On the other hand, the widespread availability of modern sensing technology offers the possibility of including the measurement data regarding the system in its operational state and permits online and adaptive calibration of the dynamic evolution of the engineering and natural system. Although enriched with valuable information regarding the operational state of the system, the sensor data is invariably contaminated by measurement noise. The work at Carleton involves the assimilation of noisy sensor data in the high resolution computer model to calibrate or infer (in a statistical sense) the parameters of the numerical simulators and provide realistic ensemble forecast (as in weather predictions) regarding the evolution or state of the system, albeit giving due regards to uncertainty or noise ever-present in the measured data and predictive model.
The following theoretical and computational developments address some these issues:
Domain Decomposition of Uncertain Systems and Its Parallel Implementation
A novel theoretical framework for the domain decomposition of uncertain systems
defined by stochastic partial differential equations (SPDEs) has been developed. The methodology
involves a domain decomposition method in the geometric space and a functional
decomposition in the probabilistic space. The probabilistic decomposition is based on
a version of stochastic finite elements which are based on orthogonal decompositions and projections
of stochastic processes. The spatial decomposition is achieved through a Schur complement
based domain decomposition. The methodology aims to exploit the full potential
of high performance computing platforms by reducing discretization errors with high resolution
numerical model while giving due regards to uncertainty in the system. The scalability
of the algorithms is demonstrated in both shared and distributed
memory machines.
The spatial domain of the PDEs is decomposed into a number of subdomains and the solution is sought concurrently on each of these subdomains using domain decomposition methods. Each processor on a multiprocessor computer can tackle each subdomain in parallel. For a fixed mesh resolution, the total solution time cannot be reduced beyond a critical number of subdomains as the interprocessor communication overhead and memory bandwidth requirements predominate floating point operations. In order to exploit the remaining idle CPUs in multiprocessor computers, recent research initiatives have been directed towards achieving time domain parallelism in conjunction with spatial domain decomposition for SPDEs.
Data Assimilation Under Uncertainty Using Nonlinear Filters
This initiative addresses the issue on how to incorporate the available
observational data regarding the system state in the high resolution
computer model to provide better predictions.
The Extended Kalman Filter (EKF) is a popular approach for the
combined state and parameter estimation of nonlinear systems.
The EKF combines the traditional Kalman filtering techniques with
the linearization tools to tackle nonlinear problems. The method augments
the unknown system parameters to the state vector of the system state-space model.
The EKF method formulation is based on the assumption that the
noise is generated by an underlying Gaussian process and the system is
weakly nonlinear. However, numerous civil and environmental engineering problems driven by
random noise display strongly nonlinear and non-Gaussian
response. This research is focused on combined parameter estimation
and tracking the non-Gaussian response of strongly nonlinear
systems driven by non-Gaussian noise. In particular, a parallel computational algorithm is being
developed to couple the high resolution numerical method based on the domain
decomposition method for SPDEs with the nonlinear filters based on Ensemble Kalman Filter and Particle Filter.
These techniques are being applied to data assimilation problems in
civil and environmental engineering problems, for example, related to tracking contaminants, reservoir simulation and structural health
monitoring problems.
The following problems that are being tackled using the aforementioned computational framework.
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