Solving bilinear inverse problems by tensorial lifting

One of the most frequent questions in science is What caused this observed datum?. For example, when measuring temperature with a thermometer, one actually asks the question Which temperature caused the observed expansion of mercury?. The datum is then the length of the column of mercury. In order to answer this question one needs a connection between the temperature and the length of the column, i.e. a model. A quick glance at a thermometer shows that the expansion is proportional to the temperature; such a model is called linear. In a formal way, one asks: Given a known model, which parameters of the model have caused the given data?. This project strives to answer this question for models depending on a pair of parameters both of which have proportional influence on the data; in mathematical terms, to solve bilinear inverse problems. For example the process of taking pictures involves a pair of parameters, the true image and the aberration caused by the lens. The result is the obtained picture, with each parameter having proportional influence. The solution of bilinear inverse problems is especially challenging, if the problem has the tendency to produce similar data for drastically different parameters. Clearly, in such case, noise or clutter inherent to every measurement setup highly impedes the ability to recover the true parameters responsible for the data. Unfortunately, this very kind of problems is very common in cutting edge problems arising in medicine, engineering and life sciences. The projects main aim is to provide, with mathematical stringency, a general framework for the solution of such problems. This begins with the thorough study of the properties of bilinear problems and it leads to the development of tools dearly needed by the practitioners dealing with bilinear problems. The innovative core of the project is to combine three ideas: The first, main idea and the crucial method this project follows is to observe that all pairs of parameters can be considered special cases of a single parameter within a more general set of parameters. Further, it turns out that in that general set the problem becomes linear in the single parameter. This makes the problem far more accessible. We call this process tensorial lifting. However, solving the resulting problem can still be very challenging. Therefore, as the second ingredient, we incorporate a slight change of the problem, a so-called relaxation, which in the best case does not change the answer to the problem significantly. Finally, as the third part, the mathematical insights developed in this project are transferred to applications by means of novel solution schemes targeted towards the needs of practitioners of bilinear problems.

One of the most frequent questions in science is "What caused this observed datum?". For example, when measuring temperature with a thermometer, one actually asks the question "Which temperature caused the observed expansion of mercury?". The datum is then the length of the column of mercury. In order to answer this question one needs a connection between the temperature and the length of the column, i.e., a model. A quick glance at a thermometer shows that the expansion is proportional to the temperature; such a model is called linear. In a formal way, one asks: "Given a known model, which parameters of the model have caused the given data?". This project strived to answer this question for models depending on a pair of parameters both of which have proportional influence on the data; in mathematical terms, to solve bilinear inverse problems. For example, the process of taking pictures involves a pair of parameters, the true image and the aberration caused by the lens. The result is the obtained picture, with each parameter having proportional influence. The solution of bilinear inverse problems is especially challenging if the problem has the tendency to produce similar data for drastically different parameters. Clearly, in such case, noise or clutter inherent to every measurement setup highly impedes the ability to recover the true parameters responsible for the data. Unfortunately, this very kind of problems is very common in cutting edge problems arising in medicine, engineering and life sciences. The project aims were to provide, with mathematical stringency, a new general framework for the solution of such problems. This began with the thorough study of the properties of bilinear problems and led to the development of tools dearly needed by the practitioners dealing with bilinear problems. The core of the project results was to successfully combine three aspects: The first, exploiting the crucial observation that all pairs of parameters can be considered special cases of a single parameter within a more general set. Further, using that in that general set, the problem becomes linear in the single parameter, making it far more accessible. This process is called tensorial lifting and the main technique used to obtain the project's results. As solving the resulting problems still could be very challenging, as the second aspect, we incorporated a slight change, the so-called relaxation, from which is known that in many situations, it does not significantly change the answer to the problem. Finally, as the third aspect, the mathematical insights developed in this project were transferred into algorithms and applications resulting in novel computational schemes targeted towards the needs of practitioners working with bilinear problems.