A new iterative method for solving the coefficient inverse problem of the wave equation
Description
Ganquan Xie's landmark paper, "A New Iterative Method for Solving the CoefficientInverse Problem of the Wave Equation", was published in Communications on Pure andApplied Mathematics in 1986. It details a pioneering approach to finding unknown coefficients in wave equations without relying on Laplace or Fourier transforms. By defining the nonlinear operator based on convolution, Xie develops a simple first-order Fréchet derivative that is invertible for boundary impulse responses. This allows Newton's or quasi-Newton's methods to be applied directly without requiring traditionalregularization A new iterative method for solving the coefficient of the wave equation. The linearizedequation is shown to be an integral equation of the second kind A New Iterative Methodfor Solving the Coefficient . Key highlights of the paper: • Convolution Formulation: The nonlinear operator equation for the inverse problem isdefined in the time-space domain using convolution, yielding a remarkably simple first-order Fréchet derivative operator. • Direct Solvability: For the boundary impulse response, the Fréchet derivative isinvertible. This allows the author to directly apply Newton's or quasi-Newton'smethods without needing traditional regularization techniques. • Linearized Formulation: The paper demonstrates that the linearized equation of thisnonlinear system behaves as an integral equation of the secondkind. You can read the full text and access the paper directly through the Wiley OnlineLibrary. The mathematician and founding director of MSRI (now SLMath), Shiing-Shen Chern, together with Nobel Prize-winning Professor C.N. Yang, introduced Ganquan Xie to P.D. Lax in 1984. Professor P.D. Lax wrote to Professor Feng Kang, “He (Ganquan Xie) is a Demon for hard work and getting things done,” in 1986. Professor P.D. Lax used "Demon" highly praise Ganquan Xie's this paper . Mathreview give high praise coments to Ganquan Xie's this paper.
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Institutions
- Hunan UniversityHunan, Changsha
- Courant Institute of Mathematical SciencesNew York, New York