Chapter 1. The basic concepts of inverse and ill-posed problems
§1 Inverse problems. The examples
§2 Well-posed and ill-posed problems
§3 Operator equations of the first and the second kinds
§4 Some aspects of setting and solving inverse problems
§5 Exercises for readers
Chapter 2. Inverse problems for ordinary differential equations
§1 Problems of determination of the right side and coefficients of linear differential equations and systems by their solutions
§2 Inverse problems for the first order linear equation with a parameter
§3 Inverse problems for linear differential equations of the second order with a parameter
§4 Inverse problems for nonlinear ordinary differential equations
§5 Exercises for readers
Chapter 3. Linear inverse problems for partial differential equations
§1 Inverse problems for the heat conductivity equation
§2 Inverse problems for the vibration equation
§3 Inverse problems for the Laplace and Poisson equations
§4 Exercises for readers
Chapter 4. Inverse coefficient problems for partial differential equations
§1 Inverse problems for first order equations
§2 The problem of determining the coefficient of the hyperbolic equation
§3 Reduction of inverse coefficient problems to inverse problems for ordinary differential equations with a parameter
§4 The problem of determining the coefficient dependent on the solution
§5 Exercises for readers
Chapter 5. Problems of determining the function from the values of integrals
§1 Determining the function of one variable from the integral of this function. The moment problem
§2 Inversion of the Radon transformation. The problems of computer tomography
§3 Determining the function of two variables from its integrals along the circles
§4 Exercises for readers
Chapter 6. Methods of solution of inverse problems
§1 Solving equations of the first kind on compact sets. The quasisolution method
§2 The Tikhonov regularization method
§3 The iterative method of solving equations of the first kind
§4 The projective method for solving equations of the first kind
§5 Methods for solving Fredholm integral equations of the first kind
§6 Method of solution of the Volterra integral equation of the first kind
§7 The quasi-inversion method
§8 Exercises for readers
Bibliography

##### Citation preview

INVERSE A N D ILL-POSED PROBLEMS SERIES

Elements of the Theory of Inverse Problems

Also available in the Inverse and Ill-Posed Problems Series: Volterra Equations and Inverse Problems A.L Bughgeim Small Parameter Method in Multidimensional Inverse Problems A.S. Barashkov Regularization, Uniqueness and Existence of Volterra Equations of the First Kind A. Asanov Methods for Solution of Nonlinear Operator Equations V.P. Tanana Inverse and Ill-Posed Sources Problems Yu.E Anikonov, B.A. Bubnov and G.N. Erokhin Methods for Solving Operator Equations V.P. Tanana Nonclassical and Inverse Problems for Pseudoparabolic Equations A. Asanov and E.R. Atamanov Formulas in Inverse and Ill-Posed Problems Yu.E. Anikonov Inverse Logarithmic Potential Problem V.G. Cherednichenko Multidimensional Inverse and Ill-Posed Problems for Differential Equations Yu.E. Anikonov Ill-Posed Problems with A Priori Information V.V. Vasin and A. L Ageev Integral Geometry of Tensor Fields V.A. Sharafutdinov Inverse Problems for Maxwell's Equations V.G. Romanov and S.I. Kabanikhin

Related Journal: Journal of Inverse and Ill-Posed Problems Editor-in-Chief: M M . Lavrent'ev

INVERSE AND ILL-POSED PROBLEMS SERIES

Elements of the Theory of Inverse Problems AM. Denisov

/// VSPIII Utrecht, The Netherlands, 1999

VSP BV P.O. Box 346 3700 AH Zeist The Netherlands

Tel: +31 30 692 5790 Fax: +31 30 693 2081 E-mail: [email protected] Home Page: http://www.vsppub.com

© V S P BV 1999 First published in 1999 ISBN 90-6764-303-3

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Printed in The Netherlands

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Preface The development of the theory of inverse problems is mainly motivated by applied researches. Inverse problems arise when processing and interpreting observations in different branches of science, in particular, in geophysics. Various applied problems cause the mathematical researches which enlarge the theory of inverse problems. On the other hand, being a mathematical discipline, the theory of inverse problems can develop independently as well. Although intensive research of inverse problems began quite recently, a lot of fundamental results have already been obtained. The publications in the theory of inverse problems are very numerous. Many results are presented in the monographs designed mostly for those who have had ample scientific experience. At the same time, the number of the educational books introducing the readers to the theory of inverse problems is rather small. The basis of this book is the course of lectures given by the author to the students of the Moscow State University. The first chapter may be considered as an introduction. In this chapter, some examples of inverse and ill-posed problems are given; methods of their solutions are described. The second chapter is devoted to the inverse problems for ordinary differential equations. Much attention is given to the inverse problems for differential equations with a parameter. The third chapter deals with the inverse problems for linear partial differential equations. In these problems we need to determine either initial conditions, boundary conditions, or the function which describes the source. These inverse problems, as a rule, may be reduced to the linear Fredholm or Volterra integral equations of the first kind. The fourth chapter is connected with the inverse coefficient problems for partial differential equations. In this chapter we determine coefficients of a partial differential equation if an additional information on its solution is given. The important peculiarity of these problems is their nonlinearity. In the fifth chapter we consider the problems of determining functions of one or

ii

A.M. Denisov. Elements of the theory of inverse

problems

two variables by integrals of these functions. The problems of this type arise often in computer tomography. The sixth chapter is devoted to the methods of solution of inverse problems, with emphasis on the methods of solution of operator and integral equations of the first kind. The reason is that the majority of inverse problems can be reduced to these equations. Each chapter has the exercises designed to help the readers to gain a better understanding of the subject matter. At the same time, some of these exercises widen the chapter material. The author had tried to make the book the most accessible to the reader. In this connection, some important results are only outlined. The bibliography at the end of the book is not complete, but it may be taken as a starting point for more detailed investigation of the theory of inverse problems. Longstanding scientific contacts with academitian A. N. Tikhonov were of inestimable value in writing of this book.

Contents

Chapter 1. T h e basic concepts of inverse and ill-posed problems 1 §1 Inverse problems. The examples 1 §2 Well-posed and ill-posed problems 5 §3 Operator equations of the first and the second kinds 12 §4 Some aspects of setting and solving inverse problems 15 §5

Chapter 2. Inverse problems for ordinary differential equations §1 Problems of determination of the right side and coefficients of linear differential equations and systems by their solutions. . §2 Inverse problems for the first order linear equation with a parameter §3 Inverse problems for linear differential equations of the second order with a parameter §4 Inverse problems for nonlinear ordinary differential equations §5 Exercises for readers

19 23 23 30 36 53 70

Chapter 3. Linear inverse problems for partial differential equations 75 §1 Inverse problems for the heat conductivity equation 75 §2 Inverse problems for the vibration equation 94 §3 Inverse problems for the Laplace and Poisson equations 106 §4 Exercises for readers 116

iv

Α. Μ. Denisov. Elements of the theory of inverse problems

Chapter 4. Inverse coefficient problems for partial differential equations 121 §1 Inverse problems for first order equations 121 §2 The problem of determining the coefficient of the hyperbolic equation 127 §3 Reduction of inverse coefficient problems to inverse problems for ordinary differential equations with a parameter 139 §4 The problem of determining the coefficient dependent on the solution 146 §5 Exercises for readers 162 Chapter 5. Problems of determining the function from the values of integrals 167 § 1 Determining the function of one variable from the integral of this function. The moment problem 167 §2 Inversion of the Radon transformation. The problems of computer tomography 176 §3 Determining the function of two variables from its integrals along the circles 189 §4 Exercises for readers 196 Chapter 6. Methods of solution of inverse problems §1 Solving equations of the first kind on compact sets. The quasisolution method §2 The Tikhonov regularization method §3 The iterative method of solving equations of the first kind. . . . §4 The projective method for solving equations of the first kind §5 Methods for solving Fredholm integral equations of the first kind. §6 Method of solution of the Volterra integral equation of the first kind §7 The quasi-inversion method §8 Exercises for readers

199

Bibliography

265

199 209 226 233 243 249 255 261

Chapter 1. The basic concepts of inverse and ill-posed problems

§1

Inverse problems. The examples.

Mathematical models are widely used for description and investigation of various processes and objects. There exists the important class of the problems for mathematical models with unknown characteristics which have to be determined using the results of the experiment. Such problems arise in the cases when the object or process themselves are inaccessible for observation, or the observation is too expensive. As an example, we may consider the problems of geophysical research, astrophysical problems of star research, the problems of medical diagnosis and many others. The main feature of the interpreting of experimental results for all these problems is that we have to derive the results by indirect manifestations of the object that can be measured experimentally. For example, we have to determine the location and strength of an earthquake by the measured vibrations in the earth surface. Thus, we are dealing with problems where we need to determine the causes if we know the results of observations. Such problems are usually called inverse problems. Now we shall give some examples. E x a m p l e 1. Suppose a particle of unit mass is moving along a strait line. The motion is stipulated by a force f(t) which depends on time. In the

2

Α. Μ.

Denisov.

Elements

of the

theory

of inverse

problems

initial moment (t — 0) the particle was situated in the origin χ = 0 and the speed of the particle was equal to zero. According to the Newton law the motion of the particle will be described by the following Cauchy problem §

=

m ,

ο < t < T ,

ar(O) = 0,

^ ( 0 ) = 0,

(l)

ήτ

(2)

where x(t) is the particle coordinate in the moment t. If we know the cause of the motion (the force f(t)), then, solving problem (1), (2), we obtain the effect x(t). Let us assume that the force acting on the particle (the function /(f)) is unknown, but we can measure in each moment of time the coordinate of the particle (the function χ(t)) and want to determine f(t). Thus, we have the following inverse problem. Determine the cause f(t), by the known effect x(t). From mathematical point of view, this inverse problem is the problem of determination of the right side f(t) of differential equation (1) by its solution x(t). Example 2. The process of chemical kinetics is described by the following mathematical model: the Cauchy problem for the system of linear differential equations dc -jj-

=

anc\{t)

Ci{t0)

=

+ ai2c2{t)

Ci,

+ ...

i — 1,2,...

+ aincn(t),

,n.

(3)

(4)

The function Ci(t) is the concentration of substance i acting in the process at the moment t. The constants atJ characterize the dependence of the change of i-substance from the concentration of other substances in our process. For the system (3), we may formulate the following inverse problem. We measure the concentrations Cj(i), i = 1,2,... , η for t G [ oo

for

η —> oo

C0[0,T]

and In

->-oo

f C[0,T]

and the third condition of well-posedness does not hold. Example 3. Consider the problem of solution of integral equation (2) when F = C[0, T ] and X = Co2[0,T] = { x € C 2 [ 0 , T ] , x(0) = X'(0) = 0, IMI C 2 [0 , t] = IMI C 2 [ 0 , t ] }· For these spaces F and X the problem of solution of integral equation (2) is well-posed. For each function χ G C q [ 0 , T ] there exists a unique solution of equation (2) f(t) = d2x/dt2. Suppose fi(t), i — 1,2 solve equation (2) for the functions x{ G Cß[0,T], i = 1,2. Then /1-/2

c[o,T]

0 for χ £ (0, χι]. Thus, y[(x,X) is strictly increasing in (0, x{\ and y[{xi, A) > 0. This contradiction shows that y[(x, A) > 0 for χ £ (0, oo], A > 0. But in this case, y\(x, A) is a positive monotone increasing function and cannot satisfy the condition 2/1(00, A) = 0. Therefore, y\(x,X) = 0 for χ £ [0, oo), and boundary-value problem (1), (6) has a unique solution. We shall prove now the second statement of the lemma. Let y2{x,X) solve (1), (6) with yo > 0. We show that y'2(x,X) < 0 for χ £ [0,oo). If 2/2(0, A) > 0, then, we may similarly obtain that y2{x, X) > 0;

y'2(x,X)>0

for

a: £ (0, 00);

therefore, y2(x, A) cannot satisfy the condition 2/2(00, A) = 0. Consequently, y'2{0, A) < 0. Assume that in a certain point Χ2 we have y'2{x2, A) = 0. Then y2(x, A) solves (1) for χ > X2 with the boundary conditions 2/2(^2, A) = 0,

2/2(00, A) = 0.

Analogously to the previous proof, we may show that y2(x, Α) ξ 0 for χ > Χ2· As y2{x, X) solves (1) for χ > 0, then y2(x, Α) ξ 0 for χ > 0. Therefore, the

Chapter 2. Inverse problems for ordinary differential equations

39

assumption that y'2(x,\) may be equal to zero is false; so, y'2{x,X) < 0 for χ £ [0, oo), λ > 0, which was to be proved. As follows from this lemma, boundary-value problem (1), (6) in the case of homogeneous boundary conditions (yo = 0) has only a trivial solution. Therefore, for each continuous function f(x) for χ > 0 such that oo j \f(x)\dx < oo, ο the boundary-value problem f(x,X)--£-y(x,X) σΔ{χ)

(7)

= -f(x),

y( 0,A) = 0,

y(oo, λ) = 0

(8)

has a unique solution oo

y(*,A)

= J

β(χ,ξ;λ)ηξ)άξ,

ο where 0 ( χ , ξ ; A) is the Green function. Some properties of the Green function are established in the following lemma. Lemma 2. The Green function for boundary-value problem (7), (8) has the following properties 1) G(x, ξ; λ) > 0 for 0 < χ, ξ < oo, A > 0; 2) if G l (x, ξ·, A) are the Green functions corresponding to coefficients Oi(x), i = l,2, then from the inequality σ\{χ) > σ·2{χ)

for

χ ε [0,oo)

follows that Gi (ζ,

λ) > σ 2 ( χ , £ ; λ )

for

0 σ2{χ) and the nonnegativeness of G\(x, ξ; A) yield that the right-hand side of (10) is nonpositive for χ G [0,00). We now show that z(x) > 0 for χ G [0, oo). Assume that in a point XQ > 0 the function z(x)

has a negative minimum. Then z(x0) < 0, z'(x0)

exists such ε > 0 that z(x)

= 0 and there

< 0 for χ G (XQ — ε,χο]· Taking into account

that the right-hand side of (10) is nonpositive, we obtain that z"(x) χ G (xo

ε,χο)·

Therefore, z'(x)

> 0 for χ G (xo — ε,χο)·

< 0 for

This inequal-

ity contradicts the assumption that XQ is the point of negative minimum for z(x).

Thus, the initial assumption was false, and z(x) > 0 for χ G [0,00). As

the values ξ, A were arbitrary, then G\(x, ξ] A) > G2{x, ζ; A) for χ, ξ G [0,00), A > 0. The lemma is proved. Now we shall find the expression for the Green function for problem (7), (8) for the constant function σ(χ)

= σ* in the half-line χ > 0.

The

fundamental system of solutions for equation (1) is as follows exp(—Χχ/σ*),

exp(A χ/σ*).

Choosing in formula (9) yi(x, A) = sinh(Aa;/a*),

y2{x,X) — exp(—\χ/σ*),

we obtain W — λ/σ* and G(x e \) = l σ*ήτύϊ(Χχ/σ*ϊ

βχρ(-Αξ/σ*)/Α, 0 < χ < ξ < oo,

I a*sinh(A^/a*) exp(—Αα;/σ*)/λ, 0 < ^ < a; < oo. Now we consider the inverse problem.

1

'

We shall take a more general

setting of inverse problem for (1). Namely, it is required to determine σ ( χ ) ,

42

A.M. Denisov. Elements of the theory of inverse problems

if, for Λ > 0, we know the function Φ(λ) = ΐ/(0,Λ)/3/(0,λ), where y(x, A) is the nontrivial solution of (1) satisfying the condition y(oo, λ) = 0. Note that this setting contains the previous setting of the inverse problem for equation (1), since, if y(0, A) = q and y'(0, λ) = φ(\), then Φ(λ) = φ(λ)/ς. We consider now the uniqueness of this inverse problem. We denote by Σ the class of functions σ(χ) satisfying the conditions: σ(χ) is continuous and positive for χ > 0; σ(χ) is constant for χ > XQ (XQ depends on σ(χ)); σ(χ) — σ(χ) for χ G [0,xo]> where σ(χ) is analytic in an interval which contains the segment [0, xo] · We shall establish the uniqueness theorem for determining σ(χ) from the function y'(0, A)/y(0, A) (Tikhonov, 1949). Theorem 1. Let y\(x, Λ) and y2(x, A) be nontrivial solutions to (1) for the functions σ\{χ) and θ2{χ) {σχ{χ),σ 0 that y[(0,A) _ y'2(0, A) y 2 (0,A)' y i (0,A)

for

A > A0,

(13)

then σι(χ) = σ2(χ) for χ G [0,oo). Proof. As yi{x, A), i = 1,2 solve homogeneous equation (1) with homogeneous condition (12), then they are determined to within a constant factor. As follows from Lemma 1, yi(0, Α) Φ 0 for all A > 0. So, yi(x, A) may be normed by the condition yj(0, A) = 1, i = 1,2. The functions yi(x, A) solve equation (1) with the coefficients σι(χ) respectively; therefore, y"(x, A) - yi,'^, A)

A2 jj—yi{x,

A2 A) + -T—-y2{x,

A)

Chapter 2. Inverse problems for ordinary differential

λ2

equations

43

λ2

af(Xjy2(X'X)

+

^)

y 2 ( X

'

X ) = 0

·

Therefore, the function z(x, A) = yi{x, X) — y2{x, A) satisfies the equation A2 z"(x, A) - - j2 — ζ(χ,λ) σ (τ)

= -F(x,

(14)

A),

with the boundary conditions z(0, A) = 0,

(15)

z(oo, A) = 0,

where F(x, A) =

1 σ2(χ)

1 y {x,X)^ σ2(χ). 2

Using the representation of solution (14), (15) by means of the Green function, we obtain oo !(x,X) = X2 J

1

Ο^χ,ξ-Χ)

1

ν2(ξ,Χ)άξ.

(16)

l_°2 ( 0 We show now that there exists xz > 0 such that σι(χ) = σ2{χ) for χ G [0, £3]. As σι,θ2 G Σ, then either their difference is equal to zero identically in a certain segment [0, £31] or it is not equal to zero in an interval (0,2:32)· In the first case we set £3 = £31. We show that the second case is impossible. Assume that this case holds, and suppose that σι(x) — σ2(χ) > 0 for χ Ε (0, ^32)· Prom the conditions imposed on the class Σ, it follows that there exist positive constants AM and OM such that cm
0 such that α

(χ)

=

~2TT σ 2 (χ)

σ12ΓΤ (χ) -

a

°

for

ζ e [25,2:4]·

44

Α. Μ. Denisov.

Elements

of the theory of inverse

problems

Representation (16) may be written as follows £4 z(x,X)

2

= X J

01(χ,ξ·λ)α(ξ)υ2(ξ,Χ)άξ

ο 00

+ λ 2 101(χ,ξ;\)α(ξ)ν2(ξ,\)άξ.

(18)

X4

From Lemmas 1 and 2 it follows that y2(x,X) is a positive monotone decreasing function for χ Ε [0, oo) and G\(χ, ξ; λ) > 0. This and (18) yield for χ 6 (0, £5) that X4

z(x,X)

>

X2y2(x^X)

J 0 (χ,ξ·,Χ)α(ξ)άξ1

LT5

00

J Gxix&X)|a(£)|d£ •

(19)

X4

Denote A — max |α(ξ)| and Gm(x,£,X), 0 0 X4

z(x, Λ) >

2

X y2(x4,X) X5

Chapter 2. Inverse problems for ordinary differential equations

45

oo —A J ^ ^ s i n h ( A χ / c m ) e x p ( — λ ξ / σ Μ ) ά ξ

X4

- λ2ρ2(Χ4 , A ) | a 0 ^ s is i n h ( A x / a m ) e x p ( - A : E 5 / a m ) - e x p ( - A x 4 / a m )

sinh(Ax/ajtf) exp(—XX^/OM)

Thus, for χ 6 (Ο,Χδ) and A > 0, we have

Φ,λ)

> P{x,X),

(20)

where P(x, X) = y 2 ( ^ 4 , A ) | a 0 a ^ s i n h ( A x / o m ) [ e x p ( - A x 5 / a m )

— exp(—Ax4/am)

— Aa\[SMH(Xx/ΣM)

exp(—XX^/UM) | ·

As z{0, A) = P ( 0 , A) = 0 for A > 0, then (20) yields

z'(0,X)>P'(0,X)

for A > 0.

(21)

From the definition of P(x, A), we have P ' ( 0 , A ) = y 2 ( x 4 , λ)

a0amAexp(-Ax5/am)

-aoamAexp(-Ax4/am) -

AaM^exp(-Xx4/aM)

As x t , / o m < X i / o M , then such Ai > 0 exists that P'(0, A) > 0 for A > λ ι · Then, from (21), we have z'(0, A) > 0 for A > Αχ. Prom (13) and the norming conditions yi(0, A) = 1, i = 1,2, it follows that z'(0, X) = 0 for all A > λο· This contradiction shows that the second case is impossible, and ct{x) — 0 for χ e [0, £3].

46

A.M.

Denisov.

Elements

So, we have shown that σχ{χ)

of the theory of inverse

=

θ2{χ)

problems

for χ 6 [Ο,α^].

Then, from

the conditions y i ( 0 , Λ) = 3/2(0, A) = 1 and (13), we obtain that yi(x,X)

=

y2{x, A) for χ G [0,2:3] and A > Ao- Therefore, for A > Ao, we have y[(x3, A) _ y'2(x3, A) 2/1(2:3, A)

2/2(2:3, A)

W e have obtained in the point £3 > 0 the equality analogous to (13). Repeating the previous proof, we obtain, as a result, that σι(χ)

= σ·2{χ) for

χ > 0. T h e theorem is proved.

Remark.

When we were proving Theorem 1, we have used, for sim-

plicity, that σ(χ)

is continuous for χ > 0.

Without essential changes in

the proof, Theorem 1 may be proved also for piecewise analytic functions (Tikhonov, 1949). This case is more interesting for practical geophysics.

T h e inverse S t u r m - L i o u v i l l e p r o b l e m .

W e shall consider the dif-

ferential equation -y"(x,

A) + q(x)y(x,

A) = Xy(x, A)

(22)

in the segment [0, π] with the boundary conditions

where q(x)

y(0, A) cos a + y'(0, A) sin α = 0,

(23)

2/(π, A) cos β + y'(7r, A) sin /? = 0;

(24)

is a real continuous function in the segment [0, π]; α, β are real

numbers, and A is a complex number. Boundary-value problem (22)-(24) is called the Sturm-Liouville problem. This problem is usually considered in the segment [0, π], since after the change x' = (x — a)n/(b — a) the segment [a, b] becomes [0,π], and the equation and the boundary conditions do not change. It is well-known that problem (22)-(24) has a countable set of real eigenvalues A n , η = 0,1,

T h e corresponding eigenfunctions y(x, A n ) form a

complete orthogonal system in the space 1955).

0, π] (Coddington and Levinson,

Chapter 2. Inverse problems

for ordinary differential

equations

47

Now we shall consider the setting of the inverse Sturm-Liouville problem. Suppose the function q(x) in (22) is unknown, and we need to find it by the given set of eigenvalues of problem (22)-(24). The first result was obtained by Ambartsumyan (1929), and is as follows. If α = β = π/2 and the eigenvalues of problem (22)-(24), λ η = η 2 , η = 0 , 1 , . . . , then q(x) = 0 for χ 6 [0, π]. However, this result is not typical because, usually, one sequence of eigenvalues is not enough to uniquely determine an arbitrary function q(x). The general uniqueness theorem was proved by Borg (1946), who proved that q(x) is determined uniquely, if the eigenvalues Λη, η = 0 , 1 , 2 , . . . of problem (22)-(24) are given, and the eigenvalues μ η , η — 0 , 1 , 2 , . . . of problem (22), (23) y(-7r, Λ) c o s 7 + υ'(π, λ ) s i n 7 =

0

(25)

are given, where 7 is such that sin(7 — β) φ 0. This theorem was established with some restrictions imposed on α, β, 7. These restrictions were removed by Chudov (1949) and Levinson (1949a). Theorem 2. Let qi(x) and q\(x) be two continuous functions in [0, π]; X , i = 1,2 be eigenvalues of problem (22)-(24), and μιη, i = 1,2 be eigenvalues of problem (22), (23), (25) for the functions qi(x), i = 1,2 respectively. Then, if λ* = λ 2 , = μ 2 for η = 0 , 1 , 2 , . . . , then qi(x) = 92(2) in the segment [0, π]. ln

It should be noted that the inverse Sturm-Liouville problem is similar to the setting of the inverse problem considered before, and may be reduced to the problem of determining q(x) by the additional information on the solution of (22), being a function of the parameter λ. Really, let v(x,X) solve equation (22) with the initial conditions ν (0, λ) = sin α,

ν' (0, Λ) = —cos a.

(26)

The solution v(x, X) of Cauchy problem (22), (26) exists for each complex λ and for a fixed χ is analytic of λ in the whole complex plane. We consider now the function g(X) = ν(π, λ) cos β + υ'(π, Λ) sin/3.

48

Α. Μ. Denisov. Elements of the theory of inverse

problems

From the definitions of g(X) and v(x,X), it follows that each zero of g(A) is an eigenvalue of problem (22)-(24) and, conversely, each eigenvalue of problem (22)-(24) is a zero of g(X)· Using the complex function theory and the properties of υ(χ, λ), we may show (Levinson, 1949a) that the analytic function g(X) is uniquely determined by its zeros. Thus, the set of eigenvalues Xn, τι — 0 , 1 , 2 , . . . of problem (22)-(24) determine uniquely the function of the complex variable g(λ). Analogously, the set of eigenvalues μη, τι = 0 , 1 , 2 , . . . of problem (22), (23), (25) uniquely determines the function p(X) = υ(π, λ) cos 7 + ν'(η, λ) sin7· Thus, we can say that the inverse Sturm-Liouville problem, to determine q(x) by two sets of eigenvalues Xn and μη, η = 0 , 1 , 2 , . . . , may be reduced to the problem of determining q(x) by two functions g{X) andp(A). Now we return to Theorem 2 on uniqueness of solution of the inverse Sturm-Liouville problem. It follows from this theorem that an arbitrary continuous function q(x) is uniquely determined if the two collections of eigenvalues Xn and μη, η — 0 , 1 , 2 , . . . are given. Here arises the problem: whether we may narrow down the class of functions q(x) which would allow us to determine uniquely the functions from this class only by the collection of An or μη, η - 0 , 1 , 2 , . . . ? We consider the class of functions q(x) continuous in [0, π] and such that q(ir — x) — q(x) for each χ G [0, π]. Denote this class as Q. For functions from this class the following uniqueness theorem holds (Levinson, 1949a). T h e o r e m 3. Let the numbers α and β in conditions (23), (24) be such that (α + β) = π, and qi(x), G Q· Then, if A* = for η = 0 , 1 , 2 , . . . , where A* and A^ are eigenvalues of the Sturm-Liouville problem (22)-(24) for q\(x) and q2{x) respectively, then q\(x) = q2{%) for χ G [Ο,π]. We see that not only qi{x) G Q, but a + β = π. Thus, it is required not only that q(x) is symmetric relative to the middle of [0, π], but also that the boundary conditions are symmetrical. It should be noted that as each eigenvalue of (22)-(24) is a spectral point of the differential operator generated by the differential expression -y"(x)

+

q(x)y{x)

with boundary conditions (23), (24), then the inverse Sturm-Liouville problem is related to the class of inverse spectral problems.

Chapter 2. Inverse problems for ordinary differential equations

49

We have considered one of the possible variants of setting of the inverse Sturm-Liouville problem and have formulated the uniqueness theorems which were proved in the first period of investigation of this problem. Then, the inverse Sturm-Liouville problem was investigated more thoroughly; in particular, the existence problems and other variants of settings were considered. Further developments may be found in Levitan (1987) and in the bibliography quoted there. The inverse problems of quantum scattering theory. One of the classical problems of quantum mechanics is the problem of a particle moving in a field with a central potential v(x). Investigation of this problem leads to the linear differential equation a, we have lk^(a,k) φ(χ, k) = — where tg 6(k) =

+

( a. Now we consider the setting of the inverse problem of quantum scattering theory. From experiments, we may obtain the phase shift S(k). The inverse problem is to find the potential v(x) from the phase shift 6(k). The great interest in this problem is generated both by theoretical problems of quantum mechanics and by the problems of data processing of experiments. One of the first results was the uniqueness theorem (Levinson, 1949b). It was proved that if S(k) is given for k > 0 and

Chapter 2. Inverse problems for ordinary differential equations

51

(32)

0 is determined uniquely. However, in the general case, we cannot determine the potential v(x) from the phase shift S(k) given for k > 0. The set of potentials was constructed which had one and the same phase shift for k > 0. This lack of uniqueness arises since there may exist a number Ν > 0 of purely imaginary values Xj = ίμ3, μ^ > 0, j = 1 , 2 , . . . , Ν such that lim ip(x,Xj)eßjX

= 1,

(33)

where φ(χ, Xj) solves problem (28), (29) with Λ = Λ j = ißj. We note that (32) fails for Ν > 0. In this case, the potential v(x) is determined uniquely if there are given S(k), for k > 0, the values Xj, j = 1 , . . . , N, and rrij such that oo 0 where φ(χ,Χj) solves (28), (29) and satisfies (33). This general uniqueness theorem was proved by Marchenko (1952). Then, the following procedure to restore the potential from given S(k), k > 0, Xj and m,j, i = 1 , 2 , . , . , Ν was suggested (Marchenko, 1955; 1986). By the phase shift S(k), we determine the function S(k) = exp{2ii(fc)} such that S(—k) = S(k) (S(k) is the complex conjugate to S(k)). Then, for χ > 0, we introduce the function

—oo Then we solve the following equation relative to K(x, y) 00 y>x. X Finally, we find the potential v(x) by the formula

52

Α. Μ. Denisov. Elements of the theory of inverse problems

vlx) =

-2—K(x,x). ax

This procedure of determining v(x) developed by V. A. Marchenko is of great importance for the theory of the inverse problem of scattering; for example, for establishing the existence theorem for the inverse problem, or for determining the initial data of the inverse problem for which the solution may be obtained exactly. There exists, also, another procedure of determining v(x) by the exactly given initial data of the inverse problem based on the Gel'fand-Levitan equation (Gel'fand and Levitan, 1951), (Chadan and Sabatier, 1989). We have considered the inverse problem for equation (28). Other settings of the inverse problem are connected with the more general equation (27). We may show that if (30) holds, then the solution φι(χ, Λ) of (27), (29), for real \ — k, has for χ —> oo the asymptotic behavior

ipi{x,k) = Ai(x,k) sin ^kx - y + S^k)^ + o(l).

In this connection we may formulate the two types of inverse problems of quantum scattering theory: 1) the phase shift S[(k) is given for a fixed I = IQ for all values of k, and it is required to determine the potential v(x); 2) the phase shift is given for a fixed k = ko, but for all values of the angular momentum Ζ = 0 , 1 , . . . , and it is required to determine the potential v(x). These two types of inverse problems of quantum scattering theory for the radial Schrödinger equation (27) have been investigated by many authors. A detailed presentation of inverse problems of quantum scattering theory is in Chadan and Sabatier (1989). In conclusion we note that inverse problems of quantum scattering theory may be set not only for (27) but for the other equations. One of the most interesting problems is the inverse problem for equation (28) considered in the whole line. (Chadan and Sabatier, 1989) This inverse problem is of great importance for investigating some nonlinear equations of mathematical physics (Marchenko, 1986; Levitan, 1987).

Chapter 2. Inverse problems for ordinary differential equations

§4

53

Inverse problems for nonlinear ordinary differential equations.

The study of inverse problems for nonlinear ordinary differential equations we begin with the problems where the initial information is the solution of the differential equation as a function of independent variable. Consider the Cauchy problem for a first order equation y'(x) =

x>xo,

(1)

y(x ο) = a.

(2)

Suppose that f(x,p) is unknown and we need to determine it by the solution of problem (1), (2). One solution of problem (l)-(2) is not sufficient to determine the function of two variables f(x,p). Really, if the solution y(x) of problem (1), (2) is known, then f(x,y(x)) = y'(x). Therefore, we may determine the function f(x,p) only in the curve (x,y(x)). To determine a continuous function f(x,p) in the set D, we need to know such a set of solutions of (1) that the set of curves which correspond to these solutions forms a set everywhere dense in D. We consider now the following example. Let y(x,a) be a set of solutions of problem (1), (2) which corresponds to an unknown function f{x,p) and different initial conditions a G [Αι, A2}· Suppose a function of two variables y(x,a) is given for χ Ε [χο,^ι] and a G [Αι, A2] and is continuous and continuously differentiate with respect to x. Then, as follows from (1), f{x,p) may be defined in the set D = | ( x , p ) :XQ

< a{X)

for

for

for

£e[0,a(A0)];

Ae[0,Ao];

a r e [0,1],

Ae[0,A0];

(11)

(12)

(13)

A)), y(x, X) satisfy (7)-(9).

For this inverse problem the following existence and uniqueness theorem holds. T h e o r e m 2. If a(X) satisfies (10), then the solution to inverse problem (7)-(9) exists and is unique. P r o o f . We assume that the pair /(£) and y(x, A) solves inverse problem (7)-(9). Then, from (7), it follows that y'{x, A) > 0 for χ € [0,1], A G (0, A0]. Dividing equation (7) by f{y{x, A)) and integrating it from 0 to 1, we have ι Γ y'{x,X)dx

=

If ( v ( x , x i ) Making the change t = y(x, A) in the integral and using conditions (8), (9), we obtain the equation for determining f(t)

58

A.M. Denisov. Elements of the theory of inverse problems

α(λ)

Im=K

0£λίν

0

Differentiating this equation by λ, we have f(a(X)) Therefore, m

= a'(a-1ßj),

= a'(X),

ξ€[0,α(λο)],

λ G [0, λο].

(14)

where a^1 (ξ) is the inverse function to a(\). Substituting this expression for / ( ξ ) into (7), we obtain ^ = Xdx. a'(a-i(y)) Taking into account the formula for the derivative of the inverse function

V

'

o'(o-'K))

and conditions (8), (10), we have a~1(y) — Xx. Therefore, y(x,X)=a(Xx),

χ € [0,1],

λ€[0,λ0].

(15)

Thus, if /(£) and y(x, Λ) solve inverse problem (7)-(9), then, for these functions, representations (14) and (15) respectively hold. Therefore, the solution of the inverse problem is unique. The existence of such a solution follows from formulas (14) and (15). By straightforward calculation, we may show that if (10) holds, the pair /(ξ) = α'(α"1(θ) satisfies (11)-(13), and f{y(x,X)), [0, λο]· This proves the theorem.

and

y(x,\)

= a(Xx)

y(x, X) satisfy (7)-(9) for χ € [0,1], Λ G

In the definition of the inverse problem we have included condition (13) - the a priori estimate of the function y(x,X). We need this condition in

Chapter 2. Inverse problems for ordinary differential

equations

59

order that the domain of definition of / ( £ ) would coincide with the range of values of y(x, Λ) for χ G [0,1], λ G [0, Λο] • We shall give another variant of definition of the solution to inverse problem (7)-(9). Definition. The pair of functions / (ξ), y{x, λ) is defined as the solution of inverse problem (7)-(9), if / ( £ ) G C ( - o o , o o ) , / ( ξ ) > 0 for ξ G [0,α(λ 0 )], φ , λ) G C71 [0,1] for λ G [0, λο]; f(y{x,X)) and y{x,X) satisfy (7)-(9) for x G [0,1], λ G [0, λο]. For this definition of solution of (7)-(9), the statement of uniqueness of solution will change. Really, as / ( £ ) is determined for all ξ, we may consider f(y(x,\)) for all values of y(x,X). However, (7), (8), and positiveness of /(C) for ξ G [Ο,α(λο)] yield y'(x,X)>

0

for

χ G [0,1],

λε(0,λ0].

The further arguments are similar to those given above. But the function / ( ξ ) is determined by formula (14) only for ξ G [0, α(λο)]. Thus, the uniqueness of / ( £ ) may be established only for ξ G [0, α(λο)], i.e., only for the range of values of y{x,\) for χ G [0,1], λ G [0, λο]. To establish the existence of continuous / ( ξ ) in the whole axis, it suffices to extend / (ξ) = α ' ( α - 1 ( £ ) ) from the segment ξ G [0, α(λο)] to the whole axis, so that the function obtained would be continuous. Evidently, such extension is not uniquely defined. We consider now the inverse problems for nonlinear equations of the second order which contain a parameter. Such problems arise in some physical processes (Kulik et α/., 1983) and inverse problems for partial differential equations (Anikonov, 1972). We shall consider the problem of determining the functions f(t) y(x, λ) satisfying the equation y"(x,\)

= λ2/(^,λ)),

0 < a: < 1,

0 < λ < λ0,

and

(16)

and the conditions 2/(0, λ) = 0,

3/'(0,λ) = 6(λ),

y(i, λ) = α(λ),

0 < λ < λο,

0 < λ < λο,

(17)

(18)

60

Α. Μ. Denisov. Elements of the theory of inverse problems

where 6(A) and a(X) are given functions. In this setting we do not select the additional condition. It may be either y'(0, λ) = b(X) or y(l, Λ) = α(λ). We assume that the functions α(λ) and b(A) satisfy the following conditions α(λ) e ^[Ο,λο], α ( 0 ) = 0 ,

b(X) = Xg(X),

α'(λ) > 0

g(A) > 0

g(X)eCl[0,Xo},

Definition. The pair of functions f(t) solution to inverse problem (16)-(18), if

y(x, A) G C 2 [0,1]

y(x, X)

and

f(y{x,

for

Ae[0,A 0 ],

for

Ae[0,A 0 ].

(19)

(20)

and y(x, A) is defined as the

/(i) G ^[Ο,οίλο)], f(t)>

f(t)eC(-οο,οο),

for

0

for

ί€[0,α(λ0)],

Ae[0,A 0 ],

A)) satisfy (16)-(18) for χ £ [0,1], A e [0, A0].

For inverse problem (16)-(18), holds (Denisov, 1990).

the following uniqueness theorem

Theorem 3. Let functions a(A) and 6(A) satisfy conditions (19) and (20) respectively. If fi{t), yi(x,X) and /2(i), y2{x,X) solve inverse problem (16)-(18), then h(t)

yi(x,X)

= f2(t)

=y2(x,X)

for

for

t e [Ο,α(Αο)]

χ G [0,1], A € [0, Aq].

Chapter 2. Inverse problems for ordinary differential

equations

61

Proof. Let f(t) and y(x, Λ) be a solution of inverse problem (16)-(18). Then, (16), (17), and the positiveness of f(t) for t G [0, α(λο)] yield that, for Λ G (0, λο], we have 0 < ρ(χ,λ) < α(λ),

y'(x, A) > 0

for

χ €[0,1].

Multiplying equation (16) by y'(x,\) and integrating it from 0 to x, we obtain y(x,X)

= 2λ2 J

{y'ixA))2-{y'(0,λ))2

ί(ξ)άξ.

v( ο,λ) Taking into account condition (17), we have y(x,\) 62(Λ) + 2λ2 J

y'(x,X)

^

/(ο« J

1/2

=i.

Integrating this equality from 0 to 1 and making in the integral the change of variable t = y(x, λ), we obtain 2/(1,λ) / J

t

\ -V2

62(Λ) + 2λ2 J/(O^J

»(ο,λ) V

ο

dt = 1. /

Finally, taking into account (17), (18), and (20), we have α(λ) / ί ν -1/2 J lg2(\) + 2 j !(ξ)άξ\ dt = λ, ο

V

ο

0 0

α'(0) > 0;

for

λ e [0, Λ0],

b{X) = ο'(0)Λ.

Then the following pair of functions α"(α~ι(ίή,ί£[0,α(Χ0)],

' m

=

«"(0),

t < 0,

α"(λ 0 ),

t > a{Xo)

(25)

and y(x, λ) = a(Xx) solves the inverse problem. Really, all the properties

64

Α. Μ. Denisov. Elements of the theory of inverse

problems

of the functions f(t), y(x, X) from the definition follow from the properties of α(λ). Substituting y(x, X) and f(y(x,X)) into (16)-(18), we see that they satisfy these equations for χ £ [0,1], Λ £ [0, λο]. R e m a r k . In the definition of the solution of inverse problem (16)-(18) we do not include an a priori estimate on the solution y(x, λ), and f(t) is considered as given and continuous in the whole axis. In this connection, in the formula (25) for the function f(t), we have chosen one of the possible continuous extensions of f ( t ) from the segment [0, λο] into the whole axis t £ (—oo, oo). Now we consider another inverse problem. We have to find the functions k(t) and y(x, A) satisfying the equation k(y{x, \)jy'{x, X) ' = X2y(x, λ),

0 < χ < 1,

0 < λ < λ0,

(26)

and the conditions j/(0, A) = 0,

y'(0,X) = b(X),

ί/(1,λ) = ο(λ),

0 < λ < λ0,

(27) (28)

0 < λ < λο,

where a(X) and b(X) are given functions satisfying conditions (19) and (20) respectively. Definition. The pair of functions k(t) and y(x,X) solution of inverse problem (26)-(28), if k(t) ECl(-oo,oo);

k(t) > 0

y{x,X) £ C 2 [0,1]

for

for

is defined as the

ί£[0,α(λ0)];

λ£[0,λ0];

satisfy (26)-(28) for

χ £ [0,1], λ £ [0, λ 0 ].

Now we shall prove the uniqueness theorem for inverse problem (26)-(28) (Denisov, 1990).

Chapter 2. Inverse problems for ordinary differential equations

65

Theorem 4. Let α(λ) and 6(A) satisfy (19) and (20) respectively. If ki(t), y\{x·, λ) and /^(i), V2{x, A) solve inverse problem (26)-(28) and fci(0) = £2(0), then ki(t) = ^ ( i ) for t £ [0,α(λο)] and yi(x,A) = y2(x,\) for χ £ [0,1],λ£[0,λ 0 ]. Proof. Let k(t) and y(x, A) solve inverse problem (26)-(28). We obtain now the integral equation for the function k(t). Integrating (26), we have X

2

k (y(x, A)) y'(x, A) = k (y(0, A)) y'(0, A) + A J y(ξ, A R . ο This relation and positiveness of y'(0, A) for A £ (0, Aq] yield that

0 < y(x,A) < a(A),

y'(x, A) > 0 for λ £ ( 0 , λ 0 ] ,

a; £[0,1].

Multiplying equation (26) by k(y(x, X))y'(x, A) and integrating it from 0 to χ, we obtain X

I

A)]'

A))y'(ξ, λ)άξ

χ 2

= A

Jy((,\)y'(t,\)k(yti,X)) 0, λ>ο

η

x>0

and oo.

Here, yn(x, λ) and y(x, A) solve problem (1), (3), (4) with the functions ση(χ) and σ(χ) respectively. Exercise 8. We q(x) — q (a constant)

consider

the

Sturm-Liouville

-y"{x, A) + qy(x, A) = Ay(x, A),

problem

0 < χ < π,

with

(6)

(7)

y(0,X)=y(w,\)=0.

To prove that the constant q is uniquely determined if two eigenvalues Ajt and Afc+i {k is unknown) of problem (6), (7) are known. May we determine q uniquely if two eigenvalues A^ and Am (k,m are unknown) are given? Exercise 9.

We consider the differential equation

y"(x) - 36e 6 *(e 6 * +

- 7x)

[ / ( y ^ ) ) - (7χ + l ) 5 ( y ( ^ ) ) ] ,

(8)

0 < χ < 1 with unknown continuous functions f(t) and g(t). g(t) for t Ε [2,8] if the solutions yi(x)

of equation (8) are given.

= 7x + l,

y2(x) = e6x

To determine f(t)

+1

and

74

A.M. Denisov.

Exercise 10. conditions

Elements of the theory of inverse

problems

We know a function α(λ), which satisfies the following

a Ε C^O, λο], α ( 0 ) = 0 , α'(Λ) > 1, Λε[0,Λ 0 ]. Prove that there exists a unique pair of functions f(t) and y(x, X) such that feC

Ο,α(λο) ,

/ ( ί ) > 0,

te

Ο,α(λο)

y(x,Ä)eC1{[0,l]x[0,A0]};

^ ( χ , λ ) = A/(y(z,A)), 2 / ( 0 , Λ) = Λ,

ϊ /(1,λ)

0 < a: < 1, = α(λ))

0 < λ < λο,

0 < λ < λο-

Chapter 3. Linear inverse problems for partial differential equations In this chapter we consider inverse problems for partial differential equations: the heat conductivity equation, the vibration equation, the Laplace equation, and the Poisson equation. We consider the following inverse problems for these equations: to determine initial or boundary condition, or the function which describes the source, if a certain additional information on solution is known. Such problems arise in geophysics, ecology, plasma physics, and many other sciences. Inverse problems which we consider here, as a rule, are reduced to linear Fredholm or Volterra integral equations. In order to simplify the presentation, we set the inverse problems for equations with constant coefficients. In more general cases similar inverse problems for partial differential equations were investigated, for example, by Tikhonov (1935), Pucci (1955), Lavrentiev (1956), Aronszain (1957), Mizohata (1958), Landis (1959), John(1960), Fridman (1964), Ptashnic (1984), Lavrentiev et oZ.(1986a), Isakov (1990), Engl et o/.(1994).

§1

Inverse problems for the heat conductivity equation.

The inverse problems which will be considered here are either the problems of determining the initial condition, or the boundary condition, or the function, which characterizes the action of heat sources.

76

A.M. Denisov. Elements of the theory of inverse problems We consider the Cauchy problem for the heat conductivity equation ut = a2uxx,

—oo < χ < oo,

u(x, 0) = φ(χ),

t > 0,

(1)

—oo < χ < oo.

(2)

The solution of this problem may be found by the formula (Tikhonov and Samarskii, 1963)

{x t)=

" '

J

2

mdi

I vhrM -^r} ·

(3)

—oo We consider the following inverse problem. We need to find the function ψ(χ) if the solution of problem (1), (2) for t — Τ > 0 u(x,T) = g(x),

—oo 0 for y € [0,6]; therefore,

Chapter 3. Linear inverse problems for partial differential equations

111

w(y) < w(0){b - y)/b + w{b)y/b.

This inequality and the definition of w(y) yield

In(p(y) + |9|)
a

Finally, we obtain the following formula for the potential a R> a. ο

(13)

Chapter 3. Linear inverse problems for partial differential equations

115

Now, using this formula, we shall analyze the inverse problem in its general setting. We shall show that the solution of this problem is not unique. Really, let all the bodies be balls with center in the origin and radius less than A. Suppose that the potential u(x, y, z) of these bodies is known outside the ball of the radius A and the center in the origin. We consider the ball of radius α and constant density po- Formula (13) yields for R > A that (14) Therefore, the potential will not change if we change the radius and the density, so that poa 3 remains without changing. Thus, the problem of simultaneous determination of the shape of the body and its density, even if the density is constant, does not have a unique solution. Note that formula (14) and non-uniqueness of the inverse problem, which follows from it, have a clear physical sense. Really, the value 4προα 3 /3 is equal to the mass of the ball, i.e., u(R) = M/R. Therefore, the balls with different densities and radii but with equal masses will have equal potentials for R > A. As the inverse problem in the general setting is not unique, there arise two other settings. The first one may be formulated as follows. We need to determine the density of the mass distribution p(x, y, z) in a body of given shape, if we know the external potential of this body. The second problem is to determine the shape of a body with a known density, if the external potential is given. We shall consider the first problem. It is easy to show that it also does not have a unique solution. Really, we may use formula (13) for the potential of a ball with a spherically symmetric density. If for R> α we are given the potential of this ball u(R), then (13) yields α R> α. ο Thus, all the information relative to the function p(r), being obtained from initial data of the inverse problem, is that the integral α (15)

ο

116

A.M. Denisov. Elements of the theory of inverse

problems

is given. Evidently, we cannot determine p(r) from this equation. For example, for a = 1, each function p(r) — cr + d, where c and d are positive constants such that c/4 + dj 3 = G, satisfy condition (15). The problem of determining the shape of a body with a known density has been investigated by many authors. First, the uniqueness was established by Novikov (1938). Then the similar inverse problems were considered by Sretenskii (1954), Ivanov (1956), Prilepko (1965, 1973), Strakhov and Brodsky (1986), Isakov (1990) and others. This problem is nonlinear. For certain conditions, we may reduce the problem to a nonlinear integral equation. We assume that the body Τ is starshaped with respect to a point M, i.e., each ray outgoing from Μ intersects the boundary Σ of Τ only in one point. We suppose, also, that p(x, y, ζ) = po is known. Taking Μ as the origin, we may represent the surface Σ as follows Γ = σ(φ,θ),

φ £ [0, π],

0€[Ο,2τγ],

Then, the formula for the potential u(x, y, z) will be as follows π2πσ(ψ,θ) [ f [ , Ροτ^ϊηφάτ 2 J J J v ( x — rsin(p-Hz))

- o o < 2 < oo.

(21)

We show now that equation (21) has a unique solution b G C(—οο,οο), b(z) ^ Ψϊ\ < ζ < oo. For this purpose, we show that for each zo > 0 there exists a unique function b(z)

e

C[-ZQ,

Z0],

b(z)

>

\

- z

0