Variational Data Assimilation via Sparse Regularization

This paper studies the role of sparse regularization in a properly chosen basis for variational data assimilation (VDA) problems. Specifically, it focuses on data assimilation of noisy and down-sampled observations while the state variable of interest exhibits sparsity in the real or transformed domain. We show that in the presence of sparsity, the $\ell_{1}$-norm regularization produces more accurate and stable solutions than the classic data assimilation methods. To motivate further developments of the proposed methodology, assimilation experiments are conducted in the wavelet and spectral domain using the linear advection-diffusion equation

Fixing Nonconvergence of Algebraic Iterative Reconstruction with an Unmatched Backprojector

We consider algebraic iterative reconstruction methods with applications in image reconstruction. In particular, we are concerned with methods based on an unmatched projector/backprojector pair; i.e., the backprojector is not the exact adjoint or transpose of the forward projector. Such situations are common in large-scale computed tomography, and we consider the common situation where the method does not converge due to the nonsymmetry of the iteration matrix. We propose a modified algorithm that incorporates a small shift parameter, and we give the conditions that guarantee convergence of this method to a fixed point of a slightly perturbed problem. We also give perturbation bounds for this fixed point. Moreover, we discuss how to use Krylov subspace methods to efficiently estimate the leftmost eigenvalue of a certain matrix to select a proper shift parameter. The modified algorithm is illustrated with test problems from computed tomography

Blind Minimax Estimation

We consider the linear regression problem of estimating an unknown, deterministic parameter vector based on measurements corrupted by colored Gaussian noise. We present and analyze blind minimax estimators (BMEs), which consist of a bounded parameter set minimax estimator, whose parameter set is itself estimated from measurements. Thus, one does not require any prior assumption or knowledge, and the proposed estimator can be applied to any linear regression problem. We demonstrate analytically that the BMEs strictly dominate the least-squares estimator, i.e., they achieve lower mean-squared error for any value of the parameter vector. Both Stein's estimator and its positive-part correction can be derived within the blind minimax framework. Furthermore, our approach can be readily extended to a wider class of estimation problems than Stein's estimator, which is defined only for white noise and non-transformed measurements. We show through simulations that the BMEs generally outperform previous extensions of Stein's technique.Comment: 12 pages, 7 figure

Variational Downscaling, Fusion and Assimilation of Hydrometeorological States via Regularized Estimation

Improved estimation of hydrometeorological states from down-sampled observations and background model forecasts in a noisy environment, has been a subject of growing research in the past decades. Here, we introduce a unified framework that ties together the problems of downscaling, data fusion and data assimilation as ill-posed inverse problems. This framework seeks solutions beyond the classic least squares estimation paradigms by imposing proper regularization, which are constraints consistent with the degree of smoothness and probabilistic structure of the underlying state. We review relevant regularization methods in derivative space and extend classic formulations of the aforementioned problems with particular emphasis on hydrologic and atmospheric applications. Informed by the statistical characteristics of the state variable of interest, the central results of the paper suggest that proper regularization can lead to a more accurate and stable recovery of the true state and hence more skillful forecasts. In particular, using the Tikhonov and Huber regularization in the derivative space, the promise of the proposed framework is demonstrated in static downscaling and fusion of synthetic multi-sensor precipitation data, while a data assimilation numerical experiment is presented using the heat equation in a variational setting

Structure-Based Subspace Method for Multi-Channel Blind System Identification

In this work, a novel subspace-based method for blind identification of multichannel finite impulse response (FIR) systems is presented. Here, we exploit directly the impeded Toeplitz channel structure in the signal linear model to build a quadratic form whose minimization leads to the desired channel estimation up to a scalar factor. This method can be extended to estimate any predefined linear structure, e.g. Hankel, that is usually encountered in linear systems. Simulation findings are provided to highlight the appealing advantages of the new structure-based subspace (SSS) method over the standard subspace (SS) method in certain adverse identification scenarii.Comment: 5 pages, Submitted to IEEE Signal Processing Letters, January 201