RECOVERY OF A DENSITY FROM THE EIGENVALUES OF A NONHOMOGENEOUS MEMBRANE

Transcription

1 Proceedings of ICIPE rd International Conference on Inverse Probles in Engineering: Theory and Practice June -8, 999, Port Ludlow, Washington, USA : RECOVERY OF A DENSITY FROM THE EIGENVALUES OF A NONHOMOGENEOUS MEMBRANE C. Maeve McCarthy Departent of Matheatics & Statistics Murray State University Murray, Kentucky 7 Eail: aeve.ccarthy@urraystate.edu ABSTRACT The vibrating elastic ebrane is a classical proble in Matheatical Physics which arises in a wide variety of physical applications. Since the geoetry of the ebrane is usually well defined for a particular proble, deterination of the nature of any nonhoogeneity is critical. The eigenvalues of particular ebranes are often quite accessible experientally and so a ethod for the deterination of the nonhoogeneity based on the available eigenvalues is of practical iportance. Projection of the boundary value proble and its coefficients onto appropriate vector spaces leads to a atrix inverse proble. Although the atrix inverse proble is of nonstandard for, it can be solved by a fixed-point iterative ethod. Convergence of the ethod for a rectangular ebrane is discussed and nuerical evidence of the success of the ethod is presented. Introduction The odes of vibration or eigenvalues of any physical object depend on a variety of factors. Typically the geoetry and boundary conditions governing a particular object can be deterined by inspection of the object and its circustances. For exaple, when a violin string is pulled taut over its bridge, it can be treated as a one-diensional string of fixed length. A dru is typically described by a claped ebrane whose geoetry is described by the shape of the dru. Thin plates and shells also have specific geoetries and boundary conditions under different circustances. and Consider the situation where the geoetry, boundary conditions, and frequencies or eigenvalues are known and the paraeters governing the aterial s constitution are sought. This class of probles are known as inverse spectral probles. Is it possible to recover the paraeters that led to those eigenvalues? If an infinite nuber of eigenvalues is available, it ay be possible. In general, if the nuber of eigenvalues is finite, the answer is no. However in certain circustances it is possible to recover an approxiation to the unknown nonhoogeneity, although this recovery ay not be unique. We begin by posing the inverse eigenvalue proble for the claped ebrane. A atrix inverse eigenvalue proble based on this is forulated. The solution of the atrix inverse proble via a fixed-point ethod is discussed. A nuber of nuerical exaples are presented. The nonhoogeneous ebrane inverse proble Consider a claped dru or nonhoogeneous ebrane over a -diensional region Ω The vibrations of the ebrane satisfy the boundary value proble p u qu λru x y Ω ( u x y on Ω ( The reader is referred to Love (9 or Courant & Hilbert (95 for further details. We seek to recover the unknown paraeters p q r fro the eigenvalues of (-. There is extensive literature for one-diensional inverse spectral probles, McLaughlin (986. Much of the literature Copyright 999 by ASME

2 on the two-diensional proble concerns the recovery of a potential q when p r The first general uniqueness result for the two-diensional potential proble was not discovered until 988 when Nachan, Sylvester and Uhlann (988 established the q is uniquely deterined by the Dirichlet eigenvalues and the noral derivatives of the eigenfunctions on the boundary. Barcilon (99 showed that when Ω is the unit disk and the boundary conditions are of Neuann type (i.e. the noral derivative is zero on Ω, the potential q can be recovered fro the eigenvalues and eigenfunction data. El Badia (989 established a uniqueness result for q independent of one direction on the unit square. Seidan (988 established an approxiation ethod for the recovery of rotationally syetric q Knobel & McLaughlin (99 extended a one-diensional technique due to Hald (978 to the two-diensional case with syetric potential. In this paper, we seek to generalize Hald s technique further to the recovery of a positive density on a known rectangle R π a value proble, π That is, given the eigenvalues of the boundary u λρu on R ( u on R ( The density ρ is assued to be syetric with respect to the idlines of the rectangle. Ultiately, the goal is to apply this approach to the general proble in future work. we seek to recover the density function ρ x y Given scalars λ i i find a vector α R such that the atrix A α given by a i j α i δ i j k α k π π has λ i i as its sallest eigenvalues. cos kx sin ix sin jxdx (7 The solution α of the atrix inverse proble yields an approxiation ˆq x k α k coskx (8 to the unknown potential q This ethod was extended to the two-diensional potential proble by Knobel and McLaughlin (99. Following their work closely, we can extend the ethod to the density case. The eigenvalues and L -orthonoral eigenfunctions of the case ρ are given by φ i x y λ i a n i i (9 a π sin an i x sin a i y ( where n i and i are positive integers, ordered such that Forulation of the Matrix Inverse Proble Given the lowest eigenvalues λ i ρ i of the boundary value proble (-, we seek to recover an approxiation ˆρ to the unknown syetric density ρ With finite data, it sees pragatic to seek a finite-diensional fraework in which to proceed. Of particular interest is the ethod developed by Hald (978 based on the conversion of the one-diensional inverse potential proble to a finite diensional atrix inverse proble. Hald considered the one-diensional potential proble u qu λu x π (5 u u π (6 for potentials q even with respect to the idpoint x π Using n the eigenfunctions of the q case, sin jx j as a basis for an n-diensional vector space V the eigenfunction u was projected onto V Using the even functions cos jx j as a basis for an -diensional vector space W the unknown potential q was projected onto W Using these projections in a Rayleigh-Ritz forulation of the boundary value proble (-, Hald arrived at the atrix inverse proble: The set φ i n i λ λ fors a basis for an n-diensional vector space V The eigenfunction u is projected onto V In order to take advantage of the structure of this particular proble, let us reforulate the boundary value proble (-. Let γ λ and consider instead the boundary value proble ρ x y u γ u on R ( u on R ( The projection of u onto the n-diensional vector space V leads to the finite-diensional atrix eigenvalue proble DU γλ nu where u is an n-vector, Λ n diag λ λ n and D i j R ρφ iφ j The k-th eigenvalue of this syste gives a lower bound approxiation to γ k ρ Copyright 999 by ASME

3 * We seek approxiations to syetric density functions which are sall perturbations of ρ Let us consider ˆρ x y r x y i α i ψ i where ψ i i are the syetric L functions ψ i x y x y ( a π sin n i ax sin i y where n i i are the integers used in (9. Essentially, we are using ψ i i as a basis for an -diensional vector space W and projecting ρ onto W Use of this projection leads to the finite-diensional atrix inverse proble: Given scalars γ i i find a vector α R such that the atrix B r given by b i j α λ i δ i j α k k has γ i i as its largest eigenvalues. ψ k φ i φ j ( R The solution α of the atrix inverse proble yields the approxiation ( to the unknown density ρ Let B j be the sub-atrix of B fored by deleting the jth row and jth colun fro B and let V j be the n -colun vector fored fro the jth colun of B by deleting b j j Then the atrix " b B $# j j Vj T V j B j % is siilar to B and has the sae eigenvalues. For the reainder of this paper we shall assue that the eigenvalues of the boundary value proble (- with ρ are siple. Thus the inial eigenvalue separation defined by ν is strictly positive, i.e. ν α * νπλ a in & i' j& ( If and λ i λ j γ j λ j then it can be shown that B j γ j is nonsingular. It follows that γ j is an eigenvalue of " B when b j j γ j V T j ν B j γ j,+ V j j (5 Solution of the Matrix Inverse Proble Generally speaking atrix inverse probles have ultiple solutions. In fact, when n there ay be as any as n n solutions to the proble. There are any algoriths for the solution of the special cases of the inverse eigenvalue proble, see Chu s survey article (998. Ji (998 developed an algorith for the construction of all of the solutions by treating the proble as a ulti-paraeter eigenvalue proble. For exaples of the any Newton-based iterative ethods that have been developed, see Biegler-König (98 and Friedland, Nocedal and Overton (987. In the case where n least-square forulations are typically used to generalize other algoriths. The approach used here is to reforulate the proble as a syste of non-linear equations and to use a fixed-point approach. By writing the non-linear syste in fixed-point for β F β existence of sall solutions and convergence as n! of the atrix inverse proble to solutions of the inverse boundary value probles originally posed can be established. Proof of the convergence of this algorith will appear in a ore general fraework (McCarthy, 999. here for copleteness. The relevant assuptions are stated where The solution of the atrix eigenvalue proble ( satisfies α Ψ i j Ψ+ - Λ Γ G α. (6 R ψ jφ i Λ diag λ λ Γ γ γ T G G T G G j α V T j α B j α γ j + V j α In order to establish existence of a solution (, the proble ust be restricted to a sufficiently sall ball in R that the function F α Ψ+ - Λ Γ G α I. (7 Copyright 999 by ASME

4 aps onto itself and is a contraction. If then F : B δ! //Γ Λ + / / π 8 * Ψ+ * λ a λ B δ where B δ is a ball of radius δ If κ λ λ π * Ψ+ * ν fro the first eigenvalues of the boundary value proble (. Figure shows the function ρ The eigenvalues of (- with ρ ρ are given in Table. Figures, and show the recovery of ρ fro, 8 and 6 eigenvalues respectively. Notice that the use of fewer eigenvalues gives the better result since 8 guarantees existence and uniqueness of sall solutions to the atrix inverse proble. Increasing aplifies the error in the approxiation. Figure 5 shows the best possible projection by this ethod by showing ρ ˆr where ˆr is the truncated Fourier sine series of r the perturbation of the density fro. then F is also a contraction. It should be noted that for the choice of ψ and hence the atrix Ψ in this paper, these conditions aount to the nuber of eigenvalues being easured and hence the nuber of ters being recovered for the approxiation ( being sall. In fact we ust restrict 8 in order to guarantee the existence and uniqueness of sall solutions to the atrix inverse proble. Each atrix inverse proble has a solution α n associated with it that depends on the diension n of the underlying syste. Convergence of these solutions can be proved under the assuption that n is large enough..5.5 Nuerical Results Each of the following exaples is on the rectangle R π a π with a 7 which for the boundary value proble ( with ρ has a inial eigenvalue separation ν for the first 56 eigenvalues. The coputations are were all carried out with n 6 basis functions for the vector space V giving 6 6 atrices. The nuber of basis functions for the vector space W is varied in order to show the affects of allowing the theoretical assuptions guaranteeing our contractive ap to break down. Let Figure. The function ρ d h x y h x π a y π.5 and define r h x y 65 e+ 7 d h8 x9 y: if d x y otherwise The data was generated using Matlab s PDE Toolbox which uses a finite eleent ethod with a piecewise linear triangulation, and the recovery algorith was ipleented in Matlab..5 Exaple We seek to recover an approxiation to the density ρ x y r π7 8 x y Figure. The recovery of ρ using eigenvalues Copyright 999 by ASME

5 Figure. The recovery of ρ using 8 eigenvalues Figure 5. ˆr, where ˆr is the Fourier sine series of ρ with 8 ters Figure. The recovery of ρ using 6 eigenvalues Figure 6. The function ρ Exaple We seek to recover an approxiation to the density ρ x y r π7 8 x y 8 5r π7 fro the first eigenvalues of the boundary value proble (. Figure 6 shows the function ρ The eigenvalues of (- with ρ ρ are given in Table. Figures 7, 8 and 9 show the recovery of ρ fro, 8 and 6 eigenvalues respectively. In this case, the higher nuber of prescribed eigenvalues gives the better result. Both recoveries locate the perturbation, but neither one recovers the aplitude of the perturbation well. Figure shows the best possible projection by this ethod by showing ρ ˆr where ˆr is the truncated Fourier sine series of r the perturbation of the density fro. Conclusions The recovery of an approxiation to a syetric density for a nonhoogeneous ebrane is possible given a liited nuber of eigenvalues. It ay be possible to apply the ethod to nonsyetric densities through an appropriate choice of the vector space W The success of the ethod relies on the assuption of a sall perturbation fro the density ρ The issue of the nonexistence of ultiple eigenvalues for the ρ case reains unaddressed. Knobel and McLaughlin relied heavily on the syetry of q to address this issue, and it is anticpated that their result can be applied to this proble. REFERENCES Barcilon, V., 99, A two-diensional inverse eigenvalue proble, Inverse Probles, Vol. 6, pp Copyright 999 by ASME

6 Figure 7. The recovery of ρ using eigenvalues Figure 9. The recovery of ρ using 6 eigenvalues Figure 8. The recovery of ρ using 8 eigenvalues Figure. ˆr, where ˆr is the Fourier sine series of ρ with 8 ters Biegler-König, F. W., 98, A Newton iteration process for inverse eigenvalue probles, Nuer. Math., Vol. 7, pp.9-5. Chu, M. T., 998, Inverse eigenvalue probles, SIAM Review, Vol., pp. -9. Courant, R. and Hilbert, D., 95, Methods of Matheatical Physics, Vol., Interscience. El Badia, A., 989, On the uniqueness of a bi-diensional inverse spectral proble, C. R. Acad. Sci. Paris Ser. I Math, Vol. 8, pp Friedland, S., Nocedal, J. and Overton, M., 987, The forulation and analysis of nuerical ethods for inverse eigenvalue probles, SIAM J. Nuer. Anal., Vol., pp Hald, O. H., 978, The Inverse Stur-Liouville Proble and the Rayleigh-Ritz Method, Matheatics of Coputation, Vol., pp Ji, X., 998, On atrix inverse eigenvalue probles, Inverse Probles, Vol., pp Knobel, R. and McLaughlin, J. R., 99, A reconstruction ethod for a two-diensional inverse eigenvalue proble, Z. Angew. Math. Phys., Vol. 5, pp Love, A. E. H., 9, A Treatise on the Matheatical Theory of Elasticity, Dover. McCarthy, C. M., 999, The inverse eigenvalue proble for a weighted Helholtz equation, in preparation. McLaughlin, J. R., 986, Analytical ethods for recovering coefficients in differential equations fro spectral data, SIAM Review, Vol. 8, pp Nachan, A., Sylvester, J. and Uhlann, G., 988, An n-diensional Borg-Levinson Theore, Counications in Matheatical Physics, Vol. 5, pp Copyright 999 by ASME

7 Table. Eigenvalue Data i ; n i< i= λ i λ i ; ρ = λ i ; ρ = (, (, (, (, (, (, (, (, (, (, (, (, (5, (, (, (5, Seidan, T., 988, An inverse eigenvalue proble with rotational syetry, Inverse Probles, Vol., pp Copyright 999 by ASME