The inverse problem for the 1-dimensional acoustic wave equation is discussed to deter-mine propagation velocity from impulse response. A relation between the propagation velocityand the wavefield can be established f...The inverse problem for the 1-dimensional acoustic wave equation is discussed to deter-mine propagation velocity from impulse response. A relation between the propagation velocityand the wavefield can be established from the analysis of propagation of discontinuities forhyperbolic equations. As a result, the inverse problem discussed in this paper is reduced to aparticular initial value problem of a semilinear system of P. D. E.. The Picard iteration forsolving this initial value problem is constructed and the convergence of iteration is proved.The main results are the following: (i) the propagation velocity can always be recovered fromthe impulse response, unless the inverse problem contains a singular point, where the propa-gation velocity is infinite or zero, or its total variation in the neighborhood of the singularpoint is infinite; (ii) the stability behaviour of the solutions of this inverse problem is es-sentially dependent on the total variation of logarithm of propagation velocity.展开更多
We consider a three-point boundary value problem for operators such as the one-dimensional p-Laplacian, and show when they have solutions or not, and how many. The inverse operators are given by various formulas invol...We consider a three-point boundary value problem for operators such as the one-dimensional p-Laplacian, and show when they have solutions or not, and how many. The inverse operators are given by various formulas involving zeros of a real-valued function. They are shown to be orderpreserving, for some parameter values, and non-singleton valued for others. The operators are shown to be m-dissipative in the space of continuous functions.展开更多
基金Project supported by National Natural Science Foundation of China.
文摘The inverse problem for the 1-dimensional acoustic wave equation is discussed to deter-mine propagation velocity from impulse response. A relation between the propagation velocityand the wavefield can be established from the analysis of propagation of discontinuities forhyperbolic equations. As a result, the inverse problem discussed in this paper is reduced to aparticular initial value problem of a semilinear system of P. D. E.. The Picard iteration forsolving this initial value problem is constructed and the convergence of iteration is proved.The main results are the following: (i) the propagation velocity can always be recovered fromthe impulse response, unless the inverse problem contains a singular point, where the propa-gation velocity is infinite or zero, or its total variation in the neighborhood of the singularpoint is infinite; (ii) the stability behaviour of the solutions of this inverse problem is es-sentially dependent on the total variation of logarithm of propagation velocity.
文摘We consider a three-point boundary value problem for operators such as the one-dimensional p-Laplacian, and show when they have solutions or not, and how many. The inverse operators are given by various formulas involving zeros of a real-valued function. They are shown to be orderpreserving, for some parameter values, and non-singleton valued for others. The operators are shown to be m-dissipative in the space of continuous functions.
