On Optimal and Simultaneous Stochastic Perturbations with Application to Estimation of High-Dimensional Matrix and Data Assimilation in High-Dimensional Systems

This chapter is devoted to different types of optimal perturbations (OP), deterministic, stochastic, OP in an invariant subspace, and simultaneous stochastic perturbations (SSP). The definitions of OPs are given. It will be shown how the OPs are important for the study on the predictability of behavior of system dynamics, generating ensemble forecasts as well as in the design of a stable filter. A variety of algorithm-based SSP methodology for estimation and decomposition of very high-dimensional (Hd) matrices are presented. Numerical experiments will be presented to illustrate the efficiency and benefice of the perturbation technique

A Comparison Study on Performance of an Adaptive Filter with Other Estimation Methods for State Estimation in High-Dimensional System

In this chapter, performance comparison between the adaptive filter (AF) and other estimation methods, especially with the variational method (VM), is given in the context of data assimilation problem in dynamical systems with (very) high dimension. The emphasis is put on the importance of innovation approach which is a basis for construction of the AF as well as the choice of a set of tuning parameters in the filter gain. It will be shown that the innovation representation for the initial dynamical system plays essential role in providing stability of the assimilation algorithms for stable and unstable system dynamics. Numerical experiments will be given to illustrate the performance of the AF

Reduced rank filtering in chaotic systems with application in geophysical sciences

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, 2008.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student submitted PDF version of thesis.Includes bibliographical references.Recent technological advancements have enabled us to collect large volumes of geophysical noisy measurements that need to be combined with the model forecasts, which capture all of the known properties of the underlying system. This problem is best formulated in a stochastic optimization framework, which when solved recursively is known as Filtering. Due to the large dimensions of geophysical models, optimal filtering algorithms cannot be implemented within the constraints of available computation resources. As a result, most applications use suboptimal reduced rank algorithms. Successful implementation of reduced rank filters depends on the dynamical properties of the underlying system. Here, the focus is on geophysical systems with chaotic behavior defined as extreme sensitivity of the dynamics to perturbations in the state or parameters of the system. In particular, uncertainties in a chaotic system experience growth and instability along a particular set of directions in the state space that are continually subject to large and abrupt state-dependent changes. Therefore, any successful reduced rank filter has to continually identify the important direction of uncertainty in order to properly estimate the true state of the system. In this thesis, we introduce two efficient reduced rank filtering algorithms for chaotic system, scalable to large geophysical applications. Firstly, a geometric approach is taken to identify the growing directions of uncertainty, which translate to the leading singular vectors of the state transition matrix over the forecast period, so long as the linear approximation of the dynamics is valid.The singular vectors are computed via iterations of the linear forward and adjoint models of the system and used in a filter with linear Kalman-based update. Secondly, the dynamical stability of the estimation error in a filter with linear update is analyzed, assuming that error propagation can be approximated using the state transition matrix of the system over the forecast period. The unstable directions of error dynamics are identified as the Floquet vectors of an auxiliary periodic system that is defined based on the forecast trajectory. These vectors are computed by iterations of the forward nonlinear model and used in a Kalman-based filter. Both of the filters are tested on a chaotic Lorenz 95 system with dynamic model error against the ensemble Kalman filter. Results show that when enough directions are considered, the filters perform at the optimal level, defined by an ensemble Kalman filter with a very large ensemble size. Additionally, both of the filters perform equally well when the dynamic model error is absence and ensemble filters fail. The number of iterations for computing the vectors can be set a priori based on the available computational resources and desired accuracy. To investigate scalability of the algorithms, they are implemented in a quasi-geostrophic ocean circulation model. The results are promising for future extensions to realistic geophysical applications, with large models.by Adel Ahanin.Ph.D

Covariance Localization in Strongly Coupled Data Assimilation

The recent development of accurate coupled models of the Earth system and enhanced computation power have enabled numerical prediction with the coupled models in weather, sub-seasonal, seasonal, and interannual time scales as well as climate projection. In the shorter timescales, the initial condition, or the estimate of the present state of the system, is essential for accurate prediction. Coupled data assimilation (DA) based on an ensemble of forecasts seems to be a promising approach for this state estimate due to its inherent ability to estimate flow-dependent error covariance. Strongly coupled DA tries to incorporate more observations of the other subsystems into an analysis (e.g., ocean observations into the atmospheric analysis) using the coupled error covariances; the covariance is estimated with a finite ensemble, and spurious covariance must be eliminated by localization. Because the coupling strength between subsystems of the Earth is not a simple function of a distance, we develop a better localization strategy than the distance-dependent localization. Based on the estimated benefit of each observation into each analysis variable, we first propose the correlation-cutoff method, where localization of strongly coupled DA is guided by ensemble correlations of an offline DA cycle. The method achieves improved analysis accuracy when tested with a simple coupled model of the atmosphere and ocean. As a related topic, error growth and predictability of a coupled dynamical system with multiple timescales are explored using a simple chaotic model of the atmosphere and ocean. A discontinuous response of the attractor's characteristics to the coupling strength is reported. The characteristic of global atmosphere-ocean coupled error correlation is investigated using two sets of ensemble DA systems. This knowledge is essential for effectively implementing global strongly coupled atmosphere-ocean DA. We report and discuss common and uncommon features, and the importance of ocean model resolution is stressed. Finally, the correlation-cutoff method is realized for global atmosphere-ocean strongly coupled DA with neural networks. The combination of static information provided by the neural networks and flow-dependent error covariance estimated by the ensemble improves the atmospheric analysis in our proof-of-concept experiment. The neural networks' ability to reproduce the error statistics, computation cost in a DA system, as well as analysis quality are evaluated

Global convection in Earth's mantle : advanced numerical methods and extreme-scale simulations

The thermal convection of rock in Earth's mantle and associated plate tectonics are modeled by nonlinear incompressible Stokes and energy equations. This dissertation focuses on the development of advanced, scalable linear and nonlinear solvers for numerical simulations of realistic instantaneous mantle flow, where we must overcome several computational challenges. The most notable challenges are the severe nonlinearity, heterogeneity, and anisotropy due to the mantle's rheology as well as a wide range of spatial scales and highly localized features. Resolving the crucial small scale features efficiently necessitates adaptive methods, while computational results greatly benefit from a high accuracy per degree of freedom and local mass conservation. Consequently, the discretization of Earth's mantle is carried out by high-order finite elements on aggressively adaptively refined hexahedral meshes with a continuous, nodal velocity approximation and a discontinuous, modal pressure approximation. These velocity--pressure pairings yield optimal asymptotic convergence rates of the finite element approximation to the infinite-dimensional solution with decreasing mesh element size, are inf-sup stable on general, non-conforming hexahedral meshes with "hanging nodes,'' and have the advantage of preserving mass locally at the element level due to the discontinuous pressure. However, because of the difficulties cited above and the desired accuracy, the large implicit systems to be solved are extremely poorly conditioned and sophisticated linear and nonlinear solvers including powerful preconditioning techniques are required. The nonlinear Stokes system is solved using a grid continuation, inexact Newton--Krylov method. We measure the residual of the momentum equation in the H⁻¹-norm for backtracking line search to avoid overly conservative update steps that are significantly reduced from one. The Newton linearization is augmented by a perturbation of a highly nonlinear term in mantle's rheology, resulting in dramatically improved nonlinear convergence. We present a new Schur complement-based Stokes preconditioner, weighted BFBT, that exhibits robust fast convergence for Stokes problems with smooth but highly varying (up to 10 orders of magnitude) viscosities, optimal algorithmic scalability with respect to mesh refinement, and only a mild dependence on the polynomial order of high-order finite element discretizations. In addition, we derive theoretical eigenvalue bounds to prove spectral equivalence of our inverse Schur complement approximation. Finally, we present a parallel hybrid spectral--geometric--algebraic multigrid (HMG) to approximate the inverses of the Stokes system's viscous block and variable-coefficient pressure Poisson operators within weighted BFBT. Building on the parallel scalability of HMG, our Stokes solver demonstrates excellent parallel scalability to 1.6 million CPU cores without sacrificing algorithmic optimality.Computational Science, Engineering, and Mathematic

Estimation and tracking of rapidly time-varying broadband acoustic communication channels

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution February 2006This thesis develops methods for estimating wideband shallow-water acoustic communication channels. The very shallow water wideband channel has three distinct features: large dimension caused by extensive delay spread; limited number of degrees of freedom (DOF) due to resolvable paths and inter-path correlations; and rapid fluctuations induced by scattering from the moving sea surface. Traditional LS estimation techniques often fail to reconcile the rapid fluctuations with the large dimensionality. Subspace based approaches with DOF reduction are confronted with unstable subspace structure subject to significant changes over a short period of time. Based on state-space channel modeling, the first part of this thesis develops algorithms that jointly estimate the channel as well as its dynamics. Algorithms based on the Extended Kalman Filter (EKF) and the Expectation Maximization (EM) approach respectively are developed. Analysis shows conceptual parallels, including an identical second-order innovation form shared by the EKF modification and the suboptimal EM, and the shared issue of parameter identifiability due to channel structure, reflected as parameter unobservability in EKF and insufficient excitation in EM. Modifications of both algorithms, including a two-model based EKF and a subspace EM algorithm which selectively track dominant taps and reduce prediction error, are proposed to overcome the identifiability issue. The second part of the thesis develops algorithms that explicitly find the sparse estimate of the delay-Doppler spread function. The study contributes to a better understanding of the channel physical constraints on algorithm design and potential performance improvement. It may also be generalized to other applications where dimensionality and variability collide.Financial support for this thesis research was provided by the Office of Naval Research and the WHOI Academic Program Office