Ill-posed equations with transformed argumentdownloads.hindawi.com/journals/aaa/2003/572142.pdf · lution of ill-posed equations with closed operators, we present a new view on the

ILL-POSED EQUATIONS WITHTRANSFORMED ARGUMENT

SIMONE GRAMSCH AND EBERHARD SCHOCK

Received 16 December 2002

We discuss the operator transforming the argument of a function in the L2-setting. Here this operator is unbounded and closed. For the approximate so-lution of ill-posed equations with closed operators, we present a new view onthe Tikhonov regularization.

1. Introduction

In theory and applications, many kinds of equations occur with transformedargument, that is, with a transformation operator Tρ, defined on function spacesby Tρx = x ◦ ρ. Examples are differential or integral equations with delay, thealgebraic approach of Przeworska-Rolewicz [6] with involutions, reflections orrotations, control problems, and so forth.

In the spaces C(K) of continuous functions on a compact set K , the transfor-mation operator Tρ is completely discussed (see [10]). Here we will consider thequestion of (approximate) solvability of an equation

Tρx = y (1.1)

in the Hilbert space L2(K). The transformation operator in general is not con-tinuous and the range is not closed. Equations of type (1.1) are ill posed. We willuse Tikhonov regularization for the approximate solution of (1.1). For this, wehave to develop a theory of Tikhonov regularization for unbounded operatorsin Hilbert spaces.

2. The transformation operator

Let K ⊂Rn be a compact subset and X = L2(K). Let ρ : K → K be a continuoussurjective mapping with the property

(P) the image and the preimage of sets of measure zero are of measure zero.

Copyright © 2003 Hindawi Publishing CorporationAbstract and Applied Analysis 2003:13 (2003) 785–7912000 Mathematics Subject Classification: 65J22, 39B22, 47A05URL: http://dx.doi.org/10.1155/S1085337503303021

http://dx.doi.org/10.1155/S1085337503303021

786 Ill-posed equations with transformed argument

Remark 2.1. If ρ is continuously differentiable and the set Sρ of critical points of ρis of measure zero, then ρ has property (P). If ρ is continuously differentiable (orLipschitzian), it maps zero sets into zero sets. In every connecting component ofK \ Sρ, the map ρ is a diffeomorphism and ρ−1 maps zero sets into zero sets.

Let Tρ be the following transformation operator: for x ∈D(Tρ), we define

(Tρx

)(t)= x

(ρ(t)

), (2.1)

where the domain of Tρ is

D(Tρ)=

{x ∈ L2(K) :

∫K

∣∣x(ρ(t))∣∣2

<∞}. (2.2)

Then Tρ is a linear not necessarily bounded operator.

Theorem 2.2. If ρ satisfies property (P), then the operator Tρ : D(Tρ)→ L2(K) iswell defined, injective, and closed.

Proof. If two functions x, y differ by a function of measure zero, then

Tρx−Tρy = Tρ(x− y)= 0 (2.3)

by property (P), thus Tρ is well defined. If Tρx = 0, then x is equivalent to a zerofunction, hence x = 0 and Tρ is injective.

Let (xn) be a sequence in D(Tρ) with limn→∞ xn = x and limn→∞Txn = y.Then, there exists a subsequence (xnk ) with the property

limxnk (t)= x(t) a.e., limxnk(ρ(t)

)= y(t) a.e. (2.4)

The set M := {t ∈ K : limxnk (ρ(t)) �= x(ρ(t))} is of measure zero since M =ρ−1(M′) with M′ := {s∈ K : limxnk (s) �= x(s)}. Since M′ is of measure zero andρ has property (P), the set M is of measure zero; hence

limxnk(ρ(t)

)= x(ρ(t)

)a.e., x

(ρ(t)

)= y(t) a.e. (2.5)

So we see that x ∈D(Tρ) and Tρx = y. �

It is easy to see that the characteristic functions of measurable subsets of Kbelong to D(Tρ), hence the set of step functions is dense in D(Tρ). Now wewill show that the transformation operator is symmetric only in the trivial caseρ = id, and in opposite to the case of C(K) the transformation operator is notisometric except for ρ = id.

Theorem 2.3. Let Tρ : D(Tρ)→ X be a transformation operator with property (P).Then,

(a) Tρ is symmetric if and only if ρ = id;(b) Tρ is isometric if and only if ρ = id.

S. Gramsch and E. Schock 787

Proof. If Tρ is symmetric, then for every x, y ∈ D(Tρ) we have 〈Tρx, y〉 =〈x,Tρ y〉. Since the unit function x = 1∈D(Tρ), we get with Tρx = x

∫Ky(t)dt =

∫Ky(ρ(t)

)dt. (2.6)

If there exists a t0 ∈ K with ρ(t0) �= t0, then there is a nonzero step function ywith the value one in an open cube C containing t0, such that C∩ ρ(C) =∅,and the value zero outside of C. Then,

∫Ky(t)dt =meas(K ∩C) �= 0,

∫Ky(ρ(t))dt = 0. (2.7)

Hence ρ(t) = t for all t ∈ K . If Tρ is isometric, then for every x, y ∈ D(Tρ) wehave 〈Tρx,Tρ y〉 = 〈x, y〉. If we again use the unit function x and the step func-tion y, then we have the desired result. �

3. Tikhonov regularization of equations with closed operators

Here we discuss the convergence and the speed of convergence for equations inHilbert spaces

Tx = y, (3.1)

where T : D(T)→ X2 is a densely defined closed operator with D(T) ⊂ X1. Wewill see that the results and the proofs are very similar to the case of continuousoperators in Hilbert spaces, especially we will choose a different method as in[9]. Our investigations strongly depend on the following result of von Neumann[11].

Theorem 3.1. Let T : D(T) → X be a closed and densely defined operator in aHilbert space X . Then the operators

B = (I +T∗T)−1

, C = T(I +T∗T

)−1,

A= T∗T(I +T∗T

)−1 (3.2)

are continuous and bounded by

‖B‖ ≤ 1, ‖C‖ ≤ 12, ‖A‖ ≤ 1. (3.3)

Proof. The continuity and the bound of B is shown by von Neumann, also thecontinuity of C, see also [8]. The equation A+B = I is easy to verify. Since B ispositive definite, ‖B‖ ≤ 1, we get

0≤A= I −B ≤ I, (3.4)


hence ‖A‖ ≤ 1. Finally from

C∗C = BA= B−B2, (3.5)

we obtain, with 0≤ B ≤ I ,

‖BA‖ = sup0≤β≤1

(β−β2)= 1

4, ‖C‖ ≤ 1

2. (3.6)

�

By easy calculations, we can show the following corollary.

Corollary 3.2. LetAα = T∗T(αI +T∗T)−1, Bα = (αI +T∗T)−1, andCα = T(αI+T∗T)−1. Then for all positive reals α, the operators Aα, Bα, and Cα are continuouswith

∥∥Aα

∥∥≤ 1,∥∥Bα

∥∥≤ 1α,

∥∥Cα

∥∥≤ 12√α. (3.7)

If T is not surjective, then equation

Tx = y (3.8)

is ill posed, also in the case when y ∈ RangeT . In this case, Tikhonov regulariza-tion is a widely used method for a stable approximation of the solution of (1.1)(see, e.g., [2]).

For every α > 0, we determine the approximation xα of the solution x of (3.8)using the equation

(αI +T∗T

)xα = T∗y. (3.9)

In the next theorem, we show that xα converge to x.

Theorem 3.3. Let T be injective, densely defined, and closed. Then for every y ∈RangeT , the elements xα defined by

xα = T∗(αI +TT∗

)−1y (3.10)

converge to the solution x of (3.8) if α tends to zero.

Proof. With the notation of Corollary 3.2, we have

xα = C∗α y = C∗α Tx = Aαx, xα− x = αBαx. (3.11)

The family of continuous operators αBα, α > 0, is uniformly bounded by one.Since A is injective, RangeAν is dense in X for 0 < ν ≤ 1. For x ∈ RangeAν, weobtain

xα− x = αBαAνu. (3.12)


Now we estimate

xα− x = αBαAνu= αν

(T∗Tα

)ν(I +

T∗Tα

)−1

Bνu (3.13)

and (if ν < 1)

∥∥xα− x∥∥≤ αννν(1− ν)1−ν

∥∥Bνu∥∥

≤ αννν(1− ν)1−ν‖u‖ (3.14)

(resp., ‖xα − x‖ ≤ αν‖u‖ = α‖u‖ if ν = 1). Therefore, we have convergence ona dense set, by the uniform boundedness principle we have convergence for allx ∈ X . �

Checking this proof, we see the following corollary.

Corollary 3.4. If x ∈ RangeAν, 0≤ ν≤ 1, then the speed of convergence

∥∥xα− x∥∥= �

(αν). (3.15)

In the ill-posed cases, that is, if T is not surjective, then (3.8) is not solvablefor a set of second category. Let {yδ,δ > 0} be a family of elements in X2 with‖y− yδ‖ ≤ δ. Then, we have to discuss the behaviour of xα,δ defined by

xα,δ = T∗(αI +TT∗

)−1yδ. (3.16)

Theorem 3.5. Let y ∈ Range(T), yδ ∈D(T∗), and ‖y− yδ‖ ≤ δ. Then,

∥∥xα− xα,δ∥∥≤ δ

2√α. (3.17)

Proof. We have

∥∥xα− xα,δ∥∥= ∥∥C∗α (y− yδ

)∥∥≤ ∥∥C∗α ∥∥ · δ ≤ δ

2√α. (3.18)

�

If we additionally assume x∈ RangeAν, then we have the following corollary.

Corollary 3.6. Let x ∈ RangeAν, 0 < ν≤ 1, then

∥∥x− xα,δ∥∥= �

(δ2ν/(2ν+1)). (3.19)


Remark 3.7. This speed of convergence is the optimal speed with a priori in-formation. Of course this information is in general not available. But about thechoice of the parameter α in the Tikhonov regularization with perturbed datafor equations with unbounded operators, some results exist, for example, Chengand Yamamoto [1], Hegland [3], Ivanov et al. [4], Liskovets [5], and Ramm [7].

4. Computational remarks

SinceD(Tρ) is dense in L2(K), the adjoint operatorT∗ρ is well defined, but only inthe case where ρ is a (simple) diffeomorphism of K , T∗ρ can be given explicitly.But if in the case when one solves (1.1) by the Galerkin method, this is not adisadvantage: the Galerkin method consists in solving the equation

(αI +PnT

∗ρ TρPn

)xGα = PnT

∗ρ y, (4.1)

where xGα is contained in a finite-dimensional subspace Xn of X with an or-thoprojection Pn : X → Xn. If u1, . . . ,un is an orthonormal basis of Xn, then theFourier coefficients ξj of xGα can be determined by the equations

αξj +∑

ξk⟨Tρuk,Tρuj

⟩= ⟨y,Tρuj⟩

(4.2)

for j = 1,2, . . . ,n. So the form of the operator T∗ρ is not necessary.

References

[1] J. Cheng and M. Yamamoto, One new strategy for a priori choice of regularizing param-eters in Tikhonov’s regularization, Inverse Problems 16 (2000), no. 4, L31–L38.

[2] H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Math-ematics and Its Applications, vol. 375, Kluwer Academic Publishers, Dordrecht,1996.

[3] M. Hegland, Variable Hilbert scales and their interpolation inequalities with applica-tions to Tikhonov regularization, Appl. Anal. 59 (1995), no. 1-4, 207–223.

[4] V. K. Ivanov, V. V. Vasin, and V. P. Tanana, Theory of Linear Ill-Posed Problems and ItsApplications, Nauka, Moscow, 1978 (Russian).

[5] O. A. Liskovets, Regularization of equations with a closed linear operator, DifferentialEquations 7 (1970), 972–976.

[6] D. Przeworska-Rolewicz, Equations with Transformed Argument, PWN—Polish Sci-entific Publishers, Warsaw, 1973.

[7] A. G. Ramm, Regularization of ill-posed problems with unbounded operators, J. Math.Anal. Appl. 271 (2002), no. 2, 542–545.

[8] F. Riesz and B. Sz.-Nagy, Functional Analysis, Frederick Ungar Publishing, New York,1955, Translated from the 2nd French edition.

[9] E. Schock and V. Q. Phong, Regularization of ill-posed problems involving unboundedoperators in Banach spaces, Hokkaido Math. J. 20 (1991), no. 3, 559–569.

[10] Z. Semadeni, Banach Spaces of Continuous Functions. Vol. I, Monografie Matematy-czne, vol. 55, PWN—Polish Scientific Publishers, Warsaw, 1971.


[11] J. von Neumann, Uber adjungierte Funktionaloperatoren, Ann. of Math. (2) 33(1932), 294–310 (German).

Simone Gramsch: Department of Mathematics, University of Kaiserslautern, 67663Kaiserslautern, Germany

E-mail address: [email protected]

Eberhard Schock: Department of Mathematics, University of Kaiserslautern, 67663Kaiserslautern, Germany

E-mail address: [email protected]

mailto:[email protected]

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Ill-posed equations with transformed argumentdownloads.hindawi.com/journals/aaa/2003/572142.pdf · lution of ill-posed equations with closed operators, we present a new view on the