Consider an inverse problem of reconstructing the coefficient in a linearwave equation on an inhomogeneous slab with density ρ(z) and wave velocity c(z). The inversioninput information is the reflection and transmiss...Consider an inverse problem of reconstructing the coefficient in a linearwave equation on an inhomogeneous slab with density ρ(z) and wave velocity c(z). The inversioninput information is the reflection and transmission data corresponding to a point source. Byapplying the characteristic theory for hyperbolic equations, we establish an integral system fromwhich ρ(z) and c(z) can be recovered simultaneously. In contrast to some known results, our inverseapproach is carried out for depth variable, rather than for travel-time variable. Thereforeinversion results in this paper are more appropriate for the physical interpretation of a mediumslab.展开更多
This paper considers a model for the growth of a solid tumor with a single anticancer agent application.The model is a free boundary problem of a nonlinear reaction-diffusion-advection equation,where the free boundary...This paper considers a model for the growth of a solid tumor with a single anticancer agent application.The model is a free boundary problem of a nonlinear reaction-diffusion-advection equation,where the free boundary is the surface of the tumor.Since multicellular spheroids are routinely used as in vitro(i.e.,outside live organisms)models of cancer growth and they can be observed and controlled in the laboratory,the following inverse problem is studied:given observed dynamics of tumor growth,a certain parameter is determined.The Lipschitz stability of solutions to the above-mentioned inverse problem is established,and this inverse problem is solved by control theory.Numerical methods for solving the inverse problem are also given.展开更多
文摘Consider an inverse problem of reconstructing the coefficient in a linearwave equation on an inhomogeneous slab with density ρ(z) and wave velocity c(z). The inversioninput information is the reflection and transmission data corresponding to a point source. Byapplying the characteristic theory for hyperbolic equations, we establish an integral system fromwhich ρ(z) and c(z) can be recovered simultaneously. In contrast to some known results, our inverseapproach is carried out for depth variable, rather than for travel-time variable. Thereforeinversion results in this paper are more appropriate for the physical interpretation of a mediumslab.
基金Natural Science Foundation of Shanghai,China(No.09ZR1401200)
文摘This paper considers a model for the growth of a solid tumor with a single anticancer agent application.The model is a free boundary problem of a nonlinear reaction-diffusion-advection equation,where the free boundary is the surface of the tumor.Since multicellular spheroids are routinely used as in vitro(i.e.,outside live organisms)models of cancer growth and they can be observed and controlled in the laboratory,the following inverse problem is studied:given observed dynamics of tumor growth,a certain parameter is determined.The Lipschitz stability of solutions to the above-mentioned inverse problem is established,and this inverse problem is solved by control theory.Numerical methods for solving the inverse problem are also given.
