Numerical solution of forward and inverseSturm–Liouville problems with an angularmomentum singularity
Lidia Aceto1, Paolo Ghelardoni1 and Marco Marletta2
1 Dipartimento di Matematica Applicata ‘U.Dini’, Universita di Pisa, Via F. Buonarroti 1/c,56127 Pisa, Italy2 School of Mathematics, Cardiff University, PO Box 926, Cardiff CF24 4YH, Wales, UK
Received 1 August 2007, in final form 18 October 2007Published 7 December 2007Online at stacks.iop.org/IP/24/015001
AbstractThis paper considers analytical and numerical-analytical issues encounteredin the solution of an inverse Sturm–Liouville problem with a Bessel-typesingularity. The main results are as follows: (i) there is a wide class ofnumerical methods for eigenvalue calculation which are not adversely affectedby this type of singularity; (ii) a simple least-squares technique proposed byRohrl in the regular case can actually be implemented in an even simpler way,and still work just as effectively; (iii) in some cases, a ‘unique local minimizer’result of the type obtained by Rohrl is still available; (iv) finite spectral dataand noise remain the most serious obstacles to accurate reconstruction.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
In this paper, we consider a numerical solution of forward and inverse problems for the singularSturm–Liouville equation
−y ′′ +
(q(x) +
m(m + 1)
x2
)y = λy, x ∈ (0, 1]. (1)
There are several versions of the problem. In the first version, one supposes that m is known andthat two spectra are known corresponding to different boundary conditions at x = 1; one wishesto determine the potential q ∈ L2[0, 1]. This is one of the problems for which the existence anduniqueness theory was considered by Carlson [7]. Carlson reduces the two-spectra problemto the second version in which one knows one spectrum and one set of the so-called normingconstants. While these two versions of the problem are mathematically equivalent, the secondis less amenable to a numerical solution because the accurate calculation of the norming
constants requires accurate calculation of the principal solution of (1) (the solution which is‘small’ near x = 0). This solution cannot be determined by initial values (y(0), y ′(0)) as, fornon-zero m(m + 1), this vector is either zero or infinite. The principal solution must thereforebe calculated by truncating the interval and imposing an approximate boundary condition, aprocedure which can result in an effective order reduction for the numerical integration. Inthe first version of the problem, this difficulty is avoided because we only require accurateeigenvalues rather than eigenfunctions. Using a simple argument we are able to show that anunexpectedly wide class of discretization methods are able to achieve the same accuracy forthe singular problem as for the regular one, at least for non-negative integer m. This is becausethe exact eigenfunctions are smooth enough for the classical order of a local truncation errorto be preserved.
The second version of the problem is not considered in this paper, although the interestedreader can find existence and uniqueness results for this case in [25].
A third version involves a fixed boundary condition at x = 1 and the knowledge of twospectra corresponding to different values of m. For this case Shubin Christ [23] proved that theintersection of the isospectral manifolds for the problems with m = 0 and m = 1 is compact,the problem with m = 0 being equipped with a Dirichlet boundary condition at the origin.Carlson and Shubin [9] and Rundell and Sacks [20] produced existence and uniqueness resultscovering a number of important cases. It seems to be widely believed that the problem shouldbe uniquely solvable knowing the spectra for two distinct values of m � 0, with the Dirichletboundary condition at x = 0 for the case m = 0. However, no published work yet treats sucha general case. In this paper we shall make the assumption that knowing the spectra for twointeger values, say m2 > m1 � 1, with a fixed self-adjoint boundary condition at x = 1, thefunction q ∈ L2(0, 1) is uniquely determined. We shall develop numerical methods for thiscase. The reason for the assumption that m1 and m2 are integers is that we need some resultsof Carlson [7] in order to establish that the functional which we use for the numerical recoveryis well defined.
Having listed the three different versions of the inverse problem for the Schrodingerequation with the Bessel-type singularity, we should also mention some other closely relatedwork. Very recently, Serier [22] has considered the ‘second version’ problem for (1) and hasshown that the eigenvalues and norming constants form a real analytic coordinate system forL2
R(0, 1) and that the isospectral sets form a smooth manifold with respect to this coordinate
system. Serier’s work is the generalization to general m of the work of Guillot and Ralston[11] for m = 1. Another interesting aspect of Serier [22] is the introduction of transformatoroperators. In the context of more general equations, Albeverio et al [1] have considered the‘second version’ problem for differential equations of the form
−(
d
dx− κ
x− ν(x)
) (d
dx+
κ
x+ ν(x)
),
where κ ∈ N and ν ∈ Lp(0, 1), p � 1, as well as for Dirac operators [2].Before proceeding with the rest of this paper, we should say something about the numerical
methods used for the inverse problem. There are several different approaches for the regularcase, in particular those of Rundell and Sacks [19], McCarthy and Rundell [15] and a methoddeveloped in [5] from an idea of Knowles [14]. However here we use an approach due toRohrl [18] which, it turns out, is also very popular in the chemical spectroscopy literatureand closely related to the Rydberg–Klein–Rees method. We chose it partly because for theregular case it appears to work just as well as the other approaches, and partly because wethought it would be possible to simplify the implementation quite substantially compared tothat described in [18] by an appropriate representation of the regular part of the potential.
We shall use certain notational conventions throughout this paper. For a Sturm–Liouvilleproblem (SLP for short) consisting of (1) together with a self-adjoint boundary condition atx = 1, the set of eigenvalues will be denoted as (λk)k∈N0 , where N0 = {0, 1, 2, . . .} is theset of non-negative integers. The indexing convention is that the eigenvalues are arranged inan ascending order, so that by the oscillation theory the eigenfunction associated with λk hasprecisely k zeros in the open interval (0, 1). (This is well known in the case when x = 0 is aregular endpoint and is also true in the singular case when, for every real λ, all solutions ofthe differential equation have finitely many zeros in (0, 1), as in the case here.) We shall notusually need an explicit notation for a boundary condition at x = 1; however any self-adjointboundary condition at x = 1 may be expressed in the form
cos(β)y(1) − sin(β)y ′(1) = 0,
for some β ∈ (0, π ].The layout of the rest of this paper is as follows. Section 2 considers the behaviour
of numerical methods for the singular forward problem, without which there is no hope oftackling the inverse problem. Section 3 analyses a Rohrl-type functional for our inverseproblem, using some ideas of Carlson [7]. Section 4 presents numerical results on the inverseproblem.
2. Discretization schemes for the singular eigenvalue problem, eigenvalue andeigenfunction convergence
2.1. Preliminaries
We shall study some different discretization schemes.
First-type schemes. By ‘first-type schemes’, we mean schemes which replace the Sturm–Liouville problem by a matrix eigenvalue problem
Ay = λy,
in which A is Hermitian.
Second-type schemes. By ‘second-type schemes’, we mean schemes which replace theSturm–Liouville problem by a matrix pencil
Ay = λBy,
in which, in general, neither A nor B is Hermitian.As pointed out by Keller [13] in a slightly different context, the quality of eigenvalue
approximations obtained from the first approach will depend only on the local truncationerror, thanks to the following result, which is an immediate consequence of the spectraltheorem for self-adjoint operators.
Lemma 2.1. Let A be a Hermitian matrix, ε > 0 a real number. Suppose that there existsa vector y, ‖y‖2 = 1, where ‖ · ‖2 is the Euclidean norm, and a real number λ such thatthe vector τ := Ay − λy satisfies ‖τ‖2 < ε. Then A possesses an eigenvalue µ such that|µ − λ| � ‖τ‖2 < ε.
For more general methods from the second-type schemes this approach is not availableexcept in some special cases, for instance if A is Hermitian and B is Hermitian and positive
definite. Otherwise, assuming that B is invertible, we use a less elegant local analysis basedon the matrix M = B−1A. For these cases, we shall need the following two lemmas.
Lemma 2.2. Let y be any unit vector and λ any (real or complex) number. Define a vector τ
by My − λy = τ. Then there exists a matrix E with ‖E‖ � ‖τ‖ such that λ is an eigenvalueof M + E.
Proof. If λ is an eigenvalue of M, then take E = 0 and the result is proved. Otherwise,since y = (M − λI)−1τ has norm 1, it follows that ‖(M − λI)−1‖ � ‖τ‖−1, and so λ liesin the ‖τ‖-pseudospectrum of M. The result then follows from one of the definitions of thepseudospectrum—see, e.g., [24]—or by the simple observation that one can take E to be therank-1 matrix given by E = −τy∗ or by E = −τyT if y has real components. �
Lemma 2.3. Let M(s) = M + sE, s ∈ [0, 1]. Suppose that all the eigenvalues of M(s)
are simple for all s ∈ [0, 1]. Let µ0 be an eigenvalue of M(0) with the right eigenvector x0
and left eigenvector z0. Then there exist differentiable functions x(s), µ(s) and z(s) such thatM(s)x(s) = µ(s)x(s), z(s)T M(s) = µ(s)z(s)T , x(0) = x0, z(0) = z0, µ(0) = µ0, and
µ′(s) = z(s)T Ex(s)
z(s)T x(s).
Proof. This result is a particular instance of more general results which may be found in [12,Chapter 2, section 1]. �
2.2. Two simple standard methods
The two simplest standard methods for discretizing the problem
where B is the (N − 1) × (N − 1) tridiagonal matrix:
B = 1
12
10 1 0 · · · · · · 0
1 10 1. . .
...
0 1. . .
. . .. . .
...
.... . .
. . .. . . 1 0
.... . . 1 10 1
0 . . . . . . 0 1 10
.
The classical orders of accuracy for these schemes (i.e. the orders achieved on problems withsmooth coefficients) are O(h2) for the standard three-point scheme (3) and O(h4) for theNumerov scheme (4).
It is clear that lemma 2.1 can be applied to the standard three-point scheme (3) by choosingA = −h−2T + Q. However it is also possible, in fact, to apply lemma 2.1 to the Numerovscheme, thanks to the following observation.
Proposition 2.4. The Numerov scheme is equivalent to a ‘first-type scheme’ with a realsymmetric matrix
A = −h−2B−1T + Q.
Proof. It is clear that the eigenvalues of the Numerov scheme are precisely the eigenvalues ofA. The fact that A is real symmetric follows from the observation that B and T are symmetricand commute, so that B−1T is symmetric. �
Thus, an analysis of the accuracy of the Numerov scheme only requires attention to thelocal truncation error. We deal with this in some detail, partly in order to establish the notationto be used later.
2.2.1. The regular case. Suppose that the function p in (2) is smooth enough to achieve aclassical order of accuracy O(h4) for the Numerov method. More precisely, suppose that y isthe true eigenfunction, with the true eigenvalue λ, which is approximated by the eigenvectorY with the eigenvalue µ. Let y be the vector of values of y at the meshpoints. The localtruncation error τ is usually defined by
τ = −h−2T y + BQy − λBy
and is O(h4) either when measured pointwise or when measured with respect to the scaledL2-norm associated with the inner product
〈f, g〉h :=N−1∑j=1
hfigi, (5)
in an obvious notation. We want to show that |λ − µ| � O(h4).
Following our earlier observations, we can write
τB = Ayh − λyh,
where yh = √hy has a Euclidean norm of order 1 (because 〈f, f〉h = ‖√hf‖2
2 for any vectorf) and where τB = B−1
√hτ. The vector
√hτ also has a Euclidean norm of order O(h4). The
matrix B−1 has bounded inverse; in fact it may be shown that
‖B−1‖ � 32
5
Inverse Problems 24 (2008) 015001 L Aceto et al
independent of h. It follows that τB has a Euclidean norm of order O(h4). By an applicationof lemma 2.1, we therefore immediately obtain that the matrix A = −h−2B−1T + Q has aneigenvalue µ such that |λ − µ| � ‖τB‖2 = O(h4).
2.2.2. The singular case. Consider now the case
p(x) = q(x) +m(m + 1)
x2, (6)
in which q is smooth and m is a positive integer. In this case, the boundary condition y(0) = 0selects the unique solution of (2) such that y(x) ∼ Cxm+1.
Consider first the classical three-point scheme (3). This scheme never requires the valueof p(0) and so it is still well defined.
Suppose that y is an eigenfunction with the eigenvalue µ. Let τ be the local truncationerror defined by
τ = −h−2T y + Qy − λy,
where Q = diag(p(h), p(2h), . . . , p(1 −h)). Because y satisfies the differential equation (2)with λ = µ, we have
τ = −h−2T y + y′′,
where y′′ denotes the vector of values of y ′′ at the meshpoints. Because m is a positive integer,y is analytic at 0, and so one still has, in terms of the norm associated with the inner productin (5), the estimate ‖τ‖h � O(h2), which is the classical order of accuracy. Consequently, inspite of the singularity, by lemma 2.1 and our previous reasoning there still exists an eigenvalueµ of −h−2T + Q such that |λ − µ| � O(h2).
Proposition 2.5. The classical three-point scheme (3) applied to the singular problem (2) withp(x) = q(x) + m(m + 1)/x2 and m ∈ N yields O(h2) accurate eigenvalue approximations.
Note that if m is not an integer, then y is no longer entire and so a loss of accuracy mayoccur, particularly for small m. Some other schemes, which we now discuss, exhibit possibleloss of accuracy in a more interesting way.
Consider the Numerov scheme, which we write explicitly as
− 1
h2(Yn+1 − 2Yn + Yn−1) +
1
12(p((n + 1)h)Yn+1 + 10p(nh)Yn + p((n − 1)h)Yn−1)
= µ
12(Yn+1 + 10Yn + Yn−1), n = 1, 3, . . . , N − 1.
When n = N − 1 the terms involving YN are zero, to discretize the condition y(1) = 0. Whenn = 1 the term Y0 is zero; however p(0) is undefined. The Numerov scheme in the form (4)takes p(0)Y0 to be zero.
The exact solutions of (2) satisfy y(x) ∼ Cxm+1 and so for m � 2, one has
limx→0
p(x)y(x) = 0.
The Numerov scheme in the form (4) therefore suffers no loss of order for integer m � 2.
Proposition 2.6. The Numerov scheme (4) applied to the problem (2) with p(x) = q(x) +m(m + 1)/x2, q smooth, m ∈ N,m � 2, yields O(h4) accurate eigenvalue approximations.
In the case m = 1, limx→0 p(x)y(x) = 0. Using the Numerov scheme in the form (4) cantherefore be expected to result in a loss of order of accuracy for the case m = 1, but not forany other non-negative integer!
Example. Consider two SLPs of the form (2) with p(x) as in (6) where we have fixedq(x) = sin x and q(x) = x2 − x3, respectively.
The order of accuracy for the eigenvalue approximations, obtained using the Numerovmethod, is estimated by the quantity
Rλ = log2
( |λ(h) − λ(h/2)||λ(h/2) − λ(h/4)|
);
here, λ(h) denotes the numerical value of λ computed considering a discretization having auniform stepsize h.
In tables 1 and 2 we report the rate corresponding to λk, k = 1, 2, . . . , 5, with m = 1, 2and h = 1
50 for both the SLPs considered.
2.3. Non-symmetric methods
We consider a discretization of the differential equation
−y ′′ +
(q(x) +
m(m + 1)
x2
)y = λy, y(0) = 0 = y(1) (7)
of the form
− 1
h2AY + BQY = λBY, (8)
in which the stepsize h = 1N
is uniform and A,B are (N − 1) × (N − 1) matrices.If λ is an eigenvalue and y is the corresponding eigenfunction of (7), then we define a
vector y by
yj = y(jh), j = 1, 2, . . . , N − 1,
and we define the local truncation error τ by
− 1
h2Ay + BQy = λBy + τ.
More generally, if y is any twice continuously differentiable function with y(0) = 0 = y(1)
and f := −y ′′ + (q(x) + (m(m + 1))/x2)y ∈ L2[0, 1] then defining a vector f by
fj = f (jh), j = 1, 2, . . . , N − 1,
we can define the local truncation error τ by
− 1
h2Ay + BQy = Bf + τ.
7
Inverse Problems 24 (2008) 015001 L Aceto et al
Assuming that B is invertible, the matrix Mh whose eigenvalues are supposed to approximatethose of (7) is
Mh = − 1
h2B−1A + Q.
We shall assume that Mh is diagonalizable. The original problem has only simple eigenvalues,so it seems appropriate to reject out of hand any scheme for which Mh does not have simpleeigenvalues.
Theorem 2.7. Let Vh be the matrix of eigenvectors of Mh normalized to be of unit length. Letλ, y, y and τ be as above. Then
infµ∈σ(Mh)
|µ − λ| � cond(Vh)‖B−1‖‖τ‖‖y‖ .
Proof. Let h be the diagonal matrix of eigenvalues of Mh so that Mh = VhhV−1h . The
definition of τ immediately yields
B−1τ = (Mh − λI)y = (VhhV−1h − λI)y,
and hence
V −1h B−1τ = (h − λI)V −1
h y.
Since h is diagonal and since the spectra σ(h) and σ(Mh) coincide, we have
‖V −1h B−1τ‖ � inf
µ∈σ(Mh)|µ − λ|‖V −1
h y‖.
But ‖y‖ � ‖Vh‖‖V −1h y‖ and so
‖V −1h B−1τ‖ � inf
µ∈σ(Mh)|µ − λ|‖Vh‖−1‖y‖,
which yields
infµ∈σ(Mh)
|µ − λ| � ‖Vh‖‖V −1h B−1τ‖
‖y‖ .
The result follows since cond(Vh) = ‖Vh‖‖V −1h ‖. �
Remark 1. This result shows that if the condition number of the matrices Vh can be boundedindependently of h, and if the matrices B have the property that ‖B−1‖ is boundedly invertibleindependently of h, then any eigenvalue λ of the original singular problem will be approximatedwith an error which depends only on the local truncation error associated with the method andwith the exact eigenfunction. In short, one may then expect to see the same performance asfor the three-point scheme and Numerov scheme.
In practice many linear multistep methods, particularly boundary value methods (BVMs)[6], are sufficiently complicated to make it rather difficult to prove a priori bounds on cond(Vh),
for instance. However this need not be a problem if one is interested in some fixed finite h,
as cond(Vh) can then be estimated numerically. In all the examples which we tried, cond(Vh)
always seemed to be bounded.
As an example, we report the results obtained using two BVMs introduced in [3]generalizing the three-point method and the Numerov method, respectively. The first BVM isa fourth-order method given by the formulae1
12 (−11y0 + 20y1 − 6y2 − 4y3 + y4) = −h2f1
112 (yn − 16yn+1 + 30yn+2 − 16yn+3 + yn+4) = −h2fn+2, n = 0, 1, . . . , N − 4,
In figure 1 the condition number of the corresponding matrix of eigenvectors Vh, computed ash varies, is plotted for the methods (9) and (10).
We now consider a ‘local’ result which does not require an estimation of the conditionnumber of the whole matrix Vh. The basic idea behind this result is to obtain a usable errorbound in the spirit of lemma 2.3.
Theorem 2.8. Let y, λ and τ be as in theorem 2.7, so that (by lemma 2.2) y is an exacteigenfunction and λ is an exact eigenvalue, for the matrix Mh,τ = Mh − (τy∗)/‖y‖2
2. Let λh bethe eigenvalue of Mh which approximates λ (a priori, possibly with a poor order of accuracy)and let Yh be the left eigenvector of Mh corresponding to λh. Then, with ‖ · ‖ denoting thenorm associated with the inner product 〈·, ·〉, which may be either the usual Euclidean innerproduct or the weighted inner product introduced in (5),
|λ − λh| � ‖τ‖‖Yh‖|〈Yh, y〉| . (11)
Proof. For simplicity, we give the proof using the usual Euclidean inner product. The case of〈·, ·〉h then follows immediately as it simply involves rescaling the numerator and denominatoron the right-hand side of (11). Observe that
Y∗hMh = λhY∗
h,
9
Inverse Problems 24 (2008) 015001 L Aceto et al
0 500 1000 15001.66
1.68
1.7
1.72
1.74
1.76
1.78
N
cond(V
h)
0 500 1000 15001.134
1.136
1.138
1.14
1.142
1.144
1.146
1.148
1.15
N
cond(V
h)
Figure 1. The condition number of Vh as N varies for methods (9) (left) and (10) (right).
(Mh − τy∗
‖y‖22
)y = λy.
Multiplying the second equation on the left by Y∗h yields
Y∗hMhy − Y∗
hτ = λY∗hy;
multiplying the first equation on the right by y yields
Y∗hMhy = λhY∗
hy.
Subtracting yields
(λh − λ)Y∗hy = Y∗
hτ.
The result is immediate. �
Remark 2. The reader may object that this result is not usable, since it involves the vector yof values at the meshpoints of the unknown eigenfunction. However we do at least know apriori that this vector will be approximated, though perhaps with rather poor accuracy, by theright eigenvector yh of Mh corresponding to λh. As long as the scalar product 〈Yh, yh〉 doesnot tend to zero with h, the denominator in (11) can be estimated, and the local truncationerror τ will determine the order of accuracy, just as in the symmetric case.
To confirm the previous remark, in figure 2 we report, for some values of i, the trend asN → +∞ (h → 0) of the scalar products 〈Yi , yi〉; Yi and yi are the ith approximated left andright eigenfunctions, respectively, computed using methods (9) and (10). We chose m = 1and q(x) = sin(x) for these experiments.
This concludes our discussion of numerical methods for the forward problem with anangular momentum singularity m(m + 1)/x2. Our aim is to apply these methods to the inverseproblem. The algorithm for the inverse problem is described in the following section.
3. Rohrl’s method: calculation of the functional and its gradient
In [18], Rohrl proposed a very simple functional to minimize in order to determine the potentialq, in the regular case m = 0 where the two spectra correspond to different choices of boundaryconditions.
We describe the analogous functional here, establish some of its properties and mentionsome of its advantages in the context of a singular problem.
10
Inverse Problems 24 (2008) 015001 L Aceto et al
0 100 200 3000.975
0.98
0.985
0.99
0.995
1
1.005
N
< Yi,y i >
0 100 200 3000.998
0.9985
0.999
0.9995
1
1.0005
< Yi,y i >
N
i=0i=1i=2i=6i=9
i=0i=1i=2i=6i=9
Figure 2. Scalar products 〈Yi , yi〉 for methods (9) (left) and (10) (right).
3.1. Knowing spectra for two distinct positive integers m and
This is the more difficult case, so we describe it first. Let λk(q,m) denote the kth eigenvaluefor the problem consisting of (1) together with a fixed self-adjoint boundary condition atx = 1, say
cos(β)y(1) − sin(β)y ′(1) = 0 (12)
for some β ∈ (0, π ]. Suppose that we know the sequences (λk(q,m))k∈N0 and (λk(q, ))k∈N0 ,
for some distinct positive integers m and . Given any function Q ∈ L2[0, 1] having the samemean value as q, i.e.∫ 1
0Q =
∫ 1
0q, (13)
and any bounded sequences (ωk,m)k∈N0 and (ωk,)k∈N0 of strictly positive numbers, we(formally) define
G(Q) =∞∑
k=0
ωk,m|λk(Q,m) − λk(q,m)|2 +∞∑
k=0
ωk,|λk(Q, ) − λk(q, )|2. (14)
Remark 3. The purpose of the ‘same mean value’ hypothesis (13) is to ensure that the sumsconverge, as our next theorem shows. It is unclear how one might determine the mean valuefrom the spectral data. We shall fill this gap in corollary 3.2, after the following theorem.
Theorem 3.1. Suppose that the spectra (λk(q,m))k∈N0 and (λk(q, ))k∈N0 ,m > � 1,
uniquely determine q. Then the functional G is well defined and has a unique global minimizerwhen Q = q.
Proof. If G is well defined (i.e. the sums converge) then the fact that it has a unique globalminimizer is immediate from the unique solvability hypothesis, together with the fact that allthe weights in the sum defining G are strictly positive.
It therefore remains only to show that G is well defined. Since the weights ωk,m andωk, are bounded, this will be true provided the sequences (λk(Q,m) − λk(q,m))k∈N0 and(λk(Q, ) − λk(q, ))k∈N0 lie in 2(N0).
The proof of this result can be recovered with a little work from the 1993 results ofCarlson [7]. It is necessary to consider separately the following cases: (m even, Dirichlet
11
Inverse Problems 24 (2008) 015001 L Aceto et al
conditions at 1), (m odd, Dirichlet conditions at 1), (m even, non-Dirichlet conditions at 1), (modd, non-Dirichlet conditions at 1). However the analysis in each case is similar, so we treatjust one case: m = 2N > 0 even and Dirichlet conditions y(1) = 0.
Consider the eigenvalues λk(q,m) for m ∈ N fixed. Theorem 1.1 of Carlson states that
λk(q,m) = (k + 1 + m/2)2π2 + C(q) + rk, (15)
where (rk)k∈N0 ∈ 2(N0).
To establish that (λk(Q,m) − λk(q,m))k∈N0 and (λk(Q, ) − λk(q, ))k∈N0 lie in 2(N0),we therefore need to establish that C(Q) = C(q). In the regular case (m = 0), it is wellknown that C(q) is the mean value of q:
C(q) =∫ 1
0q,
and so the fact that q and Q are assumed to have the same mean value yields the result. Nowwe must consider the case m = 2N > 0. In this case, Carlson shows that there exists afunction q0 with the following properties.
• The spectrum of the Dirichlet (at both ends) problem with potential q0 and m = 0 isrelated to the spectrum of the problem with m = 2N and potential q by the formula
λk(q, 2N) = λk+N(q0, 0), k ∈ N0. (16)
(Lemma 5.1 of [7].) In view of the known asymptotics of Dirichlet eigenvalues of regularproblems—put m = 0 in (15) and replace k by k + N—it follows that
λk+N(q0, 0) = (k + N + 1)2π2 +∫ 1
0q0 + ρk+N, (17)
(ρk)k∈N0 ∈ 2(N0), and so
C(q) =∫ 1
0q0.
• Our potential q is the mth function qm generated by the recurrence relation
qj+1 = H0(qj ),
in which the mapping H0 is given by
H0(qj ) = 2λ0(qj , j) − qj + 2
(y ′
0
y0
)2
− 2(j + 1)2
x2, (18)
where y0 is the zeroth eigenfunction for the problem
−y ′′ +
(qj (x) +
j (j + 1)
x2
)y = λy, y ′(1) = 0.
(Note the Neumann boundary condition at x = 1.) The term
2
(y ′
0
y0
)2
− 2(j + 1)2
x2
appearing in (18) lies in L2[0, 1] by Theorem 4.2 of [7].
Of course we can repeat the same argument with q replaced by Q: for some Q0 we haveC(Q) = ∫ 1
0 Q0 and Q = Qm, where Qj+1 = H0(Qj ). Thus to prove that C(q) = C(Q), it issufficient to show that∫ 1
0Q0 =
∫ 1
0q0.
12
Inverse Problems 24 (2008) 015001 L Aceto et al
The proof proceeds by establishing that∫ 1
0q =
∫ 1
0q0 + m(m + 1),
∫ 1
0Q =
∫ 1
0Q0 + m(m + 1). (19)
Since we have assumed that Q and q have the same mean value this means that∫ 1
0 q0 = ∫ 10 Q0,
in other words that C(q) = C(Q), as required.We now address (19). Some simple algebra shows that(
y ′0
y0
)2
= y ′′0
y0−
(y ′
0
y0
)′,
and so
H0(qj ) = 2λ0(qj , j) − qj + 2
(y ′′
0
y0−
(y ′
0
y0
)′)− 2
(j + 1)2
x2
= 2λ0(qj , j) − qj + 2
(qj − λ0(qj , j) +
j (j + 1)
x2−
(y ′
0
y0
)′)− 2
(j + 1)2
x2
= qj − 2(j + 1)
x2− 2
(y ′
0
y0
)′.
According to Carlson [7, p 132],
y ′0(x)
y0(x)= j + 1
x(1 + xγ (x)) ,
where γ ∈ C[0, 1]; from [7, p 133],(y ′
0(x)
y0(x)
)2
=(
j + 1
x
)2
+ r(x),
where r ∈ L2[0, 1]. However squaring the first expression shows that r(x) = 2(j +1)2x−1γ (x) + (j + 1)2γ (x)2, from which it follows that γ (0) = 0 is a necessary condition forr ∈ L2[0, 1].
Integrating the expression for H0(qj ) over (ε, 1] yields∫ 1
ε
qj+1 =∫ 1
ε
H0(qj ) =∫ 1
ε
qj + 2(j + 1) + 2(j + 1)γ (ε),
whence, since γ is continuous and γ (0) = 0, it follows that∫ 1
0qj+1 =
∫ 1
0qj + 2(j + 1)
and thus ∫ 1
0qj =
∫ 1
0q0 + j (j + 1).
Hence for our potential q,∫ 1
0q =
∫ 1
0q0 + m(m + 1).
Repeating the argument for Q completes the proof of (19) and hence of our result, for the caseof m even and Dirichlet boundary conditions. �
Corollary 3.2. The mean value of q is determined by∫ 1
0q = C + m(m + 1),
13
Inverse Problems 24 (2008) 015001 L Aceto et al
where C is the unique constant such that
λk(q,m) = (k + 1 + m/2)2π2 + C + rk,
where (rk)k∈N0 ∈ 2(N0).
Rohrl [18] calculates the gradient of his functional. His calculations can be repeated in astraightforward way for the singular case to show that
∇G[h](Q) = 2∞∑
k=0
ωk,m(λk(Q,m) − λk(q,m))
∫ 1
0yk(Q,m; x)2h(x) dx
+ 2∞∑
k=0
ωk,(λk(Q, ) − λk(q, ))
∫ 1
0yk(Q, ; x)2h(x) dx,
where yk(Q,m; x)2 denotes the kth eigenfunction of (1) with q = Q and L2-norm equal to 1.Rohrl proves that under certain restrictions on the boundary conditions (which exclude, for
instance, periodic boundary conditions), the squared eigenfunctions are linearly independent;thus ∇G[·](Q) is zero if and only if all the eigenvalue differences are zero, which happens ifand only if Q = q.
In the singular case the proof of this result breaks down completely, not because ofthe presence of the singularity but because the eigenfunctions satisfy different differentialequations (m = ) and hence their Wronskians are no longer constant. We argue that thepractical significance of this breakdown is limited.
• In practice, the sums which appear in the defining equation (14) must be truncated. Assoon as they are truncated, the resulting functional has infinitely many global minimizersbecause the inverse problem has infinitely many solutions. This happens both in theregular and in the singular cases, and it means that any minimization algorithm can nowget stuck in a global minimum which is not the desired solution. One may not be toodistressed by this if one has a weak stability result of the type available in [16] for theregular case, though unfortunately no such result is presently available in the singularcase.
• If the minimization algorithm converges to a function Q at which G(Q) is unacceptablylarge but ∇G[·](Q) is close to zero, the weights ωk,m and ωk, appearing in (14) can bechanged and the minimization procedure restarted. With the changed weights, ∇G[·](Q)
will no longer be zero, and the minimization should make further progress.• In spite of the non-convexity of G, it still has some important advantages over the
functional proposed for regular problems by Knowles [14] in the case of singular problems.Firstly, the Knowles functional assumes a knowledge of one spectral sequence and oneset of ‘norming constants’. In practice this requires that one be able to calculate,for all 0 < x � 1, the value of a suitably normalized solution of (1) satisfying theasymptotics y(x) ∼ xm+1 for small x. This is more difficult numerically than calculatingthe eigenvalues, as it involves solving a singular initial value problem, with possible orderreduction issues which do not arise in many cases for the eigenvalue problem, as we haveshown in section 2.
3.2. Knowing the spectra for fixed m and two different boundary conditions at x = 1
Here we suppress the dependence on m in the notation. We let λk(q) be the eigenvalues of
where β ∈ (0, π) is fixed. It is known from Carlson’s work that the spectra of these twoproblems uniquely determine q. Formally, we define
G(Q) =∞∑
k=0
ωk|λk(Q) − λk(q)|2 +∞∑
k=0
ωk|µk(Q) − µk(q)|2, (20)
in which the weights (ωk)k∈N0 and (ωk)k∈N0 are strictly positive and bounded.The following version of theorem 3.1 still holds.
Theorem 3.3. Suppose that the functions Q and q have the same mean value. Then thefunctional G is well defined and has a unique global minimizer when Q = q.
The proof is essentially the same as the proof of theorem 3.1.In this case, one has the following added bonus.
Theorem 3.4. The gradient ∇G[h](Q) is given by
∇G[h](Q) = 2∞∑
k=0
ωk(λk(Q) − λk(q))
∫ 1
0yk(Q; x)2h(x) dx
+ 2∞∑
k=0
ωk(µk(Q) − µk(q))
∫ 1
0zk(Q; x)2h(x) dx, (21)
where yk are the eigenfunctions satisfying yk(Q; 1) = 0, with unit L2-norm, and zk arethe eigenfunctions satisfying z′
k(Q; 1) + bzk(Q; 1) = 0, with unit L2-norm. Moreover,G[·](Q) = 0 if and only if G(Q) = 0.
Proof. The proof is essentially identical to that of Rohrl [18]. Some care is needed tocheck that various integrals still converge; the asymptotics xm+1 and x−m for solutions in aneighbourhood of x = 0 must be used. �
4. Numerical results for the inverse problem
In this section we apply the techniques previously exposed to three test functions q(x), both insingular and in regular cases. The test functions chosen are those used by Rohrl [18], Brownet al [5] and Rundell and Sacks [19], the latter being the first paper in which they were used.The difference is that here we allow an angular momentum singularity as well.
We should start by making a remark about the choice of weights in the definition of thefunctional. Unless the sequence of weights is summable, G(Q) will not even be finite unlessQ and q have the same mean value. One could follow [5] and [19] and commit an inversecrime by assuming that the mean value of q is known and insisting that all approximationsQ have this mean value. In practice, however, like Rohrl, we were able to get away withoutdoing this. The truncation of the sum in the definition of G(Q) is equivalent to choosingweights of which only finitely many are nonzero, and such a sequence is certainly summable.
15
Inverse Problems 24 (2008) 015001 L Aceto et al
4.1. The singular case
We considered problems of type (1) with
q(x) = x2 − x3; (22)
q(x) =
−35.2x2 + 17.6x 0 � x < 0.25
35.2x2 − 35.2x + 8.8 0.25 � x < 0.75
−35.2x2 + 52.8x − 17.6 0.75 � x � 1;(23)
q(x) =
0 0 � x < 0.1
7x − 0.7 0.1 � x < 0.3
3.5 − 7x 0.3 � x < 0.5
0 0.5 � x < 0.7
4 0.7 � x < 0.9
2 0.9 � x � 1
. (24)
To construct the approximated potential Q(x) we chose Chebyshev polynomials φj (x), j =0, 1, . . . ,M, deg(φj ) = j, so that for some real constants cj ,Q(x) = ∑M
j=0 cjφj (x).
Setting C = (c0, c1, . . . , cM), the functionals (14) and (20) induce functionals of thevector C. Our goal is to determine the minimizing vector.
Clearly the sums (14) and (20) must be truncated, say at k = L, for some suitable L. Therelationship between the choice of L and the number of basis functions M is one which we donot study here: it would require a finite data stability result of the type obtained by Marlettaand Weikard [16] for the regular case, and as yet there is no such result for the singular case.
We have chosen the weights of the two functionals to be equal to 1 for all of the plots,except the example of figure 7, where they are fixed equal to (k + 1)−2. We shall comment onthis later.
We used the Polack–Ribiere flavour of conjugate gradients [17] to compute searchdirections to minimize the functionals (14) and (20).
To apply this minimization method, we had to compute the eigenvalues of differentialproblems of type (1) with the approximated potential Q(x) replacing the exact potentialq(x). These problems were solved using the methods (9) and (10) and the eigenvalues werecomputed as generalized eigenvalues of problems of the type (8).
As is well known, the algebraic technique for approximating the eigenvalues of Sturm–Liouville problems yields a poor accuracy at a high index. In our calculations throughoutthis paper, we used 30 eigenvalues (L = 29) which is small enough and as such this wasnot a serious issue. Andrew and Paine [4] introduced a simple correction technique: inour examples, the eigenvalues were corrected extending this classical procedure to the useof BVMs as suggested in [10]. We should emphasize, however, that there is no rigorousjustification of this procedure in the singular case, and we used it simply because it appearedto give smoother functionals and hence speed up the minimization procedure.
In every example, we considered two sets of 30 eigenvalues corresponding to two differentvalues of the parameter m. The ‘exact’ eigenvalues were computed using the SLEIGN routine.The results are presented in figures 3–6. Each figure consists of three plots: the first onepresents the graph of the exact potential q(x) (dashed line) and of the approximated potentialQ(x) (solid line); the second plot shows the values of the functional by the line-search number;the third one gives the graph of the L2-norm ‖Q − q‖2 with respect to the same line-searchnumber.
16
Inverse Problems 24 (2008) 015001 L Aceto et al
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0 20 40 60 80 10010
−8
10−7
10−6
10−5
10−4
10−3
10−2
0 20 40 60 80 10010
−3
10−2
10−1
Figure 3. Method (9), h = 11000 , M = 10,m1 = 2,m2 = 3, �λ 3.17 × 10−5.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5
0
0.5
1
1.5
2
2.5
0 20 40 60 8010
−4
10−3
10−2
10−1
100
101
102
0 20 40 60 8010
−2
10−1
100
Figure 4. Method (10), h = 11000 , M = 20,m1 = 2,m2 = 3,�λ 7.66 × 10−3.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
0
1
2
3
4
5
6
7
8
9
0 20 40 60 80 10010
−4
10−3
10−2
10−1
100
0 20 40 60 80 100
10−0.3
10−0.2
10−0.1
100
100.1
Figure 5. Method (10), h = 11000 , M = 100,m1 = 2, m2 = 3,�λ = 9.67 × 10−3.
17
Inverse Problems 24 (2008) 015001 L Aceto et al
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 20 40 60 80 10010
−5
10−4
10−3
10−2
10−1
100
0 20 40 60 80 100
10−0.7
10−0.6
10−0.5
10−0.4
10−0.3
10−0.2
10−0.1
100
Figure 6. Method (10), h = 11000 ,M = 100,mi = i + 1, ωk,mi
= 1, i = 1, 2, �λ 4.54 × 10−3.
The captions of figures show the BVM applied, the values of the stepsize h, the values m1
and m2 of the parameter m, the maximum degree M of the Chebyshev polynomials and thevalue
�λ = max0�k�Lj=m1,m2
|λk(Q, j) − λk(q, j)|.
Starting from the zero potential, in figures 3 and 4 the approximations Q(x) of the exactpotentials q(x) given in (22) and (23), respectively, are drawn. As one can see, they presenta perturbation around the singular point x = 0; in relative terms, however, this is small and itdiminishes rapidly as x increases.
The situation with figures 5 and 6 (for q given by (24)) is more complicated. While theoscillations around the discontinuities were to be expected, we did not expect the oscillationsat the end of the interval which turned up in figure 5, and we were concerned that the causeof the problem might be poor accuracy in the forward eigenvalue computations due to thenon-smooth potential. What has actually happened here, however, is that the algorithm hasbecome stuck in a local minimum. Figure 6 shows what happens with a better choice of astarting point for the minimization, with respect to figure 5. In particular, we have fixed asa starting point for the minimization the potential Q(x) = ∑M
j=0 cjφj (x) of a linear type infigure 6 (c0 = 0.82, c1 = 1, cj = 0, j = 2, 3, . . . ,M) and with cj = 0.08,∀j, in figure 5.
Figure 7 shows a repeat of the experiment of figure 6 but with the kth weight in thefunctional chosen to be (k + 1)−2 instead of 1, k ∈ N0. The effect of the decaying weightsseems to be to smooth the functional and give a more rapid convergence of the conjugategradient algorithm. However, the quality of the final approximation is unchanged.
We also tested the robustness of our algorithm when the exact eigenvalues have a randomperturbation with a maximum modulus less than 10−2; figure 8 shows the approximationobtained.
4.2. The case of fixed m, varying boundary conditions
We considered Sturm–Liouville problems of the type (1) with the parameter m fixed: m = 2;we endeavoured to reconstruct the potential from two sets of eigenvalues corresponding totwo different boundary conditions. For the purpose of the numerics, we chose
y(0) = 0, y(1) = 0,
y(0) = 0, y(1) − y ′(1) = 0.
18
Inverse Problems 24 (2008) 015001 L Aceto et al
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 20 40 60 80 10010
−7
10−6
10−5
10−4
10−3
10−2
10−1
0 20 40 60 80 100
10−0.7
10−0.6
10−0.5
10−0.4
10−0.3
10−0.2
10−0.1
100
Figure 7. Method (10), h = 11000 , M = 100, mi = i + 1, ωk,mi
= (k + 1)−2, i = 1, 2, �λ 6.28 × 10−3.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Figure 8. Method (10), h = 11000 , M = 100,m1 = 2, m2 = 3,�λ 9.35 × 10−4.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2
0
2
4
6
8
10
0 20 40 60 80 10010
−14
10−12
10−10
10−8
10−6
10−4
10−2
100
102
0 20 40 60 80 100
10−0.5
10−0.4
10−0.3
10−0.2
10−0.1
100
100.1
Figure 9. Method (9), h = 11000 , M = 100,m = 2, �λ 3.18 × 10−7.
Figures 9 and 10 show the result obtained. The starting points of the minimizations were thesame as for figures 5 and 6, respectively.
19
Inverse Problems 24 (2008) 015001 L Aceto et al
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 20 40 60 80 10010
−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
101
0 20 40 60 80 100
10−0.7
10−0.6
10−0.5
10−0.4
10−0.3
10−0.2
10−0.1
100
Figure 10. Method (9), h = 11000 , M = 100,m = 2,�λ 7.76 × 10−5.
It is worth observing that the results for the fixed-m problem are not significantly differentfrom those obtained for the two-m problem, in spite of the fact that for the fixed-m problemone has the uniqueness result of theorem 3.4 which is not available in the two-m case. Finitespectral data, rather than the lack of a ‘unique local minimizer’ theorem, are therefore theprime suspect as an explanation for the fact that in figure 5 the minimization gets stuck in alocal non-global minimum.
5. Conclusions
Further work is required on stability results for all of these problems, in particular for finitespectral data. To date, the best results available are those of Savchuk and Shkalikov [21]. Herethe potential is assumed to lie in Wθ
2 for θ � −1; in other words, after one integration of thepotential, the resulting function is in L2. These do not quite cover the case we have discussedhere, which probably requires a special approach exploiting the closeness to a Bessel equation.
Issues also remain concerning the convergence of the conjugate gradient algorithm and,in particular, strategies for coping when the algorithm gets stuck in a local minimum. Fromour experiments here it seems that this is more of an issue when recovering the potential fromthe spectra for two different values of m, as one might expect from the fact that it seems to beimpossible to obtain a result of the type in theorem 3.4 for this case. Ad hoc techniques canbe used, such as changing the weights in the sum or running every minimization from at leasttwo different starting points, but a satisfactory resolution of the problem is likely to involve amore sophisticated choice of functional rather than a change of numerical procedures.
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