Solution of the Cauchy problem for the wave equation using iterative regularization

IF 1.1 4区工程技术 Q3 ENGINEERING, MULTIDISCIPLINARY

Inverse Problems in Science and Engineering Pub Date : 2021-12-23 DOI:10.1080/17415977.2021.1949590

M. Alosaimi, D. Lesnic, B. Johansson

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引用次数: 4

Abstract

We propose a regularization method based on the iterative conjugate gradient method for the solution of a Cauchy problem for the wave equation in one dimension. This linear but ill-posed Cauchy problem consists of finding the displacement and flux on a hostile and inaccessible part of the medium boundary from Cauchy data measurements of the same quantities on the remaining friendly and accessible part of the boundary. This inverse boundary value problem is recast as a least-squares minimization problem that is solved by using the conjugate gradient method whose iterations are stopped according to the discrepancy principle for obtaining stable reconstructions. The objective functional associated is proved Fréchet differentiable and a formula for its gradient is derived. The well-posed direct, adjoint and sensitivity problems present in the conjugate gradient method are solved by using a finite-difference method. Two numerical examples to illustrate the accuracy and stability of the proposed numerical procedure are thoroughly presented and discussed.

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用迭代正则化方法求解波动方程的柯西问题

提出了一种基于迭代共轭梯度法的正则化方法，用于求解一维波动方程的柯西问题。这个线性但不适定的柯西问题包括从柯西数据测量的剩余友好和可达部分上的相同量的介质边界的敌对和不可达部分上的位移和通量。将该反边值问题转化为最小二乘最小化问题，采用共轭梯度法求解，并根据差异原理停止迭代以获得稳定的重构。证明了目标函数的可微性，并推导了目标函数的梯度公式。用有限差分法解决了共轭梯度法中存在的适定的直接、伴随和灵敏度问题。文中还详细讨论了两个数值算例，以说明所提出的数值计算方法的准确性和稳定性。

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来源期刊

Inverse Problems in Science and Engineering 工程技术-工程：综合

自引率

0.00%

发文量

审稿时长

6 months

期刊介绍： Inverse Problems in Science and Engineering provides an international forum for the discussion of conceptual ideas and methods for the practical solution of applied inverse problems. The Journal aims to address the needs of practising engineers, mathematicians and researchers and to serve as a focal point for the quick communication of ideas. Papers must provide several non-trivial examples of practical applications. Multidisciplinary applied papers are particularly welcome. Topics include: -Shape design: determination of shape, size and location of domains (shape identification or optimization in acoustics, aerodynamics, electromagnets, etc; detection of voids and cracks). -Material properties: determination of physical properties of media. -Boundary values/initial values: identification of the proper boundary conditions and/or initial conditions (tomographic problems involving X-rays, ultrasonics, optics, thermal sources etc; determination of thermal, stress/strain, electromagnetic, fluid flow etc. boundary conditions on inaccessible boundaries; determination of initial chemical composition, etc.). -Forces and sources: determination of the unknown external forces or inputs acting on a domain (structural dynamic modification and reconstruction) and internal concentrated and distributed sources/sinks (sources of heat, noise, electromagnetic radiation, etc.). -Governing equations: inference of analytic forms of partial and/or integral equations governing the variation of measured field quantities.