# Spectral Regularization Methods for an Abstract Ill-Posed Elliptic Problem

Authors: Nadjib Boussetila; Salim Hamida; Faouzia Rebbani
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We study an abstract elliptic Cauchy problem associated with an unbounded
self-adjoint positive operator which has a continuous spectrum. It is well-known that such a problem is
severely ill-posed; that is, the solution does not depend continuously on the Cauchy data. We propose two
spectral regularization methods to construct an approximate stable solution to our original problem.
Finally, some other convergence results including some explicit convergence rates are also established
under a priori bound assumptions on the exact solution.


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## Body

## 1. Introduction

Throughout this paperH denotes a complex Hilbert space endowed with the inner product (·,·), and the norm ∥·∥, ℒ(H) stands for the Banach algebra of bounded linear operators on H.LetA be a linear unbounded operator with dense domain D(A). Assume that A is self-adjoint, positive definite in H, which has a continuous spectrum σ(A)=[γ,+∞[, γ=inf(σ(A))>0.We consider the elliptic Cauchy problem (ECP) of findingu:[0,L]→H such that
(1)uyy(y)=Au(y),0<y<L,u(0)=f,uy(0)=0,
where f is some prescribed data in the Hilbert space H.This problem is an abstract version of Cauchy problem, which generalizes Cauchy problem for second-order elliptic partial differential equations in a cylindrical domain; for example, we mention the following problem.Example 1.
An example of (1) is the Cauchy problem for the modified Helmholtz equation in the infinite strip ℝ×(0,1) [1]:
(2)uyy(x,y)+uxx(x,y)-γu(x,y)=0,00000000000000x∈ℝ,y∈(0,L),u(x,0)=f(x),uy(x,0)=0,x∈ℝ,
where the operator Aγ is given by
(3)Aγ=-∂2∂x2+γI,D(Aγ)=H2(ℝ)⊂H=L2(ℝ).
It is well known that this operator is self-adjoint with continuous spectrum
(4)σ(Aγ)=σ(A0)+γ=[0,+∞[+γ=[γ,+∞[.
We note here that the discrete eigenfunctions expansion method cannot be used, but we can use the Fourier diagonalization method to deal with this kind of problems.Such problem arises in many practical situations, nondestructive testing techniques [2], geophysics [3], cardiology [4], and other applications. There are many various monographs about the historical development of this topic, for more details, we refer the reader to [5, 6]. Recently there has been an excellent topic review [7] of this problem.Because problem (1) is severely ill-posed; that is, a small perturbation in the given Cauchy data may result in a very large error on the solution. In order to overcome this instability character, the regularization methods are required.Some regularization methods for the Cauchy problem for elliptic equations have been proposed by many authors. For instance: Tikhonov regularization method [8], the quasi-reversibility method [9], the quasi-boundary-value method [10–13], Kozlov-Maz’ya iteration method [14], and the mollification method [15].This work is mainly devoted to theoretical aspects of the spectral regularization methods to problem (1) in the abstract setting, by considering more general self-adjoint operators when A is positive and induces the elliptic case, that is, has the following properties: for any λ∈(-∞,0], the resolvent R(λ;A)=(A-λI)-1 exists and satisfies the estimates
(5)∃M>0:∀λ≥0,∥(A+λI)-1∥≤M(1+λ)-1.In the case whenA is a linear positive self-adjoint operator with compact inverse, problem (1) has been treated by a different method and there is a large literature in this direction. However, in the case where A has a continuous spectrum the literatures are quite scarce.In the present paper we shall use two spectral regularization methods to construct a stable solution to our original ill-posed problem.


## 2. Preliminaries and Basic Results

In this section we present the notation and the functional setting which will be used in this paper and prepare some material which will be used in our analysis.


### 2.1. Spectral Theorem and Properties

By the spectral theorem, for each positive self-adjoint operatorA, there is a unique right continuous family {Eλ}λ∈[0,∞[:[0,∞[→ℒ(H) of orthogonal projection operators such that A=∫0∞λdEλ with
(6)D(A)={v∈H:∫0∞λ2d(Eλv,v)<∞}.In our case, we haveA=∫γ∞‍λdEλ because A≥γI, γ>0.Theorem 2 (see [16, Theorem 6, XII.2.5, pages 1196–1198]).
Let{Eλ,λ≥γ>0} be the spectral resolution of the identity associate to A and let ϕ be a complex Borel function defined E-almost everywhere on the real axis. Then ϕ(A) is a closed operator with dense domain. Moreover(i)
D(ϕ(A)):={v∈H:∫γ∞‍|ϕ(λ)|2d(Eλv,v)<∞},(ii)
(ϕ(A)h,y)=∫γ∞‍ϕ(λ)d(Eλh,y), h∈D(ϕ(A)), y∈H,(iii)
∥ϕ(A)h∥2=∫γ∞‍|ϕ(λ)|2d(Eλh,h), h∈D(ϕ(A)),(iv)
ϕ(A)*=ϕ¯(A). In particular, if ϕ is real Borel function, then ϕ(A) is self-adjoint,(v)
the operatorϕ(A) is bounded if and only if ϕ(λ) is bounded on σ(A)=[γ,+∞[. In this case, ∥ϕ(A)∥=supλ∈[γ,+∞[|ϕ(λ)|.We denote byS(y)=e-yA=∫γ∞e-yλdEλ∈ℒ(H), y≥0, the C0-semigroup generated by -A. Some basic properties of S(y) are listed in the following theorem.Theorem 3 (see [17, chapter 2, Theorem 6.13, page 74]).
For this family of operators one has:(1)
∥S(y)∥≤1, ∀y≥0;(2)
the functiony↦S(y), y>0, is analytic;(3)
for every realr≥0 and y>0, the operator S(y)∈ℒ(H,𝒟(Ar/2));(4)
for every integerk≥0 and y>0, ∥S(k)(y)∥=∥Ak/2S(y)∥≤c(k)y-k;(5)
for everyx∈𝒟(Ar/2), r≥0, one has S(t)Ar/2x=Ar/2S(y)x.Theorem 4.
Fory>0, S(y) is self-adjoint and one to one operator with dense range (S(y)=S(y)*, ℛ(S(y))¯=H).Proof.
Letϕy:[γ,+∞[→ℝ, s↦ϕy(s)=e-ys. Then by virtue of (iv) of Theorem 2, we can write S(y)*=ϕy¯(A)=ϕy(A)=e-yA=S(y).
Leth∈N(S(y0)), y0>0, then S(y0)h=0, which implies that S(y)S(y0)h=S(y+y0)h=0, y≥0. Using analyticity, we obtain that S(y)h=0, y≥0. Strong continuity at 0 now gives h=0. This shows that N(S(y0))={0}.
Thanks to(7)R(S(y0))¯=N(S(y0))⊥={0}⊥=H,
we conclude that R(S(y0)) is dense in H.Remark 5.
Fory=pL, p≥1, this Theorem ensures us that S(pL) is self-adjoint and one to one operator with dense range R(S(pL)). Then we can define its inverse S(pL)-1=epLA, which is an unbounded self-adjoint strictly positive definite operator in H with dense domain
(8)D(S(pL)-1)=R(S(pL))={∫γ+∞‍h∈H:∥epLAh∥200000000000=∫γ+∞‍e2pLλd∥Eλh∥2<+∞}.Definition 6 (Hilbert scales).
LetB be an unbounded self-adjoint strictly positive definite operator in H. Following the definition in [18], one introduces the Hilbert scale {𝒢θ}θ≥0 according to definition
(9)𝒢0=H,𝒢θ=D(Bθ).
Here 𝒢θ is a Hilbert space with inner product (ξ,ζ)θ=(Bθξ,Bθζ) and norm ∥ξ∥θ2=(Bθξ,Bθξ).
In our setting we takeB=S(L)-1=eLA. In this case, one has
(10)𝒢θ={∫γ+∞‍h∈H:∥h∥θ2=∥eθLAh∥2=..h∈H:∥h∥θ2k=∫γ+∞‍e2Lθλd∥Eλh∥2<+∞}.
Following [15], one defines the following.Definition 7 (mollification operator).
Forα>0 and p≥1, one introduces the Yosida approximation of identity
(11)Mα=(I+αepLA)-1=∫γ+∞(1+αepLλ)-1dEλ.Remark 8.
The idea of the mollification method is very simple and natural: if the data are given inexactly, then we try to find a sequence of mollification operators which map the improper data into well-posed classes of the problem (mollify the improper data). Within these mollified data our problem becomes well-posed.Theorem 9.
One has(1)
Mα∈ℒ(H) and ∥Mα∥≤1,(2)
for allh∈H, Mαh∈𝒢p,(3)
for allh∈H, limα→0∥Mαh-h∥=0.Proof.
(1)∥Mα∥=supλ≥γ1/(1+αepLλ)=1/(1+αepLγ)<1.
(2)∥Mαh∥p2=∫γ+∞(epLλ/(1+αepLλ))2d∥Eλh∥2≤(1/α)2∥h∥2<∞⇔Mαh∈𝒢p.
(3) Assume thath∈𝒢p. Then
(12)∥Mαh-h∥2=∫γ+∞(11+αepLλ-1)2d∥Eλh∥2=∫γ+∞(αepLλ1+αepLλ)2d∥Eλh∥2≤α2∥h∥p2,
and thus, limα→0∥Mαh-h∥=0.
Now forh∈H, there exists hε∈𝒢p such that ∥h-hε∥≤ε, ε>0 (since 𝒢p is dense in H (see Remark 5)). We have
(13)∥Mαh-h∥=∥Mαh-Mαhε+Mαhε-hε+hε-h∥≤∥Mα(h-hε)∥+∥Mαhε-hε∥+∥hε-h∥≤2ε+∥Mαhε-hε∥.
Thus, limα→0∥Mαh-h∥≤2ε, ∀ε>0, and so limα→0∥Mαh-h∥=0.Using the change of variables(14)U(y)=12(u(y)-A-1/2u′(y)),W(y)=12(u(y)+A-1/2u′(y)).Cauchy’s problem (1) reduces to the two Cauchy problems
(15)U′(y)=-A1/2U(y),0<y<L,U(0)=12(u(0)-A-1/2u′(0))=12f,(16)W′(y)=A1/2W(y),0<y<L,W(0)=12(u(0)+A-1/2u′(0))=12f.
Thus, the solution of the original problem (1) can be written in the form
(17)u(y)=U(y)+W(y),0≤y≤L.
It is well known that the operator -A1/2 generates a strongly continuous analytic semigroup S-1/2(y)=∫γ+∞e-yλdEλ. In addition, the spectral radius of the semigroup ∥S-1/2(y)∥<1 for any y>0. Hence, it follows that the Cauchy problem (15) is well-posed and its solution may be written in the form
(18)U(y)=S-1/2(y)U(0)=12e-yAf=12∫γ+∞e-yλdEλf.
As opposed to problem (15), Cauchy problem (15) (backward parabolic equation) is not correctly posed, and its (unique) formal solution is given by
(19)W(y)=S+1/2(y)W(0)=12eyAf=12∫γ+∞eyλdEλf.Remark 10 (see [19, page 375]).
The uniqueness solvability of problem (16) results from the logarithmic convexity of the function y↦∥W(y)∥:
(20)∀y∈[0,L],∥W(y)∥≤∥W(0)∥1-y/L∥W(L)∥y/L.
A useful characterization of the admissible set for which problem (16) has a solution is as follows.Lemma 11.
Problem (16) has a solution W∈C([0,L];H) if and only if f∈𝒢1, and its unique solution is represented by
(21)W(y)=S+1/2(y)W(0)=12eyAf=12∫γ+∞eyλdEλf.Proof.
Suppose that problem (16) has a solution W∈𝒞([0,L];H). Then
(22)W(y)=12∫γ+∞eyλdEλf.
The function W∈𝒞([0,L];H) if and only if
(23)supy∈[0,L]∥W(y)∥2=14∫γ+∞e2yλd∥Eλf∥2<∞.
Observing that the function y↦∥W(L)∥2 is increasing, then
(24)supy∈[0,L]∥W(y)∥2=∥W(L)∥2=14∫γ+∞e2Lλd∥Eλf∥2<∞.
This last inequality is exactly equivalent to f∈𝒢1.As a consequence, we have the following corollary.Corollary 12.
Problem (1) has a solution u∈C([0,L];H) if and only if f∈𝒢1, and its unique solution is represented by
(25)u(y)=U(y)+W(y)=12∫γ+∞(eyλ+e-yλ)dEλf.


## 2.1. Spectral Theorem and Properties

By the spectral theorem, for each positive self-adjoint operatorA, there is a unique right continuous family {Eλ}λ∈[0,∞[:[0,∞[→ℒ(H) of orthogonal projection operators such that A=∫0∞λdEλ with
(6)D(A)={v∈H:∫0∞λ2d(Eλv,v)<∞}.In our case, we haveA=∫γ∞‍λdEλ because A≥γI, γ>0.Theorem 2 (see [16, Theorem 6, XII.2.5, pages 1196–1198]).
Let{Eλ,λ≥γ>0} be the spectral resolution of the identity associate to A and let ϕ be a complex Borel function defined E-almost everywhere on the real axis. Then ϕ(A) is a closed operator with dense domain. Moreover(i)
D(ϕ(A)):={v∈H:∫γ∞‍|ϕ(λ)|2d(Eλv,v)<∞},(ii)
(ϕ(A)h,y)=∫γ∞‍ϕ(λ)d(Eλh,y), h∈D(ϕ(A)), y∈H,(iii)
∥ϕ(A)h∥2=∫γ∞‍|ϕ(λ)|2d(Eλh,h), h∈D(ϕ(A)),(iv)
ϕ(A)*=ϕ¯(A). In particular, if ϕ is real Borel function, then ϕ(A) is self-adjoint,(v)
the operatorϕ(A) is bounded if and only if ϕ(λ) is bounded on σ(A)=[γ,+∞[. In this case, ∥ϕ(A)∥=supλ∈[γ,+∞[|ϕ(λ)|.We denote byS(y)=e-yA=∫γ∞e-yλdEλ∈ℒ(H), y≥0, the C0-semigroup generated by -A. Some basic properties of S(y) are listed in the following theorem.Theorem 3 (see [17, chapter 2, Theorem 6.13, page 74]).
For this family of operators one has:(1)
∥S(y)∥≤1, ∀y≥0;(2)
the functiony↦S(y), y>0, is analytic;(3)
for every realr≥0 and y>0, the operator S(y)∈ℒ(H,𝒟(Ar/2));(4)
for every integerk≥0 and y>0, ∥S(k)(y)∥=∥Ak/2S(y)∥≤c(k)y-k;(5)
for everyx∈𝒟(Ar/2), r≥0, one has S(t)Ar/2x=Ar/2S(y)x.Theorem 4.
Fory>0, S(y) is self-adjoint and one to one operator with dense range (S(y)=S(y)*, ℛ(S(y))¯=H).Proof.
Letϕy:[γ,+∞[→ℝ, s↦ϕy(s)=e-ys. Then by virtue of (iv) of Theorem 2, we can write S(y)*=ϕy¯(A)=ϕy(A)=e-yA=S(y).
Leth∈N(S(y0)), y0>0, then S(y0)h=0, which implies that S(y)S(y0)h=S(y+y0)h=0, y≥0. Using analyticity, we obtain that S(y)h=0, y≥0. Strong continuity at 0 now gives h=0. This shows that N(S(y0))={0}.
Thanks to(7)R(S(y0))¯=N(S(y0))⊥={0}⊥=H,
we conclude that R(S(y0)) is dense in H.Remark 5.
Fory=pL, p≥1, this Theorem ensures us that S(pL) is self-adjoint and one to one operator with dense range R(S(pL)). Then we can define its inverse S(pL)-1=epLA, which is an unbounded self-adjoint strictly positive definite operator in H with dense domain
(8)D(S(pL)-1)=R(S(pL))={∫γ+∞‍h∈H:∥epLAh∥200000000000=∫γ+∞‍e2pLλd∥Eλh∥2<+∞}.Definition 6 (Hilbert scales).
LetB be an unbounded self-adjoint strictly positive definite operator in H. Following the definition in [18], one introduces the Hilbert scale {𝒢θ}θ≥0 according to definition
(9)𝒢0=H,𝒢θ=D(Bθ).
Here 𝒢θ is a Hilbert space with inner product (ξ,ζ)θ=(Bθξ,Bθζ) and norm ∥ξ∥θ2=(Bθξ,Bθξ).
In our setting we takeB=S(L)-1=eLA. In this case, one has
(10)𝒢θ={∫γ+∞‍h∈H:∥h∥θ2=∥eθLAh∥2=..h∈H:∥h∥θ2k=∫γ+∞‍e2Lθλd∥Eλh∥2<+∞}.
Following [15], one defines the following.Definition 7 (mollification operator).
Forα>0 and p≥1, one introduces the Yosida approximation of identity
(11)Mα=(I+αepLA)-1=∫γ+∞(1+αepLλ)-1dEλ.Remark 8.
The idea of the mollification method is very simple and natural: if the data are given inexactly, then we try to find a sequence of mollification operators which map the improper data into well-posed classes of the problem (mollify the improper data). Within these mollified data our problem becomes well-posed.Theorem 9.
One has(1)
Mα∈ℒ(H) and ∥Mα∥≤1,(2)
for allh∈H, Mαh∈𝒢p,(3)
for allh∈H, limα→0∥Mαh-h∥=0.Proof.
(1)∥Mα∥=supλ≥γ1/(1+αepLλ)=1/(1+αepLγ)<1.
(2)∥Mαh∥p2=∫γ+∞(epLλ/(1+αepLλ))2d∥Eλh∥2≤(1/α)2∥h∥2<∞⇔Mαh∈𝒢p.
(3) Assume thath∈𝒢p. Then
(12)∥Mαh-h∥2=∫γ+∞(11+αepLλ-1)2d∥Eλh∥2=∫γ+∞(αepLλ1+αepLλ)2d∥Eλh∥2≤α2∥h∥p2,
and thus, limα→0∥Mαh-h∥=0.
Now forh∈H, there exists hε∈𝒢p such that ∥h-hε∥≤ε, ε>0 (since 𝒢p is dense in H (see Remark 5)). We have
(13)∥Mαh-h∥=∥Mαh-Mαhε+Mαhε-hε+hε-h∥≤∥Mα(h-hε)∥+∥Mαhε-hε∥+∥hε-h∥≤2ε+∥Mαhε-hε∥.
Thus, limα→0∥Mαh-h∥≤2ε, ∀ε>0, and so limα→0∥Mαh-h∥=0.Using the change of variables(14)U(y)=12(u(y)-A-1/2u′(y)),W(y)=12(u(y)+A-1/2u′(y)).Cauchy’s problem (1) reduces to the two Cauchy problems
(15)U′(y)=-A1/2U(y),0<y<L,U(0)=12(u(0)-A-1/2u′(0))=12f,(16)W′(y)=A1/2W(y),0<y<L,W(0)=12(u(0)+A-1/2u′(0))=12f.
Thus, the solution of the original problem (1) can be written in the form
(17)u(y)=U(y)+W(y),0≤y≤L.
It is well known that the operator -A1/2 generates a strongly continuous analytic semigroup S-1/2(y)=∫γ+∞e-yλdEλ. In addition, the spectral radius of the semigroup ∥S-1/2(y)∥<1 for any y>0. Hence, it follows that the Cauchy problem (15) is well-posed and its solution may be written in the form
(18)U(y)=S-1/2(y)U(0)=12e-yAf=12∫γ+∞e-yλdEλf.
As opposed to problem (15), Cauchy problem (15) (backward parabolic equation) is not correctly posed, and its (unique) formal solution is given by
(19)W(y)=S+1/2(y)W(0)=12eyAf=12∫γ+∞eyλdEλf.Remark 10 (see [19, page 375]).
The uniqueness solvability of problem (16) results from the logarithmic convexity of the function y↦∥W(y)∥:
(20)∀y∈[0,L],∥W(y)∥≤∥W(0)∥1-y/L∥W(L)∥y/L.
A useful characterization of the admissible set for which problem (16) has a solution is as follows.Lemma 11.
Problem (16) has a solution W∈C([0,L];H) if and only if f∈𝒢1, and its unique solution is represented by
(21)W(y)=S+1/2(y)W(0)=12eyAf=12∫γ+∞eyλdEλf.Proof.
Suppose that problem (16) has a solution W∈𝒞([0,L];H). Then
(22)W(y)=12∫γ+∞eyλdEλf.
The function W∈𝒞([0,L];H) if and only if
(23)supy∈[0,L]∥W(y)∥2=14∫γ+∞e2yλd∥Eλf∥2<∞.
Observing that the function y↦∥W(L)∥2 is increasing, then
(24)supy∈[0,L]∥W(y)∥2=∥W(L)∥2=14∫γ+∞e2Lλd∥Eλf∥2<∞.
This last inequality is exactly equivalent to f∈𝒢1.As a consequence, we have the following corollary.Corollary 12.
Problem (1) has a solution u∈C([0,L];H) if and only if f∈𝒢1, and its unique solution is represented by
(25)u(y)=U(y)+W(y)=12∫γ+∞(eyλ+e-yλ)dEλf.


## 3. Regularization and Error Estimates

### 3.1. The Truncation Method

From (25) we can see that the term eyλ is the cause of unstability. In order to overcome the ill-posedness of problem (1), we modify the solution by filtering the high frequencies using a suitable method and instead consider (25) only for λ≤β(δ), where β(δ) is some constant which satisfies limδ→0β(δ)=+∞.According to spectral theory of self-adjoint operators [20], for any bounded Borel set Δβ={γ≤t≤β}⊆σ(A)=[γ,+∞[, we can define the orthogonal projection
(26)00000000001Δβ=∫γ+∞1Δβ(λ)dEλ=Eβ,∀h∈H,hβ=Eβh⟶h,β⟶+∞.To solve (1) in a stable way we approximate f by its projection fβ, and instead of considering (1) with f we take its projected version
(27)uβ(y)=cosh(yA)fβ=12∫γ+∞(eyλ+e-yλ)1[γ,β]dEλf,
where 1[a,b] is the characteristic function of the interval [a,b] for a<b. The quantity β is referred to as a cut-off frequency.Letf (resp., fδ) be the exact (resp., the measured data) at y=0, such that ∥f-fδ∥≤δ.The approximated solutionvβδ corresponding to the measured data fδ is denoted by
(28)vβδ(y)=12∫γ+∞(eyλ+e-yλ)1[γ,β]dEλfδ.
For simplicity, we denote the solution of problem (1) by u(y), and the regularized solution associated to the data fδ by vβδ(y).Our first main theorem is the following theorem.Theorem 13.
The solution defined in (27) depends continuously in 𝒞([0,L];H) on the data f; that is, if uβ1 and uβ2 are two regularized solutions corresponding to f1 and f2, respectively, then one has
(29)∥uβ1(y)-uβ2(y)∥≤eyβ∥f1-f2∥.This inequality implies that the solution of the regularized problem (27) depends continuously on the data f.Now we compute the difference between the original solutionu=u(y;f) and the approximate solution vβδ=vβδ(y;fδ).Theorem 14.
Letu∈C([0,L];H) be a solution of problem (1) with the exact data f∈H; then the following estimate holds:
(30)∥u(y)-uβ(y)∥≤2e(L-y)β∥u(L)∥.Proof.
From relations (25) and (27) we have
(31)u(y)-uβ(t)=∫β+∞cosh(yλ)dEλf=∫γ+∞cosh(yλ)1[β,+∞]dEλf,
then
(32)u(y)-uβ(y)=∫γ+∞cosh(yλ)cosh(Lλ)1[β,+∞]cosh(Lλ)dEλf,∥u(y)-uβ(y)∥2≤∫γ+∞(cosh(yλ)cosh(Lλ)1[β,+∞])2cosh(Lλ)2d∥Eλf∥2.
Using the inequality(33)(cosh(yλ)cosh(Lλ)1[β,+∞])2≤4e2(L-y)β,∫β+∞cosh2(Lλ)d∥Eλf∥2≤∥u(L)∥2,
we derive
(34)∥u(y)-uβ(y)∥2≤4e2(L-y)β∥u(L)∥2.
Using (29), (30) and the triangle inequality, we obtain
(35)∥u(y)-vβδ(y)∥≤∥u(y)-uβ(y)∥+∥uβ(y)-vβδ(y)∥≤2e(L-y)β∥u(L)∥+eyβδ.
This completes the proof.Remark 15.
If we chooseβ=(1/L)log(M/δ), where ∥u(L)∥=M, then we have the error bound
(36)∥u(y)-vβδ(y)∥≤3M(L-y)/Lδy/L.
From (36) we see that (28) is an approximation of the exact solution u(y). The approximation error depends continuously on the measurement error for fixed 0<y<L. However, as y→L, the accuracy of the regularized solution becomes progressively lower. Consequently, we have not any information about the continuous dependence of the solution if y is close to L.
In the theory of ill-posed Cauchy problems, we can often obtain continuous dependence on the data for the closed interval[0,L] by assuming additional smoothness and using a stronger norm.
Now we show two error estimates under the following conditions:(H1)
u(L)∈D(Ap),(H2)
u(L)∈𝒢p, p>0.Remark 16.
In practice, we know that it is very difficult to verify the conditions (H1) and (H2), so we give different assumptions on the given dataf as follows:
(37)u(L)∈D(Ap)⟺∫γ+∞λ2pcosh2(Lλ)d∥Eλf∥2<∞⟺∫γ+∞λ2pe2Lλd∥Eλf∥2<∞,u(L)∈𝒢p⟺∫γ+∞e2pLλcosh2(Lλ)d∥Eλf∥2<∞⟺∫γ+∞e2(1+p)Lλd∥Eλf∥2<∞⟺f∈𝒢p+1.Theorem 17.
If∫γ+∞λpe2Lλd∥Eλf∥2<E12 (resp., ∫γ+∞e2(1+q)Lλd∥Eλf∥2<E22), p>0, q>0, then one has the following estimates:
(38)∥u(y)-vβδ(y)∥≤(La)pE1log(1δ)-p+δ1-ya/L,0<a≤1,∥u(y)-vβ(y)∥≤e-qβE2+eyβδ.Proof.
From the expansions(39)u(y)=∫γ∞cosh(yλ)dEλf,uβ(y)=∫γ∞cosh(yλ)1[γ,β]dEλf,
we have
(40)u(y)-uβ(y)=∫γ∞cosh(yλ)1[β,+∞]dEλf.
Then(41)∥u(y)-uβ(y)∥2=∫γ∞‍(λ-p/21[β,+∞])2cosh2(yλ)λpd∥Eλf∥2≤∫γ∞‍(λ-p/21[β,+∞])2e2Lλλpd∥Eλf∥2≤β-p∫γ∞‍e2Lλλpd∥Eλf∥2≤β-2pE12.
Using Theorem 17 and the triangle inequality, we can write
(42)∥u(y)-vβδ(y)∥≤∥u(y)-uβ(y)∥+∥uβ(y)-vβδ(y)∥≤β-pE1+eyβδ.
By choosing β=(a/L)log(1/δ), we obtain the desired inequality.
Using the same techniques we have(43)u(y)-uβ(y)=∫γ∞e-qλcosh(yλ)eqλ1[β,+∞]dEλf,
hence
(44)∥u(y)-uβ(y)∥2=∫γ∞(e-qλ1[β,+∞])2(cosh(yλ)e2qλ)2d∥Eλf∥2≤e-2qβ∫γ∞e2qλd∥Eλu(y)∥2≤e-2qβE22.
Using (29) and the triangle inequality, we obtain
(45)∥u(y)-vβδ(y)∥≤∥u(y)-uβ(y)∥+∥uβ(y)-vβδ(y)∥≤e-qβE2+eyβδ.
By choosing β=(a/L)log(1/δ), we obtain
(46)∥u(y)-vβδ(y)∥≤δaq/LE2+δ1-ay/L.


### 3.2. The Mollification Method

Now, we approximate the original problem (1) by the sequence of problems
(47)uyy=Au,0<y<L,u(0)=fα=Mαf,uy(0)=0.Theorem 18.
Iff∈H the approximate Cauchy problem (47) admits a unique solution uα, which depends continuously upon the data f with respect to uniform topology of C([0,L];H).Proof.
From the representation(48)uα(y)=cosh(yA)fα=∫γ+∞cosh(yλ)(1+αepLλ)-1dEλf,
we have
(49)∥uα(y)∥2=∫γ+∞{cosh(yλ)1+αepLλ}2d∥Eλf∥2≤∫γ+∞{eLλ1+αepLλ}2d∥Eλf∥2.(i)
Ifp=1, we obtain
(50)supy∈[0,L]∥uα(y)∥2≤1α∥f∥.(ii)
Ifp>1, the function M(s)=eLs/(1+αepLs) with s=λ≥γ achieves its maximum at s*=(1/pL)log(1/α(p-1)), p>1, from which we deduce
(51)M∞=M(s*)=c(p)(1α)1/p,c(p)=p-1(p-1)1-1/p≤1.
From this bound, we derive(52)supy∈[0,L]∥uα(y)∥≤(1α)1/p∥f∥.
From the linear property of our problem, stability estimate of problem (47) may be written precisely in the following corollary.Corollary 19.
Ifuα,1(y;f1) (resp., uα,2(y;f2)) is the approximate solution corresponding to f1 (resp., f2), then
(53)supy∈[0,L]∥uα,1(y)-uα,2(y)∥≤(1α)1/p∥f1-f2∥.Remark 20.
We have(54)N(s)=sreTs1+αepTs≤1αK(s)=1αsre-(p-1)Ts,p>1,s=λ≥γ.
It is easy to show that
(55)K(s)≤K(s*=rL(p-1))=(rL(p-1))re-r=κ(r,p,L)<∞.
This remark shows thatuα(y)∈D(Ar/2) for all y∈[0,L].Proof.
The inclusionuα(y)∈D(Ar/2) is equivalent to ∥Ar/2uα(y)∥<∞. We have
(56)∥Ar/2uα(y)∥2=∫γ+∞λr{cosh(yλ)1+αepLλ}2d∥Eλf∥2≤(1α)2∫γ+∞{λre-(p-1)Lλ}2d∥Eλf∥2≤(1α)2κ(r,p,L)2∥f∥2<∞,
where κ(r,p,L)=supλ≥γλre-(p-1)Lλ=(r(p-1)L)re-r.Theorem 21.
Iff∈𝒢1, then
(57)supy∈[0,L]∥u(y)-uα(y)∥⟶0,α⟶0.Proof.
We compute(58)∥u(y)-uα(y)∥2=∫γ+∞(1-Mα(λ))2cosh2(yλ)d∥Eλf∥2≤∫γ+∞(1-Mα(λ))2cosh2(Lλ)d∥Eλf∥2≤∫γ+∞(1-Mα(λ))2e2Lλd∥Eλf∥2=∥(I-Mα)f^∥2,
where f^=eLAf and ∥f^∥2=∫γ+∞e2Lλd∥Eλf∥2<∞.
This implies thatsupy∈[0,L]∥u(y)-uα(y)∥≤∥(I-Mα)f^∥ and by virtue of (3) of Theorem 9, we conclude the desired convergence.The following technical lemmas play the key role in our analysis and calculations.Lemma 22.
Let(59)[ν,+∞[∋s⟼Q({a,r,q,L};s)=1αsr+ae-qLs,
where a>0, α>0, ν>0, q>0, L>0, and r≥1. Then one has
(60)Q({a,r,q,L};s)≤1α(k1log(k2(1/α)))r,
where k1(r,q,L)=rqL, k2(q,r,L,a)=qrLr-1a/r.Proof.
Differentiating the expression and setting the derivative equal to zero, we find(61)ddsQ({a,r,q,L};s)=-1(αsr+ae-qLs)2(αrsr-1-qae-qLs)=0.
The function (d/ds)Q({a,r,q,L};s)=0 admits a unique solution
(62)s^={s⟼αrsr-1}∩{s⟼qae-qLs}.
Therefore
(63)Q({a,r,q,L};s)≤Q({a,r,q,L};s^)≤1αs^r+ae-qLs^≤1αs^r.
We have
(64)(αrs^r-1-qae-qLs^)=0⟺sr-1eqLs^=qarα.
By using the inequality (et≥t, t≥0), then for t=qLs^, we obtain eqLs^≥qLs^ and we can write
(65)qarα=s^r-1eqLs^≤eqLs^(eqLs^qL)r-1=(1qL)r-1erqLs^,
which implies that s^≥(1/rqL)log((qrLr-1a/r)(1/α)). Hence, we obtain
(66)Q({a,r,q,L};s)≤1αs^r≤1α(k1log(k2(1/α)))r,
where k1(r,q,L)=rqL, k2(q,r,L,a)=qrLr-1a/r.Lemma 23.
Let(67)[ν,+∞[∋s⟼R({p,q,L};s)=epLs(1+αepLs)eqLs=1e(q-p)Ls+αeqLs,
where p≥1, q>0, α>0, ν>0, and L>0. Then one has the following.
If1≤p≤q, then
(68)R({p,q,L};s)≤e-(q-p)Ls≤e-(q-p)Lν≤1.
If0<q<p, p≥1, 0<α≤(p-q)/q, then
(69)R({p,q,L};s)≤k3(1α)(p-q)/p,k3(p,q)=qp(p-qp)(p-q)/p≤1.Proof.
By a simple differential calculus, we show that the functionR({p,q,L};s) achieves its maximum at s^=(1/pL)log((p-q)/αq). Consequently
(70)R({p,q,L};s)≤R({p,q,L};s^)=k3(1α)(p-q)/p.Now we assume the following a priori bounds hold:(71)u(L)∈D(Ar/2)⟺∫γ+∞λ2re2Lλd∥Eλf∥2≤E12<∞,(72)u(L)∈𝒢q⟺∫γ+∞e2L(1+q)λd∥Eλf∥2≤E22<∞.Theorem 24.
Letu (resp., uα) be the solution of problem (1) (resp., (47)) with the exact data f. If (71) (resp., (72)) is satisfied, then one has the following error estimates:
(73)∥u(y)-uα(y)∥=O(1log(1/α))r,(74)∥u(y)-uα(y)∥={O(α),if1≤p≤q,O(αq/p),if0<q<p,p≥1.Proof.
Putting(75)B1(λ)={epLλ1+αepLλ}1λr=1λre-pLλ+αλr≤B2(λ)=1γre-pLλ+αλr,B3(λ)={epLλ1+αepLλ}1eqLλ=1e(q-p)Lλ+αeqLλ.
Using the change of variables s=λ, we obtain the new expressions
(76)B^2(s)=1γre-pLs+αsr,(77)B^3(s)=1e(q-p)Ls+αeqLs.
By virtue of Lemma22 (inequality (60) and Lemma 23 (inequalities (68) and (69)), we can write
(78)B^2(s)≤1α(k1log(k2(1/α)))r,
where k1(r,p,L)=rqL, k2(p,r,L,γr)=qrLr-1γr/r. Consider
(79)B^3(s)≤{1,if1≤p≤q,(1α)(p-q)/p,if0<q<p,p≥1.
We have(80)∥u(y)-uα(y)∥2=∫γ+∞{αepLλ1+αepLλ}2cosh2(yλ)d∥Eλf∥2≤α2∫γ+∞{B2(λ)}2λ2re2Lλd∥Eλf∥2≤α2(sups≥γB^2(s))2E12,∥u(y)-uα(y)∥2=∫γ+∞{αepLλ1+αepLλ}2cosh2(yλ)d∥Eλf∥2≤α2∫γ+∞{B3(λ)}2eL(1+q)λd∥Eλf∥2≤α2(sups≥γB^3(s))2E22.
Using (78) and (79), we derive
(81)∥u(y)-uα(y)∥≤α1α(k1log(k2(1/α)))r=O(1log(1/α))r,∥u(y)-uα(y)∥≤{α,if1≤p≤q,αq/p,if0<q<p,p≥1.Combining (53), (73), and (74) with the help of triangle inequality
(82)∥u(y)-uαδ(y)∥≤∥u(y)-uα(y)∥+∥uα(y)-uαδ(y)∥=Δ1+Δ2,
we deduce the following corollary.Corollary 25.
Letu(y;f) (resp., uαδ(y;fδ)) be the solution of problem (1) (resp., (47)) with the exact data f (resp., the inexact data fδ) such that ∥f-fδ∥≤δ. If (71) (resp., (72)) is satisfied, then one has the following error estimates:
(83)(caser≥1)∥u(y)-uαδ(y)∥=O(θ1(α))+(1α)1/pδ,(84)(case1≤p≤q)∥u(y)-uαδ(y)∥=O(θ2(α))+(1α)1/pδ,(85)(case0<q<p,p≥1)∥u(y)-uαδ(y)∥=O(θ3(α))+(1α)1/pδ,
where
(86)θ1(α)=O(1log(1/α))r,θ2(α)=O(α),θ2(α)=O(αq/p).If we chooseα=α(δ)=δp/ω with ω>1, then we have
(87)δ(1δp/ω)1/p=δ(ω-1)/w,(88)θ1(α)=O(1log(1/δp/ω))r,(89)θ2(α)=δp/ω,θ3(α)=O(δq/ω).


### 3.3. Example: Cauchy Problem for the Modified Helmholtz Equation

In this paragraph, we give a concrete example to see how to apply the theoretical results developed in this study.Let us consider the Cauchy problem (modified Helmholtz equation) in the infinite stripℝ×(0,1):
(90)uyy(x,y)+uxx(x,y)-γu(x,y)=0,hhhhhhhhhhx∈ℝ,y∈(0,1),u(x,0)=f(x),uy(x,0)=0,x∈ℝ,
where γ is a real positive constant.Letu^(ξ,y)=(ℱu)(ξ,y) be the Fourier transform of u(x,y):
(91)u^(ξ,y)=12π∫ℝe-iξxu(x,y)dx.
With the help of the Fourier transformation, problem (1) can be transformed to an equivalent problem in the frequency domain:
(92)u^yy(ξ,y)-ξ2u^(ξ,y)-γu^(ξ,y)=0,dddddddiddξ∈ℝ,y∈(0,1),u^(ξ,0)=f^(ξ),u^y(ξ,0)=0,ξ∈ℝ.
It is easy to check that the formal solution of problem (92) has the form
(93)u^(ξ,y)=cosh(y(ξ2+γ))f^(ξ),
or equivalently, the formal solution of problem (90) is given by
(94)u(x,y)=(ℱ-1u^)(x,y)=12π∫ℝeixξu^(ξ,y)dξ=12π∫ℝeixξcosh(y(ξ2+γ))f^(ξ)dξ.PuttingΘ(ξ)=(ξ2+γ). Then Θ(ξ)→+∞ as |ξ|→+∞. From this remark, it is easy to see that a small perturbation in the data f^(ξ) may cause a dramatically large error in the solution u^(ξ,ξ). In addition, the magnifying factor is Θ(ξ)~e|ξ|, hence, the problem is severely ill-posed.Since the dataf(·) are based on (physical) observations and are not known with complete accuracy, we assume that f and fδ satisfy
(95)∥f-fδ∥≤δ,
where f and fδ belong to L2(ℝ), fδ denotes the measured data, and δ denotes the noise level.For this problem, we define the regularized solutions with noisy datafδ:
(96)uNδ(x,y)=12π∫ℝeixξcosh(y(ξ2+γ))f^δ(ξ)1[-N,N](ξ)dξ=12π∫-NNeixξcosh(y(ξ2+γ))f^δ(ξ)dξ,
where 1[-N,N] is the characteristic function of the interval [-N,N]. Consider
(97)uαδ(x,y)=12π∫ℝeixξ(cosh(y(ξ2+γ))1+αep(ξ2+γ))f^δ(ξ)dξ,
where p≥1. The quantities α=α(δ) and N=N(δ) are the parameters which were defined in Sections 3.1 and 3.2.


## 3.1. The Truncation Method

From (25) we can see that the term eyλ is the cause of unstability. In order to overcome the ill-posedness of problem (1), we modify the solution by filtering the high frequencies using a suitable method and instead consider (25) only for λ≤β(δ), where β(δ) is some constant which satisfies limδ→0β(δ)=+∞.According to spectral theory of self-adjoint operators [20], for any bounded Borel set Δβ={γ≤t≤β}⊆σ(A)=[γ,+∞[, we can define the orthogonal projection
(26)00000000001Δβ=∫γ+∞1Δβ(λ)dEλ=Eβ,∀h∈H,hβ=Eβh⟶h,β⟶+∞.To solve (1) in a stable way we approximate f by its projection fβ, and instead of considering (1) with f we take its projected version
(27)uβ(y)=cosh(yA)fβ=12∫γ+∞(eyλ+e-yλ)1[γ,β]dEλf,
where 1[a,b] is the characteristic function of the interval [a,b] for a<b. The quantity β is referred to as a cut-off frequency.Letf (resp., fδ) be the exact (resp., the measured data) at y=0, such that ∥f-fδ∥≤δ.The approximated solutionvβδ corresponding to the measured data fδ is denoted by
(28)vβδ(y)=12∫γ+∞(eyλ+e-yλ)1[γ,β]dEλfδ.
For simplicity, we denote the solution of problem (1) by u(y), and the regularized solution associated to the data fδ by vβδ(y).Our first main theorem is the following theorem.Theorem 13.
The solution defined in (27) depends continuously in 𝒞([0,L];H) on the data f; that is, if uβ1 and uβ2 are two regularized solutions corresponding to f1 and f2, respectively, then one has
(29)∥uβ1(y)-uβ2(y)∥≤eyβ∥f1-f2∥.This inequality implies that the solution of the regularized problem (27) depends continuously on the data f.Now we compute the difference between the original solutionu=u(y;f) and the approximate solution vβδ=vβδ(y;fδ).Theorem 14.
Letu∈C([0,L];H) be a solution of problem (1) with the exact data f∈H; then the following estimate holds:
(30)∥u(y)-uβ(y)∥≤2e(L-y)β∥u(L)∥.Proof.
From relations (25) and (27) we have
(31)u(y)-uβ(t)=∫β+∞cosh(yλ)dEλf=∫γ+∞cosh(yλ)1[β,+∞]dEλf,
then
(32)u(y)-uβ(y)=∫γ+∞cosh(yλ)cosh(Lλ)1[β,+∞]cosh(Lλ)dEλf,∥u(y)-uβ(y)∥2≤∫γ+∞(cosh(yλ)cosh(Lλ)1[β,+∞])2cosh(Lλ)2d∥Eλf∥2.
Using the inequality(33)(cosh(yλ)cosh(Lλ)1[β,+∞])2≤4e2(L-y)β,∫β+∞cosh2(Lλ)d∥Eλf∥2≤∥u(L)∥2,
we derive
(34)∥u(y)-uβ(y)∥2≤4e2(L-y)β∥u(L)∥2.
Using (29), (30) and the triangle inequality, we obtain
(35)∥u(y)-vβδ(y)∥≤∥u(y)-uβ(y)∥+∥uβ(y)-vβδ(y)∥≤2e(L-y)β∥u(L)∥+eyβδ.
This completes the proof.Remark 15.
If we chooseβ=(1/L)log(M/δ), where ∥u(L)∥=M, then we have the error bound
(36)∥u(y)-vβδ(y)∥≤3M(L-y)/Lδy/L.
From (36) we see that (28) is an approximation of the exact solution u(y). The approximation error depends continuously on the measurement error for fixed 0<y<L. However, as y→L, the accuracy of the regularized solution becomes progressively lower. Consequently, we have not any information about the continuous dependence of the solution if y is close to L.
In the theory of ill-posed Cauchy problems, we can often obtain continuous dependence on the data for the closed interval[0,L] by assuming additional smoothness and using a stronger norm.
Now we show two error estimates under the following conditions:(H1)
u(L)∈D(Ap),(H2)
u(L)∈𝒢p, p>0.Remark 16.
In practice, we know that it is very difficult to verify the conditions (H1) and (H2), so we give different assumptions on the given dataf as follows:
(37)u(L)∈D(Ap)⟺∫γ+∞λ2pcosh2(Lλ)d∥Eλf∥2<∞⟺∫γ+∞λ2pe2Lλd∥Eλf∥2<∞,u(L)∈𝒢p⟺∫γ+∞e2pLλcosh2(Lλ)d∥Eλf∥2<∞⟺∫γ+∞e2(1+p)Lλd∥Eλf∥2<∞⟺f∈𝒢p+1.Theorem 17.
If∫γ+∞λpe2Lλd∥Eλf∥2<E12 (resp., ∫γ+∞e2(1+q)Lλd∥Eλf∥2<E22), p>0, q>0, then one has the following estimates:
(38)∥u(y)-vβδ(y)∥≤(La)pE1log(1δ)-p+δ1-ya/L,0<a≤1,∥u(y)-vβ(y)∥≤e-qβE2+eyβδ.Proof.
From the expansions(39)u(y)=∫γ∞cosh(yλ)dEλf,uβ(y)=∫γ∞cosh(yλ)1[γ,β]dEλf,
we have
(40)u(y)-uβ(y)=∫γ∞cosh(yλ)1[β,+∞]dEλf.
Then(41)∥u(y)-uβ(y)∥2=∫γ∞‍(λ-p/21[β,+∞])2cosh2(yλ)λpd∥Eλf∥2≤∫γ∞‍(λ-p/21[β,+∞])2e2Lλλpd∥Eλf∥2≤β-p∫γ∞‍e2Lλλpd∥Eλf∥2≤β-2pE12.
Using Theorem 17 and the triangle inequality, we can write
(42)∥u(y)-vβδ(y)∥≤∥u(y)-uβ(y)∥+∥uβ(y)-vβδ(y)∥≤β-pE1+eyβδ.
By choosing β=(a/L)log(1/δ), we obtain the desired inequality.
Using the same techniques we have(43)u(y)-uβ(y)=∫γ∞e-qλcosh(yλ)eqλ1[β,+∞]dEλf,
hence
(44)∥u(y)-uβ(y)∥2=∫γ∞(e-qλ1[β,+∞])2(cosh(yλ)e2qλ)2d∥Eλf∥2≤e-2qβ∫γ∞e2qλd∥Eλu(y)∥2≤e-2qβE22.
Using (29) and the triangle inequality, we obtain
(45)∥u(y)-vβδ(y)∥≤∥u(y)-uβ(y)∥+∥uβ(y)-vβδ(y)∥≤e-qβE2+eyβδ.
By choosing β=(a/L)log(1/δ), we obtain
(46)∥u(y)-vβδ(y)∥≤δaq/LE2+δ1-ay/L.


## 3.2. The Mollification Method

Now, we approximate the original problem (1) by the sequence of problems
(47)uyy=Au,0<y<L,u(0)=fα=Mαf,uy(0)=0.Theorem 18.
Iff∈H the approximate Cauchy problem (47) admits a unique solution uα, which depends continuously upon the data f with respect to uniform topology of C([0,L];H).Proof.
From the representation(48)uα(y)=cosh(yA)fα=∫γ+∞cosh(yλ)(1+αepLλ)-1dEλf,
we have
(49)∥uα(y)∥2=∫γ+∞{cosh(yλ)1+αepLλ}2d∥Eλf∥2≤∫γ+∞{eLλ1+αepLλ}2d∥Eλf∥2.(i)
Ifp=1, we obtain
(50)supy∈[0,L]∥uα(y)∥2≤1α∥f∥.(ii)
Ifp>1, the function M(s)=eLs/(1+αepLs) with s=λ≥γ achieves its maximum at s*=(1/pL)log(1/α(p-1)), p>1, from which we deduce
(51)M∞=M(s*)=c(p)(1α)1/p,c(p)=p-1(p-1)1-1/p≤1.
From this bound, we derive(52)supy∈[0,L]∥uα(y)∥≤(1α)1/p∥f∥.
From the linear property of our problem, stability estimate of problem (47) may be written precisely in the following corollary.Corollary 19.
Ifuα,1(y;f1) (resp., uα,2(y;f2)) is the approximate solution corresponding to f1 (resp., f2), then
(53)supy∈[0,L]∥uα,1(y)-uα,2(y)∥≤(1α)1/p∥f1-f2∥.Remark 20.
We have(54)N(s)=sreTs1+αepTs≤1αK(s)=1αsre-(p-1)Ts,p>1,s=λ≥γ.
It is easy to show that
(55)K(s)≤K(s*=rL(p-1))=(rL(p-1))re-r=κ(r,p,L)<∞.
This remark shows thatuα(y)∈D(Ar/2) for all y∈[0,L].Proof.
The inclusionuα(y)∈D(Ar/2) is equivalent to ∥Ar/2uα(y)∥<∞. We have
(56)∥Ar/2uα(y)∥2=∫γ+∞λr{cosh(yλ)1+αepLλ}2d∥Eλf∥2≤(1α)2∫γ+∞{λre-(p-1)Lλ}2d∥Eλf∥2≤(1α)2κ(r,p,L)2∥f∥2<∞,
where κ(r,p,L)=supλ≥γλre-(p-1)Lλ=(r(p-1)L)re-r.Theorem 21.
Iff∈𝒢1, then
(57)supy∈[0,L]∥u(y)-uα(y)∥⟶0,α⟶0.Proof.
We compute(58)∥u(y)-uα(y)∥2=∫γ+∞(1-Mα(λ))2cosh2(yλ)d∥Eλf∥2≤∫γ+∞(1-Mα(λ))2cosh2(Lλ)d∥Eλf∥2≤∫γ+∞(1-Mα(λ))2e2Lλd∥Eλf∥2=∥(I-Mα)f^∥2,
where f^=eLAf and ∥f^∥2=∫γ+∞e2Lλd∥Eλf∥2<∞.
This implies thatsupy∈[0,L]∥u(y)-uα(y)∥≤∥(I-Mα)f^∥ and by virtue of (3) of Theorem 9, we conclude the desired convergence.The following technical lemmas play the key role in our analysis and calculations.Lemma 22.
Let(59)[ν,+∞[∋s⟼Q({a,r,q,L};s)=1αsr+ae-qLs,
where a>0, α>0, ν>0, q>0, L>0, and r≥1. Then one has
(60)Q({a,r,q,L};s)≤1α(k1log(k2(1/α)))r,
where k1(r,q,L)=rqL, k2(q,r,L,a)=qrLr-1a/r.Proof.
Differentiating the expression and setting the derivative equal to zero, we find(61)ddsQ({a,r,q,L};s)=-1(αsr+ae-qLs)2(αrsr-1-qae-qLs)=0.
The function (d/ds)Q({a,r,q,L};s)=0 admits a unique solution
(62)s^={s⟼αrsr-1}∩{s⟼qae-qLs}.
Therefore
(63)Q({a,r,q,L};s)≤Q({a,r,q,L};s^)≤1αs^r+ae-qLs^≤1αs^r.
We have
(64)(αrs^r-1-qae-qLs^)=0⟺sr-1eqLs^=qarα.
By using the inequality (et≥t, t≥0), then for t=qLs^, we obtain eqLs^≥qLs^ and we can write
(65)qarα=s^r-1eqLs^≤eqLs^(eqLs^qL)r-1=(1qL)r-1erqLs^,
which implies that s^≥(1/rqL)log((qrLr-1a/r)(1/α)). Hence, we obtain
(66)Q({a,r,q,L};s)≤1αs^r≤1α(k1log(k2(1/α)))r,
where k1(r,q,L)=rqL, k2(q,r,L,a)=qrLr-1a/r.Lemma 23.
Let(67)[ν,+∞[∋s⟼R({p,q,L};s)=epLs(1+αepLs)eqLs=1e(q-p)Ls+αeqLs,
where p≥1, q>0, α>0, ν>0, and L>0. Then one has the following.
If1≤p≤q, then
(68)R({p,q,L};s)≤e-(q-p)Ls≤e-(q-p)Lν≤1.
If0<q<p, p≥1, 0<α≤(p-q)/q, then
(69)R({p,q,L};s)≤k3(1α)(p-q)/p,k3(p,q)=qp(p-qp)(p-q)/p≤1.Proof.
By a simple differential calculus, we show that the functionR({p,q,L};s) achieves its maximum at s^=(1/pL)log((p-q)/αq). Consequently
(70)R({p,q,L};s)≤R({p,q,L};s^)=k3(1α)(p-q)/p.Now we assume the following a priori bounds hold:(71)u(L)∈D(Ar/2)⟺∫γ+∞λ2re2Lλd∥Eλf∥2≤E12<∞,(72)u(L)∈𝒢q⟺∫γ+∞e2L(1+q)λd∥Eλf∥2≤E22<∞.Theorem 24.
Letu (resp., uα) be the solution of problem (1) (resp., (47)) with the exact data f. If (71) (resp., (72)) is satisfied, then one has the following error estimates:
(73)∥u(y)-uα(y)∥=O(1log(1/α))r,(74)∥u(y)-uα(y)∥={O(α),if1≤p≤q,O(αq/p),if0<q<p,p≥1.Proof.
Putting(75)B1(λ)={epLλ1+αepLλ}1λr=1λre-pLλ+αλr≤B2(λ)=1γre-pLλ+αλr,B3(λ)={epLλ1+αepLλ}1eqLλ=1e(q-p)Lλ+αeqLλ.
Using the change of variables s=λ, we obtain the new expressions
(76)B^2(s)=1γre-pLs+αsr,(77)B^3(s)=1e(q-p)Ls+αeqLs.
By virtue of Lemma22 (inequality (60) and Lemma 23 (inequalities (68) and (69)), we can write
(78)B^2(s)≤1α(k1log(k2(1/α)))r,
where k1(r,p,L)=rqL, k2(p,r,L,γr)=qrLr-1γr/r. Consider
(79)B^3(s)≤{1,if1≤p≤q,(1α)(p-q)/p,if0<q<p,p≥1.
We have(80)∥u(y)-uα(y)∥2=∫γ+∞{αepLλ1+αepLλ}2cosh2(yλ)d∥Eλf∥2≤α2∫γ+∞{B2(λ)}2λ2re2Lλd∥Eλf∥2≤α2(sups≥γB^2(s))2E12,∥u(y)-uα(y)∥2=∫γ+∞{αepLλ1+αepLλ}2cosh2(yλ)d∥Eλf∥2≤α2∫γ+∞{B3(λ)}2eL(1+q)λd∥Eλf∥2≤α2(sups≥γB^3(s))2E22.
Using (78) and (79), we derive
(81)∥u(y)-uα(y)∥≤α1α(k1log(k2(1/α)))r=O(1log(1/α))r,∥u(y)-uα(y)∥≤{α,if1≤p≤q,αq/p,if0<q<p,p≥1.Combining (53), (73), and (74) with the help of triangle inequality
(82)∥u(y)-uαδ(y)∥≤∥u(y)-uα(y)∥+∥uα(y)-uαδ(y)∥=Δ1+Δ2,
we deduce the following corollary.Corollary 25.
Letu(y;f) (resp., uαδ(y;fδ)) be the solution of problem (1) (resp., (47)) with the exact data f (resp., the inexact data fδ) such that ∥f-fδ∥≤δ. If (71) (resp., (72)) is satisfied, then one has the following error estimates:
(83)(caser≥1)∥u(y)-uαδ(y)∥=O(θ1(α))+(1α)1/pδ,(84)(case1≤p≤q)∥u(y)-uαδ(y)∥=O(θ2(α))+(1α)1/pδ,(85)(case0<q<p,p≥1)∥u(y)-uαδ(y)∥=O(θ3(α))+(1α)1/pδ,
where
(86)θ1(α)=O(1log(1/α))r,θ2(α)=O(α),θ2(α)=O(αq/p).If we chooseα=α(δ)=δp/ω with ω>1, then we have
(87)δ(1δp/ω)1/p=δ(ω-1)/w,(88)θ1(α)=O(1log(1/δp/ω))r,(89)θ2(α)=δp/ω,θ3(α)=O(δq/ω).


## 3.3. Example: Cauchy Problem for the Modified Helmholtz Equation

In this paragraph, we give a concrete example to see how to apply the theoretical results developed in this study.Let us consider the Cauchy problem (modified Helmholtz equation) in the infinite stripℝ×(0,1):
(90)uyy(x,y)+uxx(x,y)-γu(x,y)=0,hhhhhhhhhhx∈ℝ,y∈(0,1),u(x,0)=f(x),uy(x,0)=0,x∈ℝ,
where γ is a real positive constant.Letu^(ξ,y)=(ℱu)(ξ,y) be the Fourier transform of u(x,y):
(91)u^(ξ,y)=12π∫ℝe-iξxu(x,y)dx.
With the help of the Fourier transformation, problem (1) can be transformed to an equivalent problem in the frequency domain:
(92)u^yy(ξ,y)-ξ2u^(ξ,y)-γu^(ξ,y)=0,dddddddiddξ∈ℝ,y∈(0,1),u^(ξ,0)=f^(ξ),u^y(ξ,0)=0,ξ∈ℝ.
It is easy to check that the formal solution of problem (92) has the form
(93)u^(ξ,y)=cosh(y(ξ2+γ))f^(ξ),
or equivalently, the formal solution of problem (90) is given by
(94)u(x,y)=(ℱ-1u^)(x,y)=12π∫ℝeixξu^(ξ,y)dξ=12π∫ℝeixξcosh(y(ξ2+γ))f^(ξ)dξ.PuttingΘ(ξ)=(ξ2+γ). Then Θ(ξ)→+∞ as |ξ|→+∞. From this remark, it is easy to see that a small perturbation in the data f^(ξ) may cause a dramatically large error in the solution u^(ξ,ξ). In addition, the magnifying factor is Θ(ξ)~e|ξ|, hence, the problem is severely ill-posed.Since the dataf(·) are based on (physical) observations and are not known with complete accuracy, we assume that f and fδ satisfy
(95)∥f-fδ∥≤δ,
where f and fδ belong to L2(ℝ), fδ denotes the measured data, and δ denotes the noise level.For this problem, we define the regularized solutions with noisy datafδ:
(96)uNδ(x,y)=12π∫ℝeixξcosh(y(ξ2+γ))f^δ(ξ)1[-N,N](ξ)dξ=12π∫-NNeixξcosh(y(ξ2+γ))f^δ(ξ)dξ,
where 1[-N,N] is the characteristic function of the interval [-N,N]. Consider
(97)uαδ(x,y)=12π∫ℝeixξ(cosh(y(ξ2+γ))1+αep(ξ2+γ))f^δ(ξ)dξ,
where p≥1. The quantities α=α(δ) and N=N(δ) are the parameters which were defined in Sections 3.1 and 3.2.


## 4. The Nonlocal Boundary Value Problem Method and Some Extensions

In this section we give the connection between the mollification method and the nonlocal boundary value problem method; also we give some extensions to our investigation.


### 4.1. The Nonlocal Boundary Value Problem Method

Consider the following problem:(98)P(a(a≥1);α,β){uyy=Au,0<y<aL,u(0)+αu(aL)+βA-1/2uy(aL)=f,uy(0)=0.(i)
The caseβ=0 coincides with the method treated by Hào et al. in [10].(ii)
In the caseα=β, the solution of P(a;α,α) is uα(y)=cosh(yA)(I+αeaL)-1f coincides with the solution resulting from the mollification method.(iii)
The error estimates obtained in our analysis by using the mollification method are similar to those obtained in [8, 10].This shows that our study framework is more general and includes many results obtained in this direction.


### 4.2. Generalization

Let us consider(99)P{uyy=Au,0<y<L,u(0)=f,uy(0)=0,
where A is a self-adjoint, linear unbounded operator in H and changes the sign with 0∈ρ(A) (A-1 exists and A-1∈ℒ(H)).We assumeσ(A)=]-∞,-γ]∪[γ,+∞[, γ>0. The spectral theory of self-adjoint operators enables us to write
(100)h=∫ℝdEλh=∫-∞-γdEλh+∫γ+∞dEλh=h-+h+,h∈H;that is, the Hilbert spaceH decomposes into the direct sum H=H-⊕H+, and
(101)A=∫ℝλdEλ=∫-∞-γλdEλ+∫γ+∞λdEλ=A-+A+.This decomposition gives us two problems: one is elliptic (ill-posed) and the other is hyperbolic (well-posed). Consider(102)Elliptic  Problem{Uyy=A+U,0<y<L,U(0)=f+,Uy(0)=0,Hyperbolic  Problem{Vyy=A-V,0<y<L,V(0)=f-,Vy(0)=0.The formal solution of problem (99) is
(103)u(y)=U(y)+V(y),U(y)=cosh(yA+)f+,V(y)=cos(y-A-)f-.We define(104)uα=cos(y-A-)f-m+cosh(yA+)(I+αepLA+)-1f+=∫-∞-γcos(y-λ)dEλfm+∫γ+∞cosh(yλ)(1+αepLλ)-1dEλf,vα=cos(y-A-)f-+cosh(yA+)Eβf+=∫-∞-γcos(y-λ)dEλfm+∫γ+∞cosh(yλ)1[γ,β]dEλf.
We follow the same methodology developed in the previous Sections 3.1 and 3.2, we show that uα and vα are two stable approximations to problem (99), and we establish the same results of error estimates.Remark 26.
We define the mollification operatorMα=(I+αΘp(A))-1, where Θp:[γ,+∞[→∞ satisfies
(105)∀λ∈[γ,+∞[,Θp(λ)≥epLλ,p≥1.
Under certain conditions on Θp(·) and with the help of a technical calculation, we can extend the results obtained in Section 3.2.


## 4.1. The Nonlocal Boundary Value Problem Method

Consider the following problem:(98)P(a(a≥1);α,β){uyy=Au,0<y<aL,u(0)+αu(aL)+βA-1/2uy(aL)=f,uy(0)=0.(i)
The caseβ=0 coincides with the method treated by Hào et al. in [10].(ii)
In the caseα=β, the solution of P(a;α,α) is uα(y)=cosh(yA)(I+αeaL)-1f coincides with the solution resulting from the mollification method.(iii)
The error estimates obtained in our analysis by using the mollification method are similar to those obtained in [8, 10].This shows that our study framework is more general and includes many results obtained in this direction.


## 4.2. Generalization

Let us consider(99)P{uyy=Au,0<y<L,u(0)=f,uy(0)=0,
where A is a self-adjoint, linear unbounded operator in H and changes the sign with 0∈ρ(A) (A-1 exists and A-1∈ℒ(H)).We assumeσ(A)=]-∞,-γ]∪[γ,+∞[, γ>0. The spectral theory of self-adjoint operators enables us to write
(100)h=∫ℝdEλh=∫-∞-γdEλh+∫γ+∞dEλh=h-+h+,h∈H;that is, the Hilbert spaceH decomposes into the direct sum H=H-⊕H+, and
(101)A=∫ℝλdEλ=∫-∞-γλdEλ+∫γ+∞λdEλ=A-+A+.This decomposition gives us two problems: one is elliptic (ill-posed) and the other is hyperbolic (well-posed). Consider(102)Elliptic  Problem{Uyy=A+U,0<y<L,U(0)=f+,Uy(0)=0,Hyperbolic  Problem{Vyy=A-V,0<y<L,V(0)=f-,Vy(0)=0.The formal solution of problem (99) is
(103)u(y)=U(y)+V(y),U(y)=cosh(yA+)f+,V(y)=cos(y-A-)f-.We define(104)uα=cos(y-A-)f-m+cosh(yA+)(I+αepLA+)-1f+=∫-∞-γcos(y-λ)dEλfm+∫γ+∞cosh(yλ)(1+αepLλ)-1dEλf,vα=cos(y-A-)f-+cosh(yA+)Eβf+=∫-∞-γcos(y-λ)dEλfm+∫γ+∞cosh(yλ)1[γ,β]dEλf.
We follow the same methodology developed in the previous Sections 3.1 and 3.2, we show that uα and vα are two stable approximations to problem (99), and we establish the same results of error estimates.Remark 26.
We define the mollification operatorMα=(I+αΘp(A))-1, where Θp:[γ,+∞[→∞ satisfies
(105)∀λ∈[γ,+∞[,Θp(λ)≥epLλ,p≥1.
Under certain conditions on Θp(·) and with the help of a technical calculation, we can extend the results obtained in Section 3.2.



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