An inversion algorithm for recovering a coefficient of Sturm-Liouville operator (Analysis of inverse problems through partial differential equations and related topics)

In this paper, an efficient algorithm for recovering a density of Sturm-Liouville operator from given two spectra is investigated. Based on Lidskii's theorem and Mercer's theorem, we build a sequence of trace formulae which bridge explicitly the density and eigenvalues in terms of nonlinear Fredholm integral equations. Due to intrinsic difficulties on ill-posedness of an inverse spectral problem, a truncated Fourier series regularization method is utilized for reconstructing the unknown density. Moreover, shifted Legendre polynomials are carried to balance the different order of trace formulae. Numerical results are presented to illustrated the effectiveness and stability of the proposed reconstruction algorithm

Solving inverse Sturm-Liouville problems: theory and practice

Theoretical results on the solution of inverse Sturm-Liouville problems generally consider only idealized problems requiring much more data than is available in real applications. Typical theorems describe problems where infinitely many eigenvalues are known exactly, but in most applications we know only approximations of a finite, and usually small, number of eigenvalues. This paper considers how idealized theoretical results may assist practical numerical computation. It also reviews recent progress on a class of numerical methods for inverse Sturm-Liouville problems, it discusses some open questions, and it announces a new convergence result. References L. Aceto, P. Ghelardoni and C. Magherini. Boundary value methods for the reconstruction of Sturm–Liouville potentials. Appl. Math. Comp., 219:2960–2974, 2012. doi:10.1016/j.amc.2012.09.021 A. M. Akhtyamov, V. A. Sadovnichy and Ya. T. Sultanaev. Generalizations of Borg's uniqueness theorem to the case of non- separated boundary conditions. Eurasian Math. J., 3(4):10–22, 2012. http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=emj&paperid=101&option_lang=eng H. Altundag, C. Bockmann and H. Taseli. Inverse Sturm–Liouville problems with pseudospectral methods. Int. J. Comput. Math., 92:1373–1384, 2015. doi:10.1080/00207160.2014.939646 A. L. Andrew. Asymptotic correction of Numerov's eigenvalue estimates with natural boundary conditions. J. Comp. Appl. Math., 125:359–366, 2000. doi:10.1016/S0377-0427(00)00479-9 A. L. Andrew. Numerov's method for inverse Sturm–Liouville problems. Inverse Problems, 21:223–238, 2005. doi:10.1088/0266-5611/21/1/014 A. L. Andrew. Asymptotic correction and inverse eigenvalue problems: an overview. ANZIAM J., 46(E):C1–C14, 2005. doi:10.21914/anziamj.v46i0.943 A. L. Andrew. Computing Sturm–Liouville potentials from two spectra. Inverse Problems 22:2069–2081, 2006. doi:10.1088/0266-5611/22/6/010 A. L. Andrew. Finite difference methods for half inverse Sturm–Liouville problems. Appl. Math. Comp., 218:445–457, 2011. doi:10.1016/j.amc.2011.05.085 A. L. Andrew. Inverse Sturm–Liouville problems: some recent developments. ANZIAM J., 52(E):C287–C302, 2011. doi:10.21914/anziamj.v52i0.3888 A. L. Andrew. Convergence of Numerov's method for an inverse Sturm–Liouville problem. In preparation. A. L. Andrew and J. W. Paine. Correction of Numerov's eigenvalue estimates. Numer. Math., 47:289–300, 1985. MR 86j:65101. doi:10.1007/BF01389712 L. Efremova and G. Freiling. Numerical solution of inverse problems for Sturm–Liouville operators with discontinuous potentials. Cent. Eur. J. Math. 11:2044–2051, 2013. doi:10.2478/s11533-013-0301-1 Q. Gao. Decent [descent] flow methods for inverse Sturm–Liouville problem. Appl. Math. Modelling 36:4452–4465, 2012. doi:10.1016/j.apm.2011.11.070 Q. Gao, X. Cheng and Z. Huang. Modified Numerov's method for inverse Sturm–Liouville problems. J. Comp. Appl. Math. 253:181–199, 2013. doi:10.1016/j.cam.2013.04.025 Q. Gao, X. Cheng and Z. Huang. On a boundary value method for computing Sturm–Liouville potentials from two spectra. Internat. J. Comp. Math. 91:490–513, 2014. doi:10.1080/00207160.2013.790535 Q. Gao, Z. Huang and A. Cheng. A finite difference method for an inverse Sturm–Liouville problem in impedance form. Numer. Algorithms 70:669–690, 2015. doi:10.1007/s11075-015-9968-7 Q. Gao, Q. Zhao, X. Zheng and Y. Liung. Convergence of Numerov's method for inverse Sturm–Liouville problems. Appl. Math. Comp., 293:1–17, 2017. doi:10.1016/j.amc.2016.08.007 O. H. Hald. The inverse Sturm–Liouville problem and the Rayleigh–Ritz method. Math. Comp., 32:687–705, 1978. doi:10.1090/S0025-5718-1978-0501963-2 C.-S. Liu. Solving an inverse Sturm–Liouville problem by a Lie-group method. Bound. Value Probl. 2008:749865, 2008. doi:10.1155/2008/749865 C.-S. Liu and S. N. Atluri. A novel fictitious time integration method for solving the discretized inverse Sturm–Liouville problems, for specified eigenvalues. CMES Comput. Model. Eng. Sci. 36:261–285, 2008. doi:10.3970/cmes.2008.036.261 B. D. Lowe, M. Pilant and W. Rundell. The recovery of potentials from finite spectral data. SIAM J. Math. Anal. 23:482–504, 1992. doi:10.1137/0523023 J. W. Paine, F. R. de Hoog and R. S. Anderssen. On the correction of finite difference eigenvalue approximations. Computing 26:123–139, 1981. doi:10.1007/BF02241779 J. D. Pryce. Numerical solution of Sturm–Liouville problems. Oxford University Press, 1993. M. Rafler and C. Bockmann. Reconstructive method for inverse Sturm–Liouville problems with discontinuous potentials. Inverse Problems 23:933–946, 2007. doi:10.1088/0266-5611/23/3/006 A. Rattana and C. Bockmann. Matrix methods for computing eigenvalues of Sturm–Liouville problems of order four. J. Comp. Appl. Math. 249:144–156, 2013. doi:10.1016/j.cam.2013.02.024 W. Rundell and P. E. Sacks. Reconstruction techniques for classical Sturm–Liouville problems. Math. Comp. 58:161–183, 1992. doi:10.1090/S0025-5718-1992-1106979-0 A. M. Savchuk. Reconstruction of the potential of the Sturm–Liouville operator from a finite set of eigenvalues and normalizing constants. Math. Notes 99:715–228, 2016. doi:10.1134/S0001434616050102 A. M. Savchuk and A. A. Shkalikov. On the eigenvalues of the Sturm–Liouville operator with potentials from Sobolev spaces. Math. Notes 80:814–832, 2006. doi:10.1007/s11006-006-0204-