has non-trivial solutions. For convenience, we have chosen to work with the Liouville normal form with Dirichlet boundary conditions. The inverse Sturm-Liouville ...
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Algebraic Correction Methods for Inverse Sturm-Liouville Problems Chris Dun and Bob Anderssen
1. Introduction
For the potential form of the Sturm-Liouville eigenvalue problem, the goal is to nd, for a given q(x), the set of eigenvalues k 2 R such that the equation ?y (x) + q(x)y(x) = y(x) ; x 2 (0; ) ; (1.1) y (0) = 0 = y ( ) ; has non-trivial solutions. For convenience, we have chosen to work with the Liouville normal form with Dirichlet boundary conditions. The inverse Sturm-Liouville problem (simply called the inverse problem ) is: nd a unique potential q(x) satisfying the above equation, given a set of eigenvalues fk g as data. The focus of this paper is the numerical solution of the inverse problem under the additional simplifying assumption that the potential q(x) is symmetric in (0; ). The inverse problem in the above form was rst studied by Borg [3] in 1946. He showed that if a solution exists for a certain set of eigenvalues, then it is unique. In 1951, Gel'fand and Levitan [5] proved that a solution to the inverse problem exists, and determined how it could be constructed from the eigenvalue data. Unfortunately, their method is impractical to implement. A numerical approximation to the solution of the inverse problem is obtained by imposing a discretisation on (1.1), and then solving the resulting inverse matrix eigenvalue problem. The solution of the inverse matrix problem must then be identi ed with the solution to the original inverse problem. The diculty associated with this method of solution is determining to what extent the discrete problem and the continuous problem relate. For example, the data available for the solution of the inverse problem is the set of di erential eigenvalues fk g. To solve the inverse matrix problem, we require a set of algebraic eigenvalues, say k(N ) (where N denotes the size of the discretisation.) For standard discretisation schemes, it is known (cf. [1],[2],[7]) that the di erential eigenvalues k and the algebraic eigenvalues k(N ) di er substantially for large k. Thus, using the di erential eigenvalues as approximations to the algebraic eigenvalues will give very poor results [10]. In 1984, Paine [8] overcame this diculty by using an algebraic correction procedure developed by Paine, de Hoog and Anderssen [7]. The idea is to apply a correction to the di erential eigenvalues to make them better approximations to the algebraic eigenvalues. However, Paine found that a straight forward application of this idea failed to give the expected convergence. Paine then considered a modi cation to his algorithm which gave improved results. Paine did not investigate why his modi ed method gave improved results, nor did he give a mathematical justi cation for the consistency of his algorithm. Recently, two papers, [4] and [9], have independently considered Paine's original algorithm, and developed alternative methods based on his idea which achieve con00
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C.R. Dun and R.S. Anderssen.
vergence. Both methods identify the diculty which underlies Paine's algorithm. Dun and Anderssen choose a slight modi cation of Paine's algorithm, whereas Pirovino follows an idea proposed by Hald [6]. In this paper, we rst describe Paine's original algorithm, and then compare the two alternatives. We nd that the two methods achieve comparable accuracy, although the method due to Dun and Anderssen is faster.
2. The Paine Method for the Solution of the Inverse Problem.
Paine considered the inverse problem corresponding to (1.1), under the additional assumption that the potential q(x) is symmetric q (x) = q ( ? x) ; x 2 (0; ) : On the uniform grid G = fxi : xi = ih; i = 0; 1; 2; : : : ; N + 1; h = =(N + 1)g ; Paine invoked a central di erence discretisation of the problem to obtain the following matrix counterpart of (1.1) (?A + Q) u(N ) = (N )u(N ) ; (2.1) where A = 1 tridiag (1; ?2; 1) ; (2.2) h2
Q = diag (q(x1); q(x2); : : : ; q(xN )) :
The algebraic correction used by Paine was originally developed in the context of the numerical solution of the forward problem; ie., calculating the di erential eigenvalues fk g of (1.1), given the potential q(x). The motivation for the method follows a classic technique in numerical analysis; to estimate the error in the computed solution, and then use this as a \correction" to improve the result. Paine, de Hoog and Anderssen [7] found that the asymptotic form of the error k ? k(N ) is not sensitive to changes in the potential q (x). Since the di erential eigenvalues are known in closed form for the null potential q(x) 0, they considered the following two counterparts of (1.1) and (2.1) ?y (x) = y(x) ; x 2 (0; ) ; y (0) = 0 = y ( ) ; and ?Au(N ) = (N )u(N ) : In the case of the central di erences discretisation, the algebraic eigenvalues k(N ) are also known in closed form. Paine et al. [7] showed that k ? k(N ) = k ? k(N ) + O(kh2 ) ; for 1 k N , < 1 independent of N . Thus, the di erence "k(N ) = k ? k(N ) can be used as an estimate of the error k ? k(N ) for general potentials. In addition, it
00
Inverse Sturm-Liouville Problems.
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can be used to correct the algebraic eigenvalues, and hence improve the uniformity of the approximations to the corresponding di erential eigenvalues; namely ~ k k(N ) + "k(N ) = k + O(kh2 ) ; where the ~k denote the corrected algebraic eigenvalues. Paine [8] noted that "k(N ) could equivalently be used to correct the di erential eigenvalues to make them approximate the algebraic eigenvalues; namely, for 1 k N , < 1 independent of N , ~ k(N ) k ? "k(N ) = k(N ) + O(kh2 ) ; (2.3) where the ~k(N ) are simply called the corrected eigenvalues. Thus, the Paine method is: assuming that N di erential eigenvalues are known, correct them using (2.3) to make them approximate the algebraic eigenvalues of (2.1); and then apply a matrix reconstruction algorithm to the corrected eigenvalues, to generate the approximation ?A + Q de ned by ?A + Q v(N ) = ~(N )v(N ) ;
f
f
where A has the form (2.2), and Q is the approximation to the matrix Q. Then identify the constructed matrix with the assumed form of the discretisation, Qi;i ! Qi;i = q(xi) ; (2.4) to construct the approximation to the potential q(x). The reason for the poor convergence of Paine's algorithm is that the corrected eigenvalues, ~k(N ), are not uniform approximations to the algebraic eigenvalues for all k [4]. Notice that the relation (2.3) only holds for k N , < 1. Thus, we have no guarantee that we can control the size of the error in the last few corrected eigenvalues. As an example of this, Figure 1. shows the error between the corrected eigenvalues and the exact algebraic eigenvalues for the potential 6 cos(2x) used by Paine in his numerical calculations. In order to clarify that increasing the size of the approximation does not improve the accuracy of the corrected eigenvalues, we have plotted the error for approximations of size N = 20; 40; 80. It can be shown numerically that small non-uniform perturbations to the exact algebraic spectrum can lead to signi cant changes in the reconstructed matrix. As an example of this, we perturbed the exact algebraic eigenvalues of (2.1), for the potential q(x) = 6 cos 2x, by adding 1 to the last eigenvalue. We then applied a matrix reconstruction algorithm to the perturbed spectrum, and calculated the error in the diagonal and o -diagonal entries of the resulting matrix. The results of this can be seen in Figure 2. Thus, the non-uniformity of the corrected eigenvalues is the cause of the non-convergence of the Paine method. It is this non-uniformity in the corrected eigenvalues that the methods due to Dun and Anderssen, and Pirovino aim to compensate for. f
f
3. The Dun and Anderssen Method
Dun and Anderssen [4] follow Paine's algorithm quite closely; ie. given N di erential eigenvalues, they correct them using (2.3), and then apply a matrix reconstruction algorithm to the corrected eigenvalues. They then identify the constructed
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2 N=20 N=40 N=80
1.5
Error
1
0.5
0
-0.5
-1 0
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30 40 50 Eigenvalue number
60
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Error between the corrected and exact algebraic eigenvalues.
Figure 1.
3 diagonal entries off-diagonal entries 2
Error
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-1
-2
-3 0
Figure 2.
5
10
15
20 Matrix Entry
25
30
35
40
Error in diagonal and o -diagonal entries of perturbed matrix
Inverse Sturm-Liouville Problems.
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matrix with a consistent discretisation of the problem (1.1). However, they found that the e ect of the non-uniformity of the corrected eigenvalues was to spread the information about the potential q(x) into the o -diagonal entries of the constructed matrix Q. Consequently, they proposed a discretisation of the original problem, in which the matrix representing the potential is tridiagonal. It gave improved results because the tridiagonal discretisation allowed them to recover the information lost to the o -diagonal entries. They showed that the corrected eigenvalues are in fact uniform approximations to the eigenvalues of the modi ed discretisation, and hence, removed the diculty in the Paine method. Note that in [8], Paine proposed a second method, closely related to his rst, which implicitly exploits the information lost to the o -diagonal elements of the reconstructed matrix. Paine states the method without proof of convergence, although a close examination shows that the analysis of Dun and Anderssen [4] could be modi ed to establish convergence for this method. Although similar, Paine's second method is more cumbersome and complicated than the Dun and Anderssen method. Dun and Anderssen assume that the form of the matrix representing the potential is Q = tridiag q(xi? 12 ); (1 ? 2 )q(xi); q(xi+ 21 ) : They observed that the sum of the rows of Q, denoted qi q (xi? 21 ) + (1 ? 2 )q (xi) + q (xi+ 21 ) ; converges to q(xi) as h ! 0, and proved that qi = q (xi) + O(h2 ) : (3.1) Thus in the Dun and Anderssen approach, Paine's identi cation (2.4) is replaced by Qi;j ! Qi;j ?sum ?! qi = q(xi) + O(h2) ; ?! qi is obtained by summing the rows of Qi;j . Note where the identi cation Qi;j ?sum that to carry out this identi cation, the value of need not be known. The exact role of is fully explained in Dun and Anderssen [4]. f
f
4. The Pirovino Method
Pirovino [9] solves the inverse problem using a Newton iteration method. His method follows an idea originally proposed by Hald [6]. Hald notes that since the potential q(x) is symmetric, the problem (2.1) represents an over determined system of equations, since we have N equations in only n = [ N 2+1 ]y unknowns. Thus, it will not have a solution whenever the system is inconsistent. Hald overcomes this by posing the following minimisation problem F (q1; : : : ; qn ) =
N X j =1
j(N ) ? ^ j(N )
2
;
where the ^j(N ) are the eigenvalues corresponding to the problem ?A + Q w(N ) = ^(N )w(N ) ;
y
c
where [x] denotes the greatest integer less than or equal to x.
(4.1)
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C.R. Dun and R.S. Anderssen.
where Q = diag (q1; : : : ; qn; : : : ; q1). Hald proposes a Gauss-Newton method for the minimisation of (4.1) JT J q(+1) ? q() = JT (N ) ? ^ (N;) ; c
where J is the N n matrix representing the Frechet derivative of the mapping q 7! ^ (N;) given by @ ^ j(N ) J(q)j;i = @q ; i
and q() = q1(); : : : ; qN() is the th approximation to the minimising vector of (4.1). Hald shows that his Gauss-Newton method has a unique solution, q. He then uses the identi cation
T
qi ! Qi;i
to reconstruct the matrix Q. Notice that the Hald method is a matrix reconstruction algorithm; ie. he assumes that the exact algebraic eigenvalues k(N ) are known, and then reconstructs the matrix ?A + Q from them. Thus, if the corrected eigenvalues are used as an approximation to the exact algebraic eigenvalues in the minimisation of (4.1), the method will fail, since the algebraic approximation breaks down for j close to N . Hald also observes that his method can be simpli ed by minimising the functional G(q1 ; : : : ; qn ) =
n X j =1
j(N ) ? ^ j(N )
2
:
(4.2)
Because one has gone to a square system, the Gauss-Newton minimisation can be reduced to a Newton minimisation (4.3) J q(+1) ? q() = (N ) ? ^ (N;) ; where now J is an n n matrix. Hald conjectures that J is invertible, but does not prove it. Pirovino [9] proves a stability result which shows, at least for small potentials, that the matrix J is invertible. Thus, the Pirovino method is to correct all of the di erential eigenvalues using (2.3), but then only use the rst half of them in the minimisation of (4.2). The reason the Pirovino method converges is that this choice of half spectrum overcomes the non-uniformity of the corrected eigenvalues. Note that setting = 1=2 in (2.3) implies that the rst half of the corrected spectrum will be a uniform approximation to the rst half of the algebraic spectrum.
5. Discussion
We implemented the above methods and ran the programs on appropriate test problems. We found that the two methods gave similar numerical results. However, the Dun and Anderssen method had certain clear advantages. Firstly, we found that, on occasion, the Dun and Anderssen method gave a higher order of convergence. For example, the former gave O(h2 )-convergence for the potential q1(x) below, whereas the Pirovino method only gave O(h1:5)-convergence.
Inverse Sturm-Liouville Problems.
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At this point, it is dicult to identify the regularity under which the Dun and Anderssen method will give this improved convergence. Secondly, the Dun and Anderssen method has a lower complexity. Their method is O(N 2), which is determined by the matrix reconstruction step. The Pirovino method involves solving a system of n equations (4.3) to determine the next vector in the Newton iteration scheme. The solution of (4.3) must be recalculated at each iteration, so the order of Pirovino's algorithm is O(IN 3), where I := the number of iterations applied. To calculate the order of convergence of each method, we followed Pirovino's error analysis, and introduced the discrete norm p kxkh = hxT x ; x 2 RN : The relative error between the exact potential evaluated at the grid points, qexact, and the constructed potential, q, is then given by
kh ; err = kqkq?(xq)exact k2
(5.1)
L
where kkL2 is the usual L2(0; ) norm. For the potential q1 (x) = 6 cos 2x ; we calculated the error (5.1) generated by each algorithm, for grid sizes N = 10; 20; 40; 80. The results are presented in Table 1. N Dun and Anderssen Pirovino 2 err err/h err err/h1:5 10 9:81525 10?3 1:20334 10?1 4:30140 10?3 2:81822 10?2 20 2:16169 10?3 9:65902 10?2 1:23723 10?3 2:13824 10?2 40 4:74333 10?4 8:07889 10?2 4:01610 10?4 1:89346 10?2 80 1:07873 10?4 7:17107 10?2 1:35338 10?4 1:77183 10?2 Table 1. Comparison of results for q1 (x). Notice that in this case, the Dun and Anderssen method is achieving O(h2)convergence, whereas the Pirovino method has only O(h1:5)-convergence. Pirovino tested his algorithm on potentials which were discontinuous, and potentials with discontinuous derivatives. For completeness, we compared the two methods for the potentials p 4 3 x ? 1 x 2 0; 2 ; q2(x) = p 3 ? 4 x + 3 x 2 2 ; ; and 1 x 2 0; 4 ; q3 (x) = ?1 x 2 4 ; 34 ; 1 x 2 34 ; : These results are displayed in Tables 2 and 3, respectively. 8 > < > :
8 > > > > < > > > > :
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C.R. Dun and R.S. Anderssen.
N
10 20 40 80 N
10 20 40 80
Dun and Anderssen Pirovino err err/h1:5 err err/h1:5 3:59563 10?2 2:35580 10?1 1:16755 10?2 7:64964 10?2 1:42228 10?2 2:45804 10?1 4:91844 10?3 8:50025 10?2 5:29304 10?3 2:49549 10?1 1:81339 10?3 8:54953 10?2 1:91403 10?3 2:50582 10?1 6:52389 10?4 8:54101 10?2 Table 2. Comparison of results for q2 (x). Dun and Anderssen Pirovino 0 :5 err err/h err err/h0:5 1:22149 10?1 2:28566 10?1 1:30774 10?1 2:44704 10?1 9:32839 10?2 2:41180 10?1 9:93414 10?2 2:56841 10?1 6:61541 10?2 2:38987 10?1 7:07439 10?2 2:55568 10?1 4:68600 10?2 2:37941 10?1 5:01875 10?2 2:54838 10?1 Table 3. Comparison of results for q3 (x).
As the results in Tables 1, 2 and 3 indicate, the Pirovino method gives more accurate results than the Dun and Anderssen method, at least for small N . A possible explanation for this lies in Pirovino's use of the rst half of the corrected spectrum, which clearly has the better uniformity of approximation to the relevant algebraic eigenvalues.
References
1. Andrew,A.L., Paine,J.W., \Correction of Numerov's Eigenvalue Estimates." Numer. Math. 47 (1985) 289{300 2. Andrew,A.L., Paine,J.W., \Correction of Finite Element Estimates for Sturm-Liouville Eigenvalues." Numer. Math. 50 (1986) 205{215. 3. Borg,G., \Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe." Acta Math. 78 (1946) 1{96. 4. Dun,C.R., Anderssen,R.S., \A Modi cation of Paine's Algebraic Correction Method for Inverse Sturm-Liouville Problems." in preparation. 5. Gel'fand,I.M., Levitan,B.M., \On the Determination of a Di erential Equation from it's Spectral Function." Am. Math. Soc. Transl. 1(2) (1955) 253{304. 6. Hald,O.H., \On Discrete and Inverse Sturm-Liouville Problems." PhD thesis, New York University, 1972. 7. Paine,J.W., de Hoog,F.R., Anderssen,R.S., \On the Correction of Finite Di erence Eigenvalues for Sturm-Liouville Systems." Computing 26 (1981) 123{139. 8. Paine,J., \A Numerical Method for the Inverse Sturm-Liouville Problem." SIAM J. Sci. Stat. Comput. 5 (1984) 149{156. 9. Pirovino,M.H., \Das Strum-Liouville-Problem als direktes und inverses Eigenwertproblem und seine numerische Behandlung durch nite Di erenzen." Dissertation, ETH Zurich, 1992. 10. Pirovino,M.H., \The Inverse Strum-Liouville Problem and Finite Di erences." Research Report No. 93-04 Seminar fur Angewandte Mathematik, ETH, Zurich, 1993.
