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Inverse spectral problem for the Sturm–Liouville equation
View the table of contents for this issue, or go to the journal homepage for more
2003 Inverse Problems 19 235
(http://iopscience.iop.org/0266-5611/19/1/314)
Home Search Collections Journals About Contact us My IOPscience
Inverse spectral problem for the Sturm–Liouvilleequation
B M Brown1, V S Samko1, I W Knowles2 and M Marletta3,4
1 Department of Computer Science, University of Wales, Cardiff, PO Box 916,Cardiff CF2 3XF, UK2 Department of Mathematics, University of Alabama at Birmingham, 452 Campbell Hall, 1300University Boulevard, Birmingham, AL 35294-1170, USA3 Department of Mathematics and Computer Science, University of Leicester, University Road,Leicester LE1 7RH, UK
Received 24 June 2002, in final form 18 December 2002Published 17 January 2003Online at stacks.iop.org/IP/19/235
AbstractThis paper discusses a new numerical approach to computing the potential qin the Sturm–Liouville problem −y ′′ + qy = λy on a compact interval. It isshown that an algorithm to recover q from eigenvalues and multiplier constantscan be derived. Examples of some test problems, and questions of efficiencyare discussed.
1. Introduction
Recently [1, 6, 16, 19] there has been much interest in the inverse spectral problem for theSturm–Liouville equation
Uniqueness results for Q have been proved, in terms of the associated Titchmarsh–Weylfunction, some of which provide a local version of the celebrated Borg–Marchenko uniquenesstheorem, first mentioned by Borg in 1946 [2]. As Levitan says in the preface to hisbook [10, p 1], Borg was the first to undertake a systematic investigation of this problem. Inparticular he showed that knowledge of one spectrum is, in general, insufficient to determinethe potential Q. There are many variations on this result which include the determination ofQ from one spectrum when the potential is known to be symmetric or when it is known onone half of the interval.
Marchenko takes the spectral measure as his starting point for the analysis, while Borg,and later Levinson, develop their ideas in terms of the m function and the so-called multiplier
4 Present address: School of Mathematics, Cardiff University, PO Box 926, CF24 4YH, UK.
constants. These approaches are equivalent. In the paper of Gel’fand and Levitan [5] it isshown that, when b is finite, Q is determined by an integral equation and this apparentlyprovides a means to determine Q numerically from two sets of spectral data. Algorithmsbased on this result are known [17] as is a method based on finite elements, (an up to dateaccount of known methods may be found in [11]). However, the recovery of a potential fromspectral data is a difficult and ill-conditioned numerical problem.
This paper presents a new approach to the numerical recovery of Q from spectral data.It is based on a method of Knowles [7] developed for the recovery of coefficients in a PDEfrom boundary data and modified for use in this inverse spectral context (see also [8]). Thedata used to recover the potential are a subset of the eigenvalues together with the multiplierconstants. The method consists of first defining a Banach space functional whose unique zerois found at Q. Starting with an arbitrary guess Q0 for Q, a gradient descent is performed onthe functional until the minimum is achieved.
Section 2 of the paper discusses the formulation of the method while section 3 reports onthe numerical results obtained for some examples. These have been chosen for comparisonwith results in [17]. Section 4 compares results obtained using minimizing procedures indifferent metrics. However, it is shown that the L2[0, b] norm in general performs best. Anappendix contains some technical analytic details.
2. Formulation of the method
2.1. Background
We assume that the eigenvalues {λi}∞i=0 of (1) with separated boundary conditions
y(0) + Ay ′(0) = 0
y(1) + By ′(1) = 0(2)
are known. This formulation excludes Neumann boundary conditions; an alternativeformulation which excludes Dirichlet boundary conditions is given by Levitan [10, p 64].The two formulations are, of course, very similar, so we shall discuss only the first of them inthis paper.
Let uq(x, λ) and vq(x, λ) be solutions of (1) with Q replaced by q , defined by
uq(0, λ) = A vq(1, λ) = B (3)
u′q(0, λ) = −1 v′
q(1, λ) = −1. (4)
(In the case of Neumann boundary conditions, one would choose uq(0, 1) = 1, u′q(0, λ) = 0,
vq(1, λ) = 1, v′q(0, λ) = 0.) Then it can be shown that there exist multiplier constants Cn ,
n � 1 such that for 0 � x � 1
uQ(x, λn) = CnvQ(x, λn). (5)
In [9] a method is given to calculate these constants from a second known spectrum of (1)together with separated boundary conditions of Sturm–Liouville type (see equation (21)).
In order to set up a recovery procedure we next define a functional G(q) such thatG(Q) = 0. Let the {λi}∞i=0 and {Ci }∞i=0 be given. Denote uq,n(x) = uq(x, λn) andvq,n(x) = vq(x, λn). Consider Q(x) > 0, q in L1[0, 1], q(x) > 0 for all x in [0, 1] whereQ(x) is the true potential and
Gn(q) =∫ 1
0(u′
q,n − Cnv′q,n)
2 + q(uq,n − Cnvq,n)2. (6)
Inverse spectral problem for the Sturm–Liouville equation 237
The formal sum G is defined by
G(q) =∞∑
n=0
Gn(q). (7)
We defer to the appendix a proof that G(q) diverges unless A = 0. We both introduceweights ωi > 0 into the sum and also truncate it to give a functional G(q) = ∑N
n=0 ωn Gn(q)
for some N > 0, with which we work (see further details in section 2.3).By definition G(Q) = 0 and G(q) > 0 for q �= Q, by the uniqueness theorem. As we
shall minimize G with respect to q by a gradient descent method we next calculate its Gateauxderivative, which will be used in this procedure.
2.2. Gateaux derivative of G
By definition
G ′n(q)[h] = lim
ε→0
Gn(q + εh) − Gn(q)
ε.
A calculation based on integrating by parts, noting the difference of two squares and using (1)to eliminate u′′
In order to calculate the gradient H of G we shall need G ′(q)[h] in the form∫ 1
0 h H . To
do this we next show how to write uq,h , vq,h and 2λ∫ 1
0 (uq,h − Cvq,h)(uq − Cvq) in the form∫ 10 h H .
The equation
y ′′(x) + q(x)y(x) = ξ(x), y(0) = 0, y ′(0) = 0 (11)
has a solution y which is
y(x) =∫ x
0G(x, t)ξ(t) dt,
where G(x, t) is the Green’s function associated with (11), G(x, x) = 0 and Gx(x, x) = 1.Also the solution of
y ′′(x) + q(x)y(x) = ξ(x), y(1) = 0, y ′(1) = 0
can be written as
y(x) = −∫ 1
xG(x, t)ξ(t) dt .
Inverse spectral problem for the Sturm–Liouville equation 239
Note that the Green’s function for a Sturm–Liouville second-order linear initial valueproblem can be written
G(x, y) = f (x)g(y) − f (y)g(x)
where f and g are any two independent solutions of the homogeneous problem with WronskianW ( f, g)(x) = ( f g′ − f ′g)(x) = 1 for any, and hence all, x .
We choose functions f and g such that f (1) = 1, f ′(1) = 0, g(1) = −B , g′(1) = 1 and− f ′′ + (q − λ) f = 0, −g′′ + (q − λ)g = 0 giving W ( f, g)(1) = 1.
Note that g(x) = −vq(x) and by (8)
uq,h(x) = −∫ x
0[ f (x)g(y) − f (y)g(x)]h(y)uq(y) dy,
uq,h(x) = − f (x)
∫ x
0g(y)h(y)uq(y) dy + g(x)
∫ x
0f (y)h(y)uq(y) dy.
(12)
So by (12) we can write
uq,h(1) =∫ 1
0(uq(y)h(y)(vq(y) − B f (y))) dy.
Similarly with G1(x, y) = r(x)s(y) − s(x)r(y), it follows that
vq,h(x) = r(x)
∫ 1
xs(y)h(y)vq(y) dy − s(x)
∫ 1
xr(y)h(y)vq(y) dy (13)
and
vq,h(0) = −∫ 1
0vq(y)h(y)(uq(y) − Ar(y)) dy.
So
2(u′q(1) + C)uq,h(1) − 2(Cv′
q(0) + 1)Cvq,h(0)
=∫ 1
0h2(uq(vq − B f )(u′
q(1) + C) + vq(uq − Ar)(Cv′q(0) + 1)C) dy
and now we can write
G ′(q)[h] =∫ 1
0h[2(uq(vq − B f )(u′
q(1) + C) + vq(uq − Ar)(Cv′q(0) + 1)C)]
+ 2λ
∫ 1
0(uq,h − Cvq,h)(uq − Cvq).
Now consider
2λ
∫ 1
0(uq,h − Cvq,h)(uq − Cvq). (14)
Substituting uq,h and vq,h with (12) and (13) gives
2λ
∫ 1
0(uq − Cvq)
[− f (x)
∫ x
0g(y)h(y)uq(y) dy + g(x)
∫ x
0f (y)h(y)uq(y) dy
− Cr(x)
∫ 1
xs(y)h(y)vq(y) dy + Cs(x)
∫ 1
xr(y)h(y)vq(y) dy
].
Now, since g(x) = −vq(x) and s(x) = −uq(x), (14) becomes
2λ
∫ 1
0(uq(x) − Cvq(x))
[f (x)
∫ x
0vq(y)h(y)uq(y) dy − vq(x)
∫ x
0f (y)h(y)uq(y) dy
+ Cr(x)
∫ 1
xuq(y)h(y)vq(y) dy − Cuq(x)
∫ 1
xr(y)h(y)vq(y) dy
]dx .
240 B M Brown et al
A rearrangement of terms and exchange of integrals gives (14) as
2λ
∫ 1
0h(y)uq(y)
[vq(y)
∫ 1
y(uq(x) − Cvq(x)) f (x) dx
− f (y)
∫ 1
y(uq(x) − Cvq(x))vq(x) dx
]dy
+ 2λC∫ 1
0h(y)vq(y)
[uq(y)
∫ y
0(uq(x) − Cvq(x))r(x) dx
− r(y)
∫ y
0(uq(x) − Cvq(x))uq(x) dx
]dy.
Consider
z =[vq(y)
∫ 1
y(uq(x) − Cvq(x)) f (x) dx − f (y)
∫ 1
y(uq(x) − Cvq(x))vq(x) dx
]. (15)
We can rewrite this as
z = −∫ 1
yη(x)[ f (y)w(x) − f (x)w(y)] dx (16)
where η(x) = uq(x)−Cvq(x) and w = −vq . Note that f (1) = 1, f ′(1) = 0 and w(1) = −B ,w′(1) = 1.
It is easily seen that z is a solution of the equation −z′′ + (q − λ)z = uq − Cvq withboundary conditions z(1) = 0, z′(1) = 0. Similarly,
t = −∫ y
0η(r(y)w(x) − r(x)w(y)) dx (17)
is given by the solution of −t ′′ + (q − λ)t = uq − Cvq with boundary conditions t (0) = 0,t ′(0) = 0, where η(x) = uq(x)−Cvq(x) and w = −uq , r(0) = 1, r ′(0) = 0 and w(0) = −A,w′(0) = 1. Hence
G ′n(q)[h] =
∫ 1
0h[2(uq(vq − B f )(u′
q(1) + C) + vq(uq − Ar)(Cv′q(0) + 1)C)
+ (uq − Cvq)2 + 2λ(−uq z + Cvq t)
],
where f , r , z, t are solutions of
− f ′′ + (q − λ) f = 0, f (1) = 1, f ′(1) = 0,
−r ′′ + (q − λ)r = 0, r(0) = 1, r ′(0) = 0,
−z′′ + (q − λ)z = uq − Cvq , z(1) = 0, z′(1) = 0,
−t ′′ + (q − λ)t = uq − Cvq , t (0) = 0, t ′(0) = 0.
Since these functions are the solutions of a second-orderODE which can only have two linearlyindependent solutions, we can write f (x) = au(x) + bv(x), r(x) = cu(x) + dv(x). A simplecalculation using the boundary conditions shows that the constants a, b, c, d are
a = 1
Bu′(1) + u(1), b = u′(1)
Bu′(1) + u(1),
c = v′(0)
Av′(0) + v(0), d = 1
Av′(0) + v(0).
So
f (x) = u(x) + v(x)u′(1)
Bu′(1) + u(1), r(x) = u(x)v′(0) + v(x)
Av′(0) + v(0).
Inverse spectral problem for the Sturm–Liouville equation 241
Now we can rewrite
G ′(q)[h] =∫ 1
0h H (18)
where
H (x) =[
2
(uq
(vq − B
uq + vq u′q(1)
Bu′q(1) + uq(1)
)(u′
q(1) + C)
+ vq
(uq − A
uqv′q(0) + vq
Av′q(0) + vq(0)
)(Cv′
q(0) + 1)C
)
+ (uq − Cvq)2 + 2λ(−uq z + Cvq t)
]
where z and t are solutions of
−z′′ + (q − λ)z = uq − Cvq , z(1) = 0, z′(1) = 0,
−t ′′ + (q − λ)t = uq − Cvq , t (0) = 0, t ′(0) = 0.
2.3. Algorithm
Next we can try to find the minimum of the functional G(q) using the gradient descentalgorithm. We can write the Gateaux derivative in the form
G ′(q)[h] =∫ 1
0h(x)H (x) dx . (19)
Then, by the Riesz representation theorem the L2 gradient is H (x) and
G(q + h) = G(q) + G ′(q)[h] + O(h2).
So, taking h(x) = −H (x), there exists an α so that G(q + αh) < G(q). Thus we can set up arecovery algorithm for Q, forming a sequence of values
The algorithm. We assume that we have a finite set of eigenvalues {λn}Nn=0 and a corresponding
set of multiplier constants {Cn}Nn=0. As we previously noted, since the higher eigenvalues are
likely to contribute less to the recovery procedure than the lower ones, we introduce a set ofweights ωn , ωn > 0, decreasing with n, which reduce the effect of the higher eigenvalues onthe procedure. These also have the effect of replacing the divergent series (7) by a convergentone.
(1) Set some initial Q0 (usually, some constant greater than 0).(2) While stopping criterion is not satisfied{(3) For each n � nmax form Hn as defined by (18) and (19).(4) Form weighted sum h = − ∑nmax
n=0 ωn Hn.(5) (optional) Convert gradient h to L∞ or to Sobolev type gradient (see section 4).(6) Minimize F(α) = G(q + αh) to get αmin .(7) Set Q j+1 = Q j + αminh, where j is the number of the iteration.}
242 B M Brown et al
The result of this algorithm is a sequence of functions Qn(x), n = 0, . . . , M where M isthe number of iterations. These functions are approximations to the true potential Q.
As for the stopping criterion, the simplest one to use would be G(Qn)−G(Qn+1) < ε, butit does not appear to be of much use because even if the difference of the functional from theprevious iteration is very small, it is possible that the function is still converging at a reasonablerate. Thus we have to make ε somehow relative to the value of G(Qn).
The obvious way is by replacing ε with εG(Qn), creating the stopping criterionG(Qn)− G(Qn+1) < G(Qn)∗ ε, so the iterative process only stops when the difference of thevalue of the functional between two iterations is small relative to the value of the functional.
By construction, G(Q0) > G(Q1) > · · · > G(QM). We need to define a methodto determine which potential from this sequence suits the initial data best of all. Thestraightforward way to do this is to recover the spectra and multiplier constants correspondingto all the potentials from the sequence, then to decide the best based on some measure of the‘distance’ of the recovered spectrum from the true spectrum.
Let {λn}Mn=0 and {Cn}M
n=0 be the initial data.For each 0 � k � N calculate M eigenvalues {λkn }M
n=0 and M multiplier constants{Ckn }M
n=0 corresponding to the potential Qk(x).Now we can define:
(a) the best recovered potential in terms of the multiplier constants is Qm , where m is aninteger such that for any 0 � l � M
Nsupn=0
|Cl,n − Cn| � Nsupn=0
|Cm,n − Cn|;
(b) the best recovered potential in terms of eigenvalues is Qm , where m is an integer such thatfor any 0 � l � M
Nsupn=0
|λl,n − λn| � Nsupn=0
|λm,n − λn|.
When using initial data which contain noise, experiments show that the potential convergesto something like a distorted version of the exact potential. The distortion appears as smalloscillations in q . The best recovered potentials both in term of eigenvalues and in terms ofmultiplier constants are produced not by later iterations but rather by iterations closer to thebeginning of the recovery process. Thus the potential converges not to the exact potential, butto one with oscillatory noise.
This approach serves as a regularization method. If the initial data contain noise (andin a numerical case they always do) after some iteration the ‘distance in terms of eigenvaluedifference’ between the recovered potential and the exact one may start to increase. So we mustdetermine which potential recovered by the algorithm is the best. A regularization criterionis necessary when working with perturbed initial data. In this case the functional G(q) mayhave local minima which may not generate the best recovered potentials in terms of multiplierconstants or in terms of eigenvalues (see the previous discussion). So using our method wechoose that intermediate result which corresponds best to the initial data.
2.4. Implementation details
All the functions f in the recovery procedure are represented by values at equally spacedpositions {x0, . . . , xn}. Thus when values of f at x �= xi are needed, interpolation with cubicsplines is used. Piecewise constant and piecewise linear interpolation can also be used insteadof the cubic splines, but in that case we cannot properly recover the potential because these
Inverse spectral problem for the Sturm–Liouville equation 243
kinds of interpolation make the function very rough and so the ODE solver (used to calculateu and v functions) produces poor results.
In order to keep improving the potential when it is already close to the exact one we haveto run the ODE solver with the maximum possible precision. This is because the error due tothe numerical integration needs to be small relative to the functional, which itself will be smallin these circumstances. In order to get a higher precision when solving (1) we transform (1)by the modified Prufer transform:
u(x) = r(x)
ssin θ(x)
u′(x) = r(x)s cos θ(x)
s = λ1/4
(20)
which, when applied to (1), yields the pair of equations
θ ′ = √λ − q√
λsin2 θ,
r ′
r= s2 sin θ cos θ
(1 +
q − λ
s2
).
These new equations are used to solve (1) numerically.
2.5. Initial data
In [9] a formula is given for computing the multiplier constants from two known spectra.Adapted to our boundary conditions (22) and (23), it is
Cn = 1
B − H
∞∏k=0
(1 − λn
µk
)(21)
where the spectra {λn}∞n=0 and {µn}∞n=0 correspond to the boundary conditions (22) and (23),respectively:
y(0) + Ay ′(0) = 0, y(1) + By ′(1) = 0, (22)
y(0) + Ay ′(0) = 0, y(1) + H y ′(1) = 0. (23)
Note that the conditions at 0 for both problems are the same.The initial data are the triple:
(1) the boundary parameters A, B (see (3) and (4));(2) the size of the given spectrum N ;(3) the spectrum {λn}N
n=0 of the problem with boundary conditions (22) together withcorresponding multiplier constants {Cn}N
n=0.
The method allows computation with several input sets. This has numerical advantagesespecially in cases when the input sets contain noise. In these cases there are several schemesfor the part of the algorithm responsible for calculating the gradient and minimization ofF(α) = G(q + αh).
Suppose we have M spectra available (each of them can have a different number ofeigenvalues) where the mth spectrum has Nm eigenvalues {λm,n}Nm
n=0 with multiplier constants{Cm,n}Nm
n=0.We have two schemes for using several sets of initial data in recovering a single potential.
• Scheme 1. Calculate gradients corresponding to each given spectrum separately and thenuse their sum to produce a composite gradient for the minimization of F(α).
• Scheme 2. For each spectrum calculate the gradient and minimize F(α). Then select thebest result.
244 B M Brown et al
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
0.2 0.4 0.6 0.8 1
1
2
3
4
0.2 0.4 0.6 0.8 1
0.5
1
1.5
2
Figure 1. True potentials.
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
0.2 0.4 0.6 0.8 1
1
2
3
4
0.2 0.4 0.6 0.8 1
0.5
1
1.5
2
Figure 2. Recovered potentials.
When using several different sets of initial data, the result is better than when using onlyone of the sets; also, if the given spectra include noise, the recovery process is much morestable when using several spectra as initial data.
3. Some examples
The program recovered a great number of the potentials to high accuracy. We shall discuss therecovery of the following three problems (see figure 1):
−35.2x2 + 17.6x, 0 � x < 0.2535.2x2 − 35.2x + 8.8, 0.25 � x < 0.75−32.5x2 + 52.8x − 17.6, 0.75 � x � 1.
These have been chosen for comparison with results in [17].
The recovered potentials using the L2 gradient can be seen in figure 2. We shall discussalternative gradients in section 4.
Inverse spectral problem for the Sturm–Liouville equation 245
20 40 60 80 100
-0.0001-0.000075
-0.00005-0.000025
0.0000250.00005
0.0000750.0001
20 40 60 80 100
-0.04
-0.02
0.02
0.04
20 40 60 80 100
-0.004
-0.002
0.002
0.004
Figure 3. Eigenvalue asymptotics.
3.1. Comparison with other methods
In [17], Rundell and Sacks present a method based on results of Gel’fand and Levitan. Thismethod requires knowledge of
∫ 10 q(t) dt (which can be obtained from the spectrum, but since
we only deal with a finite number of eigenvalues, there will be an error in this value givingerrors in the recovered potential). Knowing the exact mean value they manage to recovera very smooth potential using only a few eigenvalues, but in practice one usually does notknow the mean value and can only obtain it with some error from the spectrum. In [10, p 67]and [5, p 299], (1.14) asymptotic expansions of eigenvalues of the Sturm–Liouville problemare proved. Using only the first term of the series and the boundary conditions (2) we get
λn = π2n2 + 2
(1
B− 1
A
)+
∫ 1
0q(t) dt + O
(1
n2
), A, B �= 0 (24)
and this is used in Rundell and Sacks’s method to estimate∫ 1
0 q(t) dt . Note that similar resultscan be contained if either A or B is zero (e.g. if A = B = 0 then
λn = (n + 1)2π2 +∫ 1
0q(t) dt + O
(1
n2
),
see Naimark [12, vol 1]).In figure 3 the difference
λn −(
π2n2 + 2
(1
B− 1
A
)+
∫ 1
0q(t) dt
)
can be seen between eigenvalues of the three considered problems and their asymptotic values.
4. Using different gradients
We now discuss analogous results obtained using both L∞ and Sobolev type gradients. Theexperience of the authors in numerical experiments is that the L2 gradient produces thebest results when recovering both continuous and discontinuous potentials. The recoveredpotentials in the previous section were recovered using the L2 gradient.
4.1. L2 gradient
The results in this section (and those in section 3) are found using the L2 gradient. Someadditional data on some potentials recovered with the L2 gradient can be seen in figure 4.
The question arises as to how the error in the eigenvalues due to noise is to be measured.We choose to use the absolute (and not relative) error for this. Suppose our algorithm hadintroduced an O(ε
√n) perturbation into the recovered λn , giving incorrect large n eigenvalue
246 B M Brown et al
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
0.2 0.4 0.6 0.8 1
1
2
3
4
0.2 0.4 0.6 0.8 1
0.5
1
1.5
2
2.5
Figure 4. Potential recovered using the L2 gradient.
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
0.2 0.4 0.6 0.8 1
1
2
3
4
0.2 0.4 0.6 0.8 1
0.5
1
1.5
2
2.5
Figure 5. Potential recovered using the L∞ gradient.
asymptotics: the l2 relative eigenvalue error in this case would still be O(ε), giving a falseimpression of accuracy. Further, since the error in the potential is bounded by sup(q − q) wechoose to measure all errors in the supremum norm supM
i=0 |λi − λi |.
4.2. L∞ gradient
We define the L∞ gradient as in [8]:
∇L∞ G(q)(x) =
1, H (x) > 0
−1, H (x) < 0
0, H (x) = 0
(25)
where H is the L2 gradient.A few special cases when the L∞ gradient produces the best potential are when the exact
potential is a constant or a shift (piecewise constant with two pieces). In that case (assumingthere is no noise in the given initial data) we recover the exact potential in just a few iterations.
In other cases the L∞ gradient produces many small artificial oscillations in the potential.From the numerical evidence one can see that in most cases the L∞ gradient is not suitable forthis type of problem. Potentials recovered only using L∞ gradient are shown in figure 5.
4.3. Sobolev type gradient
We first discuss these gradients. Because of its smoothness, the Sobolev type gradient (see [13])is sometimes used to avoid oscillatory noise in the recovered potential.
Inverse spectral problem for the Sturm–Liouville equation 247
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
0.2 0.4 0.6 0.8 1
1
2
3
4
0.2 0.4 0.6 0.8 1
0.5
1
1.5
2
2.5
Figure 6. Potential recovered using the Sobolev type gradient.
Let
H1 = {g : g, g′ ∈ L2[0, 1]} (26)
where g(1) is the distributional derivative of g, since H1 is a subspace of the absolutelycontinuous functions AC[0, 1]. Suppose that we can write G ′(q)[h] = ∫ 1
0 hH dx , whereH is the L2 gradient and that this in turn can be written as G ′(q)[h] = (h,H)H1 :
( f, g)H1 =∫
( f g + f ′g′) dx .
Thus, we can find such an H from∫ 1
0(hH + h′H′
) dx =∫ 1
0hH dx
for all h ∈ H1. This is possible if
−∫ 1
0hH′′ dx + (hH′
)
∣∣∣∣1
0
+∫ 1
0hH dx =
∫ 1
0Hh dx .
Now, if we take H′(0) = H′(1) = 0 then we have∫ 1
0h(−H′′ + H) dx =
∫ 1
0Hh dx .
So we have a formula for computing the Sobolev type gradient from the known L2 gradient.H is the solution of −H′′ + H = H, H′(0) = H′(1) = 0. Both the weakness and thestrength of the Sobolev type gradient is that it is very smooth. Thus it is usually hard to recoverdiscontinuous or non-smooth potentials using the Sobolev gradient. Our experiments showthat only in the case of very smooth potentials (for example, sin( x
10 )) or when noise is presentin the initial data does the Sobolev type gradient sometimes recover the potential better thanthe L2 gradient. Figure 6 shows what can be recovered in the three considered problems usingonly the Sobolev gradient.
4.4. Comparing different gradients
When we recover the potential for the first problem using only one type of gradient we get theresults in figure 7, where G Dav , G Dmin and G Dmax are correspondingly the average, minimumand maximum values of G(Qn)
G(Qn+1)in the iterative process.
248 B M Brown et al
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
Figure 7. Potential recovered using different gradients.
4.5. Noise in initial data
In order to model the effect of noise on our results, we first present the following theorem thatquantifies the effect of changing the potential Q of (1) into Q. Given two potentials Q, Q of(1) generating eigenvalues λn , λn , respectively, then we shall show that, as n → ∞,∣∣∣∣λn − λn −
∫ 1
0(q − q) dx
∣∣∣∣ < O
(1
n
). (27)
Thus if we are comparing two sets of spectra, one without noise and the other with it, albeit thatthe latter is well modelled by the regular Sturm–Liouville problem, then they must satisfy (27).Our result is contained in the following theorem.
Theorem. For all q(x), q(x) on [0, 1], let {λn}∞n=0 and {λn}∞n=0 be the eigenvalues of theproblem (1) with Q(x) = q(x) and Q(x) = q(x), respectively.
Then
|λn − λn| =∣∣∣∣∫ 1
0(q(s) − q(s)) ds
∣∣∣∣ + O
(1
n
), n → ∞.
Proof. Since the asymptotic form of the eigenvalues is
λn = n2π2 + 2
(1
B− 1
A
)+
∫ 1
0q(s) ds + an (28)
where the sequence an = O( 1n ), (see p 299, (1.14) of [5]).
Thus
limn→∞ an = 0
and by definition, for all ε > 0, there exists an N such that for all n � N , |an| < ε and so, forall ε > 0, there exists a N such that for all n � N , |an − an| < ε.
On the other hand,
|λn − λn| =∣∣∣∣∫ 1
0q(s) ds −
∫ 1
0q(s) ds + an − an
∣∣∣∣.�
We note that, given two finite sets of real numbers {λn}Nn=0 and {Cn}N
n=0 which representspectral data with perceived noise, in general we do not know if there exists such a potentialQ which gives rise to these data, i.e. if they are the spectra of some Sturm–Liouville problem.
Inverse spectral problem for the Sturm–Liouville equation 249
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
0.2 0.4 0.6 0.8 1
1
2
3
4
0.2 0.4 0.6 0.8 1
0.5
1
1.5
2
2.5
Figure 8. Potential recovered from the perturbed data.
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
0.2 0.4 0.6 0.8 1
1
2
3
4
0.2 0.4 0.6 0.8 1
0.5
1
1.5
2
2.5
Figure 9. Potential recovered from multiple perturbed spectra.
We can only say that, if a given spectrum does not follow the asymptotic form of theeigenvalues (28), then there is no potential of a regular Sturm–Liouville problem from whichthat spectrum can be recovered. For example, if we add noise to the spectrum, directlyproportional to the eigenvalues, i.e. λn = λn + Kλn , n = 0, 1, . . . , N , then this perturbedspectrum {λn}N
n=0 does not follow the asymptotic form of the eigenvalues. We therefore testedour method on initial data with the absolute noise.
Figure 8 shows the potentials recovered from the initial data with an absolute noise 0.01.We also perturb the multiplier constants with the same noise. All results in this figure wereproduced using the L2 gradient.
In figure 9 one can see the case when the Sobolev type gradient produces a better resultthan the L2 gradient. This happens because the initial data are perturbed and the potentialrecovered with the L2 gradient has a number of small oscillations, which are smoothed bythe Sobolev type gradient. Also we use five different spectra and sets of multiplier constants,ten eigenvalues and ten constants in each set. This demonstrates that we can obtain resultsusing only ten eigenvalues, but using several spectra. In this sample all three potentials wererecovered in a small number of iterations. Although the program was run for more than100 iterations, after a certain number of iterations, the eigenvalue difference with the initialspectrum started to increase. Thus our regularization criterion chose the intermediate result.
5. Conclusions
• In practice, when using a spectrum and multiplier constants with noise added, the algorithmtends to recover a potential that is broadly similar to one having the spectrum and multiplierconstants without the noise.
• Also, in practice, when using exact initial data, the recovered potential gets closer to theexact potential after each iteration. By contrast, when using perturbed initial data, aftersome iteration the difference between the exact potential and the recovered one increases,
250 B M Brown et al
and much oscillatory noise appears in the recovered potential. In such a case the Sobolevtype gradient may be used to smooth the potential and thus to get rid of the oscillatorynoise. Again the method only works with smooth potentials.
• The algorithm is very sensitive to the error in the spectrum and multiplier constants used. Ifa piecewise constant potential is considered and thus eigenvalues and multiplier constantscan be calculated in closed form, the algorithm converges to a potential very close to theexact one (as far as is possible considering that we are using interpolation using cubicsplines and a piecewise constant potential cannot be exactly represented with such aninterpolation).
• The potential can be recovered quite well using NAG routines with the highest possibleprecision for calculating the spectrum and multiplier constants.
• In general, the L2 gradient produces the best results.• Even in the presence of noise, the algorithm recovers the shape of the exact potential.• It may make sense to specify discontinuity points to the algorithm, so it will not use cubic
spline interpolation at those points, thus making the potential discontinuous at given points(this requires knowing in advance at which points the potential is not continuous).
Appendix
We investigate convergence of the formal sum
Gn(q) =∞∑
n=0
∫ 1
0(u′ − Cnv
′)2 + q(u − Cnv)2, (29)
and show that, unless Dirichlet conditions are imposed, Gn(q) diverges.
Lemma A.1. Let y = yq(x, λ) be a solution of (1),with q replacing Q, for which yq(c, λ) = Eand y ′
q(c, λ) = F, 0 � c � 1, and E and F are independent of λ. Let s = √λ = σ + it , where
s is real when λ > 0 and t is always non-negative. Let s0 > 0 be given. Then for |s| > s0
when E = 0. Formal differentiation with respect to x is permissible, the new order terms being
O(et|x−c|/|s|), O(et|x−c|/|λ|), (32)
respectively (see [18, theorem 5.1]).
For brevity, we fix n, assume A �= 0, and set s = √λ, where λ = λQ
n ; as the eigenvaluesλQ
n are real and eventually positive, we may take t = 0 in the following. For a given q we have
uq,n(x, λ) = A cos sx − 1
ssin sx +
A
s
∫ x
0q(ω) cos(s(x − ω)) cos sω dω + O(1/λ) (33)
and
u′q,n(x, λ) = −As sin sx − cos sx + A
∫ x
0q(ω) cos(s(x − ω)) cos sω dω + O(1/s), (34)
Inverse spectral problem for the Sturm–Liouville equation 251
and a similar result for vq,n(x, λ) and v′q,n(x, λ).
Setting x = 1 in (5) we have that uQ,n(1, λ) = Cn B , so that from (33)
Cn = 1
B
(A cos s − sin s
s+
A
s
∫ 1
0Q(ω) sin s(1 − ω) cos sω dω
)+ O(1/s2). (35)
On differentiating (5) and setting x = 1 again, we see that u′Q,n(1, λ) = −Cn so that from (34)
Cn = As sin s + cos s − A cos s∫ 1
0Q(ω) cos2 sω dω
− A sin s∫ 1
0Q(ω) sin sω cos sω dω + O(1/s). (36)
Consequently, equating (35) and (36) we find that
A sin s = 1
s
(A
B− 1
)cos s +
A
scos s
∫ 1
0Q(ω) cos2 sω dω + O(1/s2). (37)
In particular, note that
sin s = O(1/s). (38)
Also, if we multiply the identity (37) by cos s, and use cos2 s = 1 − sin2 s togetherwith (38) to move all terms containing sin2 s into the O(1/s) term, we obtain
A
B− 1 = As cos s sin s − A
2
∫ 1
0Q(ω) dω + O(1/s), (39)
provided that Q is of bounded variation, so that by [20, p 426]∫ 1
0Q(ω) cos 2sω dω = O(1/s). (40)
Lemma A.2. Assuming that q and Q are of bounded variation on [0, 1], and setting s2 = λ =λQ
n , the functions uq,n(x, λ) and vq,n(x, λ) defined above satisfy
u′q,n(x, λ) − Cnv
′q,n(x, λ) = A
2
∫ 1
0(q − Q)(ω) dω cos sx + O(1/s). (41)
Proof. Noting that the A = 0 case follows directly from lemma A.1, we assume that A �= 0.Consequently, in the expression for v′
q,n(x, λ) analogous to (34), by the addition formulae forsine and cosine and (38) we can move into the O(1/s) term all terms resulting from the sineand cosine of s(x − 1) and s(ω − 1) that contain the factor sin s. We thus obtain
u′q,n(x, λ) − Cnvq,n(x, λ) =
(A
B− 1 − As cos s sin s
)cos sx +
A
2cos sx
∫ 1
0q(ω) dω
+A
2
∫ 1
0q(ω) cos 2sω dω +
A
2sin sx
∫ 1
0q(ω) sin 2sω dω + O(1/s) (42)
after using cos2 s = 1 − sin2 s and the double angle formulae. On substituting (39) into theabove, we arrive at
u′q,n(x, λ) − Cnv
′q,n(x, λ) =
(A
2
∫ 1
0(q − Q)(ω) dω
)cos sx + O
(1
s
); (43)
the proof is complete. �
252 B M Brown et al
One can also show that
uq,n(x, λ) − Cnvq,n(x, λ) = A
2s
∫ 1
0(q − Q)(ω) dω sin sx + O
(1
s2
). (44)
On combining these results one then has that, for functions q of bounded variation on [0, 1],
Gn(q) = A2
8
(∫ 1
0(q − Q)(ω) dω
)2
+ O
(1
n2
). (45)
It is also worth noting that if we define
Gn(q) =
∫ 1
0(u′
q,n − Cnv′q,n)
2 + (q(x) − λ)(uq,n − Cnvq,n)2 dx (46)
then after integrating by parts one has that
Gn(q) = [(u′
q,n − Cnv′q,n)(uq,n − Cnvq,n)]1
0,
and it follows from the asymptotic formulae that
Gn = O(1/n2).
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