Narrow Results

In the first part of this paper, we review some important results on atmospheric predictability, from the pioneering work of Lorenz to recent results with operational forecasting models. Particular relevance is given to the connection between atmospheric predictability and the theory of Lyapunov exponents and vectors. In the second part, we briefly review the foundations of data assimilation methods and then we discuss recent results regarding the application of the tools typical of chaotic systems theory described in the first part to well established data assimilation algorithms, the Extended Kalman Filter (EKF) and Four Dimensional Variational Assimilation (4DVar). In particular, the Assimilation in the Unstable Space (AUS), specifically developed for application to chaotic systems, is described in detail.

We implement a data-assimilation method based on a particle filter in the coupled climate model LOVECLIM focusing on decadal to centennial time scales. Several tests are performed with particle filtering using pseudo-observations obtained from a twin experiment with the model, as well as using real-data observations over the last century. These tests demonstrate that it is possible to obtain a model output well correlated with the observations at the large scale at a reasonable cost.

In this paper, a method to account for model error due to unresolved scales in sequential data assimilation, is proposed. An equation for the model error covariance required in the extended Kalman filter update is derived along with an approximation suitable for application with large scale dynamics typical in environmental modeling. This approach is tested in the context of a low order chaotic dynamical system. The results show that the filter skill is significantly improved by implementing the proposed scheme for the treatment of the unresolved scales.

A tumor growth model based on a parametric system of partial differential equations is considered. The system corresponds to a phenomenological description of a multi-species population evolution. A velocity field taking into account the volume increase due to cellular division is introduced and the mechanical closure is provided by a Darcy-type law. The complexity of the biological phenomenon is taken into account through a set of parameters included in the model that need to be calibrated. To this end, a system identification method based on a low-dimensional representation of the solution space is introduced. We solve several idealized identification cases corresponding to typical situations where the information is scarce in time and in terms of observable fields. Finally, applications to actual clinical data are presented.

We propose an observer-based approach to circumvent the issue of unbounded approximation errors — with respect to the length of the time window considered — in the discretization of wave-like equations in bounded domains, which covers the cases of the wave equation per se and of linear elasticity as well as beam, plate and shell formulations, and so on. Namely, taking advantage of some measurements available on the system over time, we adopt a strategy inspired from sequential data assimilation and by which the discrete system is dynamically corrected using the discrepancy between the solution and the measurements. In addition to the classical cornerstones of numerical analysis made up by stability and consistency, we are thus led to incorporating a third crucial requirement pertaining to observability — to be preserved through discretization. The latter property warrants exponential stability for the corrected dynamics, hence provides bounded approximation errors over time. Special care is needed to establish the required observability at the discrete level, in particular due to the fact that we focus on an original observer method adapted to measurements of the main variable, whereas measurements of the time-derivative — admissible, of course, albeit less frequent in practical systems — lead to a stability analysis in which existing results can be more directly applied. We also provide some detailed application examples with several such wave-like equations, and the corresponding numerical assessments illustrate the performance of our approach.

In this paper, we study the parameter estimation of interacting particle systems subject to the Newtonian aggregation and Brownian diffusion. Specifically, we construct an estimator $\widehat{\nu }$ with partial observed data to approximate the diffusion parameter $\nu$, and the estimation error is achieved. Furthermore, we extend this result to general aggregation equations with a bounded Lipschitz interaction field.

Modes and wavenumbers are the principal ingredients that characterize the pressure field in an oceanic waveguide. However, wavenumber and mode inversions are well-known to be a difficult task in underwater acoustics. Moreover, this double inversion has never been performed simultaneously from the same configuration of emitters and receivers. We present a new approach to this problem in a shallow water environment between two vertical arrays of sources and receivers. Starting from a classical modal decomposition of the pressure field, our algorithm focuses on a specific treatment of phase and amplitude variables. The key idea is to run a three-stage optimization by working separately on the phase and amplitude of the acoustic field. The high number of variables of the problem is turned into an advantage by using an adjoint code generated by an Automatic Differentiation software. Numerical results in the presence of noise show that modes and wavenumbers are estimated with a high accuracy.

Forward-backward stochastic differential equation (FBSDE) systems were introduced as a probabilistic description for parabolic type partial differential equations. Although the probabilistic behavior of the FBSDE system makes it a natural mathematical model in many applications, the stochastic integrals contained in the system generate uncertainties in the solutions which makes the solution estimation a challenging task. In this paper, we assume that we could receive partial noisy observations on the solutions and introduce an optimal filtering method to make a data informed solution estimation for FBSDEs.

Iterative algorithms for solving the data assimilation problems are considered, based on the main and adjoint equations. Spectral properties of the control operators of the problem are studied, the iterative algorithms are justified.

Data assimilation is an important technique to improve simulation results by assimilating real time sensor data into a simulation model. A data assimilation framework based on Sequential Monte Carlo (SMC) methods for wildfire spread simulation has been developed in previous work. This paper provides systematic analysis and measurement to quantify the effectiveness and robustness of the developed data assimilation method. Measurement metrics are used to evaluate the robustness of SMC methods in data assimilation for wildfire spread simulation. Sensitivity analysis is carried out to examine the influences of important parameters to the data assimilation results. This work of analysis and quantification provides information to assess the effectiveness of the data assimilation method and suggests guidelines to further improve the data assimilation method for wildfire spread simulation.

As an emerging simulation technology in the field of system modeling and simulation, the equipment symbiotic simulation has become research emphasis. In the field of equipment maintenance support, the outstanding problem of equipment remaining useful life (RUL) prediction is analyzed, i.e., the stable model parameters without self-evolution ability, which has become the primary factor that hinders self-adaptive prediction of equipment RUL. Combined with parallel systems theory, the equipment RUL prediction oriented symbiotic simulation framework is proposed on the basis of modeling analysis and Wiener state space model (SSM) is taken as the basic simulation model in the framework. Driven by the dynamic injected equipment degradation observation data, the model parameters are updated online by using expectation maximum (EM) algorithm and the data assimilation between simulation outputs and observation data is executed by using Kalman filter, so as to realize dynamic evolution of the simulation model. The simulation model evolution which makes the simulation outputs close to equipment real degradation state provides high fidelity model and data for predicting equipment RUL accurately. The framework is verified by the performance degradation data of a bearing. The simulation results show that the symbiotic simulation method can accurately simulate the equipment performance degradation process and the self-adaptive prediction of equipment RUL is realized on the basis of improving prediction accuracy, proving the feasibility and effectiveness of symbiotic simulation method.

This study assesses the effect of background error length scale (Len_scaling) and variance scale (Var_scaling) factors on the intensity and track prediction of very severe cyclonic storm Nivar over the north Indian Ocean. Multiple data assimilation simulations are performed using the three-dimensional variational data assimilation technique by varying the Len_scaling and the Var_scaling of the background error. L10V08 (keeping Len_scaling constant whereas Var_scaling is reduced by 20%) experiment has shown the best results among all closely followed by L10V10 (Len_scaling and Var_scaling both remain constant). In the L10V08 experiment, the average track error is 45.7 km, whereas the average minimum sea level pressure (maximum sustained wind) error is found 3.4 hPa (2.7 m/s) compared to the observations.

Iterative algorithms for solving the data assimilation problems are considered, based on the main and adjoint equations. Spectral properties of the control operators of the problem are studied, the iterative algorithms are justified.

In this paper we examine the links between Ensemble Kalman Filters (EnKF) and Particle Filters (PF). EnKF can be seen as a mean-field process with a PF approximation. We explore the problem of dimensionality on a toy model. To by-pass this difficulty, we suggest using Local Particle Filters (LPF) to catch nonlinearities and feed larger scale EnKF. To go one step forward we conclude with a real application and present the filtering of perturbed measurements of atmospheric wind in the domain of turbulence. This example is the cornerstone of the LPF for the assimilation of atmospheric turbulent wind. These local representation techniques will be used in further works to assimilate singular data of turbulence linked parameters in non-hydrostatic models.

We describe our ongoing efforts to understand the dynamics of the sleep-wake regulatory system to improve seizure prediction. Although mathematical models of its dynamics have been developed, the activity of this system is difficult to observe directly. Therefore we have developed a data assimilation framework that combines sparse measurements of the sleep-wake regulatory system together with a mathematical model of its dynamics to estimate the unmeasured variables. This toolset will allow us to understand the interaction between modulators of the state-of-vigilance and seizure susceptibility and may provide meaningful indicators of an impending seizure.