CC BY
42
MSC 47A75
DOI: 10.14529/mmpl50307
A NUMERICAL METHOD
FOR INVERSE SPECTRAL PROBLEMS
S.I.Kadchenko, Magnitogorsk State Technical University named after G.I. Nosov, Magnitogorsk, Russian Federation, kadchenko@masu.ru,
G.A. Zakirova, South Ural State University, Chelyabinsk, Russian Federation, zakirova81@masu.ru
Basing on the Galerkin methods, we develop a new numerical method for solving the inverse spectral problems generated by discrete lower semibounded operators. The restrictions on the perturbing operator are relaxed in comparison with the method based on the theory of regular traces. A Fredholm integral equation of the first kind enables us to recover the values of the perturbing operator at the discretization nodes. We tested the method on spectral problems for the Sturm - Liouville operator, and the results of numerous simulations demonstrate its computational efficiency.
We found simple formulas for the eigenvalues of a discrete lower semibounded operator avoiding the roots of the corresponding secular equations. The calculation of eigenvalues of these operators can start at an arbitrary index independently of the (un)availability of the eigenvalues with smaller indices. For perturbed selfadjoint operators we can calculate eigenvalues with large indices when the Galerkin method becomes difficult to apply.
Keywords: inverse spectral problem; discrete selfadjoint operators; eigenvalues; eigenfunctions; ill-posed problems.
Introduction
The inverse problems of spectral analysis amount to reconstructing differential operators given certain spectral characteristics. They are important in various branches of mathematics and have many applications in mechanics, physics, electronics, geophysics, metrology, seismology, identification of composites, and other fields of natural sciences and technology. The interest in this topic grows constantly as new inverse spectral problems describing complicated technical processes arise. Presently, the theory of inverse spectral problems is actively developing in all leading countries of the world.
The most thorough studies in the theory of inverse problems deal with the Sturm - Liouville differential operator
ly = -v + q(x)y-
The first works on the Sturm - Liouville inverse spectral problem include Ambarzumian's article [1]. He proved that for the eigenvalues Ak = k2 with k > 0 the potential q must
vanish identically. It turned out subsequently that this result is special and in general the
q
of the Sturm - Liouville operator with the same boundary conditions uniquely determine the q
semiaxis or a finite interval is uniquely determined by the spectral function. For a finite interval this corresponds to specifying the spectral data. While solving the problem of determining the density of an inhomogeneous symmetric string given its frequency spectrum, Krein showed [4] that a symmetric potential is always uniquely determined by the frequency spectrum.
Making no attempt at surveying all research directions and the numerical methods developed to solve inverse spectral problems, we mention the articles relying on the method of regular traces. The advantage of this method is its applicability to both ordinary and partial differential operators. Sadovnichii and Dubrovskii published the first article in this direction [5]. They proved the existence and uniqueness of solutions to the inverse spectral problems for selfadjoint semibounded discrete operators acting on separable Hilbert spaces and established the possibility of reconstructing the perturbing operator from the spectrum of the operator alone provided that it is "small". Dubrovskii and Nagornyi gave [6, 7] the first theoretical justification to the application of the method of regular traces to inverse spectral problems for the perturbed Laplace operator. The theory of inverse problems for the Laplace operator with potential was further advanced in the work of Sadovnichii, Dubrovskii, and their school [8, 17]. Algorithms to solve approximately the inverse spectral problems generated by a perturbed power of the Laplace operator were developed in [18].
The method of [19, 22] enables us to solve the inverse spectral problems generated by discrete lower semibounded operators. Let us recall its main idea. Take a discrete lower semibounded operator T and a bounded operator P on a separable Hilbert space H. Assume that the eigenvalues 1 and orthonormal eigenfunctions of T are available and enumerate them in the
ascending order of Ani taking their multip licities vn into account. Den о te by n0 the number of
distinct eigenvalues An lying inside the circle Tn0 of radius pno = \ n°+l2+—centered at the origin. Enumerate as {jn}%=1 the eigenvalues of the operator T + P in the ascending order of their real parts, taking into account their algebraic multiplicities. 2\\PII
If qn = tt-r—г < 1 to a 11 n € N and the system of eigenfunctions of T
\An+v„ — An |
no
constitutes an orthonormal basis of H then m0 = vn ancL we can calculate the eigenvalues
n=1
{^n}mm=°1oi T + P as
Jn = —n + ^гn, гn) + 5(n), n = 1, mo, (1)
n
where S(n) = S(n) — S(n — 1) with 5(n) = J2[jk — Jk(n)] and Jk(n) is the nth Galerkin
k=1 _
approximation to the corresponding eigenvalue Jk of T + P. For 5(n) we have the estimates
~ q2
\U(n)| < (2n — 1)pn--, q = max qn.
1 — q neN
In the case that P is the operator of multiplication by a function p(s), we can use (1) to
construct the Fredholm integral equation of the first kind
f b
/ K(x, s)p(s)ds = f (x), c < x < d,
J a
where the functions f (x) and K(x, s) satisfy
n-1
f (xk) = Jk — —k + (n) — Jk(n — 1)], K(xk,s) = г1(s), c < xk < d, к = 1~n. k=1
If the kernel K(x, s) is continuous and closed in the rectangle П = [a,b] x [c,d], while p(s) € [a, b] and f (x) € L2[c,d], then the integral equation has a unique solution, which determines the values of p at the discretization nodes of the interval [a, b].
Since we obtained (1) under the restriction \\P\\ < 0, 5\An+Vn — —n\ to all n € N, it became
P
which is done in this article.
1. Finding the Eigenvalues of Discrete Lower Semibounded Operators with a Modified Galerkin Method
Consider a discrete lower semibounded operator L от a separable Hilbert space H. We determine the eigenvalues ц, by finding the nontrivial solutions to the equation
Lu = fu (2)
satisfying certain homogeneous boundary conditions.
To find the eigenvalues of L, we use the Galerkin method. Introduce a sequence {Hn}^=1 of finite-dimensional spaces Hn С H which is complete H. Assume available an orthonormal basis for Hn consisting of some functions {фк}'П=1 satisfying all boundary conditions of the problem. Following the Galerkin method, we seek an approximate solution to the spectral problem (2) in the form
n
Un = ak(п)фк■ (3)
k=1
LH
If the system of coordinate functions {фк}fe=i constitutes a basis for H then the Galerkin method constructed from this system of functions and applied to the problem of finding the eigenvalues of the spectral problem (2) converges.
Доказательство. Express (2) as
(L — XE)ф = (f — \)ф. (4)
The discrete operator L admits the resolvent R\(L) = (L — XE)-1, which is compact in H. Acting by R\(L) from the left on both sides of (3), we obtain
ф = (f — X)R\(L^.
Basing on [23], the Galerkin method applied to the problem of finding the eigenvalues of (4), and so of (2) as well, converges. □
L
space H. If the system of coordinate functions {фк}<j=1 constitutes an orthonormal basis for H then
ftn(n) = (Lфп, фп) + Sn, (5)
n- 1
where Sn = [ftк(n — 1) — Дк(n)] and jlk(n) is the nth Galerkin approximation to the к=1
corresponding eigenvalue Цк of L. Доказательство. Inserting (3) into (2) yields
nn
ак(n)Lфk = ак(n)фк■
к=1 к=1
The coefficients {ак(n)}n=1 are determined from the requirement that the left-hand side here be orthogonal to the functions {фг}n=1l which leads to the system of linear equations
n
ак(n) j]2(и)6к,1 — (Ьфк, ф^ } =0, l = 1,n
к=1
on the coefficients {akwhere 5k,i is the Kronecker symbol. Setting its determinant equal to zero, we arrive at the equation
det (A - ]l(n)Ej = 0,
which defines the approximate values of the first n eigenvalues {jk(n)}n=i of L. Here E is the n x n identity matrix and A = (aki)n i=1 is the n x n matrix with aki = (L<k, <i)■ It is known [24] that the eigenvalues {jk(n)}n=i of A satisfy
n
Y,jk (n)= SpA, (6)
k=i
which yields
nn
y^jk (n) = ^2 akk. k=i k=i
Introducing Mk = jk(n) + £k (n), we have
nn
^M k = ^2\akk + £k (n)}. (7)
k=i k=i
Subtracting (6) for n — 1, namely,
n-i n-i
^M k = ^2\akk + £k (n — 1)}, (8)
k=i k=i
from (7), we infer that
n- i
Mn = (L<fn, <n) + Mn — jn(n) — [jk(n) — jk(n — 1)}
k=i
or
n- i
jn(n) = (L<n,<n) + (n — 1) — jk (n)}.
k=i
This justifies the theorem. □
Based on the Galerkin method, (5) enables us to find approximate eigenvalues of discrete lower semibounded operators and uses only the scalar products (L<n,<n). This considerably improves computational efficiency in comparison with the classical Galerkin method.
Observe that to obtain (5) we used the diagonal elements akk = (L<k ,<k) for k = 1,n of the square matrix A = (akii=v F°r smn the error of finding the eigenvalues {jk}'k=i can be significant; consequently, we should apply (5) with care. If the requirements of Theorem 1
n jn
jn
Ln avoided.
2. Solutions of Inverse Spectral Problems Generated by Perturbed Selfadjoint Operators
Assume that the operator L of (2) is of the form
L = T + P,
where T is a selfadjoint operator and P is the bounded operator of multiplication by a function p(s) of s € [a, b] on the separable Hilbert space L2[a,b]. Consider the problem of reconstructing the potential P from the eigenvalues {^k}k=i the operator T + P on L2[a, b]. Use (5) to construct the Fredholm integral equation of the first kind
, b
/ K(x,s)p(s)ds = f (x), c < x < d, (9)
J a
where the functions f (x) and K(x, s) satisfy
_ rb _
f (xk) = Vk — T(<pk(s))<fik(s)ds — 5k, K(xk, s) = <fik(s), c < xk < d, k = l,n.
a
If the kernel K(x, s) of (9) is continuous and closed in the rectangle n = [a, b] x [c, d], while p(s) € W2i[a,b] and f (x) € L2[c,d], then the solution to (9) is unique [25]. By the definition of f (x), its values at the points xk are known approximately. Denote by f(xk) the approximate values to f (xk) satisfying \\f (xk) — f(xk)|| < £ to all xk € [c,d]. This estimate is useful in our construction of a numerical algorithm to solve this problem.
To find a solution of the Fredholm integral equation of the first kind (9) is an ill-posed problem, but Tikhonov's regularization method yields an approximate solution p(s). The numerical solution of (9) will determine the approximate values p(s) of p(s) at the nodes si for i = 1,I, with a = si < s2 < ... < si = b. To achieve good accuracy of interpolation for p(s), we can choose
I
Choose the interval [c, d] so that we can tod the eigenvalues j!n in it using (5) with the required accuracy.
Thus, using (5), we construct the integral equation (9) whose solution enables us to find the approximate values p(s) of the operator p(s) at the discretization nodes si of the interval [a, b].
3. Numerical Experiments
We applied our method to solve the spectral problem
Lu = —u'' + p(s) u = n u, a < s <b;
cos a u'(a) + sin a u(a) = 0; (10)
cos7u'(b)+sinYu(b) = 0, a,Y € [0, 2n}.
Assume that the approximate eigenvalues V of L are available. Using them, reconstruct the approximate values p(s) of the potential p(s) at the discretization nodes sk of the interval [a, b].
To construct an orthonormal system of coordinate functions {<k}fe=i satisfying the boundary conditions (10), consider the operator T< = — <p". Furthermore, the function < must satisfy (10). It is not difficult to show that T is a selfadjoint operator whose eigenvalues {Xk}^=i are the roots of the transcendental equation
[sinasin(vXa) + VXcos acos(vXa)][sin7cos(^fXb) — vXcos7sin(vXb)] +
cos a sin (^Xa) — sin a cos(vXa)][sin 7 sin(VXb) + vX cos 7 cos(vXb)] = 0.
The corresponding eigenfunctions pk of T are
(s) = Ck{[sin a sin^\/rXka) + \/xkcos acos^^/X^a)] cos(V/^fcs)+
+ cos a sin^y/Aka) — sin a cos(\/r\ka)] sin(\/Afcs)}, k = 1, m.
We can determine the constants Ck from the normalization condition.
Let us compare the results calculating the eigenvalues ¡k (n) of the Sturm - Liouville spectral problem (10) using (5) and the Galerkin method. Enumerate the eigenvalues found in the ascending order.
Table 1 presents an example of calculating the eigenvalues of (10) for a = 1, b = 3, a = n/5, Y = n/7, and p(s) = s2 — 10s + 11 + (3s2 — 10s + 9)i. We made the calculations on assuming that Uk (n) — ¡k (n — 1) = 0 for k = 1, 51 and n = 51.
Table 1
к Ик(51) Mfc (51) \$к(51) - M51)\
1 -3, 745674 + 2, 940862i -4, 310179 + 3, 541650i 0, 824365
2 4, 637443 + 2, 247669i 4, 802715 + 1, 985491i 0, 310043
3 17,153279 + 2,110610i 17, 260760 + 2,002740i 0,152264
4 34, 487259 + 2,062302i 34, 553462 + 2,006407i 0, 086644
12 350, 385311 + 2, 006931i 350, 392348 + 2,001575i 0, 008844
13 412,071642 + 2, 005906i 412, 077633 + 2,001356i 0, 007523
14 478, 692507 + 2, 005093i 478, 679670 + 2,001177i 0, 006480
15 550, 247973 + 2, 004436i 550, 252465 + 2,001032i 0, 005639
31 2366, 259503 + 2,001039i 2366, 260553 + 2, 000234i 0, 001314
32 2521, 705853 + 2,000975i 2521, 706839 + 2, 000234i 0, 001233
33 2682, 086999 + 2,000917i 2682,087925 + 2, 000220i 0, 001159
34 2847, 402940 + 2,000815i 2847,403813 + 2, 000208i 0, 001092
48 5679, 979950 + 2,000433i 5679, 980371 + 2, 000116i 0, 000527
49 5919, 317870 + 2,000416i 5919, 318364 + 2, 000043i 0, 000062
50 6162, 590609 + 2,000399i 6163, 590374 + 2, 000449i 0, 000253
51 6412, 798139 + 2,000384i 6412, 818701 + 1, 985049i 0, 025650
k
quantities |¡k(n) — fik(n) | decrease, with the exception of the last two rows for k = 50 and k = 51.
Numerous calculations for various values of the parameters a, b ci d a, ft, and p(s) demonstrated high accuracy and computational efficiency of our formula (5) for the eigenvalues of the spectral problem (10).
Table 2 presents the results of calculations of approximate values p(sk) of the function p(s) at the nodes {sk}k=! for the parameter values a = 1, d = 2, a = n/3, y = n/5, and f(xk) = Uk — ^k for k = 1, 21 and the pertirbing operator p(s) = s2 — 3s + 16 + (5s2 — 7s + 3)i. Numerous calculations demonstrated that we can find the approximate values p(sk) of the potential p(s) at the nodes {sk}'%= \ with the prescribed accuracy of residuals HAp — /|| in a large range of
p(s)
Zk = fxk) — K (xk, s)p(s)ds|
Table 2
k Sk P(sk) C (Sk)
1 1,00 14.000005 + 0, 992328« 0 007672
2 1,05 13 952503 + 1,157641« 0 004859
3 1,10 13 910002 + 1, 347527« 0 002473
4 1,15 13 872501 + 1, 561034« 0 001465
5 1, 20 13 840001 + 1, 799400« 0 000600
6 1, 25 13 812500 + 2, 062430« 0 000070
7 1, 30 13 790000 + 2, 350377« 0 000377
8 1, 35 13 772500 + 2, 663136« 0 000636
9 1,40 13 759999 + 3, 000840« 0 000840
10 1,45 13 752499 + 3, 363440« 0 000939
11 1, 50 13 749999 + 3, 750999« 0 000998
12 1, 55 13 752499 + 4,163517« 0 001079
13 1, 60 13 759999 + 4, 600995« 0 000995
14 1, 65 13 772499 + 5, 063481« 0 000912
15 1, 70 13 789999 + 5, 550912« 0 000912
16 1, 75 13 812499 + 6, 063392« 0 000892
17 1, 80 13 839999 + 6, 600788« 0 000788
18 1, 85 13 872499 + 7,163296« 0 000796
19 1, 90 13 910000 + 7, 750670« 0 000670
20 1, 95 13 952499 + 8, 363875« 0 001379
21 2,00 13 999999 + 0, 001691« 0 001691
determines the pointwise absolute error of the solution. The residual found at the node Sk of the approximate solution p(sk) equals \\Ap — f\\ = 9, 54888210-17. In the numerical solution of the Fredholm integral equation of the first kind (9) by Tikhonov's regularization method we calculated the regularization parameter a using the method of residuals. In our case a = 1, 31490010-11.
The values Zk and \ \ Ap — f \ \ of pointwise absolute error and residual enable us to conclude that we find the approximate values p(sk) of the function p(s) at the discretization nodes {sk}k=i with good accuracy.
Conclusion
Basing on the Galerkin method, we have developed a computationally efficient numerical method for solving the inverse spectral problems generated by perturbed selfadjoint operators given their spectrum alone. In comparison with the method based on the theory of regular traces, we remove the restriction on the perturbing operator P of the form \ \P\ \ < 0, 5 \ An+Vn — An\ for all n € N. We wrote a Maple package enabling us to recover the potential p(s) from the spectrum of the operator L.
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2. Borg G. Eine umkehrung der Sturm-Liouvilleschen eigenwertaufgabe. Acta Math., 1945, vol. 78, no. 3, pp. 1-90.
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Received February 9, 2015
УДК 519.642.8 Б01: 10.14529/ттр150307
ЧИСЛЕННЫЙ МЕТОД РЕШЕНИЯ ОБРАТНЫХ СПЕКТРАЛЬНЫХ ЗАДАЧ
С.И. Кадченко, Г.А. Закирова
На основе метода Галеркина разработан новый численный метод решения обратных спектральных задач, порожденных дискретными полуограниченными снизу операторами. В отличии от метода решения обратных спектральных задач, основанного на теории регуляризованных следов дискретных полуограниченными снизу операторов, в разработанном методе ослаблены ограничения на возмущающий оператор. Получено интегральное уравнение Фредгольма первого рода, позволяющее восстанавливать значения возмущающего оператора в узловых точках дискретизации области исследования. Метод был апробирован на спектральных задачах для оператора Штурма-Диувилля. Результаты многочисленных расчетов показали вычислительную эффективность метода.
Найдены простые формулы для вычисления собственных значений дискретных полуограниченных снизу оператора, без нахождения корней соответствующего векового уравнения. Вычисление собственных значений этих операторов можно начинать с любого их номера независимо от того, известны ли собственные значения с предыдущими номерами. Можно вычислять собственные значения возмущенного самосопряженного оператора с большими номерами, когда применение метода Галеркина становится затруднительным.
Ключевые слова: обратная спектральная задача; дискретные и самосопряженные операторы; собственные числа, собственные функции; некорректно поставленные задачи.
Литература
1. Ambarzumian, V.A. Ueber eine frage der eigengwerttheorie / VA. Ambarzumian // Zeits. f. Phisik. - 1929. - № 53. - P. 690-665.
2. Borg, G. Eine umkehrung der Sturm-Liouvilleschen eigenwertaufgabe / G. Borg // Acta Math. - 1945. - V. 78, № 3. - P. 1-90.
3. Марченко, В.А. Некоторые вопросы теории дифференциального оператора второго порядка / В.А. Марченко // ДАН СССР. - 1950. - Т. 72, № 3.- С. 457-460.
4. Крейн, М.Г. Определение плотности неоднородной симметричной струны по спектру частот / М.Г. Крейн // ДАН СССР. - 1951. - Т. 76, № 3.- С. 345-348.
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Сергей Иванович Кадченко, доктор физико-математических наук, профессор, кафедра «Прикладная математика и информатика:», Магнитогорский государственный технический университет им. Г.И. Носова (г. Магнитогорск, Российская Федерация), kadchenko@masu.ru.
Галия Амрулловна Закирова, кандидат физико-математических наук, доцент, кафедра «Уравнения математической физики», Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), zakirova81@masu.ru.
Поступила в редакцию 9 февраля 2015 г.
