NUMERICS FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS OF PARABOLIC TYPE
Lead Research Organisation:
University of Leicester
Department Name: Mathematics
Abstract
Stochastic partial differential equations (SPDEs) are an essential tool in the description of complex systems affected by external or internal fluctuations arising in physics, chemistry, biology, finance. Effective numerical methods play crucial role in studying models described by SPDEs. Numerics for SPDEs is a relatively new area of stochastic numerical analysis and its further development is essential for both practice and theory. We well know how it is important to have a large arsenal of methods for deterministic equations. This is even more important in the case of SPDEs due to their higher complexity. The proposed research will develop the new approach to numerics for SPDEs and give new methods together with their rigorous analysis using novel ideas of layer methods. The proposed methods will be tested on model problems.
Organisations
People |
ORCID iD |
| Michael Tretyakov (Principal Investigator) |
Publications
Dumas W
(2010)
Computing conditional Wiener integrals of functionals of a general form
in IMA Journal of Numerical Analysis
Mattingly J
(2010)
Convergence of Numerical Time-Averaging and Stationary Measures via Poisson Equations
in SIAM Journal on Numerical Analysis
Milstein G
(2009)
Practical Variance Reduction via Regression for Simulating Diffusions
in SIAM Journal on Numerical Analysis
Milstein G
(2009)
Solving parabolic stochastic partial differential equations via averaging over characteristics
in Mathematics of Computation
Milstein G
(2016)
Monte Carlo methods for backward equations in nonlinear filtering
in Advances in Applied Probability
| Description | Stochastic partial differential equations (SPDEs) are an essential tool in the description of complex systems affected by external or internal fluctuations arising in physics, chemistry, biology. Effective numerical methods play crucial role in studying models described by SPDEs. Numerics for SPDEs is a relatively new area of stochastic numerical analysis and its further development is essential for both practice and theory. The importance of having a large arsenal of methods for deterministic equations is well known. This is even more important in the case of SPDEs due to their higher complexity. We developed the new approach to numerics for linear and nonlinear parabolic SPDEs. We exploited the method of characteristics (the averaging over the characteristic formula) and numerical integration of (ordinary) stochastic differential equations (SDEs) together with the Monte Carlo technique to propose new numerical methods for both Cauchy and Dirichlet problems for linear SPDEs. Special attention was paid to numerical algorithms for SPDEs arising in nonlinear filtering. Layer methods for both linear and nonlinear SPDEs were also constructed. All the proposed methods were accompanied by their theoretical analysis and tested on model problems. The other outcomes include: variance reduction techniques which are of crucial importance for effectiveness of any Monte Carlo procedures including those used in algorithms for the nonlinear filtering; error estimates for time-averaging estimators used in numerical approximation of the long time behaviour of SDEs; a numerical method for conditional Wiener integrals applicable to computing various quantities in quantum statistical mechanics and its generalization to the case of pinned diffusions. |
| Exploitation Route | Stochastic partial differential equations (SPDEs) of parabolic type are orginated in 1960x due to their well-known relation with the nonlinear ltering problem which is widely used for applications, in particular in Aerospace and Defence industries. Now SPDEs are used for other modeliing, for instance, for realibility-type problems in Engineering. Numerical methods proposed within the project have a potential to be used in practice for solving SPDEs arising in real-world applications. Potential beneficiaries are applied mathematicians, physicists, numerical analysts working in the theory of stochastic partial differential equations and their applications. It is expected that numerical methods constructed within this project will be an effective tool for investigating spatial distributed stochastic models from various fields of science: from combustion and fluid dynamics to finance. |
| Sectors | Aerospace, Defence and Marine,Energy |
| URL | http://www.mcs.le.ac.uk/~mtretiakov |
| Description | The findings have not been used beyond academia yet. |
| First Year Of Impact | 2014 |
| Description | Probabilistic approach to numerics for Navier-Stokes equations |
| Amount | £12,000 (GBP) |
| Funding ID | JP091142 |
| Organisation | The Royal Society |
| Sector | Charity/Non Profit |
| Country | United Kingdom |
| Start | 05/2010 |
| End | 03/2012 |
| Description | Probabilistic approach to numerics for Navier-Stokes equations |
| Amount | £12,000 (GBP) |
| Funding ID | JP091142 |
| Organisation | The Royal Society |
| Sector | Charity/Non Profit |
| Country | United Kingdom |
| Start | 05/2010 |
| End | 03/2012 |