Abstract: The linearly constrained matrix rank minimization problem is widely applicable in many fields such as control, signal processing and system identification. The tightest convex relaxation of this problem is the linearly constrained nuclear norm minimization. Although the latter can be cast as a semidefinite programming problem, such an approach is computationally expensive to solve when the matrices are large. In this paper, we propose fixed point and Bregman iterative algorithms for solving the nuclear norm minimization problem and prove convergence of the first of these algorithms. By using a homotopy approach together with an approximate singular value decomposition procedure, we get a very fast, robust and powerful algorithm, which we call FPCA (Fixed Point Continuation with Approximate SVD), that can solve very large matrix rank minimization problems (the code can be downloaded from http://www.columbia.edu/~sm2756/FPCA.htmfor non-commercial use). Our numerical results on randomly generated and real matrix completion problems demonstrate that this algorithm is much faster and provides much better recoverability than semidefinite programming solvers such as SDPT3. For example, our algorithm can recover 1000 × 1000 matrices of rank 50 with a relative error of 10−5 in about 3 min by sampling only 20% of the elements. We know of no other method that achieves as good recoverability. Numerical experiments on online recommendation, DNA microarray data set and image inpainting problems demonstrate the effectiveness of our algorithms.

Abstract: In the second edition of this classic monograph, complete with four new chapters and updated references, readers will now have access to content describing and analysing classical and modern methods with emphasis on the algebraic structure of linear iteration, which is usually ignored in other literature. The necessary amount of work increases dramatically with the size of systems, so one has to search for algorithms that most efficiently and accurately solve systems of, e.g., several million equations. The choice of algorithms depends on the special properties the matrices in practice have. An important class of large systems arises from the discretization of partial differential equations. In this case, the matrices are sparse (i.e., they contain mostly zeroes) and well-suited to iterative algorithms. The first edition of this book grew out of a series of lectures given by the author at the Christian-Albrecht University of Kiel to students of mathematics. The second edition includes quite novel approaches

Abstract: We introduce a new iterative algorithm to find sparse solutions of underdetermined linear systems. The algorithm, a simple combination of the Iterative Hard Thresholding algorithm and the Compressive Sampling Matching Pursuit algorithm, is called Hard Thresholding Pursuit. We study its general convergence and notice in particular that only a finite number of iterations are required. We then show that, under a certain condition on the restricted isometry constant of the matrix of the linear system, the Hard Thresholding Pursuit algorithm indeed finds all $s$-sparse solutions. This condition, which reads $\delta_{3 s} < 1/\sqrt{3}$, is heuristically better than the sufficient conditions currently available for other compressive sensing algorithms. It applies to fast versions of the algorithm, too, including the Iterative Hard Thresholding algorithm. Stability with respect to sparsity defect and robustness with respect to measurement error are also guaranteed under the condition $\delta_{3 s} < 1/\sqrt{3}$. We conclude with some numerical experiments to demonstrate the good empirical performance and the low complexity of the Hard Thresholding Pursuit algorithm.

Abstract: This paper presents a novel SParse Iterative Covariance-based Estimation approach, abbreviated as SPICE, to array processing. The proposed approach is obtained by the minimization of a covariance matrix fitting criterion and is particularly useful in many-snapshot cases but can be used even in single-snapshot situations. SPICE has several unique features not shared by other sparse estimation methods: it has a simple and sound statistical foundation, it takes account of the noise in the data in a natural manner, it does not require the user to make any difficult selection of hyperparameters, and yet it has global convergence properties.

Abstract: In this work a new algorithm is derived for the onboard calibration of three-axis strapdown magnetometers. The proposed calibration method is written in the sensor frame, and compensates for the combined effect of all linear time-invariant distortions, namely soft iron, hard iron, sensor nonorthogonality, and bias, among others. A maximum likelihood estimator (MLE) is formulated to iteratively find the optimal calibration parameters that best fit to the onboard sensor readings, without requiring external attitude references. It is shown that the proposed calibration technique is equivalent to the estimation of a rotation, scaling and translation transformation, and that the sensor alignment matrix is given by the solution of the orthogonal Procrustes problem. Good initial conditions for the iterative algorithm are obtained by a suboptimal batch least squares computation. Simulation and experimental results with low-cost sensors data are presented and discussed, supporting the application of the algorithm to autonomous vehicles and other robotic platforms.

Abstract: We consider the classical policy iteration method of dynamic programming (DP), where approximations and simulation are used to deal with the curse of dimensionality. We survey a number of issues: convergence and rate of convergence of approximate policy evaluation methods, singularity and susceptibility to simulation noise of policy evaluation, exploration issues, constrained and enhanced policy iteration, policy oscillation and chattering, and optimistic and distributed policy iteration. Our discussion of policy evaluation is couched in general terms and aims to unify the available methods in the light of recent research developments and to compare the two main policy evaluation approaches: projected equations and temporal differences (TD), and aggregation. In the context of these approaches, we survey two different types of simulation-based algorithms: matrix inversion methods, such as least-squares temporal difference (LSTD), and iterative methods, such as least-squares policy evaluation (LSPE) and TD (λ), and their scaled variants. We discuss a recent method, based on regression and regularization, which rectifies the unreliability of LSTD for nearly singular projected Bellman equations. An iterative version of this method belongs to the LSPE class of methods and provides the connecting link between LSTD and LSPE. Our discussion of policy improvement focuses on the role of policy oscillation and its effect on performance guarantees. We illustrate that policy evaluation when done by the projected equation/TD approach may lead to policy oscillation, but when done by aggregation it does not. This implies better error bounds and more regular performance for aggregation, at the expense of some loss of generality in cost function representation capability. Hard aggregation provides the connecting link between projected equation/TD-based and aggregation-based policy evaluation, and is characterized by favorable error bounds.

Abstract: Scaled sparse linear regression jointly estimates the regression coefficients and noise level in a linear model. It chooses an equilibrium with a sparse regression method by iteratively estimating the noise level via the mean residual square and scaling the penalty in proportion to the estimated noise level. The iterative algorithm costs little beyond the computation of a path or grid of the sparse regression estimator for penalty levels above a proper threshold. For the scaled lasso, the algorithm is a gradient descent in a convex minimization of a penalized joint loss function for the regression coefficients and noise level. Under mild regularity conditions, we prove that the scaled lasso simultaneously yields an estimator for the noise level and an estimated coefficient vector satisfying certain oracle inequalities for prediction, the estimation of the noise level and the regression coefficients. These inequalities provide sufficient conditions for the consistency and asymptotic normality of the noise level estimator, including certain cases where the number of variables is of greater order than the sample size. Parallel results are provided for the least squares estimation after model selection by the scaled lasso. Numerical results demonstrate the superior performance of the proposed methods over an earlier proposal of joint convex minimization.

Abstract: Recent applications of model-based iterative reconstruction (MBIR) algorithms to multislice helical CT reconstructions have shown that MBIR can greatly improve image quality by increasing resolution as well as reducing noise and some artifacts However, high computational cost and long reconstruction times remain as a barrier to the use of MBIR in practical applications Among the various iterative methods that have been studied for MBIR, iterative coordinate descent (ICD) has been found to have relatively low overall computational requirements due to its fast convergence This paper presents a fast model-based iterative reconstruction algorithm using spatially nonhomogeneous ICD (NH-ICD) optimization The NH-ICD algorithm speeds up convergence by focusing computation where it is most needed The NH-ICD algorithm has a mechanism that adaptively selects voxels for update First, a voxel selection criterion VSC determines the voxels in greatest need of update Then a voxel selection algorithm VSA selects the order of successive voxel updates based upon the need for repeated updates of some locations, while retaining characteristics for global convergence In order to speed up each voxel update, we also propose a fast 1-D optimization algorithm that uses a quadratic substitute function to upper bound the local 1-D objective function, so that a closed form solution can be obtained rather than using a computationally expensive line search algorithm We examine the performance of the proposed algorithm using several clinical data sets of various anatomy The experimental results show that the proposed method accelerates the reconstructions by roughly a factor of three on average for typical 3-D multislice geometries

Abstract: Through waveform diversity, multiple-input multiple-output (MIMO) radar can provide higher resolution, improved sensitivity, and increased parameter identifiability compared to more traditional phased-array radar schemes. Existing methods for target estimation, however, often fail to provide accurate MIMO angle-range-Doppler images when there are only a few data snapshots available. Sparse signal recovery algorithms, including many l1-norm based approaches, can offer improved estimation in that case. In this paper, we present a regularized minimization approach to sparse signal recovery. Sparse learning via iterative minimization (SLIM) follows an lq-norm constraint (for 0 <; q ≤ 1), and can thus be used to provide more accurate estimates compared to the l1-norm based approaches. We herein compare SLIM, through imaging examples and examination of computational complexity, to several well-known sparse methods, including the widely used CoSaMP approach. We show that SLIM provides superior performance for sparse MIMO radar imaging applications at a low computational cost. Furthermore, we will show that the user parameter q can be automatically determined by incorporating the Bayesian information criterion.

Abstract: In this paper, we study the finite-horizon optimal control problem for discrete-time nonlinear systems using the adaptive dynamic programming (ADP) approach. The idea is to use an iterative ADP algorithm to obtain the optimal control law which makes the performance index function close to the greatest lower bound of all performance indices within an -error bound. The optimal number of control steps can also be obtained by the proposed ADP algorithms. A convergence analysis of the proposed ADP algorithms in terms of performance index function and control policy is made. In order to facilitate the implementation of the iterative ADP algorithms, neural networks are used for approximating the performance index function, computing the optimal control policy, and modeling the nonlinear system. Finally, two simulation examples are employed to illustrate the applicability of the proposed method.

Abstract: Applied Iterative Methods is a self-contained treatise suitable as both a reference and a graduate-level textbook in the area of iterative algorithms. It is the first book to combine subjects such as optimization, convex analysis, and approximation theory and organize them around a detailed and mathematically sound treatment of iterative algorithms. Such algorithms are used in solving problems in a diverse area of applications, most notably in medical imaging such as emission and transmission tomography and magnetic-resonance imaging, as well as in intensity-modulated radiation therapy. Other applications, which lie outside of medicine, are remote sensing and hyperspectral imaging. This book details a great number of different iterative algorithms that are universally applicable.

Abstract: Separable models occur frequently in spectral analysis, array processing, radar imaging and astronomy applications Statistical inference methods for these models can be categorized in three large classes: parametric, nonparametric (also called “dense”) and semiparametric (also called “sparse”) We begin by discussing the advantages and disadvantages of each class Then we go on to introduce a new semiparametric/sparse method called SPICE (a semiparametric/sparse iterative covariance-based estimation method) SPICE is computationally quite efficient, enjoys global convergence properties, can be readily used in the case of replicated measurements and, unlike most other sparse estimation methods, does not require any subtle choices of user parameters We illustrate the statistical performance of SPICE by means of a line-spectrum estimation study for irregularly sampled data

Abstract: We study the design optimization of linear precoders for maximizing the mutual information between finite alphabet input and the corresponding output over complex-valued vector channels. This mutual information is a nonlinear and non-concave function of the precoder parameters, posing a major obstacle to precoder design optimization. Our work presents three main contributions: First, we prove that the mutual information is a concave function of a matrix which itself is a quadratic function of the precoder matrix. Second, we propose a parameterized iterative algorithm for finding optimal linear precoders to achieve the global maximum of the mutual information. The proposed iterative algorithm is numerically robust, computationally efficient, and globally convergent. Third, we demonstrate that maximizing the mutual information between a discrete constellation input and the corresponding output of a vector channel not only provides the highest practically achievable rate but also serves as an excellent criterion for minimizing the coded bit error rate. Our numerical examples show that the proposed algorithm achieves mutual information very close to the channel capacity for channel coding rate under 0.75, and also exhibits a large gain over existing linear precoding and/or power allocation algorithms. Moreover, our examples show that certain existing methods are susceptible to being trapped at locally optimal precoders.

Abstract: We propose a preconditioned variant of the modified HSS (MHSS) iteration method for solving a class of complex symmetric systems of linear equations. Under suitable conditions, we prove the convergence of the preconditioned MHSS (PMHSS) iteration method and discuss the spectral properties of the PMHSS-preconditioned matrix. Numerical implementations show that the resulting PMHSS preconditioner leads to fast convergence when it is used to precondition Krylov subspace iteration methods such as GMRES and its restarted variants. In particular, both the stationary PMHSS iteration and PMHSS-preconditioned GMRES show meshsize-independent and parameter-insensitive convergence behavior for the tested numerical examples.

Abstract: In this paper, we study the problem of optimal trajectory generation for a team of heterogeneous robots moving in a plane and tracking a moving target by processing relative observations, i.e., distance and/or bearing. Contrary to previous approaches, we explicitly consider limits on the robots' speed and impose constraints on the minimum distance at which the robots are allowed to approach the target. We first address the case of a single tracking sensor and seek the next sensing location in order to minimize the uncertainty about the target's position. We show that although the corresponding optimization problem involves a nonconvex objective function and a nonconvex constraint, its global optimal solution can be determined analytically. We then extend the approach to the case of multiple sensors and propose an iterative algorithm, i.e., the Gauss-Seidel relaxation (GSR), to determine the next best sensing location for each sensor. Extensive simulation results demonstrate that the GSR algorithm, whose computational complexity is linear in the number of sensors, achieves higher tracking accuracy than gradient descent methods and has performance that is indistinguishable from that of a grid-based exhaustive search, whose cost is exponential in the number of sensors. Finally, through experiments, we demonstrate that the proposed GSR algorithm is robust and applicable to real systems.

Abstract: The integral equation (IE) method is commonly utilized to model time-harmonic electromagnetic (EM) problems. One of the greatest challenges in its applications arises in the solution of the resulting ill-conditioned matrix equation. We introduce a new domain decomposition method (DDM) for the IE solution of EM wave scattering from non-penetrable objects. The proposed method is a non-overlapping/non-conformal DDM and it provides a computationally efficient and effective preconditioner for the IE matrix equations. Moreover, the proposed approach is very suitable for dealing with multi-scale electromagnetic problems since each sub-domain has its own characteristics length and will be meshed independently. Furthermore, for each sub-domain, we are free to choose the most effective IE sub-domain solver based on its local geometrical features and electromagnetic characteristics. Additionally, the multilevel fast multi-pole algorithm (MLFMA) is utilized to accelerate the computations of couplings between sub-domains. Numerical results demonstrate that the proposed method yields rapid convergence in the outer Krylov iterative solution process. Finally, simulations of several large-scale examples testify to the effectiveness and robustness of the proposed IE based DDM.

Abstract: Seismic acquisition is a trade-off between economy and quality. In conventional acquisition the time intervals between successive records are large enough to avoid interference in time. To obtain an efficient survey, the spatial source sampling is therefore often (too) large. However, in blending, or simultaneous acquisition, temporal overlap between shot records is allowed. This additional degree of freedom in survey design significantly improves the quality or the economics or both. Deblending is the procedure of recovering the data as if they were acquired in the conventional, unblended way. A simple least-squares procedure, however, does not remove the interference due to other sources, or blending noise. Fortunately, the character of this noise is different in different domains, e.g., it is coherent in the common source domain, but incoherent in the common receiver domain. This property is used to obtain a considerable improvement. We propose to estimate the blending noise and subtract it from the blended data. The estimate does not need to be perfect because our procedure is iterative. Starting with the least-squares deblended data, the estimate of the blending noise is obtained via the following steps: sort the data to a domain where the blending noise is incoherent; apply a noise suppression filter; apply a threshold to remove the remaining noise, ending up with (part of) the signal; compute an estimate of the blending noise from this signal. At each iteration, the threshold can be lowered and more of the signal is recovered. Promising results were obtained with a simple implementation of this method for both impulsive and vibratory sources. Undoubtedly, in the future algorithms will be developed for the direct processing of blended data. However, currently a high-quality deblending procedure is an important step allowing the application of contemporary processing flows

Abstract: Iterative methods for linear systems of algebraic equations arising from the finite element discretization of nonsymmetric and indefinite elliptic problems are considered. Methods previously known to work well for positive definite, symmetric problems are extended to certain nonsymmetric problems, which can also have some eigenvalues in the left half plane.This paper presents an additive Schwarz method applied to linear, second order, symmetric or nonsymmetric, indefinite elliptic boundary value problems in two and three dimensions. An alternative linear system, which has the same solution as the original problem, is derived and this system is then solved by using GMRES, an iterative method of conjugate gradient type. In each iteration step, a coarse mesh finite element problem and a number of local problems are solved on small, overlapping subregions into which the original region is subdivided. The rate of convergence is shown to be independent of the number of degrees of freedom and the number of local...

Abstract: We address a p -shift finite impulse response (FIR) unbiased estimator (UE) for linear discrete time-varying filtering (p=0), p-step prediction (p >; 0), and p-lag smoothing (p <; 0) in state space with no requirements for initial conditions and zero mean noise. A solution is found in a batch form and represented in a computationally efficient iterative Kalman-like one. It is shown that the Kalman-like FIR UE is able to outperform the Kalman filter if the noise covariances and initial conditions are not known exactly, noise is not white, and both the system and measurement noise components need to be filtered out. Otherwise, the errors are similar. Extensive numerical studies of the FIR UE are provided in Gaussian and non-Gaussian environments with outliers and temporary uncertainties.

Abstract: We introduce a new adaptive method for analyzing nonlinear and nonstationary data. This method is inspired by the empirical mode decomposition (EMD) method and the recently developed compressed sensing theory. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form {a(t )c os(θ(t))} ,w herea ≥ 0i s assumed to be smoother than cos(θ(t)) and θ is a piecewise smooth increasing function. We formulate this problem as a nonlinear L 1 optimization problem. Further, we propose an iterative algorithm to solve this nonlinear optimization problem recursively. We also introduce an adaptive filter method to decompose data with noise. Numerical examples are given to demonstrate the robustness of our method and comparison is made with the EMD method. One advantage of performing such a decomposition is to preserve some intrinsic physical property of the signal, such as trend and instantaneous frequency. Our method shares many important properties of the original EMD method. Because our method is based on a solid mathematical formulation, its performance does not depend on numerical parameters such as the number of shifting or stop criterion, which seem to have a major effect on the original EMD method. Our method is also less sensitive to noise perturbation and the end effect compared with the original EMD method.

Abstract: In this work, we exploit the fact that wavelets can represent magnetic resonance images well, with relatively few coefficients. We use this property to improve magnetic resonance imaging (MRI) reconstructions from undersampled data with arbitrary k-space trajectories. Reconstruction is posed as an optimization problem that could be solved with the iterative shrinkage/thresholding algorithm (ISTA) which, unfortunately, converges slowly. To make the approach more practical, we propose a variant that combines recent improvements in convex optimization and that can be tuned to a given specific k-space trajectory. We present a mathematical analysis that explains the performance of the algorithms. Using simulated and in vivo data, we show that our nonlinear method is fast, as it accelerates ISTA by almost two orders of magnitude. We also show that it remains competitive with TV regularization in terms of image quality.

Abstract: This paper presents a novel adaptive reduced-rank multiple-input-multiple-output (MIMO) equalization scheme and algorithms based on alternating optimization design techniques for MIMO spatial multiplexing systems. The proposed reduced-rank equalization structure consists of a joint iterative optimization of the following two equalization stages: 1) a transformation matrix that performs dimensionality reduction and 2) a reduced-rank estimator that retrieves the desired transmitted symbol. The proposed reduced-rank architecture is incorporated into an equalization structure that allows both decision feedback and linear schemes to mitigate the interantenna (IAI) and intersymbol interference (ISI). We develop alternating least squares (LS) expressions for the design of the transformation matrix and the reduced-rank estimator along with computationally efficient alternating recursive least squares (RLS) adaptive estimation algorithms. We then present an algorithm that automatically adjusts the model order of the proposed scheme. An analysis of the LS algorithms is carried out along with sufficient conditions for convergence and a proof of convergence of the proposed algorithms to the reduced-rank Wiener filter. Simulations show that the proposed equalization algorithms outperform the existing reduced- and full- algorithms while requiring a comparable computational cost.

Abstract: Our contribution deals with fast computation of dense two-component (2C) PIV vector fields using Graphics Processing Units (GPUs). We show that iterative gradient-based cross-correlation optimization is an accurate and efficient alternative to multi-pass processing with FFT-based cross-correlation. Density is meant here from the sampling point of view (we obtain one vector per pixel), since the presented algorithm, folki, naturally performs fast correlation optimization over interrogation windows with maximal overlap. The processing of 5 image pairs (1,376 × 1,040 each) is achieved in less than a second on a NVIDIA Tesla C1060 GPU. Various tests on synthetic and experimental images, including a dataset of the 2nd PIV challenge, show that the accuracy of folki is found comparable to that of state-of-the-art FFT-based commercial softwares, while being 50 times faster.