## 68Q25 Analysis of algorithms and problem complexity [See also 68W40]

If gradient based derivative algorithms are used to improve industrial products by reducing their target functions, the derivatives need to be exact.
The last percent of possible improvement, like the efficiency of a turbine, can only be gained if the derivatives are consistent with the solution process that is used in the simulation software.
It is problematic that the development of the simulation software is an ongoing process which leads to the use of approximated derivatives.
If a derivative computation is implemented manually, it will be inconsistent after some time if it is not updated.
This thesis presents a generalized approach which differentiates the whole simulation software with Algorithmic Differentiation (AD), and guarantees a correct and consistent derivative computation after each change to the software.
For this purpose, the variable tagging technique is developed.
The technique checks at run-time if all dependencies, which are used by the derivative algorithms, are correct.
Since it is also necessary to check the correctness of the implementation, a theorem is developed which describes how AD derivatives can be compared.
This theorem is used to develop further methods that can detect and correct errors.
All methods are designed such that they can be applied in real world applications and are used within industrial configurations.
The process described above yields consistent and correct derivatives but the efficiency can still be improved.
This is done by deriving new derivative algorithms.
A fixed-point iterator approach, with a consistent derivation, yields all state of the art algorithms and produces two new algorithms.
These two new algorithms include all implementation details and therefore they produce consistent derivative results.
For detecting hot spots in the application, the state of the art techniques are presented and extended.
The data management is changed such that the performance of the software is affected only marginally when quantities, like the number of input and output variables or the memory consumption, are computed for the detection.
The hot spots can be treated with techniques like checkpointing or preaccumulation.
How these techniques change the time and memory consumption is analyzed and it is shown how they need to be used in selected AD tools.
As a last step, the used AD tools are analyzed in more detail.
The major implementation strategies for operator overloading AD tools are presented and implementation improvements for existing AD tools are discussed.\
The discussion focuses on a minimal memory consumption and makes it possible to compare AD tools on a theoretical level.
The new AD tool CoDiPack is based on these findings and its design and concepts are presented.
The improvements and findings in this thesis make it possible, that an automatic, consistent and correct derivative is generated in an efficient way for industrial applications.

We propose a new discretization scheme for solving ill-posed integral equations of the third kind. Combining this scheme with Morozov's discrepancy principle for Landweber iteration we show that for some classes of equations in such method a number of arithmetic operations of smaller order than in collocation method is required to appoximately solve an equation with the same accuracy.