Now coming to the integral equation of second kind, we can write equation (2.3.4) as
Lϕ=f
where L := I −A with I is the identity. We have developed tools for analyzing uniqueness and existence of the integral equation of second kind. Due to Frigyes Riesz (1880−1956) we know that the null space of the operatorL, i.e.,
N(L) :={ϕ∈X:Lϕ= 0}
is a finite dimensional subspace and its range is a closed linear subspace. The boundedness of the inverse operator L is confirmed by the following fundamental result of the Riesz theory [22].
Theorem 2.17. Let X be the normed space and A : X → X is a compact linear operator. Then the operatorI−A is injective if and only if it is surjective. Also the inverse operator(I−A)−1:X →X is bounded provided that I−A is injective.
With this knowledge we are able to conclude the following result which ensures the uniqueness and existence of the integral equation of second kind.
Corollary 2.18. Let A :X → X be a compact linear operator on a normed space X. If the homogeneous equation
ϕ−Aϕ= 0
only has the trivial solution ϕ = 0, then for each f ∈ X the corresponding inho-mogeneous equation (2.3.4) has a unique solution ϕ∈X and this solution depends continuously onf.
Thus with the help of the Riesz theory we are able to solve integral equations of the second kind, which arise in many practical problems in the theory of fluid dynamics, electromagnetic or acoustics.
2.4 Tikhonov Regularization
In 1923, Hadamard [18] defined a well-posed problem by postulating the following three properties:
• Existence of the solution.
• Uniqueness of the solution.
• Continuous dependence of the solution on the data.
If one of the above properties is violated then we can say that the problem is ill-posed. We give the definition of a well-posed problem in the setting of an operator equation.
18 Chapter 2. Basic Tools
Definition 2.19. Consider a bounded linear operator A :X → Y from a normed space X to a normed space Y. If the operator A is bijective and its inverse is con-tinuous, then the equation
Aϕ=f is called well-posed, otherwise it is called ill-posed.
Theorem 2.20. Let X and Y be two normed spaces and let A : X → Y be a compact linear operator . Then the integral equation of the first kind
Aϕ=f (2.4.1)
is ill-posed provided that the normed spaceX is infinite dimensional.
Proof. On contrary we assume that inverse operatorA−1 is bounded then the prod-uct ofA−1A=I is compact on X (see Theorem 2.16 of [22]), which is not possible because the identity operator I is compact only on finite dimensional spaces
(com-pare Theorem 2.20 in [22]).
This theorem tells us that the linear integral equations of the first kind with continuous or weakly singular kernels are examples of ill-posed problems. The third postulate described by Hadamard for well-posed problems is difficult for integral equation of the first kind. Due to the discontinuity of the inverse operator A−1 small changes in the data leads to unstable solutions. In order to obtain a stable solution we have to consider the third condition postulated by Hadamard.
The basic idea to deal with the instability of such ill-posed problems is to find a bounded approximationRα to the unbounded operator A−1 depending on some parameterα. The strategy to find such a bounded approximation Rα is known as the regularization scheme.
Definition 2.21. A family of bounded linear operators defined on the normed spaces X and Y, such that
Rα :Y →X, α >0
is called a regularization scheme for an injective operator A:X →Y, if
α→0limRαAϕ=ϕ, ϕ∈X. (2.4.2)
The limit in equation (2.4.2) describes that Rα tends pointwise to A−1. In the following theorem we observe two fundamental properties of the regularization schemeRα for compact operators.
Theorem 2.22. Let A:X → Y be the compact operator on the normed spaces X and Y with dimX = ∞ and a regularization scheme Rα, α > 0. Then the family Rα, α >0 of bounded operators cannot be uniformaly bounded with respect toα and the operators Rα can not be norm convergent as α→0.
2.4 Tikhonov Regularization 19
Proof. Following [22], we assume on contrary basis that the regularization operator Rα is bounded such that kRαk < C for all α > 0 with some constant C. For all f ∈A(X) and in the view of equation (2.4.2) we have Rαf → A−1f when α → 0.
Due to our assumption we can deduce that A−1f
≤Ckfk, i.e., A−1 :A(X)→X is bounded. Theorem2.20 leads us to a contradiction.
We prove the second statement with the assumption that we have the norm convergence. Then there exists α > 0 such that kRαA−Ik < 1/2. Now for all
leads us to the same contradiction as above.
The regularization scheme converges pointwise such thatRαf →A−1 forα→0 holds for allf ∈A(X). On the other hand if the data is perturbed by some noise such that
fδ−f
≤δ, then for a regularization parameter α, we find an approximate solutionϕδ such that
ϕδα:=Rαfδ. To estimate the error in the solution we write,
ϕδα−ϕ = Rαfδ−ϕ
= Rαfδ−Rαf+Rαf−ϕ
= Rαfδ−Rαf+RαAϕ−ϕ.
Using the triangle inequality we obtain
kϕδα−ϕk ≤ kRαfδ−Rαfk+kRαAϕ−ϕk
≤ δkRαk+kRαAϕ−ϕk.
Thus we decomposed the error into two parts, the first term reflects the data error and the second term expresses the error between the regularization operator Rα and the inverse operator A−1. Theorem 2.22 tells us that the first term is not uniformly bounded with respect toα. It means this term increases asα→0, due to the unboundedness of the regularization operatorRα. The second term decreases as α→0 because of the limit defined in equation (2.4.2). This leads us to a difficult task how to choose the regularization parameterαsuch that we have an acceptable error level for the regularized solution. The accuracy of the approximation requires small errorkRαAϕ−ϕk, i.e., a small parameter α and at the same time for the stability of the problem we need a largeα. Thus we have some kind of compromise between the accuracy and the stability for the choice of α. The choice of the regularization parameter depending on the error levelδ is called a strategy.
20 Chapter 2. Basic Tools
Definition 2.23. A strategy is called regular if for all f ∈ A(X) and all fδ ∈ Y withkfδ−fk≤δ we have
Rα(δ)fδ→A−1f, δ →0.
In the area of inverse problems there are several strategies for the choice of regularization parameter α, for a comprehensive view see for example [9]. We can divide them into the class of a priori and a posteriori strategies. The a priori strategies would be based on some additional information about the problem, for example the information about the smoothness properties of the exact solution.
These strategies are not widely used because this kind of information is usually not available. So we mainly focused on the a posteriori strategies of which one is the followingdiscrepancy orresidual principle introduced by Morozov [31].
Definition 2.24 (Discrepancy Principle). The regularization parameter α, for the error levelδ, should be chosen such that
kARαfδ−fδk=γδ with some fixed parameter γ≥1.
The basic idea of theTikhonovregularization is to approximate the fundamental solution by an element in the range of some integral operatorAbetween two Hilbert spacesXandY. So in the Tikhonov regularization we are interested to minimize the residualkAϕ−fk for allf ∈X. For the stability of the minimization procedure a penalty termαkϕk, with a regularization parameterα >0, is added. The existence and uniqueness of the minimizerϕn inX is proved by the following theorem.
Theorem 2.25. For the Hilbert spaces X and Y, we assume that A : X → Y is a compact linear operator. We also assume that the regularization parameter α is positive. For each f ∈X there exist a unique ϕα ∈X such that,
kAϕα−fk2+αkϕαk2= inf
ϕ∈X
kAϕ−fk2+αkϕk2 . (2.4.4) The minimizerϕα is given by the unique solution of the following equation
αϕα+A∗Aϕα=A∗f and depends continuously on f.
The right hand side of equation (2.4.4) is known as Tikhonov functional. The Tikhonov regularizationscheme is explicitly stated by the following theorem
Theorem 2.26. Let A : X → Y be a compact injective linear operator for the Hilbert spaces X and Y. Then for each α >0 the operator αI+A∗A:X →Y is a boundedly invertible and the operator
Rα:= (αI+A∗A)−1A∗ describes a regularization scheme withkRαk ≤ 1
2√ α.