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TO BE OR NOT TO BE INTRUSIVE? THE SOLUTION OF PARAMETRIC AND STOCHASTIC EQUATIONS-PROPER GENERALIZED DECOMPOSITION

SIAM JOURNAL ON SCIENTIFIC COMPUTING 2015年第1期37卷 A347-A368页

作者： Giraldi, Loic Liu, Dishi Matthies, Hermann G. Nouy, Anthony Ecole Cent Nantes GeM UMR 6183 Nantes France German Aerosp Ctr DLR Inst Aerodynam & Flow Control Braunschweig ME Germany Tech Univ Carolo Wilhelmina Braunschweig Inst Comp Sci Braunschweig Germany

A numerical method is proposed to compute a low-rank Galerkin approximation to the solution of a parametric or stochastic equation in a nonintrusive fashion. The considered nonlinear problems are associated with the minimization of a parameterized differentiable convex functional. We first introduce a bilinear parameterization of fixed-rank tensors and employ an alternating minimization scheme for computing the low-rank approximation. In keeping with the idea of nonintrusiveness, at each step of the algorithm the minimizations are carried out with a quasi-Newton method to avoid the computation of the Hessian. The algorithm is made nonintrusive through the use of numerical integration. It only requires the evaluation of residuals at specific parameter values. The algorithm is then applied to two numerical examples.

关键词： parametric stochastic equation Galerkin approximation nonintrusive method low-rank approximation alternating minimization algorithm quasi-Newton method proper generalized decomposition

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Splitting Methods for Convex Clustering

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JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS 2015年第4期24卷 994-1013页

作者： Chi, Eric C. Lange, Kenneth Rice Univ Dept Elect & Comp Engn Houston TX 77005 USA Univ Calif Los Angeles Dept Biomath Los Angeles CA 90095 USA Univ Calif Los Angeles Dept Human Genet Los Angeles CA 90095 USA Univ Calif Los Angeles Dept Stat Los Angeles CA 90095 USA

Clustering is a fundamental problem in many scientific applications. Standard methods such as k-means, Gaussian mixture models, and hierarchical clustering, however, are beset by local minima, which are sometimes drastically suboptimal. Recently introduced convex relaxations of k-means and hierarchical clustering shrink cluster centroids toward one another and ensure a unique global minimizer. In this work, we present two splitting methods for solving the convex clustering problem. The first is an instance of the alternating direction method of multipliers (ADMM);the second is an instance of the alternating minimization algorithm (AMA). In contrast to previously considered algorithms, our ADMM and AMA formulations provide simple and unified frameworks for solving the convex clustering problem under the previously studied norms and open the door to potentially novel norms. We demonstrate the performance of our algorithm on both simulated and real data examples. While the differences between the two algorithms appear to be minor on the surface, complexity analysis and numerical experiments show AMA to be significantly more efficient. This article has supplementary materials available online.

关键词： alternating direction method of multipliers alternating minimization algorithm Convex optimization Hierarchical clustering k-means Regularization paths

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An integrated approach to determine prior information for improved wide-field imaging models from computational interference microscopy 22

An integrated approach to determine prior information for im...

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Conference on Three-Dimensional and Multidimensional Microscopy - Image Acquisition and Processing XXII

作者： Hossain, Md Shohag King, Sharon V. Preza, Chrysanthe Univ Memphis Dept Elect & Comp Engn Memphis TN 38152 USA

ISBN: (纸本)9781628414202

Knowledge of prior sample information, such as a refractive index (RI) map, can be used to improve image formation models enabling more accurate three-dimensional (3D) restoration in fluorescence microscopy. RI is an indicator of cell composition and structure that allows a more comprehensive representation of the 3D structure of a specimen than fluorescence alone. Due to the integral nature of sample phase, the challenge to compute the RI map is to decouple RI and thickness. Our work investigates the feasibility of determining RI of a specimen from differential interference contrast (DIC) microscopy data acquired by using different wavelengths in illumination. This spectral diversity in the data is exploited to determine sample thickness and RI. Results from simulated and experimental data of polystyrene bead samples are presented to analyze this approach. Phase images were estimated from the DIC data using an alternating minimization algorithm. This study shows that the maximum estimated phase delay is accurate within approximately 7 percent error relative to the 2D phase model. The sensitivity of this integrated approach allows RI to be computed within approximately 0.4 percent error relative to values from the literature.

关键词： Quantitative phase differential interference contrast microscopy alternating minimization algorithm rotational diversity

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alternating direction optimization algorithms for covariance completion problems

Alternating direction optimization algorithms for covariance...

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American Control Conference

作者： Zare, Armin Jovanovic, Mihailo R. Georgiou, Tryphon T. Univ Minnesota Dept Elect & Comp Engn Minneapolis MN 55455 USA

ISBN: (纸本)9781479986842

Second-order statistics of nonlinear dynamical systems can be obtained from experiments or numerical simulations. These statistics are relevant in understanding the fundamental physics, e. g., of fluid flows, and are useful for developing low-complexity models. Such models can be used for the purpose of control design and analysis. In many applications, only certain second-order statistics of a limited number of states are available. Thus, it is of interest to complete partially specified covariance matrices in a way that is consistent with the linearized dynamics. The dynamics impose structural constraints on admissible forcing correlations and state statistics. Solutions to such completion problems can be used to obtain stochastically driven linearized models. Herein, we address the covariance completion problem. We introduce an optimization criterion that combines the nuclear norm together with an entropy functional. The two, together, provide a numerically stable and scalable computational approach which is aimed at low complexity structures for stochastic forcing that can account for the observed statistics. We develop customized algorithms based on alternating direction methods that are well-suited for large scale problems.

关键词： alternating direction method of multipliers alternating minimization algorithm convex optimization low-rank approximation nuclear norm regularization state covariances structured matrix completion problems

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minimization of multi-penalty functionals by alternating iterative thresholding and optimal parameter choices

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INVERSE PROBLEMS 2014年第12期30卷 125003-125003页

作者： Naumova, Valeriya Peter, Steffen Simula Res Lab NO-1364 Fornebu Norway Tech Univ Munich Fac Math D-85748 Garching Germany

Inspired by several recent developments in regularization theory, optimization, and signal processing, we present and analyze a numerical approach to multi-penalty regularization in spaces of sparsely represented functions. The sparsity prior is motivated by the largely expected geometrical/structured features of high-dimensional data, which may not be well-represented in the framework of typically more isotropic Hilbert spaces. In this paper, we are particularly interested in regularizers which are able to correctly model and separate the multiple components of additively mixed signals. This situation is rather common as pure signals may be corrupted by additive noise. To this end, we consider a regularization functional composed by a data-fidelity term, where signal and noise are additively mixed, a non-smooth and non-convex sparsity promoting term, and a penalty term to model the noise. We propose and analyze the convergence of an iterative alternating algorithm based on simple iterative thresholding steps to perform the minimization of the functional. By means of this algorithm, we explore the effect of choosing different regularization parameters and penalization norms in terms of the quality of recovering the pure signal and separating it from additive noise. For a given fixed noise level numerical experiments confirm a significant improvement in performance compared to standard one-parameter regularization methods. By using high-dimensional data analysis methods such as principal component analysis, we are able to show the correct geometrical clustering of regularized solutions around the expected solution. Eventually, for the compressive sensing problems considered in our experiments we provide a guideline for a choice of regularization norms and parameters.

关键词： multi-penalty regularization compressed sensing support identification non-convex constraints iterative thresholding alternating minimization algorithm

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alternating minimization algorithm for quantitative differential-interference contrast (DIC) microscopy

Alternating minimization algorithm for quantitative differen...

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Conference on Computational Imaging VI

作者： O'Sullivan, Joseph A. Preza, Chrysanthe Washington Univ Dept Elect & Syst Engn St Louis MO 63130 USA Univ Memphis Dept Elect & Comp Engn Memphis TN 38152 USA

ISBN: (纸本)9780819469861

A new iterative algorithm for quantitative image reconstruction for differential-interference contrast (DIC) microscopy is presented, along with simulations demonstrating key properties of the algorithm. The algorithm is an alternating minimization (AM) algorithm based on a diffraction imaging model for DIC images. The algorithm computes a specimen's complex transmittance function (magnitude and phase) from the DIC images. This new method extends the rotational diversity (RD) method developed by Preza that is based on the assumption that the specimen does not absorb light (or that its magnitude is constant and can be ignored) and thus only the specimen's phase function is computed from rotationally-diverse DIC images. The AM approach lifts the optimization problem to a higher dimensional space that is intimately tied to the underlying physics. The variables that are introduced in this lifting process are the unknown, immeasurable phases on the data. The AM algorithm iterates between estimating the image given the phase of the data and estimating the phase given the image. The framework allows constraints and penalties on the magnitude and phase estimates to be incorporated in a principled manner. The performance of the AM method is evaluated using simulated noiseless data. Simulation results compare specimen magnitude (absorption) and phase estimated with the AM method to the true specimen parameters showing good quantitative agreement using the mean square error and difference images. The estimated phase is a significant improvement over the phase estimated with the RD method from the same DIC images as shown in a comparison between the two methods. The AM method provides a novel approach in extracting quantitative phase and amplitude information of the same field of view from a single microscope mode.

关键词： alternating minimization algorithm complex function estimation phase estimation computational DIC (Nomarski) microscopy

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A fixed-point augmented Lagrangian method for total variation minimization problems

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JOURNAL OF VISUAL COMMUNICATION AND IMAGE REPRESENTATION 2013年第7期24卷 1168-1181页

作者： Duan, Yuping Huang, Weimin ASTAR Inst Infocomm Res Singapore Singapore

In this paper, we propose a fixed-point augmented Lagrangian method (FPALM) for general convex problems arising in image processing. We can easily obtain the alternating minimization algorithm (AMA) referred to [1] from the proposed FPALM. The proof for the convergence of the FPALM is provided under some mild assumptions. We present two kinds of first-order augmented Lagrangian schemes and show their connections to first-order primal-dual algorithms [2]. Furthermore, we apply an acceleration rule to both the FPALM and AMA to achieve better convergence rates. Numerical examples on different image denosing models including the ROF model, the vectorial TVmodel, high order models and the TV-L-1 model are provided to demonstrate the efficiency of the proposed algorithms. (C) 2013 Elsevier Inc. All rights reserved.

关键词： Convex optimization Dual approaches Augmented Lagrangian method Fixed-point method alternating minimization algorithm Total variation First-order algorithm Image denoising

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Fast algorithm for multiplicative noise removal

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JOURNAL OF VISUAL COMMUNICATION AND IMAGE REPRESENTATION 2012年第1期23卷 126-133页

作者： Shi, Baoli Huang, Lihong Pang, Zhi-Feng Hunan Univ Coll Math & Econometr Changsha 410082 Hunan Peoples R China Hunan Univ Coll Math & Informat Sci Kaifeng 475004 Hunan Peoples R China Hunan Womens Univ Changsha 410000 Hunan Peoples R China

In this work, we consider a variational restoration model for multiplicative noise removal problem. By using a maximum a posteriori estimator, we propose a strictly convex objective functional whose minimizer corresponds to the denoised image we want to recover. We incorporate the anisotropic total variation regularization in the objective functional in order to preserve the edges well. A fast alternating minimization algorithm is established to find the minimizer of the objective functional efficiently. We also give the convergence of this minimization algorithm. A broad range of numerical results are given to prove the effectiveness of our proposed model. Crown Copyright (C) 2011 Published by Elsevier Inc. All rights reserved.

关键词： Multiplicative noise Anisotropic total variation Maximum a posteriori Convex function alternating minimization algorithm Proximal operator Gamma distribution Newton method

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Low complexity interference alignment algorithms for desired signal power maximization problem of MIMO channels

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EURASIP JOURNAL ON ADVANCES IN SIGNAL PROCESSING 2012年第1期2012卷 1页

作者： Sun, Cong Yang, Yunchuan Yuan, Yaxiang Chinese Acad Sci ICMSEC AMSS State Key Lab Sci & Engn Comp Beijing 100190 Peoples R China Beijing Univ Posts & Telecommun Wireless Signal Proc & Network Lab Beijing 100876 Peoples R China

In this article, we investigate the interference alignment (IA) solution for a K-user MIMO interference channel. Proper users' precoders and decoders are designed through a desired signal power maximization model with IA conditions as constraints, which forms a complex matrix optimization problem. We propose two low complexity algorithms, both of which apply the Courant penalty function technique to combine the leakage interference and the desired signal power together as the new objective function. The first proposed algorithm is the modified alternating minimization algorithm (MAMA), where each subproblem has closed-form solution with an eigenvalue decomposition. To further reduce algorithm complexity, we propose a hybrid algorithm which consists of two parts. As the first part, the algorithm iterates with Householder transformation to preserve the orthogonality of precoders and decoders. In each iteration, the matrix optimization problem is considered in a sequence of 2D subspaces, which leads to one dimensional optimization subproblems. From any initial point, this algorithm obtains precoders and decoders with low leakage interference in short time. In the second part, to exploit the advantage of MAMA, it continues to iterate to perfectly align the interference from the output point of the first part. Analysis shows that in one iteration generally both proposed two algorithms have lower computational complexity than the existed maximum signal power (MSP) algorithm, and the hybrid algorithm enjoys lower complexity than MAMA. Simulations reveal that both proposed algorithms achieve similar performances as the MSP algorithm with less executing time, and show better performances than the existed alternating minimization algorithm in terms of sum rate. Besides, from the view of convergence rate, simulation results show that the MAMA enjoys fastest speed with respect to a certain sum rate value, while hybrid algorithm converges fastest to eliminate interference.

关键词： Interference alignment Power maximization Courant penalty function alternating minimization algorithm Householder transformation

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Quantitative Phase and Amplitude Imaging Using Differential-Interference Contrast (DIC) Microscopy

Quantitative Phase and Amplitude Imaging Using Differential-...

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Conference on Computational Imaging VII

作者： Preza, Chrysanthe O'Sullivan, Joseph A. Univ Memphis Dept Elect & Comp Engn Memphis TN 38152 USA Washington Univ Dept Elect & Syst Engn St Louis MO 63130 USA

ISBN: (纸本)9780819474964

We present an extension of the development of an alternating minimization (AM) method(1) for the computation of a specimen's complex transmittance function (magnitude and phase) from DIC images. The ability to extract both quantitative phase and amplitude information from two rotationally-diverse DIC images (i.e., acquired by rotating the sample) extends previous efforts in computational DIC microscopy that have focused on quantitative phase imaging only. Simulation results show that the inverse problem at hand is sensitive to noise as well as to the choice of the AM algorithm parameters. The AM framework allows constraints and penalties on the magnitude and phase estimates to be incorporated in a principled manner. Towards this end, Green and De Pierro's "log-cosh" regularization penalty is applied to the magnitude of differences of neighboring values of the complex-valued function of the specimen during the AM iterations. The penalty is shown to be convex in the complex space. A procedure to approximate the penalty within the iterations is presented. In addition, a methodology to pre-compute AM parameters that are optimal with respect to the convergence rate of the AM algorithm is also presented. Both extensions of the AM method are investigated with simulations.

关键词： alternating minimization algorithm complex function estimation regularization penalties phase estimation computational DIC (Nomarski) microscopy

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