Many aspects critical to our lives (e.g. availability of renewable energy resources, electrical patterns in the human heart, and the evolution of global pandemics) are not available for direct measurement. Blending models with measured data lets us make reasonable inferences about the state of a system. When using partial observations with a stochastic model to make rigorous inferences about an evolving system, we call this process stochastic filtering. The choice of the state model plays a crucial role in the applicability of any given stochastic filtering methodology. The proposed research will focus stochastic partial differential equations as state models, as they are some of the most versatile models applied in diverse areas of human endeavour: physics, biology, chemistry, weather prediction, finance, renewable energy and manufacturing, etc.
Practical implementation of stochastic filtering for phenomena modelled by stochastic partial differential equations remains an outstanding challenge. One key question is how to approximate to the "true" description of the state of the system in a computationally feasible way. Particle filters (PFs) are some of the most successful methods for solving the filtering problem, offering a theoretically justified approach to inferences about the state of hidden systems. PFs involve sets of "particles": different realizations of the state model. At regular intervals, the cloud of particles is corrected using partial and noisy observations. In the language of Data Assimilation (DA), evolving particles as realizations of the model is the forecast step, whilst correction using data is the analysis step. PFs have proved immensely successful in engineering applications (for example) provided the state model has small to moderate size. They succeed by processing data sequentially: at the analysis step only the new observations are used, without the need to revisit past observations. It is frequently impossible to process the entire set of data available up to the current time.
Many challenging real world problems have large model states, with large quantities of observations becoming available at each analysis step. In numerical weather prediction, tens of millions of data measurements occur at each analysis time. This makes each individual analysis step (almost) as hard as processing data nonsequentially. Recently, the PI, CoI and colleagues have researched methodologies that can alleviate this problem, but certainly not overcome it.
We need a new paradigm for developing efficient PFs for these challenging problems. The current paradigm separates the DA mechanism into sequential forecast and analysis steps, describing the PF in a convenient and pedagogical manner. In particular, the evolution of the particles between analysis steps ignores the forthcoming data. In our new paradigm, rather than evolving the particles using a numerical approximation of the stochastic partial differential equation, we advocate "observation informed" trajectories, where the particles are "nudged" in suitably chosen directions. Currently in data assimilation, procedures that move the particles towards observations are adhoc methodologies are not theoretically justified. In contrast, we will develop methodologies that are provably consistent approximations of the filtering problem. Our nudges will perturb the trajectories of the particles to maximise the likelihood of their positions given the observation data. This introduces a bias that is eliminated through the judicious selection of particles. The new methodology will be optimized for stochastic partial differential equations, from fluid dynamics, reactiondiffusion equations, AllenCahn etc.
Our project will deliver the complete pipeline for our paradigm, from theoretical analysis to high performance software implementations that can run on large parallel computers, enabling performance evaluations on challenging benchmarks.
