Semi-Smooth Newton Methods for the Time Optimal Control of Nonautonomous Ordinary Differential Equations

AMS Subj. Classification: 49J15, 49M15

A semi-automatic computerized method to measure baroreflex-mediated heart rate responses that reduces interobserver variability

Arterial baroreflex sensitivity estimated by pharmacological impulse stimuli depends on intrinsic signal variability and usually a subjective choice of blood pressure (BP) and heart rate (HR) values. We propose a semi-automatic method to estimate cardiovascular reflex sensitivity to bolus infusions of phenylephrine and nitroprusside. Beat-to-beat BP and HR time series for male Wistar rats (N = 13) were obtained from the digitized signal (sample frequency = 2 kHz) and analyzed by the proposed method (PRM) developed in Matlab language. In the PRM, time series were low-pass filtered with zero-phase distortion (3rd order Butterworth used in the forward and reverse direction) and presented graphically, and parameters were selected interactively. Differences between basal mean values and peak BP (deltaBP) and HR (deltaHR) values after drug infusions were used to calculate baroreflex sensitivity indexes, defined as the deltaHR/deltaBP ratio. The PRM was compared to the method traditionally (TDM) employed by seven independent observers using files for reflex bradycardia (N = 43) and tachycardia (N = 61). Agreement was assessed by Bland and Altman plots. Dispersion among users, measured as the standard deviation, was higher for TDM for reflex bradycardia (0.60 ± 0.46 vs 0.21 ± 0.26 bpm/mmHg for PRM, P < 0.001) and tachycardia (0.83 ± 0.62 vs 0.28 ± 0.28 bpm/mmHg for PRM, P < 0.001). The advantage of the present method is related to its objectivity, since the routine automatically calculates the desired parameters according to previous software instructions. This is an objective, robust and easy-to-use tool for cardiovascular reflex studies.

A Newton method for pose refinement of 3D models

Different optimization methods can be employed to optimize a numerical estimate for the match between an instantiated object model and an image. In order to take advantage of gradient-based optimization methods, perspective inversion must be used in this context. We show that convergence can be very fast by extrapolating to maximum goodness-of-fit with Newton's method. This approach is related to methods which either maximize a similar goodness-of-fit measure without use of gradient information, or else minimize distances between projected model lines and image features. Newton's method combines the accuracy of the former approach with the speed of convergence of the latter.

Structured minimal-memory inexact quasi-Newton method and secant preconditioners for augmented Lagrangian optimization

Augmented Lagrangian methods for large-scale optimization usually require efficient algorithms for minimization with box constraints. On the other hand, active-set box-constraint methods employ unconstrained optimization algorithms for minimization inside the faces of the box. Several approaches may be employed for computing internal search directions in the large-scale case. In this paper a minimal-memory quasi-Newton approach with secant preconditioners is proposed, taking into account the structure of Augmented Lagrangians that come from the popular Powell-Hestenes-Rockafellar scheme. A combined algorithm, that uses the quasi-Newton formula or a truncated-Newton procedure, depending on the presence of active constraints in the penalty-Lagrangian function, is also suggested. Numerical experiments using the Cute collection are presented.

On the Newton method for solving fuzzy optimization problems

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

An adaptive Newton-method based on a dynamical systems approach

The traditional Newton method for solving nonlinear operator equations in Banach spaces is discussed within the context of the continuous Newton method. This setting makes it possible to interpret the Newton method as a discrete dynamical system and thereby to cast it in the framework of an adaptive step size control procedure. In so doing, our goal is to reduce the chaotic behavior of the original method without losing its quadratic convergence property close to the roots. The performance of the modified scheme is illustrated with various examples from algebraic and differential equations.

Uniform Convergence of the Newton Method for Aubin Continuous Maps

* This work was supported by National Science Foundation grant DMS 9404431.

The Graves Theorem Revisited II: Robust Convergence of the Newton Method

AMS subject classification: 65J15, 47H04, 90C30.

A study of a numerical solution of the steady two dimensions Navier-Stokes equations in a constricted channel problem by a compact fourth order method

We present a numerical solution for the steady 2D Navier-Stokes equations using a fourth order compact-type method. The geometry of the problem is a constricted symmetric channel, where the boundary can be varied, via a parameter, from a smooth constriction to one possessing a very sharp but smooth corner allowing us to analyse the behaviour of the errors when the solution is smooth or near singular. The set of non-linear equations is solved by the Newton method. Results have been obtained for Reynolds number up to 500. Estimates of the errors incurred have shown that the results are accurate and better than those of the corresponding second order method. (C) 2002 Elsevier B.V. All rights reserved.

A new approach to the solution of the optimal power flow problem based on the modified Newton's method associated to an augmented Lagrangian function

This paper presents a new algorithm for optimal power flow problem. The algorithm is based on Newton's method which it works with an Augmented Lagrangian function associated with the original problem. The function aggregates all the equality and inequality constraints and is solved using the modified-Newton method. The test results have shown the effectiveness of the approach using the IEEE 30 and 638 bus systems.

The composite Euler method for stiff stochastic differential equations

In this paper we present the composite Euler method for the strong solution of stochastic differential equations driven by d-dimensional Wiener processes. This method is a combination of the semi-implicit Euler method and the implicit Euler method. At each step either the semi-implicit Euler method or the implicit Euler method is used in order to obtain better stability properties. We give criteria for selecting the semi-implicit Euler method or the implicit Euler method. For the linear test equation, the convergence properties of the composite Euler method depend on the criteria for selecting the methods. Numerical results suggest that the convergence properties of the composite Euler method applied to nonlinear SDEs is the same as those applied to linear equations. The stability properties of the composite Euler method are shown to be far superior to those of the Euler methods, and numerical results show that the composite Euler method is a very promising method. (C) 2001 Elsevier Science B.V. All rights reserved.

An EM-based Semi-Parametric Mixture Model Approach to the Regression Analysis of Competing-Risks Data

We consider a mixture model approach to the regression analysis of competing-risks data. Attention is focused on inference concerning the effects of factors on both the probability of occurrence and the hazard rate conditional on each of the failure types. These two quantities are specified in the mixture model using the logistic model and the proportional hazards model, respectively. We propose a semi-parametric mixture method to estimate the logistic and regression coefficients jointly, whereby the component-baseline hazard functions are completely unspecified. Estimation is based on maximum likelihood on the basis of the full likelihood, implemented via an expectation-conditional maximization (ECM) algorithm. Simulation studies are performed to compare the performance of the proposed semi-parametric method with a fully parametric mixture approach. The results show that when the component-baseline hazard is monotonic increasing, the semi-parametric and fully parametric mixture approaches are comparable for mildly and moderately censored samples. When the component-baseline hazard is not monotonic increasing, the semi-parametric method consistently provides less biased estimates than a fully parametric approach and is comparable in efficiency in the estimation of the parameters for all levels of censoring. The methods are illustrated using a real data set of prostate cancer patients treated with different dosages of the drug diethylstilbestrol. Copyright (C) 2003 John Wiley Sons, Ltd.

Manual and semi-automatic quantification of in vivo ¹H-MRS data for the classification of human primary brain tumors

In vivo proton magnetic resonance spectroscopy (¹H-MRS) is a technique capable of assessing biochemical content and pathways in normal and pathological tissue. In the brain, ¹H-MRS complements the information given by magnetic resonance images. The main goal of the present study was to assess the accuracy of ¹H-MRS for the classification of brain tumors in a pilot study comparing results obtained by manual and semi-automatic quantification of metabolites. In vivo single-voxel ¹H-MRS was performed in 24 control subjects and 26 patients with brain neoplasms that included meningiomas, high-grade neuroglial tumors and pilocytic astrocytomas. Seven metabolite groups (lactate, lipids, N-acetyl-aspartate, glutamate and glutamine group, total creatine, total choline, myo-inositol) were evaluated in all spectra by two methods: a manual one consisting of integration of manually defined peak areas, and the advanced method for accurate, robust and efficient spectral fitting (AMARES), a semi-automatic quantification method implemented in the jMRUI software. Statistical methods included discriminant analysis and the leave-one-out cross-validation method. Both manual and semi-automatic analyses detected differences in metabolite content between tumor groups and controls (P < 0.005). The classification accuracy obtained with the manual method was 75% for high-grade neuroglial tumors, 55% for meningiomas and 56% for pilocytic astrocytomas, while for the semi-automatic method it was 78, 70, and 98%, respectively. Both methods classified all control subjects correctly. The study demonstrated that ¹H-MRS accurately differentiated normal from tumoral brain tissue and confirmed the superiority of the semi-automatic quantification method.