Fourier Spectral Method for Analytic Continuation
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.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1073381Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1105
 R. G. Airapetyan and A. G. Ramm, Numerical inversion of the Laplace transform from the real axis, J. Math. Anal. Appl., 248 (2000), pp. 572- 587.
 A. Carasso, Determining surface temperatures from interior observations, SIAM J. Appl. Math., 42 (1982), pp. 558-574.
 H.W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse problems, Kluwer Academic, Dordrecht, 1996.
 B.Y. Guo and C.L. Xu, Hermite pseudospectral method for nonlinear partial differential equations, Math. Model. Numer. Anal. 34 (2000) 859- 872.
 B.Y. Guo, J. Shen and C.L. Xu, Spectral and pseudospectral approximations using Hermite functions: application to the Dirac equation, Adv. Comput. Math., 19(2003), 35-55.
 C. L. Epstein, Introduction to the Mathematics of Medical Imaging, 2nd ed., SIAM, Philadelphia, 2008.
 C. L. Fu, F. F. Dou, X. L. Feng, and Z. Qian, A simple regularization method for stable analytic continuation, Inverse Problems, 24 (2008), 065003.
 C. L. FU, Z. L. Deng, X. L. Feng and F. F. Dou, Modified Tikhonov regularization for stable analytic continuation, SIAM J. Numer. Anal. 47(2009), pp. 2982-3000.
 D. N. Ho and H. Shali, Stable analytic continuation by mollification and the fast Fourier transform, in Method of Complex and Clifford Analysis, Proceedings of ICAM, Hanoi, 2004, pp. 143-152.
 Franklin J 1990 Analytic continuation by the fast Fourier transform SIAM J. Sci. Stat. Comput. 11 112-22
 B. Kaltenbacher,A. Neubauer, and O. Scherzer, Iterative Regularization Methods for Nonlinear Ill-Posed Problems, Radon Series on Computational and Applied Mathematics, de Gruyter, Berlin, 2008.
 A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer-Verlag, New York, 1996.
 A. G. Ramm, The ground-penetrating radar problem, III, J. Inverse Ill- Posed Problems, 8 (2000), pp. 23-30.
 I. Sabba Stefanescu, On the stable analytic continuation with a condition of uniform boundedness, J. Math. Phys., 27 (1986), pp. 2657-2686.
 U. Tautenhahn, Optimality for ill-posed problems under general source conditions, Numer. Funct. Anal. Optim., 19 (1998), pp. 377-398.