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学术报告:Multidimentional Gelfand-Levitan-Krein Equations and Inverse Hyperbolic Problems

报告题目:Multidimentional Gelfand-Levitan-Krein Equations and Inverse Hyperbolic Problems




报告摘要:Hyperbolic equations describing the wave processes are of great concern in many domains of applied mathematics. Waves come through object and deliver information about its structure to the surface. Solutions of hyperbolic equations can contain non-smooth and singular components. This leads to easier (compared with elliptic and parabolic cases) inversion of the operator. Usually inverse problems for hyperbolic equations are solved by minimizing the residual functional. Iterative method of minimizing the functional requires the solution of the direct (and, perhaps, adjoint) problem for every iteration of the method. In multidimensional case iterative methods for multidimensional inverse problems are very timeconsuming.e will consider several examples, including - Multidimensional nonlinear acoustic inverse problem (S.I. Kabanikhin, A.D. Satybaev, M.A. Shishlenin, 2004)
- Recovering of the Lame parameters and density of the medium (A.S. Alekseev, 1967; V.S. Belonosov, A.S. Alekseev, 1998)
- Solving the GLM-equations for obtaining the solution of the nonlinear Schrodinger equation  (D.A. Shapiro, 2011; R.G. Novikov, 2014; S.K. Turitsyn, 2015).
- Method of inverse scattering problem: integrating nonlinear equations (C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, 1967): KdF (1D) and Kadomtcev-Petviashvili (2D, V.E. Zakharov and A.B. Shabat, 1974).
报告人简介:谢尔盖·卡巴尼瀚,俄罗斯科学院通讯院士,现为俄罗斯科学院西伯利亚分院计算数学与数学地球物理研究所所长。长期从事数学反问题的研究工作,在电磁学、梯度方法的强收敛性、非线性抛物问题算法设计、多维Gelfand-Levitan and Krein 方程反问题等诸多问题方面,做出了开创性的工作。

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