Some Studies On The Uniqueness And Algorithms Of Inverse Problems For The Time-fractional Diffusion-wave Equations

In this thesis,we consider several inverse problems in time fractional diffusionwave equations(TFDWE),i.e.inverse initial value problem,fractional order and zeroth-order coefficient inverse problem,fractional order and advection coefficient inverse problem,and a preconditioned alternating direction method of multipliers is proposed with application to the inverse source problem.In Part 1,we study the problem of recovering two initial values for a timefractional diffusion-wave equation from boundary Cauchy data.We provide the uniqueness result for recovering two initial values simultaneously by the method of Laplace transformation and analytic continuation.And then we use a non-stationary iterative Tikhonov regularization method to solve the inverse problem and propose a finite dimensional approximation algorithm to find a good approximation to the initial values.Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed method.In Part 2,we investigate an inverse problem of recovering the zeroth-order coefficient and fractional order simultaneously in a time-fractional reaction-diffusion-wave equation by using boundary measurement data from both of uniqueness and numerical method.We prove the uniqueness of the considered inverse problem and the locally Lipschitz continuity of the forward operator.Then the inverse problem is formulated into a variational problem by the Tikhonov type regularization.Based on the continuity of the forward operator,we prove that the minimizer of the Tikhonov type functional exists and converges to the exact solution under an a priori choice rule of regularization parameter.The steepest descent method combined with Nesterov acceleration is adopted to solve the variational problem.Three numerical examples are presented to support the efficiency and rationality of our proposed method.In Part 3,the work is concerned with inverse problem of recovering the spacedependent advection coefficient and the fractional order in a one-dimensional timefractional reaction-advection-diffusion-wave equation.Based on a transformation,the original equation can be changed into a new form without an advection term.Then we can show the uniqueness of recovering the fractional order and the zeroth order coefficient which contains the information of the ”original” advection coefficient by the observation data at two end points.Under the theory of first-order ordinary differential equations,we obtain the uniqueness result of the advection coefficient.Lastly,we solve the inverse problem numerically from Bayes perspective by using iterative regularizing ensemble Kalman method,and numerical examples are presented to show the effectiveness of the proposed method.In Part 4,we propose a residual method for solving linear inverse problems.To implement this method,a preconditioned alternating direction method of multipliers(ADMM)is given.Since the observed data is always noise-contaminated,we introduce a new variable to replace the noisy data and then impose additional constraint on the new variable.Thus the inverse problem is totally converted into an optimization problem.Without taking account of the existence of the Lagrange multiplier,we provide the convergence result of the proposed method.Finally we apply this method to the inverse source problem in TFDWE.Numerical examples are presented to show the efficiency of the proposed method.