{"title":"Fourier Spectral Method for Analytic Continuation","authors":"Zhenyu Zhao, Lei You","volume":51,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":374,"pagesEnd":377,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/9760","abstract":"
The numerical analytic continuation of a function f(z) = f(x + iy) on a strip is discussed in this paper. The data are only given approximately on the real axis. The periodicity of given data is assumed. A truncated Fourier spectral method has been introduced to deal with the ill-posedness of the problem. The theoretic results show that the discrepancy principle can work well for this problem. Some numerical results are also given to show the efficiency of the method.<\/p>\r\n","references":"[1] R. G. Airapetyan and A. G. Ramm, Numerical inversion of the Laplace\r\ntransform from the real axis, J. Math. Anal. Appl., 248 (2000), pp. 572-\r\n587.\r\n[2] A. Carasso, Determining surface temperatures from interior observations,\r\nSIAM J. Appl. Math., 42 (1982), pp. 558-574.\r\n[3] H.W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse\r\nproblems, Kluwer Academic, Dordrecht, 1996.\r\n[4] B.Y. Guo and C.L. Xu, Hermite pseudospectral method for nonlinear\r\npartial differential equations, Math. Model. Numer. Anal. 34 (2000) 859-\r\n872.\r\n[5] B.Y. Guo, J. Shen and C.L. Xu, Spectral and pseudospectral approximations\r\nusing Hermite functions: application to the Dirac equation, Adv.\r\nComput. Math., 19(2003), 35-55.\r\n[6] C. L. Epstein, Introduction to the Mathematics of Medical Imaging, 2nd\r\ned., SIAM, Philadelphia, 2008.\r\n[7] C. L. Fu, F. F. Dou, X. L. Feng, and Z. Qian, A simple regularization\r\nmethod for stable analytic continuation, Inverse Problems, 24 (2008),\r\n065003.\r\n[8] C. L. FU, Z. L. Deng, X. L. Feng and F. F. Dou, Modified Tikhonov\r\nregularization for stable analytic continuation, SIAM J. Numer. Anal.\r\n47(2009), pp. 2982-3000.\r\n[9] D. N. Ho and H. Shali, Stable analytic continuation by mollification and\r\nthe fast Fourier transform, in Method of Complex and Clifford Analysis,\r\nProceedings of ICAM, Hanoi, 2004, pp. 143-152.\r\n[10] Franklin J 1990 Analytic continuation by the fast Fourier transform\r\nSIAM J. Sci. Stat. Comput. 11 112-22\r\n[11] B. Kaltenbacher,A. Neubauer, and O. Scherzer, Iterative Regularization\r\nMethods for Nonlinear Ill-Posed Problems, Radon Series on Computational\r\nand Applied Mathematics, de Gruyter, Berlin, 2008.\r\n[12] A. Kirsch, An Introduction to the Mathematical Theory of Inverse\r\nProblems, Springer-Verlag, New York, 1996.\r\n[13] A. G. Ramm, The ground-penetrating radar problem, III, J. Inverse Ill-\r\nPosed Problems, 8 (2000), pp. 23-30.\r\n[14] I. Sabba Stefanescu, On the stable analytic continuation with a condition\r\nof uniform boundedness, J. Math. Phys., 27 (1986), pp. 2657-2686.\r\n[15] U. Tautenhahn, Optimality for ill-posed problems under general source\r\nconditions, Numer. Funct. Anal. Optim., 19 (1998), pp. 377-398.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 51, 2011"}