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: Imperial College London
Department Name: Dept of Mathematics

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.

Publications

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Beskos A (2017) A stable particle filter for a class of high-dimensional state-space models in Advances in Applied Probability

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Beskos A (2014) On the stability of sequential Monte Carlo methods in high dimensions in The Annals of Applied Probability

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Crisan D (2010) On the Monte Carlo simulation of BSDEs: An improvement on the Malliavin weights in Stochastic Processes and their Applications

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Crisan D (2015) Generalised particle filters with Gaussian mixtures in Stochastic Processes and their Applications

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Crisan D (2015) Kusuoka-Stroock gradient bounds for the solution of the filtering equation in Journal of Functional Analysis

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Crisan D (2012) Particle filters with random resampling times in Stochastic Processes and their Applications

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Crisan D (2014) Conditional distributions, exchangeable particle systems, and stochastic partial differential equations in Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

 
Description A qualitative and quantitative analysis of solutions of certain classes of semilinear deterministic and stochastic partial differential equations has been developed. Smoothness results and high order cubature methods for these (stochastic) PDEs have been introduced. We have established a basis for a unified theory of approximation which encompasses particle approximations, Gaussian approximation, finite element/finite difference methods to the solution of the filtering problem.
Exploitation Route The major drive in the research area at the moment is how to come with high-dimensional systems/problems. The findings resulting from the research form a strong basis for tackling difficult high-dimensional problems. The methodology developed can be extended and applied to various atmospheric science and financial applications.
Sectors Environment,Financial Services, and Management Consultancy

 
Description Beatriu de Pinós grant
Amount € 120,000 (EUR)
Organisation Government of Catalonia 
Sector Public
Country Spain
Start 10/2012 
End 09/2014
 
Description Conference Grant
Amount £7,000 (GBP)
Organisation London Mathematical Society 
Sector Learned Society
Country United Kingdom
Start 06/2013 
End 12/2013
 
Description EPSRC Standard Grant
Amount £972,326 (GBP)
Funding ID EP/N023781/1 
Organisation Engineering and Physical Sciences Research Council (EPSRC) 
Sector Academic/University
Country United Kingdom
Start 04/2016 
End 03/2019
 
Description Royal Society/CNRS collaboration grant
Amount £12,000 (GBP)
Organisation The Royal Society 
Sector Academic/University
Country United Kingdom
Start 03/2012 
End 07/2014
 
Description Ajay Jasra and Alex Beskos 
Organisation National University of Singapore
Country Singapore 
Sector Academic/University 
PI Contribution Joint research on Sequential Monte Carlo Methods
Collaborator Contribution Joint research on Sequential Monte Carlo Methods
Impact We have joint publications as listed the relevant section.
Start Year 2011
 
Description Ajay Jasra and Alex Beskos 
Organisation University College London
Country United Kingdom 
Sector Academic/University 
PI Contribution Joint research on Sequential Monte Carlo Methods
Collaborator Contribution Joint research on Sequential Monte Carlo Methods
Impact We have joint publications as listed the relevant section.
Start Year 2011
 
Description Francois Delarue 
Organisation University of Nice Sophia-Antipolis
Country France 
Sector Academic/University 
PI Contribution Joint research on semilinear PDEs
Collaborator Contribution Joint research on semilinear PDEs
Impact We have joint publications as listed the relevant section.
Start Year 2011
 
Description Jean-François Chassagneux 
Organisation Imperial College London
Country United Kingdom 
Sector Academic/University 
PI Contribution Joint Research on Backward SDEs
Collaborator Contribution Joint Research on Backward SDEs
Impact We have joint publications as listed the relevant section.
Start Year 2012
 
Description Jie Xiong 
Organisation University of Tennessee
Country United States 
Sector Academic/University 
PI Contribution We work on joint research programme on stochastic PDEs.
Collaborator Contribution It is a collaboration. They come up with new ideas, methodology etc.
Impact We have joint publications as listed the relevant section.
Start Year 2008
 
Description Joaquin Miguez 
Organisation Charles III University of Madrid
Country Spain 
Sector Academic/University 
PI Contribution We work on joint research programme on sequential Monte Carlo methods
Collaborator Contribution It is a reasearch collaboration. They come up with new ideas, methodology etc.
Impact We have joint publications as listed the relevant section.
Start Year 2011
 
Description Nizar Touzi 
Organisation Ecole Polytechnique
Country France 
Sector Academic/University 
PI Contribution joint research on Backward SDEs
Collaborator Contribution joint research on Backward SDEs
Impact We have joint publications as listed the relevant section.
Start Year 2009
 
Description Peter Friz 
Organisation Technical University Berlin
Country Germany 
Sector Academic/University 
PI Contribution joint research on Nonlinear Filtering
Collaborator Contribution Is a collaboration. They come up with new ideas, methodology etc.
Impact We have joint publications as listed the relevant section.
Start Year 2012
 
Description Terry Lyons 
Organisation University of Oxford
Department Oxford-Man Institute of Quantitative Finance
Country United Kingdom 
Sector Academic/University 
PI Contribution Joint research of nonlinear filtering.
Collaborator Contribution Joint research of nonlinear filtering.
Impact software library (as described in the relevant section of the form) and forthcoming publication.
 
Description Thomas Kurtz 
Organisation University of Wisconsin-Madison
Country United States 
Sector Academic/University 
PI Contribution joint research of Stochastic PDEs
Collaborator Contribution It is a collaboration. They come up with new ideas, methodology etc.
Impact We have joint publications as listed the relevant section.
Start Year 2007
 
Title Library of subroutines for solving linear and semilinear PDEs 
Description We have developed a library of subroutines/programs that solve linear and semilinear PDEs to high accuracy. 
Type Of Technology Software 
Year Produced 2013 
Impact The library forms the basis of additional developments by the PI and collaborators. 
 
Description 7th International Symposium on Backward SDEs Shandong University 22-27 June 2014 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Participants in your research and patient groups
Results and Impact I disseminated the result of the research funded by the grant.

The participants requested the papers that contain the details of the research.
Year(s) Of Engagement Activity 2014
URL http://rida.sdu.edu.cn/BSDE2014/
 
Description A mathematician's view on Asimov's psychohistory (inaugural talk) 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Public/other audiences
Results and Impact My talk increased the people's awareness of research in this area.

Many people got interested in my research and findings.
Year(s) Of Engagement Activity 2013
URL http://www3.imperial.ac.uk/newsandeventspggrp/imperialcollege/eventssummary/event_11-12-2012-13-34-2...
 
Description Anniversary Conference, Faculty of Science 150 years, University of Bucharest, Romania 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Participants in your research and patient groups
Results and Impact I disseminated the result of the research funded by the grant


The participants requested the papers that contain the details of the research.
Year(s) Of Engagement Activity 2013
 
Description Data Assimilation Research Centre University of Reading 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Participants in your research and patient groups
Results and Impact I disseminated the result of the research funded by the grant


The participants requested the papers that contain the details of the research.
Year(s) Of Engagement Activity 2013
 
Description International conference on Controlled Deterministic and Stochastic Systems, Iasi Romania, 2012 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Participants in your research and patient groups
Results and Impact I disseminated the result of the research funded by the grant


The participants requested the papers that contain the details of the research.
Year(s) Of Engagement Activity 2012
 
Description Linnaeus University Workshop in Stochastic Analysis and Applications, 22-24 May 2013 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Participants in your research and patient groups
Results and Impact I disseminated the result of the research funded by the grant


The participants requested the papers that contain the details of the research.
Year(s) Of Engagement Activity 2013
 
Description Mathematical and Algorithmic Aspects of Atmosphere-Ocean Data Assimilation (Conference) 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Participants in your research and patient groups
Results and Impact I presented the results of my research. I made several research contacts through this.

I got invited to give a course in based on my findings in Bangalore, India.
Year(s) Of Engagement Activity 2012
URL http://www.mfo.de/occasion/1249/www_view
 
Description Probability and Statistics Seminar Department of Mathematics University of Bristol 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Participants in your research and patient groups
Results and Impact The participants requested the papers that contain the details of the research.
Year(s) Of Engagement Activity 2013
 
Description Talk at the European Congress of Mathematics 2012 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Participants in your research and patient groups
Results and Impact The talk generated a lot of interest and discussion.

After the talks I have received a lot of requests for the paper.
Year(s) Of Engagement Activity 2012
URL http://www.6ecm.pl/
 
Description Thalesian Society talk 2013 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Professional Practitioners
Results and Impact I disseminated the result of the research funded by the grant.

The participants requested the papers that contain the details of the research.
Year(s) Of Engagement Activity 2013
URL http://www.thalesians.com/finance/index.php/Events/Seminars#Thalesian_Seminar_.28London.29_.E2.80.94...
 
Description UK-Japan Stochastic Analysis School JSPS Core-to-Core programm Warwick Mathematics Institute 1-5 September 2014 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Participants in your research and patient groups
Results and Impact I disseminated the result of the research funded by the grant.

The participants requested the papers that contain the details of the research.
Year(s) Of Engagement Activity 2014
 
Description Workshop on "Partial Information and Filtering" Thematic Semester on "Information in Finance and Insurance" Institut Henri Poincar` e, Paris 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Participants in your research and patient groups
Results and Impact I disseminated the result of the research funded by the grant.

The participants requested the papers that contain the details of the research.
Year(s) Of Engagement Activity 2014
URL http://www.louisbachelier.org/evenement/thematic-semester-%E2%80%9C-information-finance-and-insuranc...
 
Description talk at the The 8th World Congress in Probability and Statistics 2012 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Participants in your research and patient groups
Results and Impact I disseminated the results of the research to a wide audience.

There were a lot requests for the papers that covered my findings
Year(s) Of Engagement Activity 2012
URL http://home.ku.edu.tr/~wcong2012/
 
Description talk at the Workshop on Sequential Monte Carlo Methods 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Participants in your research and patient groups
Results and Impact I disseminated the result of the research funded by the grant

The participants requested the papers that contain the details of the research.
Year(s) Of Engagement Activity 2012
URL https://project.inria.fr/SMCMESF/