Inverse and ill-posed problems

Most of our research deals with inverse problems, an active area with important applications in medicine and natural and engineering sciences. Often, the resulting linear and nonlinear problems lack some of the properties postulated by Hadamard for a problem to be "well-posed". Taking existence and uniqueness (identifiability) of a solution aside, the lack of continuous dependence is the most crucial violation, at least from a numerical analysis point of view; such problems are called "ill-posed". Numerical inversion of an ill-posed problem can lead to an unbounded propagated data error in the computed solution.

To overcome this problem solution methods (called regularization methods) should only be exact to a certain extent, typically depending on some scalar parameter. This regularization parameter is used to control the propagated data error. Depending on the application, the regularization parameter can be some discretization parameter, an intrinsic parameter of a certain method (e.g., in Tikhonov regularization), etc.

In our work, iterative regularization methods where the regularization parameter is the iteration index, have often been the method of choice. The examples include semiiterative methods (the ν-methods) and the method of conjugate gradients for linear problems, and Newton type methods for nonlinear problems, respectively. The linear theory appears to be pretty complete, and is the main subject of our monograph with H.W. Engl and A. Neubauer (Johannes Kepler University of Linz, Austria). More recently, nonlinear problems like the impedance tomography problem, have received increasing attention.