ALDER - Space-time sequential schemes and Bayesian inference methods for continuous-time stochastic
Abstract
Many real-world systems are described by either deterministic or random differential equations. Examples abound in geophysics, where, ocean and atmosphere models are constructed around relevant systems of partial differential equations (PDEs), including the Euler,,Burgers or Navier-Stokes equations. A PDE is an infinite-dimensional model where the magnitudes of interest can evolve both over con,tinuous time and over continuous space. In order to account for various sources of uncertainty, considerable attention has been turn,ed over the past decade to stochastic versions of classical PDEs and, as a result, stochastic PDE (SPDE) modelling is currently a ve,ry active area of research. Relatively simpler stochastic differential equation (SDE) models including classical SDEs and stochastic, delayed differential equations (SDDEs), are also the state of the art in many active fields, including mesoscale modelling of atmos,phere-ocean interaction systems.Practical applications involving PDEs, SPDEs or SDEs usually involve two steps, namely,- the discret,isation of continuous time and continuous-space systems, so that they can be implemented as computer models, - and the implementatio,n of parameter estimation, signal tracking and prediction methods on the discretised models.The methods for inference on the discret,ised models need to operate on very high dimensional spaces and they are often subject to the so-called "curse of dimensionality", i,.e., the computational effort needed to achieve a certain performance increases exponentially with the dimension of the signal and/o,r parameters of interest.The aim of this research is to tackle the SDE/PDE/SPDE discretisation (step a) above) and the design of inf,erence methods (step b)) in a holistic manner. We claim that, by coupling both procedures, it is possible to improve drastically the, performance of the inference algorithms and beat the curse of dimensionality --at least under assumptions that are reasonable in so,me scenarios. The final goal is, therefore, the design of efficient numerical algorithms that admit a rigorous mathematical analysis, in terms of error bounds and their dependence on the dimension of the discrete-time-and-space models. We will resort to recent adva,nces in the field of Bayesian computation and a sequential discretisation scheme for SDEs that has been developed with support of an, earlier ONR grant (award no. N00014-19-1-2226).To attain this global aim, we will pursue three specific objectives:1. The design an,d analysis of computational methods for adaptive discretisation (over time and space) and Bayesian inference on PDEs. 2. The desig,n and analysis of space-time discretisation schemes and Monte Carlo filters for real-world systems described by partially-observed S,PDEs.3. The design and analysis of a new class of stochastic filters based on deep learning schemes for real-world systems described, by partially observed SDEs.The research will be carried out by a team at Universidad Carlos III the Madrid (Spain), with collaborat,ors at Imperial College London and The Alan Turing Institute (London, UK).
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Oct 07, 2022
- Source ID
- N000142212647
Entities
People
- Joaqun Mguez Arenas
Organizations
- Office of Naval Research
- United States Navy
- Universidad Carlos III de Madrid