## Inverse problems in partial differential equations

### Abstract (summary)

A brief survey and a literature review of inverse problems in partial differential equations are given in Chapter 1.

Because most inverse problems are ill-posed in the Hadamard's sense, in Chapter 2 we apply the Tikhonov's regularization method to an abstract inverse problem to obtain an approximate solution of the inverse problem, which has the following properties: (1) If there is a solution to the inverse problem, then the approximate solution converges to a true solution to the inverse problem as a certain parameter goes to zero. (2) If the above-mentioned parameter is small enough, then the approximate solution depends continuously on data.

These results are applied to an inverse problem involving a second-order hyperbolic equation. The coefficients of the highest-order derivatives are estimated. We use spline approximations of parameter functions in a numerical implementation.

In Chapter 3 we consider an inverse problem with equality constraints on the parameters to be determined. We use the Duboviskii-Milyutin optimization lemma in a local convex topological space to obtain a necessary condition of existence for optimal parameters. We compute a descent direction cone, a feasible direction cone, a tangent direction cone, and their dual cones. The functions obtained from those dual cones satisfy the Euler-Lagrange equation, from which we can obtain a useful computational condition for existence.

The above results are applied to an inverse problem for an elliptic equation. Finite-element approximations of the parameter functions are used in this case.

In Chapter 4 we consider an approximate solution to an inverse problem involving a quasilinear parabolic equation. We compute second-order Frechet derivative of the state with respect to the parameters, give a computational method to obtain an approximate solution to the inverse problem, and prove that the sequence of the approximate solutions converges to the true solution of the inverse problem as a constant parameter goes to zero. The method is illustrated with a numerical example.