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A modified quasi-boundary value method for an abstract ill-posed biparabolic problem

  • Khelili Besma , Boussetila Nadjib EMAIL logo and Rebbani Faouzia
Published/Copyright: December 29, 2017
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Open Mathematics
From the journal Open Mathematics Volume 15 Issue 1

Abstract

In this paper, we are concerned with the problem of approximating a solution of an ill-posed biparabolic problem in the abstract setting. In order to overcome the instability of the original problem, we propose a modified quasi-boundary value method to construct approximate stable solutions for the original ill-posed boundary value problem. Finally, some other convergence results including some explicit convergence rates are also established under a priori bound assumptions on the exact solution. Moreover, numerical tests are presented to illustrate the accuracy and efficiency of this method.

Keywords: Ill-posed problems; Biparabolic problem; Regularization
MSC 2010: 47A52; 65J22

1 Formulation of the problem

Throughout this paper H denotes a complex separable Hilbert space endowed with the inner product 〈.,.〉 and the norm ∥.∥, 𝓛(H) stands for the Banach algebra of bounded linear operators on H.

Let A : 𝓓(A) ⊂ H ⟶ H be a positive, self-adjoint operator with compact resolvent, so that A has an orthonormal basis of eigenvectors (ϕn) ⊂ H with real eigenvalues (λn) ⊂ ℝ+, i.e.,

Aϕn=λnϕn,n∈N∗,〈ϕi,ϕj〉=δij=1,if i=j0,if i≠j,0<ν≤λ1≤λ2≤λ3≤…,limn→∞λn=∞,∀h∈H,h=∑n=1∞hnϕn,hn=〈h,ϕn〉.

In this paper, we consider the following inverse source problem of determining the unknown source term u(0) = f and the temperature distribution u(t) for 0 ≤ t < T, of the following biparabolic problem

B2u=(ddt+A)2u(t)=u″(t)+2Au′(t)+A2u(t)=0,0<t<T,u(T)=g,u′(0)=0, (1)

where 0 < T < ∞ and g is a given H-valued function.

To our knowledge, the literature devoted to this class of problems is quite scarce, except the papers [1, 2, 3, 4, 5, 6]. The study of this case is caused not only by theoretical interest, but also by practical necessity. In particular, the biparabolic model is used in mathematical modeling to describe specials features of the dynamics of deformable water-saturated porous media during their filtration consolidation subject to applied loads [7, 8, 9].

It is well-known that the classical heat equation does not accurately describe the conduction of heat [10, 11]. Numerous models have been proposed for better describing this phenomenon. Among them, we can cite the biparabolic model proposed in [12], the fractional biparabolic model [13], for a more adequate mathematical description of heat and diffusion processes than the classical heat equation. For physical motivation and other models we refer the reader to [14, 15, 16, 17, 18, 19, 20, 21].

This work is a continuity of the work developed recently by Lakhdari and Boussetila [2], where the strategy of regularization which will be used is completely different that used in [2]. Our new strategy is motivated by the simplicity of the method, as well as the numerical results obtained, which are better compared to those obtained using a variant of an iterative regularization [2]. More precisely, we propose an improved modified quasi-boundary-value method with two parameters α > 0 and r ≥ 0, where the parameter α is introduced to filter the high frequencies, and the second parameter r to include the regularity of the solution of the original problem. The advantage of the multi-parameter regularization is such that it gives more freedom in attaining order optimal accuracy [22, 23, 24, 25, 26, 27, 28, 29].

The quasi-boundary value method, also called non-local auxiliary boundary condition, introduced and developed by Showalter [30, 31], is a regularization technique by replacing the final condition or boundary condition by a nonlocal condition such that the perturbed problem is well-posed.

The main advantage of the quasi-boundary-value method is that it gives a well-posed problem, where the differential equation has not been changed, only the boundary values have been modified. Therefore, we can exploit various numerical methods to approach the problem in question, for arbitrary geometry [0, T] × Ω, where Ω is a sub-set of ℝn, n ≥ 1.

This method has been used to solve some ill-posed problems for parabolic, hyperbolic and elliptic equations; for more details, see [22, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44] and the references therein.

2 Ill-posedness of the problem and a conditional stability result

We point out here some results established in [2].

Let use consider the following well-posed problem.

B2w=(ddt+A)2w(t)=w″(t)+2Aw′(t)+A2w(t)=0,0<t<T,w(T)=ξ,w′(0)=0, (2)

where ξ ∈ 𝓓(A).

Theorem 2.1

([2]). For any ξ ∈ 𝓓(A), problem (2) admits an unique solution

w∈C2]0,+∞[;H∩C1[0,+∞[;H∩C[0,+∞[;D(A)∩C1]0,+∞[;D(A)∩C2]0,+∞[;D(A2)

given by

w(t)=R(t;A)ξ=(I+tA)e−tAξ=∑n=1∞(1+tλn)e−tλn〈ξ,ϕn〉ϕn. (3)

Remark 2.2

It is easy to check that

∥R(t;A)∥=supλ≥λ1(1+tλ)e−tλ≤(1+tλ1)e−tλ1, (4)

sup0≤t≤T∥R(t;A)∥=sup0≤t≤T(1+tλ1)e−tλ1=1. (5)

2.1 Ill-posedness of the problem (1)

Theorem 2.3

([2]). Let g ∈ H, then the unique formal solution of the problem (1) is given by

u(t)=∑n=1∞1+tλn1+Tλne(T−t)λn〈g,ϕn〉ϕn. (6)

In this case,

f=u(0)=∑n=1∞11+TλneTλn〈g,ϕn〉ϕn. (7)

From this representation we see that u(t) is unstable in [0,T[. This follows from the high-frequency

σ(t,λn)=1+tλn1+Tλne(T−t)λn⟶+∞,n⟶+∞.

Remark 2.4

  1. In the classical backward parabolic problem

    vt+Av=0,0<t<T,v(T)=g, (8)

    the unique formal solution is given by

    v(t)=∑n=1∞θn(t,λn)〈g,ϕn〉ϕn, (9)

    where

    θn(t,λn)=e(T−t)λn⟶+∞,n⟶+∞.

    In this case, the high-frequency θn(t,λn) is equal to e(T−t)λn and the problem is severely ill-posed.

  2. In the case of biparabolic model, we have σn = rnθn, where

    rn=1+tλn1+Tλn,

    is the relaxation coefficient resulting from the hyperbolic character of the biparabolic model.

    Observe that

    tT≤rn≤1+tλ11+Tλ1≤1, (10)

    and

    u(t)=R(t)v(t), (11)

    where

    ∥R(t)∥=supn≥1{rn}=r1=1+tλ11+Tλ1. (12)

    From this remark, we observe that the degree of ill-posedness in the biparabolic model is relaxed compared to the classical parabolic case.

2.2 Conditional stability estimate

We would like to have estimates of the form

∥u(t)∥≤Ψ(∥g∥),

for some function Ψ(.) such that Ψ(s) ⟶ 0 as s ⟶ 0.

Since the problem of determining u(t) from the knowledge of {u(T) = g, u′(0) = 0} is ill-posed, an estimate such as the above will not be possible unless we restrict the solution u(t) to certain source set 𝓜 ⊂ H.

In our model, we will see that we can employ the method of logarithmic convexity to identify this source set:

Mρ={w(t)∈H:wobeys(1)and∥Aw(0)∥≤ρ<∞}. (13)

On the basis {ϕn} we introduce the Hilbert scale (Hs)s∈ℝ (resp. (𝔈s)s∈ℝ) induced by A as follows

Hs=D(As)={h∈H:∥h∥Hs2=∑n=1∞λn2s|〈h,ϕn〉|2<+∞},Es=D(esTA)={h∈H:∥h∥Es2=∑n=1∞e2Tsλn|〈h,ϕn〉|2<+∞},

We give here a result of conditional stability. The demonstration is given in the paper [2].

Theorem 2.5

The problem 1 is conditionally well-posed on the set

M={w(t)∈H:∥Aw(0)∥<∞}

if and only if

g∈E1={h∈H:∑n=1∞e2Tλn|(h,ϕn)|2<∞}.

Moreover, if u(t) ∈ Mρ, then we have the following Hölder continuity

∥u(t)∥≤Ψ(∥g∥)=γρT−tT∥g∥tT, (14)

where γ=1+Tλ1λ1T−tT.

3 Regularization and error estimates

In this work, we propose a modified quasi-boundary value method (MQBVM) to solve the inverse problem 1, i.e., replacing the final condition u(T) = g with the functional time nonlocal condition,

αAru0+uT=g, (15)

to form an approximate regularized problem

ddt+A2ut=u″t+2Au′t+A2ut=0,0<t<T,αAru0+uT=g,u′0=0, (16)

where r > 0 is a real parameter and α > 0 is the regularization parameter.

Remark 3.1

The case r = 0, corresponds to the classical quasi-boundary value method.

Denoting by uα(t) the solution of 16. By the separation of variables and the formula (3), we show the well-posedness of (16), and its solution can be expressed by

uαt=I+tAαAr+1+TAe−TA−1e−tAg=∑n=1+∞1+tλnαλnr+1+Tλne−Tλne−tλng,ϕnϕn (17)

Theorem 3.2

For all g ∈ D(A) and r > 0, the approximate problem 16 admits an unique solution uα given by

uαt=I+tAαAr+1+TAe−TA−1e−tAg=∑n=1+∞1+tλnαλnr+1+Tλne−Tλne−tλngnϕn,gn=g,ϕn (18)

Moreover, the following inequality holds

sup0≤t≤Tuαt≤uα0≤C3καg, (19)

where

κα=1α1rln1α,0<r<1,1αln1αr,r≥1, (20)

and C3 = max (C1,C2), C1 = (rT)r, C2 = r.

Proof

We compute

uαt2=∑n=1+∞1+tλne−tλnαλnr+1+Tλne−Tλn2gn2

Putting

Gλn=1+tλne−tλn≤supn≥1Gλn=1+tλ1e−tλ1≤1, (21)

and

H~λn=1αλnr+1+Tλne−Tλn. (22)

Our goal here is to prove that

supλ≥λ1H~λ=κα=1α1rln1α,0<r<1,1αln1αr,r≥1, (23)

Indeed, we have

Hλn=1+tλne−tλnαλnr+1+Tλne−Tλn≤1αλnr+1+Tλne−Tλn=H~λn.

Now to estimate H̃(λ), we proceed as follows

supλ≥λ1H~λ≤maxA,B,whereA=supλ≤λ∗H~λ,B=supλ≥λ∗H~λ,λ∗=1Trln1α,0<α<1.

For 0 < ν ≤ λ ≤ λ*, we have

H~λn≤11+Tλne−Tλn=eTλn1+Tλn

We denote s = Tλn and the function

fs=es1+s.

The function attains its maximum at λ*,

supλ≤λ∗H~λ=eTλ∗1+Tλ∗≤eTλ∗Tλ∗=eT1Trln1αT1Trln1α.

Hence, we get

supλ≤λ∗H~λ≤rα1rln1α. (24)

If λ ≥ λ∗, we can write

H~λ=1αλr+1+Tλe−Tλ≤1α1λr≤1α1λ∗r

which implies that

supλ≥λ∗H~λ≤1α11Trln1αr=rTr1αln1αr. (25)

Putting C1 = (rT)r, C2 = r, C3 = max (C1,C2).

  • If 0 < r < 1 and 0 < α < 1, we observe that

    1α1rln1α1αln1αr−1=αln1αrα1rln1α=αr−1rln1αr−1=1α1−rrln1α1−r=γα⟶+∞,asα⟶0.

    limα→0γα = +∞ ⟹ for ε = 1, ∃ α0 such as α ≤ α0 ⇒ γ(α) ≥ ε = 1. Then, for α sufficiently small, we have

    1α1rln1α≥1αln1αr.

    Therefore,

    maxA,B=C31α1rln1α. (26)

  • If r ≥ 1 and 0 < α < 1, we have limα→0γα = 0 ⟹ for ε = 1, ∃α0 such that α ≤α0 ⇒ γ(α) ≤ ε = 1. Then, for α sufficiently small, we have

    1α1rln1α≤1αln1αr,

    and thus

    maxA,B=C31αln1αr. (27)

    From (26) and (27), we obtain the desired estimate:

    sup0≤t≤Tuαt≤uα0≤C3καg,

    where

    κα=1α1rln1α,0<r<1,1αln1αr,r≥1.
□

Theorem 3.3

If u(0) ∈ H and u(0) ∈ 𝓓(A), i.e., ∥u(0)∥ + ∥Au(0)∥ < ∞, then we have

sup0≤t≤Tut−uαt+Aut−uαt≤u0−uα0+Au0−uα0→0,asα→0 (28)

Remark 3.4

We recall here that

∥u(0)∥+∥Au(0)∥<∞⟺g∈E1.

Proof

We have

u(0)=∑n=1∞11+TλneTλn〈g,ϕn〉ϕn,uα(0)=∑n=1+∞1αλnr+1+Tλne−Tλng,ϕnϕn,

and

∥u(0)−uα(0)∥2=∑n=1+∞αλnreTλn1+Tλnαλnr+1+Tλne−Tλn2gn2.

From this equality we can write

ut−uαt2=∑n=1+∞1+tλn1+TλneT−tλn−1+tλnαλnr+1+Tλne−Tλne−tλn2gn2=∑n=1+∞1+tλnαλnr+1+Tλne−TλneTλn−1+Tλn1+Tλnαλnr+1+Tλne−Tλne−tλn2gn2=∑n=1+∞1+tλnαλnreTλn1+Tλnαλnr+1+Tλne−Tλne−tλn2gn2≤∑n=1+∞αλnreTλn1+Tλnαλnr+1+Tλne−Tλn2gn2=u0−uα02,

and

Aut−uαt2=∑n=1+∞λn1+tλn1+TλneT−tλn−1+tλnαλnr+1+Tλne−Tλne−tλn2gn2=∑n=1+∞λn1+tλnαλnreTλn1+Tλnαλnr+1+Tλne−Tλne−tλn2gn2≤∑n=1+∞λnαλnreTλn1+Tλnαλnr+1+Tλne−Tλn2gn2=Au0−uα02,

thus we get

sup0≤t≤Tut−uαt+Aut−uαt≤u0−uα0+Au0−uα0.

Now, we show that

u0−uα0+Au0−uα0→0,asα→0.

We have

u0=I+TA−1eTAg=∑n=1+∞eTλn1+Tλng,ϕnϕn,

and

uα0=αAr+I+TAe−TA−1g=∑n=1+∞1αλnr+1+Tλne−Tλng,ϕnϕn,

then we get

u0−uα02=∑n=1+∞αλnreTλnαλnr+1+Tλne−Tλn1+Tλn2gn2=∑n=1+∞αλnrαλnr+1+Tλne−Tλn2eTλn1+Tλn2gn2=∑n=1+∞αλnrαλnr+1+Tλne−Tλn2un02=∑n=1+∞F~λn2un02,

where

F~λn=αλnrαλnr+1+Tλne−Tλn.

We assume that u(0) ∈ H.

u(0)∈H⟺∑n=1+∞eTλn1+Tλn2gn2=∑n=1+∞un02=u02<+∞

For ε > 0, we choose N > 0 such that ∑n=N+∞eTλn1+Tλn2gn2≤ε22. Thus

u0−uα02=∑n=1NF~λ2un02+∑n=N+∞F~λn2un02. (29)

We observe that F~λn=αλnrαλnr+1+Tλne−Tλn≤1, then we can write

∑n=N+∞F~λn2un02≤∑n=N+∞un02≤ε22.

The other quantity can be estimated as follows

∑n=1NαλnreTλnαλnr+1+Tλne−Tλn1+Tλn2gn2=∑n=1Nαλnrαλnr+1+Tλne−Tλn2un02=∑n=1NF~λn2un02≤sup1≤n≤NF~λn2∑n=1Nun02.

It is clear that

F~λn=αλnrαλnr+1+Tλne−Tλn≤αλnr1+Tλne−Tλn≤αλnreTλn1+Tλn,

and λn ≤ λN implies that

F~λn≤αλnr1+TλneTλn≤αλNreTλN.

It follows that

sup1≤n≤NF~λn≤αλNreTλN,

and consequently

∑n=1NF~λn2un(0)2≤αλNreTλN2u02.

If we choose the parameter α such that αλNreTλNu0≤ε2, we obtain

u0−uα02≤αλNreTλN2u02+ε22≤ε22+ε22=ε2.

Which shows that

uα0→u0,asα→0.

To complete the proof, it remains to show that

∥Au0−uα0∥→0,asα→0.

We compute

Au0−uα02=∑n=1+∞αλnr+1eTλnαλnr+1+Tλne−Tλn1+Tλn2gn2=∑n=N+∞F~λn2λnun02.

We have

u(0)∈D(A)⟺Au02=∑n=1+∞λneTλn1+Tλn2gn2<+∞.

For ε > 0, we choose N > 0 such that

∑n=N+∞λnun02=∑n=N+∞λneTλn1+Tλn2gn2<ε22.

Then, we can write

∑n=N+∞F~λn2λnun02≤∑n=N+∞λnun02≤ε22.

and

∑n=1NF~λn2λnun02≤sup1≤n≤NF~λn∑n=1Nλnun02≤αλNreTλN2Au02.

If we choose the parameter α such that αλNreTλNAu0≤ε2, we get

Au0−uα02≤αλNreTλN2Au02+ε22≤ε22+ε22=ε2.

Which shows that

Au0−uα0→0,asα→0.

In conclusion,

sup0≤t≤Tut−uαt+Aut−uαt≤u0−uα0+Au0−uα0→0,α→0 (30)

□

Theorem 3.5

If u(0) ∈ 𝓓(A(θ+1)) such that ∥A(θ+1)u(0)∥ ≤ Eθ, and 1 ≤ r ≤ θ, then we have the following estimate

sup0≤t≤Tut−uαt+Aut−uαt≤u0−uα0+Au0−uα0≤C5lnC4αθEθ (31)

Proof

We have

u0−uα02=∑n=1+∞αλnreTλnλn−θ+1λnθ+1αλnr+1+Tλne−Tλn1+Tλn2gn2=∑n=1+∞αλnrλn−θ+1αλnr+1+Tλne−Tλn2λnθ+1eTλn1+Tλn2gn2=∑n=1+∞αλn−θ+1α+1+Tλnλn−re−Tλn2λnθ+1eTλn1+Tλn2gn2=∑n=1+∞αG^αλn2λnθ+1un02,

where

G^αλn=λn−θ+1α+1+Tλnλn−re−Tλn=1λnαλnθ+1+Tλnλnθ−re−Tλn.

If r ≤ θ, then

G^αλn≤1λ1αλnθ+1+Tλnλ1θ−re−Tλn=1λ1θ+1−rαλ1θ−rλnθ+1+Tλne−Tλn=1λ1θ+1−rH~β(λn),

where β=αλ1θ−r. Now by (23), we conclude that

supn≥11λ1θ+1−rαλ1θ−rλnθ+1+Tλne−Tλn=1λ1θ+1−rsupλ≥λ1H~β(λn)=λ1−θλ11−1θrα1θlnλ1θ−rα,0<θ<1,λ1−1αlnλ1θ−rαθ,θ≥1. (32)

If θ ≥ 1, we can write

u0−uα02≤∑n=1+∞αG^αλn2λnθ+1un02 (33)

≤λ1−1lnλ1θ−rαθ2∑n=1+∞λnθ+1un02 (34)

≤λ1−1lnλ1θ−rαθ2Aθ+1u(0)2, (35)

and

Au0−uα02=∑n=1+∞αλnrλn−θαλnr+1+Tλne−Tλn2λnθ+1eTλn1+Tλn2gn2≤∑n=1+∞ααλnθ+1+Tλnλnθ−re−Tλn2λnθ+1un02≤∑n=1+∞αλ1θ−rαλ1θ−rλnθ+1+Tλne−Tλn2λnθ+1un02≤αsupn≥11λ1θ−rαλ1θ−rλnθ+1+Tλne−Tλn2∑n=1+∞λnθ+1un02.

By virtue of (23), we obtain

Au0−uα02≤1lnλ1θ−rα2θAθ+1u02. (36)

Combining (33) and (36), we obtain

sup0≤t≤Tut−uαt+Aut−uαt≤u0−uα0+Au0−uα0≤1lnλ1θ−rαθλ1−1+1Aθ+1u0 (37)

≤C5lnC4αθEθ, (38)

where C4=λ1θ−r and C5=λ1−1+1. □

We conclude this paper by constructing a family of regularizing operators for the problem 1.

Definition 3.6

A family {Rα(t), α > 0, t ∈ [0,T]} ⊂ 𝓛(H) is called a family of regularizing operators for the problem (1) if for each solution u(t), 0 ≤ t ≤ T of (1) with final element g, and for any η > 0, there exists α(η) > 0, such that

α(η)→0,η→0, (39)

∥Rα(η)(t)gη−u(t)∥→0,η→0, (40)

for each t ∈ [0,T] provided that gη satisfies ∥gη𠈒g∥ ≤ η.

Define Rα(t) = (I + tA) [α Ar + (1 + TA) e−TA]−1e−tA. It is clear that Rα(t) ∈ 𝓛(H) (see (19)). In the following we will show that Rα(t) is a family of regularizing operators for the problem 1.

Theorem 3.7

Under the assumption g ∈ 𝔈1, the condition (40) holds.

Proof

We have

Δα(t)=∥Rα(t)gη−u(t)∥≤∥Rα(t)(gη−g)∥+∥Rα(t)g−u(t)∥=Δ1(t)+Δ2(t),

where

Δ1(t)=∥Rα(t)(gη−g)∥≤κ(α)η,Δ2(t)=∥Rα(t)g−u(t)∥.

We observe that

Δ1(t)≤ηα1rln1α,0<r<1,ηαln1αr,r≥1.

Choose α=ηr2 if 0 < r < 1, and α=η if r ≥ 1, it follows

Δ1(t)≤ηln1η1/r,0<r<1,ηln1ηr,r≥1,⟶0,as η⟶0. (41)

Now, by Theorem 3.3 we have

Δ2(t)=∥uα(η)(t)−u(t)∥→0,as η⟶0, (42)

uniformly in t. Combining (41) and (42) we obtain

sup0≤t≤T∥Rα(t)gη−u(t)∥⟶0,asη⟶0.

This shows that Rα(t) is a family of regularizing operators for the problem 1. □

4 Numerical results

In this section we give a two-dimensional numerical test to show the feasibility and efficiency of the proposed method. Numerical experiments where carried out using MATLAB.

We consider the following inverse problem

∂∂t−∂2∂x22u(x,t)=0,x∈(0,π),t∈(0,1),u(0,t)=u(π,t)=0,t∈(0,1),u(x,1)=g(x),ut(x,0)=0,x∈[0,π], (43)

where f(x) = u(x,0) is the unknown initial condition and u(x,1) = g(x) is the final condition.

It is well known that the operator

A=−∂2∂x2,D(A)=H01(0,π)∩H2(0,π)⊂H=L2(0,π),

is positive, self-adjoint with compact resolvent (A is diagonalizable).

The eigenpairs (λn, ϕn) of A are

λn=n2,ϕn(x)=2πsin⁡(nx),n∈N∗.

In this case, the formula (7) takes the form

f(x)=u(x,0)=2π∑n=1+∞11+n2en2∫0πg(x)sin⁡(nx)dxsin⁡(nx). (44)

In the following, we consider an example which has an exact expression of solutions (u(x,t), f(x)).

Example

If u(x,0)=ϕ1(x)=2πsin⁡(x), then the function

u(x,t)=∑n=1∞(1+tλn)e−tλn〈ϕ1,ϕn〉ϕn(x)=(1+tλ1)e−tλ1ϕ1(x)=2π(1+tλ1)e−tλ1sin⁡(x)

is the exact solution of the problem (43). Consequently, the data function is g(x)=u(x,1)=2π2esin⁡(x).

By using the central difference with step length h=πN+1 to approximate the first derivative ux and the second derivative uxx, we can get the following semi-discret problem (ordinary differential equation):

ddt−Ah2u(xi,t)=0,xi=ih,i=1,…N,t∈(0,1),u(x0=0,t)=u(xN+1=π,t)=0,t∈(0,1),u(xi,0)=g(xi),ut(xi,0)=0,xi=ih,i=1,…N, (45)

where 𝔸h is the discretisation matrix stemming from the operator A=−d2dx2 :

Ah=1h2Tridiag(−1,2,−1)∈MN(R)

is a symmetric, positive definite matrix. We assume that it is fine enough so that the discretization errors are small compared to the uncertainty δ of the data; this means that 𝔸h is a good approximation of the differential operator A=−d2dx2 , whose unboundedness is reflected in a large norm of 𝔸h. The eigenpairs (μk, ek) of 𝔸h are given by

μk=4N+1π2sin2kπ2(N+1),ek=sinjkπN+1j=1N,k=1…N.

Adding a random distributed perturbation (obtained by the Matlab command randn) to each data function, we obtain the vector gδ:

gδ=g+εrandn(size(g)),

where ε indicates the noise level of the measurement data and the function “randn(.)” generates arrays of random numbers whose elements are normally distributed with mean 0, variance σ2 = 1, and standard deviation σ = 1. “randn(size(g))” returns an array of random entries that is the same size as g. The bound on the measurement error δ can be measured in the sense of Root Mean Square Error (RMSE) according to

δ=∥gδ−g∥∗=1N∑i=1Ng(xi)−gδ(xi)21/2.

The discret approximation of (18) takes the form

uαδ(xj,0)=fα,δ(xj)=(αAhr+(IN+Ah)e−Ah)−1gδ(xj),j=1…N, (46)

where 𝕀N is the identity matrix.

In our numerical computations we always take N = 40 and consider only the cases when ε = 0.001, 0.01. The regularization parameter (α, r) is chosen in the following way: for any fixed r ∈ {0,1,2,3}, we try to find a satisfactory error by varying the second parameter α = εs with step length s = 0.1. We note α0 one of the best choice which gives this result. Now, for α = α0 fixed, we try to find an acceptable error by varying the first parameter r = 0, 1, 2, 3 in order to obtain the best possible convergence rate. It is important to note that this choice is of heuristic nature and the multiparameter discrepancy principle is quite scarce in the literature.

The relative error RE(f) is given by

RE(f)=∥fα,δ−f∥∗∥f∥∗.

Conclusion and discussion

Numerical results are shown in Figures 1-8 and Tables 1-2.

Fig. 1 ε (noise level) = 0.001, α (regularization parameter) = 0.015849, r (relaxation parameter) = 0
Fig. 1

ε (noise level) = 0.001, α (regularization parameter) = 0.015849, r (relaxation parameter) = 0

Fig. 2 ε (noise level) = 0.001, α (regularization parameter) = 0.015849,r (relaxation parameter) = 1
Fig. 2

ε (noise level) = 0.001, α (regularization parameter) = 0.015849,r (relaxation parameter) = 1

Fig. 3 ε (noise level) = 0.001, α (regularization parameter) = 0.015849,r (relaxation parameter) = 2
Fig. 3

ε (noise level) = 0.001, α (regularization parameter) = 0.015849,r (relaxation parameter) = 2

Fig. 4 ε (noise level) = 0.001, α (regularization parameter) = 0.015849,r (relaxation parameter) = 3
Fig. 4

ε (noise level) = 0.001, α (regularization parameter) = 0.015849,r (relaxation parameter) = 3

Fig. 5 ε (noise level) = 0.01, α (regularization parameter) = 0.025119, r (relaxation parameter) = 0
Fig. 5

ε (noise level) = 0.01, α (regularization parameter) = 0.025119, r (relaxation parameter) = 0

Fig. 6 ε (noise level) = 0.01, α (regularization parameter) = 0.025119, r (relaxation parameter) = 1
Fig. 6

ε (noise level) = 0.01, α (regularization parameter) = 0.025119, r (relaxation parameter) = 1

Fig. 7 ε (noise level) = 0.01, α (regularization parameter) = 0.025119, r (relaxation parameter) = 2
Fig. 7

ε (noise level) = 0.01, α (regularization parameter) = 0.025119, r (relaxation parameter) = 2

Fig. 8 ε (noise level) = 0.01, α (regularization parameter) = 0.025119, r (relaxation parameter) = 3
Fig. 8

ε (noise level) = 0.01, α (regularization parameter) = 0.025119, r (relaxation parameter) = 3

Table 1

The absolute errors Era for fixed α and for various value of r

N ε α r RE
40 0.001 0.015849 0 0.0920
40 0.001 0.015849 1 0.0035
40 0.001 0.015849 2 0.0019
40 0.001 0.015849 3 0.0003501

Table 2

The absolute errors Era for fixed α and for various value of r

N ε α r RE
40 0.01 0.025119 0 0.1707
40 0.01 0.025119 1 0.0103
40 0.01 0.025119 2 0.0070
40 0.01 0.025119 3 0.0061

In this paper, we have proposed an improved two-parameter regularization method (MQBVM) to solve an ill-posed biparabolic problem. The convergence and stability estimates have been obtained under a priori bound assumptions for the exact solution. Finally, some numerical tests show that our proposed regularization method is effective and stable.

According to the numerical tests, we observe the following regularizing effect:

  • In the case r = 0, ε = 0.001 and α = 0.015849 (resp. r = 0, ε = 0.01 and α = 0.025119), the approximate solution is far from the exact solution. But for the case r = 1,2,3, we observe that the solution becomes precise and very near to the exact solution (in particular for r = 2,3).

This shows that our approach has a nice regularizing effect and gives a better approximation with comparison to the classical QBV-method.

Acknowledgement

The authors thank the referees for their constructive comments which improved this paper.

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Received: 2017-8-29
Accepted: 2017-11-16
Published Online: 2017-12-29

© 2017 Besma et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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  273. Topical Issue on Metaheuristics - Methods and Applications
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https://doi.org/10.1515/math-2017-0140
Keywords for this article
Ill-posed problems; Biparabolic problem; Regularization
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