Adaptive time-stepping methods for SPDEs and Multilevel Monte Carlo techniques for inverse prob
Lead Research Organisation:
Heriot-Watt University
Department Name: S of Mathematical and Computer Sciences
Abstract
The MCMC and HMC methods are well known methods for Bayesian inference to obtain parameters from data. Recent advances have examined multi-level version of these methods. There are a number of areas we would like to explore in the thesis:
Develop novel time-stepping methods for SPDEs with non-globally Lipschitz drift with a view to extending to space-time adaptivity.
At the moment a global refinement strategy is mostly used for MLMC. One area of interest is to combine the multilevel technique with a posteriori error techniques - where a FE mesh is refined to minimize the error of a given quantity of interest or use space-time adaptivity for mesh refinement.
We plan to extend the work to examine model error in the approximation and also develop the proxy models.
A number of numerical integration techniques are used - we have ideas around the choice of integrator that should improve the sampling and hence convergence of the MCMC chain.
Develop novel time-stepping methods for SPDEs with non-globally Lipschitz drift with a view to extending to space-time adaptivity.
At the moment a global refinement strategy is mostly used for MLMC. One area of interest is to combine the multilevel technique with a posteriori error techniques - where a FE mesh is refined to minimize the error of a given quantity of interest or use space-time adaptivity for mesh refinement.
We plan to extend the work to examine model error in the approximation and also develop the proxy models.
A number of numerical integration techniques are used - we have ideas around the choice of integrator that should improve the sampling and hence convergence of the MCMC chain.
Organisations
People |
ORCID iD |
| Stuart Campbell (Student) |
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509474/1 | 01/10/2016 | 30/09/2021 | |||
| 1820832 | Studentship | EP/N509474/1 | 01/10/2016 | 30/09/2020 | Stuart Campbell |
| Description | We have developed a novel numerical method, which uses adaptive time-stepping, to solve stochastic partial differential equations (SPDEs). SPDEs are a class of equations that can be used for modelling a wide range of systems, including physical, biological and financial systems, particularly when the presence of noise or uncertainty is important to the overall systems' behaviour. Many SPDEs that are of interest for real applications, cannot be solved using standard methods. Our numerical method allows a wider range of SPDEs to be solved than standard methods, including many relevant to real world modelling. In addition, we have demonstrated that our method outperforms the existing numerical methods that are able to solve such problems. In addition, we can see that our method significantly increases performance of multi-level Monte Carlo estimators (MLMC) when compared to the alternative methods existing in the literature and this will be a focus of future work. |
| Exploitation Route | Our numerical method could potentially be of real interest to practitioners that utilise stochastic modelling in a wide range of subjects. It can solve a broad range of problems more efficiently than other competing methods. A modern approach to filtering noisy data can be re-cast as an SPDE problem, again which could be practically solved using our scheme. |
| Sectors | Aerospace, Defence and Marine,Chemicals,Digital/Communication/Information Technologies (including Software),Electronics,Energy,Financial Services, and Management Consultancy,Manufacturing, including Industrial Biotechology,Pharmaceuticals and Medical Biotechnology |
| URL | http://arxiv.org/abs/1812.09036 |