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Exploring the potential of nonparametric modelling of complex systems via stochastic partial differential equations

Carlsberg Foundation Young Researcher Fellowships


Given the ubiquitous use of mathematical models in almost all areas of our lives, it is essential that modelling is carried out with the greatest care, especially in sensitive applications such as the disposal of radioactive waste where one needs to assess the risk of scenarios like radionuclide transport. The standard approach of using partial differential equations (PDEs) to model physical processes like the groundwater flow is not always adequate (e.g., when the flow also depends in a complicated way on microscopic properties of the medium). Stochastic partial differential equations (SPDEs) consider such phenomena and mathematically generalize PDEs. This project aims at exploring the potential of SPDE models and providing basic statistical methods facilitating their use in practice.


For many mathematical models, it has proven to be very rewarding to integrate randomness, thus accounting for hidden random dynamics, measurement errors and other external influences. In various applications, randomness can easily be reflected in the model by simply adding "noise", i.e., a stochastic component. However, for a whole class of highly relevant models, namely those based on PDEs, considering noise naturally leads to SPDEs that are mathematically very complex. As a result, despite the evidence that SPDE models are highly relevant, these models are largely overlooked due to a lack of understanding and a lack of feasible tools for their statistical analysis. Our goal is to contribute to closing this gap, thereby reducing the entry barrier for SPDEs in real world scenarios.


The SPDE model is determined by characteristics that are unknown in practice. We focus on statistical methods for estimating these characteristics, i.e., we mathematically investigate general methods to learn the parameters from the available data. Furthermore, we want to identify criteria and provide a mathematical framework that will allow deciding whether modelling with SPDEs is appropriate in a given situation. As a result, we expect to provide procedures and to legitimate them by mathematically proving their efficiency in terms of common risk measures.


Our research will provide techniques leading to a better understanding of complex systems modelled via SPDEs in various scientific fields such as medicine, ecology or natural sciences in general. This will help to make more effective decisions on very important issues affecting society as a whole, such as the fight against cancer, pollution or climate change, to name but a few of many other issues. As mathematicians, we strive to provide the tools for understanding and progress in other disciplines. I see it as my personal responsibility to use my knowledge to contribute to creating accurate mathematical models that provide reliable information, especially in sensitive applications, and to increase awareness of such issues.