Project title: A novel mathematical and computational paradigm for nonlinear filtering problems
Sponsor: ORNL - Laboratory Directed Research & Developement
Project description:
Project description:
We
propose to develop a novel mathematical and computational approache for
data-intensive nonlinear filtering problems. The techniques of linear
filtering have contributed tremendously in simulating dynamical
systems, but they are only first-order approximations to nonlinear
filtering problems. Our objective is to significantly improve the
applicability and efficiency of nonlinear filtering simulations by
exploring substantially novel directions based on the theory of the
equivalence between the nonlinear filtering problems and a class of
backward stochastic differential equations (BSDEs). In our methodology,
a nonlinear filtering problem is handled by numerically solving a BSDE,
in the sense of which several fundamental challenges, e.g. massive
data, high dimensionality and non-Gaussian noise, etc., can be
addressed . In addition, a truly scalable filtering capability will be
established for applications of importance to the DOE mission, and the
proposed algorithms will be made available through a recently developed
ORNL Toolkit for Adaptive Stochastic Modeling and Non-Intrusive
ApproximatioN (TASMANIAN).
Funding period: 2014 -- 2015
Publications:
Funding period: 2014 -- 2015
Publications:
- H. Tran, C. Webster and G. Zhang, A sparse grid method for Bayesian uncertainty quantification with application to large eddy simulation turbulence models, Springer Lecture Notes on CS&E, 109 (2016), pp. 291-313. [PDF, bibtex]
- F. Bao, Y. Cao, C. Webster and G. Zhang, An efficient meshfree implicit filter for nonlinear filtering problems, International Journal for Uncertainty Quantification, 6 (2016), pp. 19-33. [PDF, bibtex]
- G. Zhang, W. Zhao, M. Gunzburger and C. Webster, Numerical methods for a class of nonlocal diffusion problems with the use of backward SDEs, Computers & Mathematics with Applications, 71 (2016), pp. 2479-2496. [PDF, bibtex]
- G. Zhang, C. Webster, M. Gunzburger and J. Burkardt, A hyper-spherical adaptive sparse-grid method for high-dimensional discontinuity detection, SIAM Journal of Numerical Analysis, 53 (2015), pp. 1508-1536. [PDF, bibtex]
- V. Reshniak, A. Khaliq, D. Voss and G. Zhang, Split-step Milstein methods for multi-channel stiff stochastic differential systems, Applied Numerical Mathematics, 89 (2015), pp. 1-23. [PDF, bibtex]
- F. Bao, Y. Cao, C. Webster and G. Zhang, A hybrid sparse-grid approach for nonlinear filtering problems based on adaptive domain of the Zakai equation approximations, SIAM/ASA Journal on Uncertainty Quantification, 2 (2014), pp. 784-804. [PDF, bibtex]
- C. Webster, G. Zhang and M. Gunzburger, An adaptive sparse-grid-based iterative ensemble Kalman filter approach for parameter field estimation, International Journal of Computer Mathematics, 91 (2014), pp. 798-817. [PDF, bibtex]
- 13th US National Congress on Computational Mechanics, San Diego, CA (July 2015)
- 45th Annual John H. Barrett Memorial Lectures, University of Tennessee, Knoxville, TN (May 2015)
- SIAM Conference on Computational Science and Engineering, Salt Lake City, UT (Mar. 2015)
- The 9th International Conference on Computational Physics, Singapore (Jan. 2015)
- American Geophysics Union Annual Meeting, San Francisco, CA (Dec. 2014)
- Department of Mathematics and Statistics, Auburn University, Auburn, AL (Nov. 2014)
- Department of Mathematics, University of Tennessee, Knoxville, TN (Oct. 2014)
- 3rd Workshop on Sparse Grid and Applications, Stuttgart, Germany (Sep. 2014)
- Guannan co-organized a mini-symposium on "High-dimensional Approximation and Integration: Analysis and Computation", at SIAM Conference on Computational Science and Engineering (Mar. 2015)
- Guannan
co-organized a mini-symposium on "Theoretical and numerical analysis
for forward-backward stochastic differential equations and stochastic
optimal control", at 2014 SIAM Conference on Uncertainty
Quantification (Apr. 2014)