Inverse Problems And Applications For Time Fractional Diffusion Models

The diffusion process is an important phenomenon in nature,with wide application background in many applied engineering areas such as life science,material science and environment protections.The diffusion parameters reconstructions based on PDE models,are essentially the mathematical analysis and numerical solutions for non-standard PDE problems.Due to the combinations of nonlinearity and ill-posedness for such kinds of problems,the regularizing techniques should be introduced for yielding the stable solutions,while the nonlinearities of the problems should also be treated carefully.This thesis aims to the researches on the media detection and reconstructions based on the diffusion processes,with the mathematical models by some inverse problems for classical parabolic equations,time fractional PDE equations and distributed order time fractional PDE equations.Our main work is to apply and develop the mathematical analysis and numerical algorithms for the inverse problems in PDEs.Firstly,suppose the internal source in the time fractional equation is F(x,t)= f(x)?(t),we consider to identify the space-dependent source function f(x)from the Neumann data on the partial boundary domain.In Chapter 3,based on the variational identity connecting the inversion input data and unknown source function,we provide a new method to consider the conditional stability for the inverse problem in the sense of weak norm.Compared with the traditional Carleman estimate,our method is also useful for the cases of general fractional order ? ?(0,1)and general time-dependent source function ? ? C[0,T ].Secondly,we consider the inverse time-dependent source problem for distributed order time fractional equation.Suppose the internal source in the government equation is F(x,t)= f(x)?(t),we consider to identify the time-dependent source function ?(t)from the interior measurement data observed on some partial internal domain.In Chapter 4,we analyze the properties of the solution to distributed order time fractional ODE,by which we can prove the well-posedness of the distributed order time fractional PDE in the sense of strong solution.On the other hand,we prove the well-posedness of its adjoint problem in the sense of weak solution.Then,similarly,under the conditions of strong solution to direct problem and weak solution to its adjoint problem,we construct the variational identity and prove the conditional stability for the inverse problem.Thirdly,we consider the recovery of the weight function in distributed time-fractional diffusion system using the interior measurement in Chapter 5,which arises in some ultraslow diffusion phenomena.Due to the nonlinear and nonlocal dependance of the measurement data on the weight function,such an inverse problem is novel and important.Based on the regularities of the direct problems for general diffusion equation with distributed time fractional derivative,we establish the theoretical framework for the optimization version of the inverse problem,including existence of the minimizer,the differentiability of the cost functional,as well as the gradient of the cost functional,in suitable functional spaces.At the same time,for the above three kinds of inverse problems,gradient type iteration algorithm is applied to carry out the numerical inversion.Numerical examples verify the validity and accuracy of our methods.Furthermore,we can also find that the reconstructions for time-dependent source function from the time-series measurements are very satisfactory,while the reconstructions for space-dependent source or fractional-order dependent weight function are relatively difficult.One possible reason is the ill-posedness among these three kinds of inverse problems are different.Another reason is the behavior of inverse problem is closely related to the way of observing inversion input data.Finally,Chapter 6 is devoted to consider the linearized diffusion model for fluorescence imaging.To quantify fluorescence imaging of biological tissues under flat excitation and detection probes,we need to solve an inverse problem for the coupled radiative transfer equations which describe the excitation and emission fields in biological tissue.By transforming the radiative transfer system into coupled diffusion equations for the averaged fields,we have a nonlinear inverse problem to identify the absorption coefficient for a fluorophore by this system with impedance boundary condition,using some time resolved measurement data.For the linearized inverse problem obtained by ignoring the absorbing effect of fluorophores on the excitation field,we firstly establish the estimates of errors on the excitation field and the solution to the inverse problem,which ensures the reasonability of the model approximation qualitatively.Some numerical verifications are presented to show the validity of such a linearizing process quantitatively.Then,based on the analytic expressions of excitation and emission fields,the identifiability of the absorption coefficient from the linearized inverse problem is rigorously analyzed for the absorption coefficient in special form,revealing the physical difficulty of the 3-dimensional imaging model by the back scattering diffusive system.