Equation-free particle-based computations in multiple dimensions and multiscale data assimilation with the ensemble Kalman filter

Multiscale phenomena are very common in various disciplines of science and engineering. In the first part of this thesis, a multiscale technique for modeling multidimensional particle systems is discussed. This modeling technique is an extension of coarse-time-stepper based approaches that have been proposed in recent years to treat multiscale phenomena where evolution equations for coarse-scale observables are usually not explicitly available. In this thesis, these time-stepper based approaches are extended to multidimensional particle systems by utilizing marginal and conditional inverse cumulative distribution functions (ICDF) of particle positions as coarse-scale observables of coarse time-steppers. Coarse projective integration and renormalization methods are subsequently applied to evolve these observables in time and to investigate their self-similar behavior.; Due to inaccurate information on initial conditions at fine scales in multiscale problems, experimental observations are expected to calibrate model predictions across different scales to improve quality of their predictions. Since usually measurements are not available at the fine scales, it is often desirable to utilize observations at coarse scales to estimate fine-scale model states. Two techniques employing the ensemble Kalman filter are constructed in the second part of this thesis to fulfill this purpose. The first technique is called the method of extended state in which the macroscale state, which is derived from the microscale state through a multiscale bridging model, is combined with the microscale state to form an extended state. The second technique is called the method of coarse time-stepper, which is based on the combined application of the coarse time-stepper and ensemble Kalman filter.; The third part of this thesis is about the error estimation arising in spatial discretization of multiscale bridging models. These bridging models typically involve integral equations over a spatial domain which are numerically solved through a discretization process. This part investigates errors in the approximation of states incurred by the spatial discretization of these bridging models. Sufficient conditions on the bridging models are given to control these errors in deterministic and stochastic multiscale problems.