In many of the modern biomedical imaging modalities, the measurable signal can be described as the solution of a partial differential equation that depends nonlinearly on the tissue properties (the "parameters") one would like to image. Consequently, there are typically no explicit solution formulas for these so-called "inverse problems" that can recover the parameters from the measurements, and the only way to generate body images from measurements is through numerical approximation.

The resulting parameter estimation schemes have the underlying partial differential equations as side-constraints, and the solution of these optimization problems often requires solving the partial differential equation thousands or hundred of thousands of times. The development of efficient schemes is therefore of great interest for the practical use of such imaging modalities in clinical settings. In this talk, the formulation and efficient solution strategies for such inverse problems will be discussed, and we will demonstrate its efficacy using examples from our work on Optical Tomography, a novel way of imaging tumors in humans and animals. The talk will conclude with an outlook to even more complex problems that attempt to automatically optimize experimental setups to obtain better images.