# Increasing the efficiency of numerical methods for estimating the state of a partially observed system. High order methods for solving parabolic PDEs

Lead Research Organisation:
University of Oxford

Department Name: Mathematical Institute

### Abstract

Many aspects of phenomena critical to our lives on this planet such as the global climate, the state of the economy, the evolution of the human foetus are not available for direct measurements. Fortunately models of these phenomena, together with more limited observations frequently allow us to make reasonable inferences about the state of the systems that affect us. The process of using partial observations and a stochastic model to make inferences about an evolving system is known as stochastic filtering. The scoop of applications of stochastic filtering is huge and ranges from non-invasive methods to identify tumours to digital recording. The practical implementation of this process to concrete classes of models raises many important mathematical questions. One key question is how to approximate to the true description of the state of the system in an optimal, computationally feasible way. Under certain conditions, the distribution of the hidden'' model solves a non-linear stochastic PDE. It is desirable to find efficient numerical methods to handle this nonlinear PDE. Particle approximations are some of the most successful methods, especially for moderate and high-dimensional models (for example, for satellite tracking one needs to solve a six-dimensional stochastic PDE). A particle approximation uses a cloud of particles that evolve in the underlying state space. The choice of the particles' trajectories has a crucial influence on the properties of the ensuing approximations. The cubature method recently introduced by Lyons and Victoir produces particle approximations for linear/deterministic PDEs. In this case the particle evolve along admissible trajectory (unlike those produced by classical methods such us the Euler methods) and branch at pre-determined time intervals. The proposed research aims to extend the cubature results and the methods to produce high order approximations for the nonlinear stochastic PDE governing the solution of the filtering problem. Moreover we aim to tackle the increase in the complexity of the computation with time. We aim to study two methods for doing this: The first method consists in the addition of a randomized selection by which the particles that follow the right paths are multiplied and those drifting away from the plausible signal trajectories are rapidly removed. The second method consists in a recombination procedure by which the population of particles is divided into subsets. Then each subset of existing particles is replaced by a single particle which inherits the position of one of the particles in the original subset. The particles are recombined in a way that keeps the accuracy of the approximation unchanged.

## People |
## ORCID iD |

Terry Lyons (Principal Investigator) |

### Publications

Boutaib Y
(2014)

*Dimension-free Euler estimates of rough differential equations*in Rev. Roumaine Math. Pures Appl.
Cass T
(2015)

*Evolving communities with individual preferences*in Proceedings of the London Mathematical Society
Cass T
(2013)

*Integrability and tail estimates for Gaussian rough differential equations*in The Annals of Probability
Cass T
(2013)

*On The Error Estimate for Cubature on Wiener Space*in Proceedings of the Edinburgh Mathematical Society
Crisan D
(2015)

*Kusuoka-Stroock gradient bounds for the solution of the filtering equation*in Journal of Functional Analysis
Crisan D
(2010)

*Probabilistic methods for semilinear partial differential equations. Applications to finance*in ESAIM: Mathematical Modelling and Numerical Analysis
Gyurk G
(2013)

*Extracting information from the signature of a financial data stream*in arXiv e-prints
Gyurkó L
(2010)

*Mathematics in Finance*
Gyurkó L
(2011)

*Stochastic Analysis 2010*
Hambly B
(2010)

*Uniqueness for the signature of a path of bounded variation and the reduced path group*in Annals of MathematicsDescription | Mathematical tools for filtering, and the methods studied in this project are useful for monitoring and controlling many different types of systems. The mathematics is very central to many types of problem - from weather prediction to engineering. This research made a useful contribution to the overall challenges, and formed a foundation for future research of the PIs. |

Exploitation Route | High order methodology is mathematically challenging but very powerful. It can permit efficient anlysis of smaller datasets than might otherwise be the case. |

Sectors | Aerospace, Defence and Marine,Digital/Communication/Information Technologies (including Software),Energy,Financial Services, and Management Consultancy,Manufacturing, including Industrial Biotechology,Pharmaceuticals and Medical Biotechnology,Retail,Security and Diplomacy,Transport,Other |

Description | Not directly, there is an useful paper which considers exemplar models and integrated the new theory with practical data analysis effectively. One hopes this will form a bridge to applications in due course. The methods did demonstrate orders of magnitude increase in quality of prediction, however they are intellectually challenging to implement and so the exemplar paper becomes important - but will probably require follow up to achieve a full transition into the wider community. However, indirectly it stimulated useful further research work on recombination and scenario management that potentially accelerates certain important cases of the simplex algorithm. |

First Year Of Impact | 2011 |

Sector | Aerospace, Defence and Marine,Digital/Communication/Information Technologies (including Software),Electronics,Energy,Financial Services, and Management Consultancy,Healthcare,Manufacturing, including Industrial Biotechology,Security and Diplomacy,Other |

Impact Types | Cultural,Economic |

Description | An international collaboration of leading early stage contibutors to BSDE and cubature |

Organisation | Government of China |

Country | China |

Sector | Public |

PI Contribution | The grant enabled the PI to invite a small group of the leading international early stage reseachers in numerical methods for cubature and BSDE to work together in Oxford and establish collaboration. One outcome is a manuscript in which each of the following Danyu Yang Arnaud Lionnet Lajos Gergely Gyurkó Weijun Xu Dejan Velu?cek Maria Tchernychova Mariko Arisawa Camilo Andrés García Trillos Paul-Eric Chaudru de Raynal has, in a single collaborative work, written a short account of a key recent area for innovation. All have written their parts and near final version exists and should soon be placed on the Arxiv. A small group of the leading international early stage reseachers in numerical methods for cubature and BSDE worked together in Oxford and established a collaboration. Danyu Yang, Arnaud Lionnet, - Lajos Gergely Gyurkó, Weijun Xu, Dejan Velu?cek, Maria Tchernychova, Mariko Arisawa, Camilo Andrés García Trillos, Paul-Eric Chaudru de Raynal - have in a single collaborative work, written an account of key recent developments. A complete version exists and should soon be placed on the Arxiv. |

Start Year | 2013 |

Description | An international collaboration of leading early stage contibutors to BSDE and cubature |

Organisation | Lubiana S.A |

Country | Poland |

Sector | Private |

PI Contribution | The grant enabled the PI to invite a small group of the leading international early stage reseachers in numerical methods for cubature and BSDE to work together in Oxford and establish collaboration. One outcome is a manuscript in which each of the following Danyu Yang Arnaud Lionnet Lajos Gergely Gyurkó Weijun Xu Dejan Velu?cek Maria Tchernychova Mariko Arisawa Camilo Andrés García Trillos Paul-Eric Chaudru de Raynal has, in a single collaborative work, written a short account of a key recent area for innovation. All have written their parts and near final version exists and should soon be placed on the Arxiv. A small group of the leading international early stage reseachers in numerical methods for cubature and BSDE worked together in Oxford and established a collaboration. Danyu Yang, Arnaud Lionnet, - Lajos Gergely Gyurkó, Weijun Xu, Dejan Velu?cek, Maria Tchernychova, Mariko Arisawa, Camilo Andrés García Trillos, Paul-Eric Chaudru de Raynal - have in a single collaborative work, written an account of key recent developments. A complete version exists and should soon be placed on the Arxiv. |

Start Year | 2013 |

Description | An international collaboration of leading early stage contibutors to BSDE and cubature |

Organisation | University of Nice Sophia-Antipolis |

Country | France |

Sector | Academic/University |

PI Contribution | The grant enabled the PI to invite a small group of the leading international early stage reseachers in numerical methods for cubature and BSDE to work together in Oxford and establish collaboration. One outcome is a manuscript in which each of the following Danyu Yang Arnaud Lionnet Lajos Gergely Gyurkó Weijun Xu Dejan Velu?cek Maria Tchernychova Mariko Arisawa Camilo Andrés García Trillos Paul-Eric Chaudru de Raynal has, in a single collaborative work, written a short account of a key recent area for innovation. All have written their parts and near final version exists and should soon be placed on the Arxiv. A small group of the leading international early stage reseachers in numerical methods for cubature and BSDE worked together in Oxford and established a collaboration. Danyu Yang, Arnaud Lionnet, - Lajos Gergely Gyurkó, Weijun Xu, Dejan Velu?cek, Maria Tchernychova, Mariko Arisawa, Camilo Andrés García Trillos, Paul-Eric Chaudru de Raynal - have in a single collaborative work, written an account of key recent developments. A complete version exists and should soon be placed on the Arxiv. |

Start Year | 2013 |