Damped Gauss-Newton Algorithm Research Articles

Efficient computational method for the dynamic responses of a floating wind turbine

ABSTRACTThis paper investigates the computational methods for calculating the dynamic responses of a floating wind turbine. The mathematical procedures for a proposed new identification method for fitting a state space model to approximate the convolution term in the motion equation of the floating wind turbine have been specified, and these mathematical procedures have been executed in a stand-alone MATLAB programme in this study. This new identification method consists of finding the vector of parameters that give the best un-weighted least squares fitting to the frequency responses, and a damped Gauss–Newton algorithm is first proposed to solve this un-weighted least squares problem. The calculation results in this paper have been systematically analysed and compared, and the accuracy and efficiency of our proposed new method have been convincingly substantiated.

Simultaneous magnetic resonance diffusion and pseudo-diffusion tensor imaging.

PurposeAn emerging topic in diffusion magnetic resonance is imaging blood microcirculation alongside water diffusion using the intravoxel incoherent motion (IVIM) model. Recently, a combined IVIM diffusion tensor imaging (IVIM‐DTI) model was proposed, which accounts for both anisotropic pseudo‐diffusion due to blood microcirculation and anisotropic diffusion due to tissue microstructures. In this article, we propose a robust IVIM‐DTI approach for simultaneous diffusion and pseudo‐diffusion tensor imaging.MethodsConventional IVIM estimation methods can be broadly divided into two‐step (diffusion and pseudo‐diffusion estimated separately) and one‐step (diffusion and pseudo‐diffusion estimated simultaneously) methods. Here, both methods were applied on the IVIM‐DTI model. An improved one‐step method based on damped Gauss–Newton algorithm and a Gaussian prior for the model parameters was also introduced. The sensitivities of these methods to different parameter initializations were tested with realistic in silico simulations and experimental in vivo data.ResultsThe one‐step damped Gauss–Newton method with a Gaussian prior was less sensitive to noise and the choice of initial parameters and delivered more accurate estimates of IVIM‐DTI parameters compared to the other methods.ConclusionOne‐step estimation using damped Gauss–Newton and a Gaussian prior is a robust method for simultaneous diffusion and pseudo‐diffusion tensor imaging using IVIM‐DTI model. Magn Reson Med 79:2367–2378, 2018. © 2017 The Authors Magnetic Resonance in Medicine published by Wiley Periodicals, Inc. on behalf of International Society for Magnetic Resonance in Medicine. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Robust frequency-domain identification of parametric radiation force models for a floating wind turbine

This paper mainly concerns the hydrodynamic responses prediction of a floating wind turbine. A new frequency-domain identification approach is first proposed to fit a parametric model (state space model) to approximate the convolution term in the Cummins equation of the floating wind turbine. This frequency-domain identification approach consists of finding the vector of parameters that give the best least squares fitting to the frequency responses, and a damped Gauss–Newton algorithm is first proposed to solve this non-linear least squares problem. The damped Gauss–Newton algorithm consists of calculating an increment vector based on the information of residuals and Jacobian values, finding a “fraction” factor by using a line search algorithm, then updating the values of the vector of parameters until the norm of the gradient vector is less than a specific tolerance value. Various kinds of hydrodynamic radiation forces (moments) and motion responses of this floating wind turbine have been calculated. For comparison purpose, the calculation results based on a time-domain identification approach for fitting the state space model are also included in this paper. This time-domain identification approach is based on finding a parametric transfer function of an auto-regressive moving-average (ARMA) filter based on prescribed time domain responses by using Prony׳s method. The accuracy and efficiency of our proposed frequency-domain identification approach have been convincingly substantiated through the comparisons of the calculation results in this article.

Regularization opportunities for the diffuse optical tomography problem

In optical tomography, the radiative properties of the medium under investigation are estimated from information contained in measurements provided by a set of light sources and sensors located on the frontier of the probed medium. Such a non-linear ill-posed inverse problem is usually solved through optimization with the help of gradient-type methods. Since it is well known that such inverse problems are ill-posed, regularization is required. This paper compares three distinct regularization strategies for two different optimization algorithms, namely the damped Gauss–Newton and BFGS algorithms, for the two-dimensional diffuse approximation to the radiative transfer equation in the frequency domain. More specifically, the mesh-based regularization is combined with the Tikhonov regularization in the damped Gauss–Newton algorithm. For the BFGS algorithm, the mesh-based regularization is combined with the Sobolev gradients method. Moreover, a space-dependent Sobolev gradients method is proposed for the first time. The performance of the proposed algorithms are compared by utilizing synthetic data. The deviation factor and correlation coefficient are used to quantitatively compare the final reconstructions. Also, three levels of noise are considered to characterize the behaviour of the proposed methods against measurement noises. Numerical results indicate that the BFGS algorithm outperforms the damped Gauss–Newton in many aspects.

Cramér-Rao-Induced Bounds for CANDECOMP/PARAFAC Tensor Decomposition

This paper presents a Cramer-Rao lower bound (CRLB) on the variance of unbiased estimates of factor matrices in Canonical Polyadic (CP) or CANDECOMP/PARAFAC (CP) decompositions of a tensor from noisy observations, (i.e., the tensor plus a random Gaussian i.i.d. tensor). A novel expression is derived for a bound on the mean square angular error of factors along a selected dimension of a tensor of an arbitrary dimension. The expression needs less operations for computing the bound, O(NR^6), than the best existing state-of-the art algorithm, O(N^3R^6) operations, where N and R are the tensor order and the tensor rank. Insightful expressions are derived for tensors of rank 1 and rank 2 of arbitrary dimension and for tensors of arbitrary dimension and rank, where two factor matrices have orthogonal columns. The results can be used as a gauge of performance of different approximate CP decomposition algorithms, prediction of their accuracy, and for checking stability of a given decomposition of a tensor (condition whether the CRLB is finite or not). A novel expression is derived for a Hessian matrix needed in popular damped Gauss-Newton method for solving the CP decomposition of tensors with missing elements. Beside computing the CRLB for these tensors the expression may serve for design of damped Gauss-Newton algorithm for the decomposition.

Convergence Analysis of Gauss–Newton Methods for the Complementarity Problem

In a recent paper (Ref. 1), the first author presented a damped Gauss–Newton algorithm to solve the complementarity problem. Although the conclusions of the convergence theorems in the paper are valid, the proofs contain minor errors. The aim of this paper is to give correct proofs and also to show that they are valid under milder conditions.

Gauss-Newton methods for the complementarity problem

Mangasarian has shown that the solution of the complementarity problem is equivalent to the solution of a system of nonlinear equations. In this paper, we propose a damped Gauss-Newton algorithm to solve this system, prove that under appropriate hypotheses one gets rapid local convergence, and present computational experience.

Comparison between structural and residual error formulations of an optimum design mechanism problem

The model formulation of the objective function is one of the most important steps for solving an optimization problem. Generally, the objective function of an optimal design mechanism problem can be mathematically formulated either with structural or residual errors. In this paper we have conducted some comparisons between the results obtained by the structural and residual error formulations utilized in solving the optimum design of a four-bar linkage. The optimization problem is defined as a nonlinear least-squares problem. The damped Gauss-Newton algorithm is utilized for computing the solution, which ensures a local convergence for almost all nonlinear least-squares problems.