numerical methods of solving inverse problems · 5 historical perspective it is well known that...

Sergey KabanikhinSobolev Institute of Mathematics

Institute of Computational Mathematicsand Mathematical GeophysicsNovosibirsk State University

Novosibirsk, Russia

e-mail: [email protected]

Workshop IP-TA 2010 in Warsaw

Numerical methods of solving Numerical methods of solving inverse problemsinverse problems

mailto:[email protected]

22

The speed of the bulletThe speed of the bullet (the convergence rate of the regularization algorithm(the convergence rate of the regularization algorithm))strongly depends on the properties of the desired subjectstrongly depends on the properties of the desired subject

(smoothness, source(smoothness, source--wise representation etc).wise representation etc).JozefJozef Brodsky, Nobel Prize WinnerBrodsky, Nobel Prize Winner

Iterative and direct methodsIterative and direct methodsExample of acoustic inverse problemExample of acoustic inverse problemUsage of Usage of a prioria priori information information –– V.V. V.V. VasinVasin , A.G. , A.G. YagolaYagolaLinear algebraic systems Linear algebraic systems –– S.K. Godunov schemeS.K. Godunov schemeMathematicsMathematicsApplicationsApplications

RemarkRemark: well: well--known approaches are described in the wellknown approaches are described in the well--known known books by books by A.N.TikhonovA.N.Tikhonov et alet al, V.K. , V.K. IvanovIvanov et alet al, M.M. , M.M. LavrentievLavrentiev et alet al, , H. H. EnglEngl et alet alWe use some new We use some new iapproachesiapproaches from the books and papers by:from the books and papers by:

VasinVasin, , AgeevAgeev--VasinVasin, , EreminEremin--VasinVasinYagolaYagola , , LeonovLeonovKabanikhinKabanikhin--ScherzerScherzer--ShishleninShishlenin--VasinVasinKabanikhinKabanikhin--SchieckSchieck

33

Iterative methods:Landweber iteration

qn+1 = qn - α [A’(qn )] *(Aqn - f)

Gradient methodsqn+1 = qn - αn J’(qn )

Newton-Kantorovichqn+1 = qn – [A’(qn )] -1 (Aqn - f)

Levenberg-Marquardt

Direct methods:Linearization: A’(q0 ) q1 = f1

Finite-Difference Scheme Inversion

Gelfand-Levitan method

Boundary Control method

Singular Value Decomposition

Ill-Posed and Inverse Problems

Differential, Integral, Operator Equations

Aq = f

A(q + δq) – Aq = A’(q) δq + o(|| δq ||)

Variational Form

J(q) = <Aq - f, Aq - f> → min

J’(q) = 2 [ A’(q) ] * (Aq – f)

44

Inverse problems in mathematicsInverse problems in mathematics

It can be said that specialists in inverse and illIt can be said that specialists in inverse and ill--posed posed problems problems study the properties of and regularization methods study the properties of and regularization methods for unstable problemsfor unstable problems..

In other words, they develop and study In other words, they develop and study stable methods for stable methods for approximating unstable mappings.approximating unstable mappings.

In terms of In terms of linear algebralinear algebra, this means developing , this means developing approximate methods of finding approximate methods of finding normal pseudonormal pseudo--solutionssolutions to to systems of linear algebraic equations with systems of linear algebraic equations with rectangular, rectangular, degenerate, or illdegenerate, or ill--conditioned matrices. conditioned matrices. In In functional analysisfunctional analysis, the main example of ill, the main example of ill--posed posed problems is represented by an operator equation problems is represented by an operator equation AqAq = = ff, , where where AA is a is a compact (completely continuous) operator. compact (completely continuous) operator. In some recent publications, certain problems of In some recent publications, certain problems of mathematical statistics are viewed as inverse problems of mathematical statistics are viewed as inverse problems of probability theory.probability theory.From the point of view of From the point of view of information theoryinformation theory, the theory of , the theory of inverse and illinverse and ill--posed problems deals with the properties of posed problems deals with the properties of maps from compact sets with high epsilonmaps from compact sets with high epsilon--entropy to tables entropy to tables with low epsilonwith low epsilon--entropy.entropy.

55

Historical PerspectiveHistorical PerspectiveIt is well known that many mathematical concepts and problem forIt is well known that many mathematical concepts and problem formulations are products of mulations are products of studying studying physical phenomenaphysical phenomena. .

This is certainly true for the theory of inverse and illThis is certainly true for the theory of inverse and ill--posed problems.posed problems.

PlatoPlato’’ss philosophical allegory about echo and shadows on the cave wallsphilosophical allegory about echo and shadows on the cave walls (i.e., the data of an (i.e., the data of an inverse problem) being the only reality available to human cogniinverse problem) being the only reality available to human cognition was a precursor to tion was a precursor to AristotleAristotle’’s solution to the problem of reconstructing the shape of the Ears solution to the problem of reconstructing the shape of the Earth from its shadow th from its shadow on the moonon the moon (projective geometry). (projective geometry).

The introduction of the The introduction of the physical concept of instantaneous speed physical concept of instantaneous speed ledled Isaac Newton to the Isaac Newton to the discovery of the derivativediscovery of the derivative, and the instability (ill, and the instability (ill--posednessposedness) of the problem of numerical ) of the problem of numerical differentiation of an approximate function is still a subject ofdifferentiation of an approximate function is still a subject of presentpresent--day research.day research.

Lord RayleighLord Rayleigh’’s research in acousticss research in acoustics led him to the question of whether it is possible led him to the question of whether it is possible to to determine the density of a nondetermine the density of a non--uniform string from its sounduniform string from its sound (the inverse problem of (the inverse problem of acoustics), which brought about the development of seismic prospacoustics), which brought about the development of seismic prospecting on one hand, and ecting on one hand, and the theory of spectral inverse problems on the other hand.the theory of spectral inverse problems on the other hand.

The study of the motions of celestial objects and the problem ofThe study of the motions of celestial objects and the problem of estimating unknown estimating unknown parameters based on measurement results that parameters based on measurement results that contain random errorscontain random errors led led Legendre and Legendre and Gauss to Gauss to overdeterminedoverdetermined systems of algebraic equations and to the method of least squarsystems of algebraic equations and to the method of least squareses. . CauchyCauchy proposed the steepest descent method for finding the minimum ofproposed the steepest descent method for finding the minimum of a multivariate a multivariate function. function. In 1948, In 1948, L.V. Kantorovich generalizedL.V. Kantorovich generalized, developed, and , developed, and appliedapplied these ideas to operator these ideas to operator equations in Hilbert spaces. At present, the steepest descent meequations in Hilbert spaces. At present, the steepest descent method together with the thod together with the conjugate gradient method are among the most popular methods forconjugate gradient method are among the most popular methods for solving illsolving ill--posed posed problems. It should be noted that problems. It should be noted that KantorovichKantorovich was the first to point outwas the first to point out that if the that if the problem is problem is illill--posedposed, then, then the method he proposed converges only with respect to the objectthe method he proposed converges only with respect to the objectiveivefunctional.functional.

66

WellWell –– PosedPosed ProblemsProblems Ill Ill –– Posed and Inverse ProblemsPosed and Inverse Problems

ArithmeticArithmetic

AlgebraAlgebra

AnalysisAnalysis

DifferentialDifferentialEquationsEquations

Inverse SturmInverse Sturm--LiouvilleLiouville problemproblem

Integral GeometryIntegral Geometry

IntegralIntegralEquationsEquations

Elliptic EquationsElliptic Equations

Parabolic Parabolic EquationsEquations

Hyperbolic Hyperbolic EquationsEquations

CoefficientCoefficientInverse ProblemsInverse Problems

baqbaq −=+= abbaqbabaq <<=≥⋅= ,,

small very is detdconditione-ill is

AAfAq =

∫+=x

dqfxf0

)()0()( ξξ )()( xfxq ′=

0)1()1(,0)0()0(

),()()()(

=′+=′+

=−′′

uHuuhu

xuxuxqxu λ

)(||||, 2 xqunn →λ

∫Γ =),(

),(),(yx

yxfdsq ηξ

∫+=x

dqxKxfxq0

)(),()()( ξξξ ∫+=x

dqxKxf0

)(),()(0 ξξξ

(mixed)Robin Newman, Dirichlet, 0=Δu boundary theofPart boundary.-Initial Cauchy, 0=Δu

0at boundary-Initial Cauchy, ==Δ tuu t),(

0yxfu

uu

t

t

=

−=Δ

=

),(),,( 2010tyfutyfu

uu

xxx

t

==

Δ=

==

boundary-InitialCauchy

part like-TimeCauchy. Newman, Dirichlet,

ttq

tq

qq

uuL

uuL

LuL

=

=

−= order second theofoperator elliptic0

Regularization Methods for Inverse Problems

The pseudoinverse and the singular value decomposition of an operator

Theorem (on the singular value decomposition of a compactoperator.) If Q and F are separable Hilbert spaces and A : Q → Fis a compact linear operator, then there exist orthonormalsequences of functions vn ⊂ Q (right singular vectors),un ⊂ F (left singular vectors), and a nonincreasing sequence ofnonnegative numbers σn (singular values) such that

Avn = σnun,

A∗un = σnvn,

span vn = R(A∗) = N(A)⊥,

span un = R(A) = N(A∗)⊥,

and the set σn has no nonzero limit points.



The sequence un is a complete orthonormal system ofeigenvectors of the operator AA∗ such that

Avn = σnun, A∗un = σnvn, n ∈ N

and the following decompositions hold:

Aq =∑n

σn〈q, vn〉un, A∗f =∑n

σn〈f , un〉vn.



Let A : Q → F be a compact linear operator, where Q and F areseparable Hilbert spaces (i.e., Hilbert spaces with countable bases).A system σn, un, vn with n ∈ N, σn ≥ 0, un ∈ F , and vn ∈ Q willbe called the singular system of the operator A if the followingconditions hold:

the sequence σn consists of nonnegative numbers such thatσ2

n is the sequence of eigenvalues of the operator A∗Aarranged in the descending order with respect to multiplicity;

the sequence vn consists of the eigenvectors of the operatorA∗A corresponding to σ2

n (and is orthogonal and complete,R(A∗) = R(A∗A) );

the sequence un is defined in terms of vn asun = Avn/‖Avn‖.



Let Qpf denote the set of all pseudo-solutions to the equation

Aq = f for a fixed f ∈ F .Then

Qpf = qp : ‖Aqp − f ‖ = inf

q∈Q‖Aq − f ‖ = q : Aq = PR(A)f ,

which implies that Qpf is nonempty if and only if

f ∈ R(A)⊕ N(A∗). In this case, Qpf is a convex closed set, and it

contains the element of minimal length, qnp — a pseudo-solutionof minimal norm (Nashed and Votruba, 1976), also called thenormal pseudo-solution to the equation Aq = f (with respect tothe zero element).



For example, the problem Aq = f is said to be weakly ill-posed ifσn = O(n−γ) for some γ ∈ R+, and strongly ill-posed otherwise(for instance, if σn = O(e−n)).The singular value decomposition of the operator A can be used toconstruct a regularization method for the problem Aq = f basedon the projections of

qδn =n∑

j=1

〈fδn, uj〉σj

vj ,

and prove that

‖qδn − qnp‖ = O(σn+1 +

δ

σn

), n →∞.

55

RtRytyhtyv

vtxRy

vyxvvyxc

nx

t

ntt

∈∈⋅=+

≡>>∈

∇⋅∇−Δ=

<

−

−

,),()(),,0()3(

;0|)2(;0,0,

,),(ln),()1(

0

1

2

δ

ρ

.,),,(),,0()4( RtRytyftyv n ∈∈=+

0 < c0 ≤ c(x,y) ( c0 = const) - velocity; 0 < ρ0 ≤ ρ(x,y) ( ρ0 = const) - density; v(x,y,t) - exceeded pressure.

Inverse Problem: find coefficients of equation (1) using additional information:

Forward (Direct) Problem

77

)(

)(|

)()(

0

0

0

tFV

tHVOV

VxBVxAVV

x

xx

t

xxxtt

=

=

≡−−=

=

=

<

δ

Projection MethodProjection Method

),(|

)()(;0|

0,0,),(ln

0

0

0

tyfu

tyhuu

txuyxuuu

x

xx

t

yyxxtt

=

=

≡

>>∇∇−+=

=

=

<

δ

ρ

ijy

jj

ijy

jj exyxetxutyxu )(),(;),(),,( ∑∑ == ρρ

),(),(

21

),(),(

),,(~)(),(),,(

yxyx

yxSyxS

tyxuxtyxStyxu

xx

ρρ

θ

=

+−=

( )NNN vvvvV ,,,,, 01 KK+−−=

A(x) and B(x) depend on ρj(x), j = - N,- N+1,…,0,…,N

99

.exp)(

;0|

;0,0,),(ln

0

0

ikytu

u

txuyxuuu

x

kx

tk

kky

kxx

ktt

δ

ρ

=

≡

>>∇∇−Δ+=

+=

<

The sequence of forward initial boundary value problems k∈Z =0,±1,±2, … or k∈Z n.

Inverse problem: find ρ(x,y)>0 using the traces of the solutions:

).,(),,0( tyftyu kk =+

The necessary condition of existence of solution to the inverse problem:

.,exp)0,( Zkikyyf k ∈−=+

GelGel’’fandfand--LevitanLevitan--KreinKrein method method –– statement of the problemstatement of the problem

∑=m

m ikyxyx exp)(),( ρρ

1010

We use the sequence of Green’s functions, which solves

0),,0(,exp)(),,0(

,0,),(ln

==

∈>∇∇−Δ+=

tywimyttyw

Rtxwyxwwwmx

m

mmy

mxx

mtt

δ

ρ

For every m∈Z[ ]

imyyyxyxS

tyxwtxtxyxStyxw

m

mmm

exp),0(),(

21),(

),,(~)()(),(),,(

ρρ

δδ

=

+−++=

The sequence w m is some kind of a bridge between the original problem and Φ m (x,t) - the solution of the system of Gel’fand-Levitan equations (Kabanikhin, 1988) :

dydy

tywtxx

o

mm ξ

ξρξπ

π∫ ∫−

=Φ),(

),,(),(

GelGel’’fandfand--LevitanLevitan--KreinKrein MethodMethod

Zkxtdyy

edssxstft

txiky

m

x

x

mkm

k ∈<−=Φ−∂∂

−Φ ∫∑ ∫−−

,,),0(

),()(),(2π

π ρ

∫−

=−Φπ

π ρρdy

yyxexx

imym

),0(),(2)0,(

1111

Zkxt

dyy

edssxstft

txiky

m

x

x

mkm

k

∈<

−=Φ−∂∂

−Φ ∫∑ ∫−−

,

),0(),()(),(2

π

π ρ

The solution of inverse problem can be obtained from the solution of Gelfand-Levitan-Krein equation by formula

imy

m

miky

imym

exxeyyx

dyyyx

exx

−

−

∑

∫

−Φ=

=−Φ

)0,(),0(1),(

1),0(),(2

)0,(

ρπρ

ρρ

π

π

Therefore in order to find solution ρ(x,y) in the depth x0 we solve GLK equation with the fixed parameter x0 and then calculate ρ(x0 ,y) .

The multidimensional analog of Gelfand-Levitan-Krein (GLK) equation

GelGel’’fandfand--LevitanLevitan--KreinKrein MethodMethod

1212

NewtonNewton--Kantorovich Method 1DKantorovich Method 1D

0),0();(),()(),(,)(

==

Δ∈⋅−=

tuxsxxuTtxuxquu

xnnn

xnnxxnttn

)(),0(),0(

0),0(;)(2

),(),(

)(),(,)()(

0

tftutw

twdxxuxxw

Ttxuxwxqww

nn

xn

x

nn

n

xnnxnnxxnttn

−=

=⋅=

Δ∈⋅−⋅−=

∫ ξξμ

μ4. Find A’(qn) μn = un(0,t) - f(t).5. Solve linear inverse problem:

Theorem. Let T > 0 and for f ∈ L2 (T) there exists solution q * of equation Aq = f. Then exist δ* > 0 and ω∈ (0,1) such that if q (0) ∈ Bδ* (q*) then NK converges to the solution q* and

|| q* - q (n) || ≤ ω n || q* - q (0) ||

1. Chose approximation q0(x).2. Let qn(x) be known.3. Solve the direct problem:

6. Put qn+1(x) = qn(x) - μn(x)

[ ] ( )fAqqAqq nnnn −−= −+

11 )('

1313

LandweberLandweber IterationsIterations

0),0();(),()(),(,)(

==

Δ∈−=

tuxsxxuTtxuxquu

xnnn

xnnxxnttn

( )

[ ] 0)(),0(2),0()0(),0(0)2,(

)(),(,)(

=−+⋅+=−

Δ∈⋅+=

tftutqtxTx

Ttxxq

nxn

n

xnnxxnttn

ψψψ

ψψψ

1. Chose the approximation q0(x)2. Let qn(x) be known3. Solve the direct problem:

4. Find the discrepancy ηn(t) = un (0,t) - f(t) 5. Solve the adjoint problem:

6. Find

[ ] ( )[ ]

7. Put qn+1(x) = qn(x) - α J’(qn)

[ ]∫∫∫ −+−−=′− T

xnnn

T

xtnxnn

xT

xxnnn dtttutqdtttudttxuxqJ ),()(

21),(),())((

2

ψψψψ

[ ] ( )fAqqAqq nnnn −−=+*

1 )('α

1414

Numerical results Numerical results –– 2D2D--acoustics. MGL. acoustics. MGL. Exact, N=10, N=50 Exact, N=10, N=50

1515

Numerical results Numerical results –– 2D acoustics. MGL. 2D acoustics. MGL. Noise=0.05, N=10, N=50Noise=0.05, N=10, N=50

1616

X

0

0.2

0.4

0.6

0.8

1

Y

-3

-2

-1

0

1

2

3

Z

1

2

3

4

X

0

0.2

0.4

0.6

0.8

1

Y

-3

-2

-1

0

1

2

3

Z

1

2

3

4

X

0

0.2

0.4

0.6

0.8

1

Y

-3

-2

-1

0

1

2

3Z

1

2

3

4

Numerical results Numerical results –– 2D2D--acoustics. MGL. acoustics. MGL. Exact, N=10, N=50 Exact, N=10, N=50


Landweber iteration

We consider as the simpliest example the following inverseproblem: find function q(x , y) in domain (0, `)× (−π, π), whichsatisfies the following conditions

utt = uxx + uyy − q(x , y)u, (x , y , t) ∈ Ω; (2)

u|t<0 ≡ 0, ux |x=0 = γδ(t); (3)

u|y=π = u|y=−π; (4)

u(0, y , t) = f (y , t), t ∈ (0, 2`). (5)

Here Ω = x , y , t : (x , t) ∈ ∆(`), y ∈ (−π, π) and∆(`) = (x , t) : 0 < x < t < 2`− x.Approximate solution of the inverse problem will be found in theform of finite Fourier series (PN -approximation):

q(x , y) ∼=N∑

n=−N

qn(x)e iny , u(x , y , t) ∼=N∑

n=−N

un(x , t)einy . (6)


Landweber iteration

Let us introduce: U(x , t) = (u−N , u−N+1, . . . , u0 . . . , uN),Q(x) = (q−N , . . . , q0 . . . , qN) and consider inverse problem invector form:

Utt = Uxx − B(x)U, (x , t) ∈ ∆(`); (7)

Ux |x=0 = 0, t ∈ (0, 2`); (8)

U(x , x) = S(x), x ∈ (0, `); (9)

U(0, t) = F (t), t ∈ (0, 2`). (10)

Here B(x) is defined as follows (n = −N, . . . ,N):[B(x)U]n = n2un(x , t) +

∑|k|≤N,|k−n|≤N

qn−k(x)uk(x , t).

S(x) and F (x) are vector functions consisting of the Fouriercoefficients of functions −γ + βq(x , y) and f (y , t) correspondingly.In inverse problem (7)–(10) we need to find the vector-functionQ(x) by known data F (t).


Landweber iteration

Let us consider inverse problem (7)–(10) as nonlinear operatorequation

A(Q) = F . (11)

Properties of operator A have been investigated in [6].Due to the uniqueness of the solution to the inverse problem(7)–(10) it is enough to find the minimum of cost functional

J(Q) = ‖A(Q)− F‖2L2(0,2`). (12)

Here‖F‖2

L2(2`) =∑|k|≤N

‖fk‖2L2(0,2`).


Landweber iteration

For solving minimization problem J(q) → inf we apply Landweberiteration

Q(n+1) = Q(n) − α[A′(Q(n))]∗(A(Q(n) − F )), n = 0, 1, . . . . (13)

Landweber iteration method can be considered as optimizationmethod with fixed descent parameter α∗. Indeed it is easy to showthat

J ′(Q) = 2[A′(Q)

]∗(A(Q)− F ).


Landweber iteration

Let us consider the following adjoint problem

Ψtt = Ψxx − [B(x)]∗Ψ, (x , t) ∈ ∆(`);

Ψx |x=0 = 2[U(0, t)− F (t)], t ∈ (0, 2`);

Ψ(x , 2`− x) = 0, x ∈ (0, `).

One can easily check that the component of the gradient of (12) isdefined by formula

[J ′(Q)]m(x) = 2β(ψmx + ψmt)(x , x)+

+∑

|k|≤N,|k−m|≤N

∫ 2`−x

xum−k(x , t)ψm(x , t)t.. (14)


Landweber iteration

The scheme of Landweber iteration method1 Let Q(0)(x) is initial guess.2 Let Q(n)(x) be known, then solve the forward problem:

U(n)tt = U

(n)xx − B(n)(x)U(n), (x , t) ∈ ∆(`);

U(n)x

∣∣∣x=0

= 0, t ∈ (0, 2`);

U(n)(x , x) = S (n)(x), x ∈ (0, `).

3 Find discrepancy η(n)(t) = U(n)(0, t)− F (t) and its norm.4 Solve the adjoint problem:

Ψ(n)tt = Ψ

(n)xx − [B(n)(x)]∗Ψ(n), (x , t) ∈ ∆(`);

Ψ(n)x |x=0 = 2[U(n)(0, t)− F (t)], t ∈ (0, 2`);

Ψ(n)(x , 2`− x) = 0, x ∈ (0, `).

Define the gradient [J ′(Q(n))](x) by formula (14)5 Find Q(n+1)(x) = Q(n)(x)− α[J ′(Q(n))](x).


Modification of algorithm

The usual convergence theorem can be proved as in [5, 6].Theorem(convergence of Landweber iteration). Let ` > 0 andF ∈ L2(`). Suppose that there exists a solution QT ∈ L2(`) of theproblem A(Q) = F . Then one can find such ν∗ ∈ (0, 1), δ∗ > 0,α∗ > 0, that if Q(0) ∈ B(QT , δ∗) and α ∈ (0, α∗) then theapproximations Q(n) of Landweber iteration converge to thesolution QT as n →∞ at the rate

‖QT − Q(n)‖2L2(`)

≤ νn∗ δ

2∗ .


Modification of algorithm

Using constant r (‖QT‖L2(`) ≤ r) we modify the Landweber

iteration as follows. First, given the approximation Q(n) wecalculate

Q(n+1) = Q(n) − α[A′(Q(n))

]∗(A(Q(n))− F

).

Then we put

Q(n+1) =

Q(n+1), if ‖Q(n+1)‖L2(`) < r ;

Q(n+1) r ‖Q(n+1)‖−1L2(`)

, if ‖Qn+1‖L2(`) ≥ r .

Theorem(convergence of modified algorithm). Suppose that thereexists a solution QT ∈ B(0, r)

⋂L2(`) of the problem A(Q) = F .

Then one can find such ν∗ ∈ (0, 1), C∗ > 0, α∗ > 0, that for anyinitial guess Q(0) and any α ∈ (0, α∗) the approximations Q(n) ofLandweber iteration converge to the solution QT as n →∞ at therate

‖QT − Q(n)‖2L2(`)

≤ ν2∗C

2∗ .


References

Ivanov V. K., Vasin V. V., Tanana V. P. Theory of LinearIll-Posed Problems and its Applications. VSP, TheNetherlands, 2002.

Kabanikhin, S.I. Projection-Difference Methods forDetermining the Coefficients of Hyperbolic Equations. (1988),Izd-vo SO Akad. Nauk SSSR, Novosibirsk (in Russian).

Kabanikhin S.I., Satybaev A.D., Shishlenin M.A. DirectMethods of Solving Inverse Hyperbolic Problems. VSP, TheNetherlands, 2004.

Azamatov J. S., and Kabanikhin S. I., Nonlinear Volterraoperator equations. L2 – theory. Journal of Inverse andIll-Posed Problems (1999) Vol.7(6), The Netherlands, Utrecht.


References

S. I. Kabanikhin, R. Kowar, and O. Scherzer, On theLandweber iteration for the solution of parameter identificationproblem in a hyperbolic partial differential equation of secondorder. JIIPP (1998). Vol.6, No.5.

Kabanikhin S.I., Scherzer O. and Shishlenin M.A. Iterationmethod for solving a two dimensional inverse problem for ahyperbolic equation. JIIPP, 11 (1), 2003.

Kabanikhin S.I., Bakanov G.B. and Shishlenin M.A.Comparative analysis of methods of finite inversion scheme,Newton-Kantorovich and Landweber in inverse problem forhyperbolic equation. Novosibirsk, Novosibirsk State University,Preprint 12, 2001 (in Russian).


References

V.G. Romanov and S.I. Kabanikhin, Inverse Problems ofGeoelectrics (Numerical Methods of Solution). PreprintNo. 32. Inst. Math., Siberian Branch of the USSR Acad. Sci.,Novosibirsk (1989) (in Russian).

S. He and S.I. Kabanikhin, An optimization approach to athree-dimensional acoustic inverse problemin in the timedomain, J. Math. Phys. 36, 8 (1995).

V.G. Romanov, Local solvability of some multidimensionalinverse problems for equations of hyperbolic type. DifferentialEquations, 25, 2 (1989) (in Russian).

numerical methods of solving inverse problems · 5 historical perspective it is well known that...

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