a solution for any fixedδ>0 , lim_(δ→0)a(δ)=0, and
Another version of theorem 10 :
theorem 10 ' Assume that
i)A is a bounded linear operator in a Hilbert spaceⅡ,
ii)Thc equation Au = f solvable , y is the unique minimalnorm solution ,
iii) ||f_δ-f||≤δ,||f_δ||≥Cδ.C = const>1.
then
a)The equationhas a solution tor any fixedδ>0, where u_(δ,(?)) satisfied F(u_(δ,(?)))≤m + (C~2-1-b)δ~2,F(u) := ||A(u)-f_δ||_2+∈||u||_2,m=m(δ.∈) := inf_uF(u), b=canst>0,C~2>1+b,
b) If (?) = (?)(δ) solved ||Au_(δ,(?))-f_δ||=Cδ,u_δ:=u_δ.(?)(δ),then lim_(δ→0)||u_δ-y||=0.
Furthermore,we estimate the convergence of solution for the discrepancy principle:
theorem 11 Assume that
i)A is a bounded linear operator in a Hilbert space H,
ii)The equation Au = f solvable , y is the unique minimalnorm solution ,
iii) ||(?)_δ-(?)||≤δ,||(?)_δ||≥Cδ,C = const>1.
then
c)If t = (?)(δ) solved ||Au_(δ,(?))-f_δ||= Cδ,u_δ:= u_(δ,(?))(δ),then ||u_δ-y||=O(δ~(1/2)).
d)If ||u_δ-y||=o(δ~(1/2)), then R(A) has to be finite dimension, that is, orders of convergence O(δ~(1/2))is optimal.
引文
[1] Andreas Kirsch. An introduction to the mathematical theory of inverse problems[M]. New York: Springer-Vcrlag, 1999, 1-122. .
[2] Morozov V.,Methods of Solving Incorrectly Posed Problems,Springcr Verlag.New York,1984.
[3] A. G. Ramm.Dynamical systems method solving operator equation[J], Communic.in Nonlinear Sci.and Numer.Simulation. 2004, 9(2). 383-402.
[4] A. G. Ramm, Discrepancy principle for the Dynamical systems,Communic.iu Nonlinear Sci.andNumer.Simulation, 10(N1)(2005),95-101.
[5] A. G. Ramm, Discrepancy principle for the Dynamical systems ILCommunic.in Nonlinear Sci.and Numer.Simulation,(2007)at press.
[6] A. G. Ramm, A new Discrepancy principle,J Math Anal Appl 7(2005),342-345.
[7] T. Kato, Perturbation theory for linear operators, Springer Vcrlag, New York,1984.
[8] O. Liskovetz, Regularization of equations with a closed linear operator, Diff. Equations.7, (1970), 972-976.
[9] A. G. Ramm, Inverse problems. Springer, New York, 2005.
[10] H W Engl, M Hanke,Neubauer.Regularization of inverse problems[M]. Dordrecht: Kluwer.1996.
[11] Lars Elden.An algorithm for the regularization of ill-conditioned .banded least squares problems[J].Siam.Sci.Stat.Comput,1984,5(1),237-254.
[12] A Neubauer.On accelerated iteration for nonlinear ill-posed problems in Hilbert scalcs[J].Numer. Math, 2000,85(2),309-328.
[13] Tikhonov,A.N.and Arsenin, V.Y. Solution of ILL-Posed Problems,Wiley,New York, 1997.
[14] Engl.Ⅱ.W.and Groetsch.C.W.Invcrsc and ILL-Poscd Problems,Acad.Prcs. Orlando, 1987.
[15] F R Gantmacher. The Theory of Matrices[M]. New York: Chelsea, 1959 .
[16] Laudwcbcr,L.An iteration formula for Fredholm integral equation of the first kind.Amer.J.Math. 73(1951),615-624
[17] NemirovskiiA.S.and Polyak.B.T. Iteration methods for solving linear ill-posed problems under precise information.I,Engry,Cybernetics.22(1981) 1-11.
[18] Brakhage.H. On ill-posed problem and the methods of conjugate gradients, I, Engry, Cybernetics, 22(1984)165-175.
[19] He Guoqiang and Liu Linxian, A kind of implicit iteration methods for ill-posed operator equations,J.of Comp.Math,17(1993)76-85.
[20] Hanke,M. Accelerated Landweber iterations for the solution of ill-posed equationm Numer, Math, 60(1991),341-373
[21] Vaininkko G. On the Optimality of Methods for ILL-Posed Problems, Z. Anal. Anw. 1987, 6(4),351-362.
[22] 肖庭轩，于慎根，王彦飞等．反问题的数值解法[M]．北京：科学出版社，2003， 1-170．
[2] Morozov V.,Methods of Solving Incorrectly Posed Problems,Springcr Verlag.New York,1984.
[3] A. G. Ramm.Dynamical systems method solving operator equation[J], Communic.in Nonlinear Sci.and Numer.Simulation. 2004, 9(2). 383-402.
[4] A. G. Ramm, Discrepancy principle for the Dynamical systems,Communic.iu Nonlinear Sci.andNumer.Simulation, 10(N1)(2005),95-101.
[5] A. G. Ramm, Discrepancy principle for the Dynamical systems ILCommunic.in Nonlinear Sci.and Numer.Simulation,(2007)at press.
[6] A. G. Ramm, A new Discrepancy principle,J Math Anal Appl 7(2005),342-345.
[7] T. Kato, Perturbation theory for linear operators, Springer Vcrlag, New York,1984.
[8] O. Liskovetz, Regularization of equations with a closed linear operator, Diff. Equations.7, (1970), 972-976.
[9] A. G. Ramm, Inverse problems. Springer, New York, 2005.
[10] H W Engl, M Hanke,Neubauer.Regularization of inverse problems[M]. Dordrecht: Kluwer.1996.
[11] Lars Elden.An algorithm for the regularization of ill-conditioned .banded least squares problems[J].Siam.Sci.Stat.Comput,1984,5(1),237-254.
[12] A Neubauer.On accelerated iteration for nonlinear ill-posed problems in Hilbert scalcs[J].Numer. Math, 2000,85(2),309-328.
[13] Tikhonov,A.N.and Arsenin, V.Y. Solution of ILL-Posed Problems,Wiley,New York, 1997.
[14] Engl.Ⅱ.W.and Groetsch.C.W.Invcrsc and ILL-Poscd Problems,Acad.Prcs. Orlando, 1987.
[15] F R Gantmacher. The Theory of Matrices[M]. New York: Chelsea, 1959 .
[16] Laudwcbcr,L.An iteration formula for Fredholm integral equation of the first kind.Amer.J.Math. 73(1951),615-624
[17] NemirovskiiA.S.and Polyak.B.T. Iteration methods for solving linear ill-posed problems under precise information.I,Engry,Cybernetics.22(1981) 1-11.
[18] Brakhage.H. On ill-posed problem and the methods of conjugate gradients, I, Engry, Cybernetics, 22(1984)165-175.
[19] He Guoqiang and Liu Linxian, A kind of implicit iteration methods for ill-posed operator equations,J.of Comp.Math,17(1993)76-85.
[20] Hanke,M. Accelerated Landweber iterations for the solution of ill-posed equationm Numer, Math, 60(1991),341-373
[21] Vaininkko G. On the Optimality of Methods for ILL-Posed Problems, Z. Anal. Anw. 1987, 6(4),351-362.
[22] 肖庭轩，于慎根，王彦飞等．反问题的数值解法[M]．北京：科学出版社，2003， 1-170．