On Qualitative Experimental Design for PDE Parameter Identification

Experimental design is a central task in inverse problems that concerns the planning of data collection in accordance with a specific reconstruction goal. In general, not all data are equally informative. Choices made when setting up an experiment - such as how measurements are taken or how the physical test system is designed - alongside other factors, determine whether the resulting data contain useful information for the inference problem. In practice, where only a limited amount of data can be collected, it is therefore crucial to focus on highly informative experiments in order to enable reconstruction and improve its quality. With this in mind, we hope to encourage an interdisciplinary perspective on inverse problems, in which planning, experimentation, and inference are carried out in collaboration between practitioners and mathematicians. Traditionally, experimental design has been formulated as an optimization problem: identifying the "best" experiments according to a chosen mathematical criterion. This approach is often ambitious, and the associated computational costs can be substantial. In this book, we advocate for a broader notion of experimental design that incorporates additional nuances. In particular, we adopt a qualitative perspective: rather than searching for optimal experiments, the goal is to identify experiments that enable reliable reconstruction. This viewpoint has gained increasing attention in recent years, and we review two methods that are rooted in identifiability analysis and sensitivity analysis: a theory-based approach and a sampling-based approach. Both shall be illustrated on an example from mathematical biology: chemotaxis, the directed movement of bacteria in response to chemical signals, along with the associated inverse problem of reconstructing motion parameters.