Bulletin of the Japan Society for Industrial and Applied Mathematics

We discuss the mathematical analysis in non-destructive techniques and stress the role. For the account, we take one problem of determining subboundaries by means of a stationary heat conduction process. A part of boundary is unknown to be determined from measurements of temperature and heat flux on the rest subboundary. In this problem, we distinguish two schemes: (A) Original continuous forward problem=Discretized forward problem=Inverse problem to the discretized forward problem. (B) Original continuous forward problem=Inverse problem to the continuous forward problem=Discretized inverse problem. As for the second step in the scheme (B), we show the uniqueness and stability for our inverse problem and point usefulness of such results for numerical computations. We assert that the mathematical analysis should eventually synthesize these two schemes.

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Basic characteristics, solution algorithm and current trends in the inverse problems in medicine are generally reviewed. First we deal the basic classification of medical inverse problems in relation to the difference of observation: remote sensing and remote probing typed inverse problems. Second, we describe the several characteristics of the solving method which are used in the medical inverse problems. In doing so, we pay close attention to the structure of observational equation and the method to introduce the a priori information to the inverse solution. Among these, Wiener filter method, Tikhonov regularization method are introduced as they are applied in the medical inverse problems. In the above description, several examples such as computed tomograpy, inverse magnetic encephalography and inverse ECG problems are introduced.

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An earthquake is faulting in the Earth. Seismic waves from it propagate through the Earth, and reach the Earth's surface generating seismic ground motions. Thus, seismologists observe these ground motions using seismographs, and carry out inversion of them for investigating the faulting and the Earth's structure. Hypocenter determination is the most basic inverse problem for the earthquake itself, and has developed into the centroid moment tensor inversion and source process inversion. Seismic tomography and inversion of free oscillation and surface waves are inverse problems for the Earth's structure.

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Elasticity with residual stress is a special class of anisotropic elasticity. We consider the inverse boundary value problem for this elasticity, that is, the problem of determining the residual stress and the Lame parameters of an elasitic body by measuring the displacements and the tractions at the boundary. Mathematically, these measurements made at the boundary are encoded in the so-called Dirichlet to Neumann map. We develop Stroh's formalism for this elasticity and give the formula of the surface impedance tensor, which is a principal part of the Dirichlet to Neunann map. Then we prove that all the components of residual stress and Lame parameters at the boundary can be identified from the Dirichlet to Neumann map. The derivatives of residual stress at the boundary can also be identified. Finally, by a layer stripping algorithm, we give an approximation for residual stress in the interior from measurements at the boundary.

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