Inverse Problems And Numerical Solutions Of Partial Differential Equations Based On Diffusion Process

The diffusion processes are important and have extensive applications in the fields of life science,material science,environmental science and so on.Mathematically,the diffusion processes can be modeled by the initial-boundary value problems for parabolic equations.In many cases,the physical mechanism of the diffusion process is clear(the mathematical model is diffusion equation),but some information describing the diffusion process is unknown,such as initial distribution,source term,diffusion coefficient.At this time,it is necessary to determine this unknown information from the additional information,and then determine the whole diffusion process.These problems belong to the category of inverse problems for partial differential equations.Based on the mathematical models of classical diffusion system,slow diffusion system with time fractional derivative and ultra slow diffusion system with distributed order time-fractional derivative,this thesis aims to study the(nonlinear)linear inverse source problem for classical diffusion,the inverse problem of simultaneous reconstruction of boundary impedance coefficient and space-dependent source term for a slow diffusion system and the linear inverse source problem for ultra slow diffusion.Firstly,we consider the identification of the non-smooth radiative coefficient in heat equation.This inverse problem is nonlinear,with the average measurement of temperature field in some time interval as the inversion input.The physical motivation for such an observation is that the average measurement is common in science and engineering applications for weakening the random observation errors.Meanwhile,the composite medium in heat process modeled by a parabolic equation with discontinuous zero order coefficient which has important physical background.We establish the uniqueness for this nonlinear inverse problem,based on the known uniqueness result for linear inverse source problem.To solve the inverse problem from a nonlinear operator equation,the differentiability and the tangential condition of this nonlinear map are investigated.An iterative process called two-point gradient method is proposed by minimizing data-fit term and the penalty term alternatively,with rigorous convergence analysis in terms of the tangential condition.Numerical simulations are presented to illustrate the effectiveness of the proposed method.Secondly,we consider an initial-boundary value problem for the time-fractional diffusion equation with inhomogeneous Robin boundary condition.There are sufficient research works on the direct problems with Dirichlet boundary conditions,however,the literatures are rare about the well-posedness of the direct problem with inhomogeneous Robin boundary conditions.We show the unique existence of the weak/strong solution based on the eigenfunction expansions,which ensures the well-posedness of the direct problem.The Hopf lemma for time-fractional diffusion operator,generalizing the counterpart for the classical parabolic equation is established.Based on this new Hopf lemma,the maximum principles for this time-fractional diffusion are proven,which play essential roles for further studying the uniqueness of the inverse problems corresponding to this system.Thirdly,we consider an inverse problem of simultaneous reconstruction of boundary impedance coefficient and space-dependent source term from the final measurement data for a slow diffusion system,which is governed by a diffusion equation with time-fractional order derivative and Robin boundary condition.We prove the uniqueness of this inverse problem by the maximum principle for the slow diffusion system.A regularizing scheme combining the mollification method and the Tikhonov regularization is proposed to recover the two unknowns,with a rigorous analysis on the choice strategies for the regularizing parameters and the error estimates on the regularizing solutions,revealing the error propagation effects due to recovering the boundary Robin coefficient firstly.Fourthly,we consider an inverse problem of recovering the time-dependent internal source from a nonlocal integral observation for ultra-slow diffusion process governed by distributed-order time-fractional diffusion equation.The solvability of the inverse problem is equivalent to the second kind of operator equation and the uniqueness of the source reconstruction using observation data in a finite time interval is proven based on the energy estimate.Numerically,the reconstruction problem is reformulated as a minimization problem involving a Tikhonov regularizing term.By deriving the explicit representation of the gradient of the cost functional in terms of an adjoint problem,the conjugate gradient method is applied to obtain the numerical solution.Finally,the inverse source problem for parabolic system using parametric approximations,where deep neural networks(DNNs)are used to approximate the solution of the inverse problem.In recent years,data-driven method has been widely used in solving direct or inverse problems of partial differential equations and it can effectively break the curse of dimensionality for high-dimensional space.We prove generalization error estimates depending on training errors and data noise levels by establishing conditional stability of the inverse problem.Following our analysis,we propose a new loss function involving the derivative of the residuals for PDE and measurement data.These extra fit terms adding higher regularity requirements can be understood as the regularizing penalty specified to the solution of inverse problems,dealing with the ill-posedness of the problem.Using these regularization terms,we develop reconstruction schemes and demonstrate the effectiveness of our proposed methodology on a number of test problems.