Dissertation > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations

Multi-Scale Inversion Methods for Inverse Problem of Wave Equations

Author LiuTao
Tutor ZhaoJingXia
School Harbin Engineering University
Course Applied Mathematics
Keywords inverse problem of wave equations multi-scale inversion homotopy method trust region technique
CLC O175
Type Master's thesis
Year 2009
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Inverse problem of wave equations is applied widely in many areas. It has nonlinear and ill-posed difficulties in essence and heavy computation in practice. Thus, the survey of inverse problem of wave equation and its numerical inversion method has meaningful signifycance in theory and value in practice.Facing the characters of inverse problem of wave equations and the difficulties of its numerical inversion method, this paper consider inverse problem of 2-D wave equations as concrete model. By combining with multigrid method and introducing it to the course of numerical inversion of 2-D wave equations, the performance of mono-scale inversion methods is improved. Multi-scale inversion methods are constructed which can decrease computation cost greatly.The inverse problem of 2-D wave equations can be transferred to a nonlinear optimization problem by employing Tikhonov regularization method for solving ill-posed problem. But the calculation quantity of the mono-scale inversion method which solves it is great.The paper studies the multigrid so as to reduce the computation cost and enhance the capability for solving large-scale inverse problem of wave equations of numerical inversion methods. Based on information of gradient, widely convergent multi-scale inversion algorithms are formed by using the proposed mono-scale methods as smoothness methods on some fix grid.Based on the formed widely convergent multi-scale inversion algorithms, a widely convergent multi-scale trust region inversion algorithm which can reduce the computational work to some extent is formed by using trust region method and adjusting stopping criterion for iteration of mono-scale methods. And convergence of these multi-scale inversion algorithms is discussed in theory.Numerical simulations of 2-D wave equations inversion are carried out by constructed inversion algorithms, where the point epicenter is applied to layered medium and mediums with mono-abnormity and multi-abnormity. The constructed algorithm is analyzed on the base of practical computational results.The analysis of results illuminate the convergence and efficiency of constructed multi-scale inversion methods and indicate that the methods have better adaptation to get over the many difficulties of wave equations inversion to some extent and obvious innovation in theory. And the comparison between results of two constructed multi-scale inversion methods indicates that the new methods can further reduce the computational work.Because of agility and practicality of the multi-scale inversion algorithms, the survey result in the paper is popular and easy to adapt for many other fields.

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