A Number Of Digital Physics Inverse Problem

The inverse problems in mathematical physics originated from various practical problems in physics, biology, medicine and geography, etc.. By mathematical mod-eling, they have become a promising inter-disciplinary region which is characterized by its variety and numerous theories included. Most of inverse problems belong to ill-posed problems. The complete theory and method for inverse problems are not available. Many problems depend on their direct problem studies. In this paper, we deal with some problems which belong to two areas of inverse problems as follows: the unknown boundary detection and the biophysics problems.The whole paper is outlined as follows:1．The conditional stability of unique continuation along an line for an elliptic equation2．The conditional stability in line unique continuation for a wave equation and an inverse wave source problem3．The forward problems to EEG and MEG4．The inverse MEG problem in ovoid geometryIn section 1, we mainly discuss the conditional stability of the unique continu-ation along a line for a kind of high-order elliptic equation with analytic coefficients by using holomorphic extension in C. Such conditional stability means that under an assumption of the priori bound of the solution we can estimate the values of the solution to the elliptic equation on a larger part of the line from the values of the solution on a smaller part of the line. This problem is a kind of new inverse problem originated from the inverse problem of determining the unkonwn boundary. As an application, we apply such a stability result to two equations of elasticity : an isotropic Lamésystem with constant Lamécoefficients and the Kirchhoff plate equation with analytic coefficients. At last, we implement a numerical experiment for a harmonic function in two dimensional space. We intend to verify how such a conditional stability affects our numerical result. As a reult, we obtain an approxi-mate solution with better accuracy. And the numerical result shows that the smaller distance between the point on the long segment and the right end point of the short segment, the higher accuracy of the approximate solution we have。This accords with our theoretical result of the conditional stability of the unique continuation.In section 2, we investigate the corresponding problems for a wave equation and an inverse wave source problem. We obtain a condition stability of the logarithmic type for a wave equation in line by using the Fourier-Bros-Iagolnitzer transformation, and transforming the wave equation problem to Laplace problem. Then we apply it to an inverse wave source problem of determining a spatially varing source term on its extended line by observations of a segment and establish the conditional stability.In section 3, we deal with the forward problems of EEG(electroencephalography) and MEG(magnetoencephalography) which belong to medicine imaging methods. The solution to the forward problem is an important component for stuyding the inverse problem for determining the distribution of the neural sources by the mea-surements of EEG and MEG. The forward problem involves computing the scalp potentials or external magnetic configuration. Under the quasistatic approxima-tion of Maxwell equation, we present some analytic expressions for EEG and MEG problems in nonspherical model and a unified frame for numerical solutions of the forward EEG and MEG problems by weighted residual BEM: collocation and col-location of least square. Then, for a nonspherical model-an ovoid geometry, the numerical experiments in multi-directions arc implemented in a single layer ovoid head model and three-layer ovoid head model by two methods above. We make a detail analysis about the relative errors of EEG and MEG according to the positions and directions of the dipole. And the stable and reliable results about the error of EEG and MEG are obtained in all cases.The discussion of inverse MEG problem is included in section 4．. We approxi-mate a human head by use of a homogeneous ovoid geometry which is more realistic than sphere and ellipsoid, and more easily to deal with than ellipsoid mathemati-cally. By Geselowitz equation, we investigate the relations between the components of primary current and the external magnetic fields using the decomposition of pri-mary current in different coordinates in the case of a special MEG sensor position, which is on z-axis (over the head). In the case of general MEG sensor position, using decomposition of the primary current in spherical coordinates, we obtain an analytic result about the unique determination of the primary current from the mea-surements of scalp potentials and the external magnetic fields by combining MEG with EEG.