## Industrial Mathematics - A Key Technology for the Future

Mathematics, as the language of the sciences, has always played an important role in technology, and now is applied also to a variety of problems in commerce and the environment.

More and more companies recognize that computer simulations may replace experiments in their product design to give both reduced costs and flexibility. These simulations leading to the optimization of the manufacturing process itself, require mathematical skills if they are to be used effectively. They involve problem identification, a mathematical formulation, and mathematical/ numerical analysis to reduce the problem to its simplest form for computation; a procedure called *Mathematical Modelling*.

These new demands on mathematics have stimulated academic interest in Industrial Mathematics and many mathematical groups world-wide are committed to interaction with industry as part of their research activities.

In Austria, already in 1988 a Chair for Industrial Mathematics has been set up at the Johannes Kepler Universität Linz. Since 1997 the Chair for Industrial Mathematics is an own Institute. Its founder and first head until 2007 was Prof. Heinz Engl.

## Basic Research: Inverse Problems

Many problems in practice are so-called inverse problems: they are concerned with determining causes for a desired or an observed effect (e.g., computerized tomography, parameter identification, inverse heat conduction or diffusion problems).

Inverse problems most often are *ill-posed*. A consequence is that arbitrarily small changes in the data may lead to arbitrarily large changes in the solution. As in the numerical treatment of inverse problems data errors are inevitable, one has to use stabilizing procedures for successfully dealing with ill-posed problems, so-called *regularization methods*.

The Industrial Mathematics Institute has a long tradition in analysing regularization schemes. While in the eighties the emphasis lay on the analysis of regularization methods for linear ill-posed problems, in the last years results have been obtained for regularization methods for nonlinear ill-posed problems, covering both theoretical analysis and practical applications.