Augmented Lagrangian methods under the constant positive linear dependence constraint qualification

Two Augmented Lagrangian algorithms for solving KKT systems are introduced. The algorithms differ in the way in which penalty parameters are updated. Possibly infeasible accumulation points are characterized. It is proved that feasible limit points that satisfy the Constant Positive Linear Dependence constraint qualification are KKT solutions. Boundedness of the penalty parameters is proved under suitable assumptions. Numerical experiments are presented.

Skull Modularity in Neotropical Marsupials and Monkeys: Size Variation and Evolutionary Constraint and Flexibility

An organism is built through a series of contingent factors, yet it is determined by historical, physical, and developmental constraints. A constraint should not be understood as an absolute obstacle to evolution, as it may also generate new possibilities for evolutionary change. Modularity is, in this context, an important way of organizing biological information and has been recognized as a central concept in evolutionary biology bridging on developmental, genetics, morphological, biochemical, and physiological studies. In this article, we explore how modularity affects the evolution of a complex system in two mammalian lineages by analyzing correlation, variance/covariance, and residual matrices (without size variation). We use the multivariate response to selection equation to simulate the behavior of Eutheria and Metharia skulls in terms of their evolutionary flexibility and constraints. We relate these results to classical approaches based on morphological integration tests based on functional/developmental hypotheses. Eutherians (Neotropical primates) showed smaller magnitudes of integration compared with Metatheria (didelphids) and also skull modules more clearly delimited. Didelphids showed higher magnitudes of integration and their modularity is strongly influenced by within-groups size variation to a degree that evolutionary responses are basically aligned with size variation. Primates still have a good portion of the total variation based on size; however, their enhanced modularization allows a broader spectrum of responses, more similar to the selection gradients applied (enhanced flexibility). Without size variation, both groups become much more similar in terms of modularity patterns and magnitudes and, consequently, in their evolutionary flexibility. J. Exp. Zool. (Mol. Dev. Evol.) 314B:663-683, 2010. (C) 2010 Wiley-Liss, Inc.

A collocation method for solving singular integro-differential equations

This paper describes a collocation method for numerically solving Cauchy-type linear singular integro-differential equations. The numerical method is based on the transformation of the integro-differential equation into an integral equation, and then applying a collocation method to solve the latter. The collocation points are chosen as the Chebyshev nodes. Uniform convergence of the resulting method is then discussed. Numerical examples are presented and solved by the numerical techniques.

A numerical method for solving the dynamic three-dimensional Ericksen-Leslie equations for nematic liquid crystals subject to a strong magnetic field

A finite difference technique, based on a projection method, is developed for solving the dynamic three-dimensional Ericksen-Leslie equations for nematic liquid crystals subject to a strong magnetic field. The governing equations in this situation are derived using primitive variables and are solved using the ideas behind the GENSMAC methodology (Tome and McKee [32]; Tome et al. [34]). The resulting numerical technique is then validated by comparing the numerical solution against an analytic solution for steady three-dimensional flow between two-parallel plates subject to a strong magnetic field. The validated code is then employed to solve channel flow for which there is no analytic solution. (C) 2009 Elsevier B.V. All rights reserved.

An approach for solving the lot-sizing problem of a market-driven foundry

Foundries can be found all over Brazil and they are very important to its economy. In 2008, a mixed integer-programming model for small market-driven foundries was published, attempting to minimize delivery delays. We undertook a study of that model. Here, we present a new approach based on the decomposition of the problem into two sub-problems: production planning of alloys and production planning of items. Both sub-problems are solved using a Lagrangian heuristic based on transferences. An important aspect of the proposed heuristic is its ability to take into account a secondary practice objective solution: the furnace waste. Computational tests show that the approach proposed here is able to generate good quality solutions that outperform prior results. Journal of the Operational Research Society (2010) 61, 108-114. doi:10.1057/jors.2008.151

An implicit technique for solving 3D low Reynolds number moving free surface flows

This paper describes the development of an implicit finite difference method for solving transient three-dimensional incompressible free surface flows. To reduce the CPU time of explicit low-Reynolds number calculations, we have combined a projection method with an implicit technique for treating the pressure on the free surface. The projection method is employed to uncouple the velocity and the pressure fields, allowing each variable to be solved separately. We employ the normal stress condition on the free surface to derive an implicit technique for calculating the pressure at the free surface. Numerical results demonstrate that this modification is essential for the construction of methods that are more stable than those provided by discretizing the free surface explicitly. In addition, we show that the proposed method can be applied to viscoelastic fluids. Numerical results include the simulation of jet buckling and extrudate swell for Reynolds numbers in the range [0.01, 0.5]. (C) 2008 Elsevier Inc. All rights reserved.

Outer Trust-Region Method for Constrained Optimization

Given an algorithm A for solving some mathematical problem based on the iterative solution of simpler subproblems, an outer trust-region (OTR) modification of A is the result of adding a trust-region constraint to each subproblem. The trust-region size is adaptively updated according to the behavior of crucial variables. The new subproblems should not be more complex than the original ones, and the convergence properties of the OTR algorithm should be the same as those of Algorithm A. In the present work, the OTR approach is exploited in connection with the ""greediness phenomenon"" of nonlinear programming. Convergence results for an OTR version of an augmented Lagrangian method for nonconvex constrained optimization are proved, and numerical experiments are presented.

A Bayesian Semiparametric Approach for Solving the Discrete Calibration Problem

In this article, we introduce a semi-parametric Bayesian approach based on Dirichlet process priors for the discrete calibration problem in binomial regression models. An interesting topic is the dosimetry problem related to the dose-response model. A hierarchical formulation is provided so that a Markov chain Monte Carlo approach is developed. The methodology is applied to simulated and real data.

Efficient solutions to robust, semi-implicit discretizations of the immersed boundary method

The immersed boundary method is a versatile tool for the investigation of flow-structure interaction. In a large number of applications, the immersed boundaries or structures are very stiff and strong tangential forces on these interfaces induce a well-known, severe time-step restriction for explicit discretizations. This excessive stability constraint can be removed with fully implicit or suitable semi-implicit schemes but at a seemingly prohibitive computational cost. While economical alternatives have been proposed recently for some special cases, there is a practical need for a computationally efficient approach that can be applied more broadly. In this context, we revisit a robust semi-implicit discretization introduced by Peskin in the late 1970s which has received renewed attention recently. This discretization, in which the spreading and interpolation operators are lagged. leads to a linear system of equations for the inter-face configuration at the future time, when the interfacial force is linear. However, this linear system is large and dense and thus it is challenging to streamline its solution. Moreover, while the same linear system or one of similar structure could potentially be used in Newton-type iterations, nonlinear and highly stiff immersed structures pose additional challenges to iterative methods. In this work, we address these problems and propose cost-effective computational strategies for solving Peskin`s lagged-operators type of discretization. We do this by first constructing a sufficiently accurate approximation to the system`s matrix and we obtain a rigorous estimate for this approximation. This matrix is expeditiously computed by using a combination of pre-calculated values and interpolation. The availability of a matrix allows for more efficient matrix-vector products and facilitates the design of effective iterative schemes. We propose efficient iterative approaches to deal with both linear and nonlinear interfacial forces and simple or complex immersed structures with tethered or untethered points. One of these iterative approaches employs a splitting in which we first solve a linear problem for the interfacial force and then we use a nonlinear iteration to find the interface configuration corresponding to this force. We demonstrate that the proposed approach is several orders of magnitude more efficient than the standard explicit method. In addition to considering the standard elliptical drop test case, we show both the robustness and efficacy of the proposed methodology with a 2D model of a heart valve. (C) 2009 Elsevier Inc. All rights reserved.

ENO adaptive method for solving one-dimensional conservation laws

In this work an efficient third order non-linear finite difference scheme for solving adaptively hyperbolic systems of one-dimensional conservation laws is developed. The method is based oil applying to the solution of the differential equation an interpolating wavelet transform at each time step, generating a multilevel representation for the solution, which is thresholded and a sparse point representation is generated. The numerical fluxes obtained by a Lax-Friedrichs flux splitting are evaluated oil the sparse grid by an essentially non-oscillatory (ENO) approximation, which chooses the locally smoothest stencil among all the possibilities for each point of the sparse grid. The time evolution of the differential operator is done on this sparse representation by a total variation diminishing (TVD) Runge-Kutta method. Four classical examples of initial value problems for the Euler equations of gas dynamics are accurately solved and their sparse solutions are analyzed with respect to the threshold parameters, confirming the efficiency of the wavelet transform as an adaptive grid generation technique. (C) 2008 IMACS. Published by Elsevier B.V. All rights reserved.