From the introduction: The goal of this work is the development of techniques for the efficient numerical solution of inverse problems, based on adaptive finite element methods. After the statement of the problem in Chapter 1, we derive a posteriori error estimates for inverse problems in Chapter 2, both for natural “energy type” quantities as well as for general functionals, and demonstrate their efficiency.
A second, new aspect of this work is the inclusion of bounds into the solution process in Chapter 3. In practical applications, physical upper and lower bounds on possible values of the unknown coefficients are usually available, either from prior knowledge of the particular case under investigation, or from extremal material properties existing in nature. In practice, such bounds are usually mugh tighter, and alike bounds are available for other properties as well, such as elasticity coefficients. The efficient inclusion of such bounds is discussed in Chapter 3 where we develop an active set method in a continuous setting and show its efficiency in enhancing stability of identified coefficients.
In Chapter 4, we extend the problem under consideration to the case that more than just one measurement is available. This can be favorably used to suppress the effects of measurement noise, and examples of this are shown. It also allows to solve certain classes of problems in which one measurement is not sufficient to identify the unknown coefficient. Beyond the already high computational requirements for distributed parameter identification in partial differential equations, multiple measurements increase it even more. This requires using specialized algorithms tailored to the problem. However, their structure allows for efficient parallelization strategies, for example using clusters of computers. The work required for each of the subproblems associated with one measurement is thus distributed to different computers. The structure of a program doing this will be introduced in Chapter 4.
The techniques developed thus far at the Laplace equation will be applied to parameter identification problems for the Helmholtz equation in Chapter 5. Since Helmholtz’s equation is the frequency domain version of the wave equation, parameter estimation for this type of problems has many applications in geophysics. It is shown that adaptive techniques and error estimation work in this context as well, and that they lead to very efficient schemes. The most complex problems of this work is considered in this chapter.