Abstract: We consider linear inverse problems where the solution is assumed to have a sparse expansion on an arbitrary preassigned orthonormal basis. We prove that replacing the usual quadratic regularizing penalties by weighted p-penalties on the coefficients of such expansions, with 1 ≤ p ≤ 2, still regularizes the problem. Use of such p-penalized problems with p < 2 is often advocated when one expects the underlying ideal noiseless solution to have a sparse expansion with respect to the basis under consideration. To compute the corresponding regularized solutions, we analyze an iterative algorithm that amounts to a Landweber iteration with thresholding (or nonlinear shrinkage) applied at each iteration step. We prove that this algorithm converges in norm. © 2004 Wiley Periodicals, Inc.

Abstract: We address the problem of joint downlink beamforming in a power-controlled network, where independent data streams are to be transmitted from a multiantenna base station to several decentralized single-antenna terminals. The total transmit power is limited and channel information (possibly statistical) is available at the transmitter. The design goal: jointly adjust the beamformers and transmission powers according to individual SINR requirements. In this context, there are two closely related optimization problems. P1: maximize the jointly achievable SINR margin under a total power constraint. P2: minimize the total transmission power while satisfying a set of SINR constraints. In this paper, both problems are solved within a unified analytical framework. Problem P1 is solved by minimizing the maximal eigenvalue of an extended crosstalk matrix. The solution provides a necessary and sufficient condition for the feasibility of the SINR requirements. Problem P2 is a variation of problem P1. An important step in our analysis is to show that the global optimum of the downlink beamforming problem is equivalently obtained from solving a dual uplink problem, which has an easier-to-handle analytical structure. Then, we make use of the special structure of the extended crosstalk matrix to develop a rapidly converging iterative algorithm. The optimality and global convergence of the algorithm is proven and stopping criteria are given.

Abstract: In this paper a fast sweeping method for computing the numerical solution of Eikonal equations on a rectangular grid is presented. The method is an iterative method which uses upwind difference for discretization and uses Gauss-Seidel iterations with alternating sweeping ordering to solve the discretized system. The crucial idea is that each sweeping ordering follows a family of characteristics of the corresponding Eikonal equation in a certain direction simultaneously. The method has an optimal complexity of O(N) for N grid points and is extremely simple to implement in any number of dimensions. Monotonicity and stability properties of the fast sweeping algorithm are proven. Convergence and error estimates of the algorithm for computing the distance function is studied in detail. It is shown that 2 n Gauss-Seidel iterations is enough for the distance function in n dimensions. An estimation of the number of iterations for general Eikonal equations is also studied. Numerical examples are used to verify the analysis.

Abstract: We propose a method of iterative phase retrieval that uses measured intensities in the diffraction plane to solve the phase problem in a way that bypasses the problem of lens aberration, leading to greatly improved spatial resolution. This method is stable, easy to implement experimentally, and can be used to view a large area of the specimen when that is desired.

Abstract: We propose an iterative phase retrieval method that uses a series of diffraction patterns, measured only in intensity, to solve for both amplitude and phase of the image wave function over a wide field of view and at wavelength-limited resolution. The new technique requires an aperture that is scanned to two or more positions over the object wave function. A simple implementation of the method is modeled and demonstrated, showing how the algorithm uses overlapping data in real space to resolve ambiguities in the solution. The technique opens up the possibility of practical transmission lensless microscopy at subatomic resolution using electrons, x rays, or nuclear particles.

Abstract: This paper proposes an algebraic solution for the position and velocity of a moving source using the time differences of arrival (TDOAs) and frequency differences of arrival (FDOAs) of a signal received at a number of receivers. The method employs several weighted least-squares minimizations only and does not require initial solution guesses to obtain a location estimate. It does not have the initialization and local convergence problem as in the conventional linear iterative method. The estimated accuracy of the source position and velocity is shown to achieve the Crame/spl acute/r-Rao lower bound for Gaussian TDOA and FDOA noise at moderate noise level before the thresholding effect occurs. Simulations are included to examine the algorithm's performance and compare it with the Taylor-series iterative method.

Abstract: An advanced random phase-shifting algorithm to extract phase distributions from randomly phase-shifted interferograms is proposed. The algorithm is based on a least-squares iterative procedure, but it copes with the limitation of the existing iterative algorithms by separating a frame-to-frame iteration from a pixel-to-pixel iteration. The algorithm provides stable convergence and accurate phase extraction with as few as three interferograms, even when the phase shifts are completely random. The algorithm is simple, fast, and fully automatic. A computer simulation is conducted to prove the concept.

Abstract: The extrinsic information transfer (EXIT) chart pioneered by Stephan ten Brink is introduced in a tutorial way for readers not yet familiar with this powerful technique which is used to analyze the behavior of iterative so-called turbo techniques. Simple examples and preliminary analytical results are given as well as some typical applications.

Abstract: In recent years, soft iterative decoding techniques have been shown to greatly improve the bit error rate performance of various communication systems. For multiantenna systems employing space-time codes, however, it is not clear what is the best way to obtain the soft information required of the iterative scheme with low complexity. In this paper, we propose a modification of the Fincke-Pohst (sphere decoding) algorithm to estimate the maximum a posteriori probability of the received symbol sequence. The new algorithm solves a nonlinear integer least squares problem and, over a wide range of rates and signal-to-noise ratios, has polynomial-time complexity. Performance of the algorithm, combined with convolutional, turbo, and low-density parity check codes, is demonstrated on several multiantenna channels. The results for systems that employ space-time modulation schemes seem to indicate that the best performing schemes are those that support the highest mutual information between the transmitted and received signals, rather than the best diversity gain.

Abstract: 1 Notations and definitions 2 Iterative methods THEORY 3 Toeplitz systems 4 Circulant preconditioners 5 Non-circulant type preconditioners 6 Ill-conditioned Toeplitz systems 7 Structured systems APPLICATIONS 8 Applications to ordinary and partial differential equations 9 Applications to queuing networks 10 Applications to signal processing 11 Applications to image processing 12 Applications to integral equations

Abstract: This paper presents the Cholesky factor--alternating direction implicit (CF--ADI) algorithm, which generates a low-rank approximation to the solution X of the Lyapunov equation AX+XAT = -BBT. The coefficient matrix A is assumed to be large, and the rank of the right-hand side -BBT is assumed to be much smaller than the size of A. The CF--ADI algorithm requires only matrix-vector products and matrix-vector solves by shifts of A. Hence, it enables one to take advantage of any sparsity or structure in A. This paper also discusses the approximation of the dominant invariant subspace of the solution X. We characterize a group of spanning sets for the range of X. A connection is made between the approximation of the dominant invariant subspace of X and the generation of various low-order Krylov and rational Krylov subspaces. It is shown by numerical examples that the rational Krylov subspace generated by the CF--ADI algorithm, where the shifts are obtained as the solution of a rational minimax problem, often gives the best approximation to the dominant invariant subspace of X.

Abstract: The convergence of the iterative identification algorithm for the Hammerstein system has been an open problem for a long time. In this paper, a detailed study is carried out and various convergence properties of the iterative algorithm are derived. It is shown that the iterative algorithm with normalization is convergent in general. Moreover, it is shown that convergence takes place in one step (two least squares iterations) for finite-impulse response Hammerstein models with i.i.d. inputs.

Abstract: In the class of quasi-contractive operators satisfying Zamfirescu's conditions, the most used fixed point iterative methods, that is, the Picard, Mann, and Ishikawa iterations, are all known to be convergent to the unique fixed point. In this paper, the comparison of the first two methods with respect to their convergence rate is obtained.

Abstract: For the nonlinear eigenvalue problem T(λ)x=0 we propose an iterative projection method for computing a few eigenvalues close to a given parameter. The current search space is expanded by a generalization of the shift-and-invert Arnoldi method. The resulting projected eigenproblems of small dimension are solved by inverse iteration. The method is applied to a rational eigenvalue problem governing damped vibrations of a structure.

Abstract: We study the problem of global stabilization by smooth output feedback, for a class of n-dimensional homogeneous systems whose Jacobian linearization is neither controllable nor observable. A new output feedback control scheme is proposed for the explicit design of both homogeneous observers and controllers. While the smooth state feedback control law is constructed based on the tool of adding a power integrator, the observer design is new and carried out by developing a machinery, which makes it possible to assign the observer gains one-by-one, in an iterative manner. Such design philosophy is fundamentally different from that of the traditional "Luenberger" observer in which the observer gain is determined by observability. In the case of linear systems, our design method provides not only a new insight but also an alternative solution to the output feedback stabilization problem. For a class of high-order nonhomogeneous systems, we further show how the proposed design method, with an appropriate modification, can still achieve global output feedback stabilization. Examples and simulations are given to demonstrate the main features and effectiveness of the proposed output feedback control schemes.

Abstract: In this paper, we introduce a new iterative learning control (ILC) method, which enables learning from different tracking control tasks. The proposed method overcomes the limitation of traditional ILC in that, the target trajectories of any two consecutive iterations can be completely different. For nonlinear systems with time-varying and time-invariant parametric uncertainties, the new learning method works effectively to nullify the tracking error. To facilitate the learning control system design and analysis, in the paper we use a composite energy function (CEF) index, which consists of a positive scalar function and L/sup 2/ norm of the function approximation error.

Abstract: We investigate turbo equalization, or iterative equalization and decoding, as a receiver technology for systems where data is protected by an error-correcting code, shuffled by an interleaver, and mapped onto a signal constellation for transmission over a frequency-selective channel with unknown time-varying channel impulse response. The focus is the concept of soft iterative channel estimation, which is to improve the channel estimate over the iterations by using soft information fed back from the decoder from the previous iteration to generate "extended training sequences" between the actual transmitted training sequences.

Abstract: Results for wave-equation migration in the frequency domain using the constant-density acoustic two-way wave equation have been compared to images obtained by its one-way approximation. The two-way approach produces more accurate reflector amplitudes and provides superior imaging of steep flanks. However, migration with the two-way wave equation is sensitive to diving waves, leading to low-frequency artifacts in the images. These can be removed by surgical muting of the input data or iterative migration or high-pass spatial filtering. The last is the most effective. Iterative migration based on a least-squares approximation of the seismic data can improve the amplitudes and resolution of the imaged reflectors. Two approaches are considered, one based on the linearized constantdensity acoustic wave equation and one on the full acoustic wave equation with variable density. The first converges quickly. However, with our choice of migration weights and high-pass spatial filtering for the linearized case, a real-data migration result shows little improvement after the first iteration. The second, nonlinear iterative migration method is considerably more difficult to apply. A real-data example shows only marginal improvement over the linearized case. In two dimensions, the computational cost of the twoway approach has the same order of magnitude as that for the one-way method. With our implementation, the two-way method requires about twice the computer time needed for one-way wave-equation migration.

Abstract: The problem of estimating and predicting Origin-Destination (OD) tables is known to be important and difficult. In the specific context of Intelligent Transportation Systems (ITS), the dynamic nature of the problem and the real-time requirements make it even more intricate. We consider here a least-square modeling approach for solving the OD estimation and prediction problem, which seems to offer convenient and flexible algorithms. The dynamic nature of the problem is represented by an autoregressive process, capturing the serial correlations of the state variables. Our formulation is inspired from Cascetta et al. (1993) and Ashok and Ben-Akiva (1993). We compare the Kalman filter algorithm to LSQR, an iterative algorithm proposed by Paige and Saunders (1982) for the solution of large-scale least-squares problems. LSQR explicitly exploits matrix sparsity, allowing to consider larger problems likely to occur in real applications. We show that the LSQR algorithm significantly decreases the computation effort needed by the Kalman filter approach for large-scale problems. We also provide a theoretical number of flops for both algorithms to predict which algorithm will perform better on a specific instance of the problem.