Samarskii A.A., Vabishchevich P.N. Numerical Methods for Solving Inverse Problems of Mathematical Physics

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266 Chapter 7 Evolutionary inverse problems

According to (7.86), improved stability can be gained in two ways. In the ﬁrst case,

stability can be improved due to increased energy (By, y) of the operator B (left-hand

side of inequality (7.86)) or, alternatively, due to decreased energy of the operator A

(right-hand side of inequality (7.86)). Consider ﬁrst the opportunities related with

addition of operator terms to the operators B and A. In this case, we will speak of

additive regularization.

We can most naturally start from an additive perturbation applied to the operator B,

i.e., from the transition B → B +α R, where R is the regularizing operator and α is the

regularization parameter. Taking the fact into account that in the generating scheme of

interest we have B = E, we put:

B = E + α R. (7.87)

To retain the ﬁrst approximation order in the scheme (7.85), (7.87), it sufﬁces for us to

choose α = O(τ ).

Consider two typical choices of the regularizing operator:

R = , (7.88)

R = 

. (7.89)

We can directly establish that the regularized difference scheme (7.85), (7.87) is

stable in H

provided that α ≥ τ/2 in the case of (7.88) and α ≥ τ

/16 in the case

of (7.89).

The regularized scheme (7.85), (7.87), (7.88) corresponds to the case in which we

use the standard weighted scheme

n+1

− y

+ (σ y

n+1

+ (1 − σ)y

) = 0 n = 0, 1,...,N

− 1,

with α = στ.

In the standard approach for the construction of stable schemes, additive regular-

ization is used. The alternative approach uses multiplicative perturbation of difference

operators in the generating scheme. Within the framework of the latter approach, con-

sider simplest examples part of which can be considered as a new interpretation of the

above regularized schemes.

In multiplicative regularization of B, apply, for instance, the change B → B(1 +

α R) or B → (1 + α R)B. With such a perturbation, we still remain in the class of

schemes with self-adjoint operators provided that R = R

∗

. In this case, we have the

previously examined regularized scheme (7.85), (7.87).

An example of more complex regularization is given by the transformation

B → (E + α R

∗

)B(E + α R).

In the case of R = A, the condition for stability is α ≥ τ/8. In another interesting

example, the alternate triangular method is used, in which A = R

∗

+ R and α ≥ τ/2.

Section 7.2 Regularized difference schemes 267

In a similar manner, multiplicative regularization due to perturbed operator A can

be applied. With allowance for the inequality (7.86), we can use the transformation

A → A(1 + α R)

−1

or A → (1 + α R)

−1

A. For simplest two-layer schemes, such

regularization can be considered as a new version of regularization applied to B.To

remain in the class of schemes with self-adjoint operators, it sufﬁces for us to choose

R = R(A). A higher potential is offered by the regularization

A → (E + α R

∗

)

−1

A(E + α R)

−1

In the latter case, the regularizing operator R can be chosen not directly related with

the operator A.

7.2.2 Inverted-time problem

In the development of computational algorithms for the approximate solution of ill-

posed problems involving evolutionary equations, broad possibilities are offered by

the regularization principle. In the rectangle , we seek the solution of the parabolic

equation

∂u

∂t



α=1

∂

∂x



k(x)

∂u

∂x



= 0, x ∈ , 0 < t < T , (7.90)

that differs from (7.77) only by the sign at the spatial derivatives, which corresponds to

the substitution of t with −t, yielding the equation with inverted time. The boundary

and initial conditions here remain the same as previously (see (7.78) and (7.79)).

We put into correspondence to the differential inverse problem (7.78), (7.78), (7.90)

a Cauchy problem for the differential-operator equation

− y = 0, 0 < t < T. (7.91)

Let us construct now absolutely stable difference schemes for (7.82), (7.91) using the

regularization principle for difference schemes.

To construct the difference scheme for the ill-posed problem under consideration,

we use the regularization principle. We start with the simplest explicit difference

scheme

n+1

− y

− y

= 0, x ∈ ω, n = 0, 1,...,N

− 1 (7.92)

supplemented with the initial condition (7.84).

In compliance with the regularization principle, we write the scheme (7.92) in the

canonical form with

A =−, B = E , (7.93)

i. e., A = A

∗

< 0.

268 Chapter 7 Evolutionary inverse problems

We use (see Lemma 3.2) the upper estimate for the difference operator :

 ≤ ME (7.94)

with the constant

M =

max

x∈ω

(1)

(x) + a

(1)

+ h

, x

)

max

x∈ω

(2)

(x) + a

(2)

, x

+ h

)

Theorem 7.8 The explicit scheme (7.84), (7.92) is -stable in H with

 = 1 + Mτ. (7.95)

Proof. The latter statement stems from the general conditions for -stability for two-

layer operator-difference schemes. The difference scheme (7.84), (7.85) with self-

adjoint, time-independent operators B > 0, A is -stable in H

in the case of (see

Theorem 4.4)

1 − ρ

B ≤ A ≤

1 + ρ

B. (7.96)

For the scheme (7.84), (7.92), we have B > 0, A < 0, and for >1 the right-hand

side of the two-sided operator inequality (7.96) is fulﬁlled for all τ>0. The left-hand

side of (7.96) assumes the form ( − 1)/τ E ≥ ; in view of (7.94), this inequality

holds with  chosen according to (7.95).

Remark 7.9 In the approximate solution of ill-posed problems, the value of the regu-

larization parameter must be matched with the input-data inaccuracy. Here, we restrict

ourselves to the construction of stable computational algorithms for ill-posed evolu-

tionary problems and to the examination of the effect of regularization parameter on

the stability of the difference scheme only. For a given value of α, the minimum value

of  is to be speciﬁed according to (7.95).

Starting from the explicit scheme (7.84), (7.92), we write the regularized scheme

for problem (7.82), (7.91) in the canonical form (7.85) with

A =−, B = E + α R. (7.97)

Theorem 7.10 The regularized scheme (7.85), (7.97) is -stable in H

with

 = 1 +

(7.98)

if the regularizing operator is chosen according to (7.88) and with

 = 1 +

√

(7.99)

if the operator (7.89) is used.

Section 7.2 Regularized difference schemes 269

Proof. To prove the theorem, it sufﬁces to check that the left-hand side of the two-

sided inequality (7.96) is fulﬁlled. For (7.97), this inequality assumes the form

 − 1

(E + α R) ≥ . (7.100)

In the case of R =  and with  chosen by (7.98), inequality (7.100) is fulﬁlled.

In the case of R = 

, inequality (7.100) can be transformed into

E + α

−

 − 1



√

α −

√

α( − 1)





1 −

4α( − 1)



E ≥ 0.

The latter inequality will be fulﬁlled with  chosen in the form of (7.99).

In a similar manner, other regularized difference schemes can be constructed. In par-

ticular, by now second-order evolutionary problems, problems with non-self-adjoint

operators, additive schemes for multi-dimensional inverse problems, etc. have been

considered. With constructed regularized difference schemes, these or those (standard

or non-standard) variants of the generalized inverse method can be related.

7.2.3 Generalized inverse method

First of all, consider methods for the approximate solution of ill-posed problems in-

volving evolutionary equations based on some perturbation of the initial equation that

makes the perturbed problem a well-posed problem. Here, the perturbation parameter

serves the regularization parameter. Such methods are known as generalized inverse

methods. Let us give a priori estimates for the solution of the perturbed problem

in various versions of the generalized inverse method for problems with self-adjoint,

positive operators. Convergence of the approximate solution u

to the exact solution u

takes place in some well-posedness classes. It makes sense to consider here the matter

of approximate solution of unstable evolutionary problems using difference methods

based on various versions of the generalized inverse method. Primary attention will be

paid to the matter of stability of the difference schemes.

First of all, let us dwell on the examination of the main version of the generalized

inverse method. In the Hilbert space H, consider an ill-posed Cauchy problem for the

ﬁrst-order evolutionary equation

− Au = 0, (7.101)

u(0) = u

. (7.102)

In the model inverse problem (7.78), (7.79), (7.90), we can put H = L

() and

Au =−



α=1

∂

∂x



k(x)

∂u

∂x



270 Chapter 7 Evolutionary inverse problems

on the set of functions satisfying (7.78). We assume that the operator A is a positive

deﬁnite self-adjoint operator in H. Let 0 <λ

≤ λ

≤···be the eigenvalues of A.

The corresponding system of eigenfunctions {w

}, w

∈ D(A), k = 1, 2,... is an

orthonormal complete system in H.

For the stable approximate solution of the ill-posed problem (7.101), (7.102), apply

the generalized inverse method. Taking into account the self-adjointness of A,we

determine u

(t) as the solution of the equation

− Au

+ αA

= 0 (7.103)

with the initial condition

(0) = u

. (7.104)

Let us formulate a typical statement about the approximate-solution stability.

Theorem 7.11 For the solution of problem (7.103), (7.104), the following estimate

holds:

u

(t)≤exp



4α



u

(0). (7.105)

Proof. To derive the estimate (7.105), we scalarwise multiply equation (7.103) in H

by u

(t). This yields the equality

u



+ αAu



= (Au

, u

). (7.106)

The right-hand side of (7.106) can be estimated as

(Au

, u

) ≤ εAu



4ε

u



. (7.107)

Substitution of (7.107) with ε = α into (7.106) yields:

u



≤

2α

u



From here, and also from the Gronwall inequality, the desired estimate (7.105) follows.

Let us brieﬂy discuss the matter of convergence of the approximate solution u

(t)

to the exact solution u(t ). From (7.101) and (7.103), for the inaccuracy v(t) = u

(t)−

u(t) we obtain the equation

− Av + αA

v =−αA

u. (7.108)

The ill-posedness of problem (7.101), (7.102) results from instability of the solution

with respect to initial data. Instead of the exact initial condition (7.104), we use the

condition

(0) = u

. (7.109)

Section 7.2 Regularized difference schemes 271

For the initial-condition inaccuracy, we can use the estimate of type

u

− u

≤δ. (7.110)

In view of this, equation (7.108) is supplemented with the initial condition

v(0) = u

− u

, (7.111)

in which v(0)≤δ.

Convergence of the approximate solution to the solution of the ill-posed problem

can be established in certain well-posedness classes. Most frequently, the a priori con-

straints on the solution of the unstable problem are related with the assumption about

solution boundedness. Concerning the problem (7.101), (7.102), we can consider the

following classes of solutions:

Au(t)≤M, (7.112)

A

u(t)≤M (7.113)

for all t ∈ [0, T ].

By examining stability with respect to initial data and right-hand side, we can derive

an estimate for the inaccuracy in the classes of a priori constraints (7.112), (7.113).

Under the constraints (7.112), we obtain:

v(t)≤δ exp



2α





exp





− 1



1/2

M. (7.114)

In the class (7.113), the corresponding estimate is

v(t)≤δ exp



2α



√



exp





− 1



1/2

αM. (7.115)

The derived estimates (7.114) and (7.115) are not optimal estimates. In particular,

these estimates do not prove convergence of the approximate solution u

(t) found

from equation (7.103) and initial condition (7.109) to the exact solution u(t) with

δ → 0. Here, a more subtle consideration is required. Let us show, for instance, the

convergence in H in the class of solutions bounded in H.

We will adhere to a scheme closely following the proof of convergence of approx-

imate solution to the exact solution in the case of approximate solution obtained with

non-locally perturbed initial condition. We write the approximate solution u

(t) for

the inaccurately given initial condition (7.109) in the operator form:

(t) = R(t,α)u

. (7.116)

In this notation, R(t, 0) is the exact solution of the problem and, hence, u(t) =

R(t, 0)u

. With the introduced settings, for the inaccuracy v we obtain

u

(t) − u(t)=R(t,α)(u

− u

) − (R(t,α)− R(t, 0))u



≤R(t,α)u

− u

+(R(t,α)− R(t, 0))u

. (7.117)

272 Chapter 7 Evolutionary inverse problems

The ﬁrst term in (7.117) refers to stability with respect to initial data (boundedness

of R(t,α)). The second term in the right-hand side of (7.117) requires that the solution

R(t,α)u

of the perturbed problem obtained with exact input data be close to the exact

solution u(t ). It is with this aim that a certain class of solutions (well-posedness class)

has been isolated.

In view of the derived initial-data stability estimate (7.105) and estimate (7.110), we

have:

R(t,α)u

− u

≤exp



4α



δ. (7.118)

Consider the second term in the right-hand side of (7.117).

The operator R(t,α)has the form

R(t,α) = exp ((A − αA

)t). (7.119)

For the approximate solution determined from (7.113), (7.109), with (7.119) we obtain

the representation

(t) =

∞



k=1

) exp ((λ

− αλ

)t)w

. (7.120)

In view of (7.120), we have

χ(t) ≡(R(t,α)− R(t, 0))u



∞



k=1

exp (2λ

t)(1 − exp (−αλ

t))

)

. (7.121)

We consider stability in the class of solutions bounded in H:

u(t)≤M, 0 ≤ t ≤ T . (7.122)

In view of (7.122), for any ε>0 there exists a number r (ε) such that

∞



k=r(ε)+1

exp (2λ

t)(u

)

≤

Hence, from (7.121) we obtain:

χ(t) ≤

r(ε)



k=1

exp (2λ

t)(1 − exp (−αλ

t))

)

∞



k=r(ε)+1

exp (2λ

t)(u

)

≤ M

r(ε)



k=1

(1 − exp (−αλ

t))

Section 7.2 Regularized difference schemes 273

For any r (ε), there exists a number α

such that in the case of α ≤ α

we have

r(ε)



k=1

(1 − exp (−αλ

t))

≤

Hence, (7.122) yields:

(R(t,α)− R(t, 0))u

≤ε/2. (7.123)

With relations (7.118) and (7.123) taken into account, from (7.117) we obtain the

estimate

u

(t) − u(t)≤exp



4α



δ +

. (7.124)

For any ε>0 there exist a number α = α(δ) ≤ α

and a sufﬁciently small δ(ε) for

which δ exp (t/(4α)) ≤ ε/2. Hence, (7.124) assumes the form

u

(t) − u(t)≤ε.

Thus, the following statement can be formulated:

Theorem 7.12 Suppose that for the initial-condition inaccuracy the estimate (7.110)

is valid. Then, the approximate solution u

(t) found as the solution of problem (7.103),

(7.109) with δ → 0, α(δ) → 0, δ exp (t/(4α)) → 0 converges to the bounded exact

solution u(t) of problem (7.101), (7.102) in H.

Remark 7.13 The proved statement admits various generalizations. For instance, in

the narrower class of a priori constraints (7.113), one can derive a direct estimate of

(R(t,α)− R(t, 0))u

 in terms of α. It follows from (7.121) and (7.113) that

(R(t,α)− R(t, 0))u



≤

∞



k=1

exp (2λ

t)α

)

≤ α

and, hence, the following inequality holds:

u

(t) − u(t)≤exp



4α



+ αtM.

In the second variant of the generalized inverse method, transition to a pseudo-

parabolic equation is used. In the case of a self-adjoint, positive operator A, instead of

the unstable problem (7.101), (7.102), we solve the equation

− Au

+ α A

= 0, (7.125)

supplemented with the initial condition (7.104).

274 Chapter 7 Evolutionary inverse problems

Theorem 7.14 For the solution of problem (7.104), (7.125), the following estimate

holds:

u

(t)≤exp





u

(0). (7.126)

Proof. To prove the theorem, we rewrite equation (7.125) as

− (E + αA)

−1

= 0. (7.127)

We scalarwise multiply (7.127) in H by u

(t); then, we obtain:

u



= ((E + αA)

−1

, u

). (7.128)

It is easy to check that for the right-hand side of (7.128) there holds the estimate

((E + αA)

−1

, u

) ≤

, u

On the operator level, the latter inequality corresponds to the inequality

(E + αA)

−1

A ≤

E. (7.129)

To prove inequality (7.129), we multiply (7.129) from the right and from the left by

(E + αA). In this way, we pass to the equivalent inequality

(E + αA)A ≤

(E + 2αA + α

which obviously holds in the case of α>0.

With inequality (7.129) taken into account, from (7.128) we obtain the inequality

u

≤

u



which yields the desired estimate (7.126).

Convergence of the approximate solution to the exact solution can be established

in the case of δ → 0, α(δ) → 0, δ exp (t/(α)) → 0 under the previously discussed

constraints.

Let us brieﬂy discuss the matter of construction and examination of numerical solu-

tion methods for ill-posed evolutionary problems with perturbed equations. As a model

problem, consider the problem (7.78), (7.79), (7.90), a retrospective heat-transfer prob-

lem in the rectangle . On discretization over space, we arrive at a Cauchy problem

for the differential-operator equation (7.82), (7.91).

Section 7.2 Regularized difference schemes 275

The difference analogue of the basic generalized inverse method (see (7.103)) has

the form

n+1

− y

− (σ

n+1

+ (1 − σ

) + α

(σ

n+1

+ (1 − σ

) = 0, (7.130)

n = 0, 1,...,N

− 1.

Let us examine stability of this weighted scheme using the results of the general

stability theory for operator-difference schemes. Here, stability criteria are formulated

as operator inequalities for difference schemes written in the canonical form.

The scheme (7.130) can be written in the canonical form (7.85) with

A =− + α

B = E − σ

τ+ σ

τα

(7.131)

Theorem 7.15 The scheme (7.85), (7.131) is ρ-stable in H = L

(ω) for any τ>0

with

ρ = 1 +

4α

τ, (7.132)

provided that σ

≤ 0, σ

≥ 1/2.

Proof. First of all, note that the choice of ρ is perfectly consistent with the estimate

(7.105) for the solution of the continuous problem. The proof of (7.132) is based on

checking the necessary and sufﬁcient conditions for ρ-stability.

First of all, we check that under the formulated constraints on the weights of (7.130)

we have: B = B

∗

> 0. With (7.131), the right-hand side of (7.96) can be transformed

as follows:

0 ≤

1 + ρ

B − A

=  − α

1 + ρ

E − (1 + ρ)σ

 + (1 + ρ)ασ



1 + ρ

E + (1 − (1 + ρ)σ

) + α((1 + ρ)σ

− 1)

Under the indicated constraints on the weights, this inequality holds for all ρ>1.

An estimate for ρ can be obtained from the left inequality of (7.96). With ρ =

1 + cτ , this inequality assumes the form

cB ≥−A. (7.133)

Substitution of (7.131) into (7.133) yields:

c(E − σ

τ+ σ

τα

) ≥  − α

=−



√

E −

√

α



4α

In the case of σ

≤ 0, σ

≥ 1/2, for the inequality to be fulﬁlled, we can put c =

1/(4α). The latter yields for ρ the expression (7.132).