A common problem in applied mathematics is that of finding a function in a Hilbert space with prescribed best approximations from a finite number of closed vector subspaces. In the present paper we study the question ...
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A common problem in applied mathematics is that of finding a function in a Hilbert space with prescribed best approximations from a finite number of closed vector subspaces. In the present paper we study the question of the existence of solutions to such problems. A finite family of subspaces is said to satisfy the Inverse Best Approximation Property (IBAP) if there exists a point that admits any selection of points from these subspaces as best approximations. We provide various characterizations of the IBAP in terms of the geometry of the subspaces. Connections between the IBAP and the linear convergence rate of the periodic projection algorithm for solving the underlying affine feasibility problem are also established. The results are applied to investigate problems in harmonic analysis, integral equations, signal theory, and wavelet frames. (C) 2009 Elsevier Inc. All rights reserved.
This communication is concerned with the development of a model-order reduction (MOR) approach for the acceleration of a source-sweep analysis using the volume electric field integral equation (EFIE) formulation. In p...
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This communication is concerned with the development of a model-order reduction (MOR) approach for the acceleration of a source-sweep analysis using the volume electric field integral equation (EFIE) formulation. In particular, we address the prohibitive computational burden associated with the repeated solution of the two-dimensional electromagnetic wave scattering problem for source-sweep analysis. The method described within is a variant of the Krylov subspace approach to MOR, that captures at an early stage of the iteration the essential features of the original system. As such these approaches are capable of creating very accurate low-order models. Numerical examples are provided that demonstrate the speed-up achieved by utilizing these MOR approaches when compared against a method of moments (MoM) solution accelerated by use of the fast Fourier transform (FFT).
Restricted complexity estimation is a major topic in control-oriented identification. Conditional algorithms are used to identify linear finite-dimensional models of complex systems, the aim being to minimize the wors...
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Restricted complexity estimation is a major topic in control-oriented identification. Conditional algorithms are used to identify linear finite-dimensional models of complex systems, the aim being to minimize the worst-case identification error. High computational complexity of optimal solutions suggests to employ suboptimal estimation algorithms. This paper studies different classes of conditional estimators and provides results that assess the reliability level of suboptimal algorithms.
The convergence of Lardy's series representation of the Moore-Penrose inverse of a closed unbounded linear operator is proved via Dykstra's alternating projection algorithm. (C) 202 Elsevier Science (USA).
The convergence of Lardy's series representation of the Moore-Penrose inverse of a closed unbounded linear operator is proved via Dykstra's alternating projection algorithm. (C) 202 Elsevier Science (USA).
Set theoretic estimates have been shown to be effective in many areas of signal processing. Unfortunately, in many problems, projection methods, which are the primary way of generating such estimates, cannot be implem...
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Set theoretic estimates have been shown to be effective in many areas of signal processing. Unfortunately, in many problems, projection methods, which are the primary way of generating such estimates, cannot be implemented due to theoretical limitations and computational complexity. In this correspondence, an adapted random search algorithm is shown to be a feasible method for the synthesis of set theoretic estimates. It circumvents the theoretical and computational shortcomings of existing deterministic methods and does not place any geometrical restrictions on the sets.
In this letter, a derivation of the normalized LMS algorithm is generalized, resulting in a family of projection-like algorithms based on an L(p)-minimized filter coefficient change. The resulting algorithms include t...
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In this letter, a derivation of the normalized LMS algorithm is generalized, resulting in a family of projection-like algorithms based on an L(p)-minimized filter coefficient change. The resulting algorithms include the simplified NLMS algorithm of Nagumo and Noda and an even simpler single-coefficient update algorithm based on the maximum absolute value datum of the input data vector. A complete derivation of the algorithm family is given, and simulations are performed to show the convergence behaviors of the algorithms.
This letter proposes a design of extremum seeking controllers that guarantees precise convergence of the control system to the unknown optimizer of a measured unknown cost function. The approach introduces an uncertai...
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This letter proposes a design of extremum seeking controllers that guarantees precise convergence of the control system to the unknown optimizer of a measured unknown cost function. The approach introduces an uncertainty estimation technique that provides an estimate of the worst case uncertainty associated with the unknown optimizer. An uncertainty set update algorithm is proposed to reduce the radius of the uncertain set as new data is received. The radius of the uncertainty set is used to reduce the required dither amplitude. The convergence to the unknown optimum and the removal of the dither signal are achieved simultaneously. Asymptotic convergence of the extremum seeking control system to the unknown minimizer is achieved in the absence of measurement noise. A simulation example is provided to demonstrate the effectiveness of the approach.
In this paper, we are concerned with the split feasibility problem (SFP) whenever the convex sets involved are composed of level sets. By applying Polyak's gradient method, we get a new and simple algorithm for su...
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In this paper, we are concerned with the split feasibility problem (SFP) whenever the convex sets involved are composed of level sets. By applying Polyak's gradient method, we get a new and simple algorithm for such a problem. Under standard assumptions, we prove that the whole sequence generated by the algorithm weakly converges to a solution. We also modify the proposed algorithm and state the strong convergence without regularity conditions on the sets involved. Numerical experiments are included to illustrate its applications in signal processing.
The notion of relaxation is well understood for orthogonal projections onto convex sets. For general Bregman projections it was considered only for hyperplanes, and the question of how to relax Bregman projections ont...
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The notion of relaxation is well understood for orthogonal projections onto convex sets. For general Bregman projections it was considered only for hyperplanes, and the question of how to relax Bregman projections onto convex sets that are not linear (i.e., not hyperplanes or half-spaces) has remained open. A definition of the underrelaxation of Bregman projections onto general convex sets is given here, which includes as special cases the underrelaxed orthogonal projections and the underrelaxed Bregman projections onto linear sets as given by De Pierro and Iusem [ J. Optim. Theory Appl., 51 ( 1986), pp. 421 440]. With this new definition, we construct a block-iterative projection algorithmic scheme and prove its convergence to a solution of the convex feasibility problem. The practical importance of relaxation parameters in the application of such projection algorithms to real-world problems is demonstrated on a problem of image reconstruction from projections.
The first and second moment behavior of one variation of the normalized LMS (ε-NLMS) algorithm is investigated for a white covariance matrix and Gaussian statistics for the data. For this model, it is shown that the ...
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The first and second moment behavior of one variation of the normalized LMS (ε-NLMS) algorithm is investigated for a white covariance matrix and Gaussian statistics for the data. For this model, it is shown that the ε-NLMS algorithm has neither behavior independent of the input data power nor a performance significantly better than the LMS algorithm for which the input power level also must be known a priori. Hence, based upon the results of the analysis, it is recommended that the algorithm not be used in place of the LMS for known input power levels or in place of the NLMS for unknown input power levels.
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