Dogleg Method Research Articles

A Sobolev gradient method for treating the steady-state incompressible Navier-Stokes equations

Abstract The velocity-vorticity-pressure formulation of the steady-state incompressible Navier-Stokes equations in two dimensions is cast as a nonlinear least squares problem in which the functional is a weighted sum of squared residuals. A finite element discretization of the functional is minimized by a trust-region method in which the trustregion radius is defined by a Sobolev norm and the trust-region subproblems are solved by a dogleg method. Numerical test results show the method to be effective.

Subspace trust-region algorithm with conic model for unconstrained optimization

In this paper, a new subspace algorithm is proposed for unconstrained optimization. In this new algorithm, the subspace technique is used in the trust region subproblem with conic model, and the dogleg method is modified to solve this subproblem. The global convergence of this algorithm under some reasonable conditions is established. Numerical experiment shows that this algorithm may be superior to the corresponding algorithm without using subspace technique especially for large scale problems.

A New Dogleg Method for Solving the Trust-Region Subproblem

Abstract: In this paper a new dogleg method for solving the trust region subproblem where convergence is based on constructing two paths is presented. The condition on the paths is incorporated into an algorithm to determine the optimum point of a smooth function. Numerical experiments with some classical problems showed that the new dogleg method is robust and efficient.

Solución de la cinemática directa de la plataforma Gough-Stewart general usando un algoritmo híbrido de optimización

The direct kinematics problem for parallel robots can be stated as follows: given values of the joint variables, the corresponding Cartesian variable values, the pose of the end-effector, must be found. Most of the times the direct kinematics problem involves the solution of a system of non-linear equations. The most efficient methods to solve such kind of equations assume convexity in a cost function which minimum is the solution of the non-linear system. In consequence, the capacity of such methods depends on the knowledge about an starting point which neighboring region is convex, hence the method can find the global minimum. This article propose a method based on probabilistic learning about an adequate starting point for the Dogleg method which assumes local convexity of the function. The proposed method efficiently avoids the local minima, without need of human intervention or apriori knowledge, thus it shows a more robust performance than the simple Dogleg method or other gradient based methods. To demonstrate the performance of the proposed hybrid method, numerical experiments and the respective discussion are presented. The proposal can be extended to other structures of closed-kinematics chains, to the general solution of systems of non-linear equations, and to the minimization of non-linear functions.

Constrained Dogleg methods for nonlinear systems with simple bounds

We focus on the numerical solution of medium scale bound-constrained systems of nonlinear equations. In this context, we consider an affine-scaling trust region approach that allows a great flexibility in choosing the scaling matrix used to handle the bounds. The method is based on a dogleg procedure tailored for constrained problems and so, it is named Constrained Dogleg method. It generates only strictly feasible iterates. Global and locally fast convergence is ensured under standard assumptions. The method has been implemented in the Matlab solver CoDoSol that supports several diagonal scalings in both spherical and elliptical trust region frameworks. We give a brief account of CoDoSol and report on the computational experience performed on a number of representative test problems.

Optimization of Lower Hybrid Current Drive Efficiency for EAST Plasma with Non-Circular Cross Section and Finite Aspect-Ratio

The two-dimensional equilibrium with flexible boundaries is solved via using a MAT-LAB Equilibrium Code (MEC), which has applied the finite element method to handle the changeable plasma shape and employed the trust-region dogleg method to solve the nonlinear partial differential equation. The corresponding driven current profile is also calculated by coupling with the lower-hybrid simulation code (LSC). The results are applied to optimize the lower hybrid current drive (LHCD) efficiency for the Experimental Advanced Superconductor Tokamak (EAST) and suggested that both elongation and triangularity have a notable effect on the efficiency because of the competition between the increase in the resonant area and in the Shafranov shift. Moreover, large aspect-ratio has a negative effect on the efficiency. These effects are studied numerically, which might be considered carefully for both good plasma confinement and high LHCD efficiency.

Numerical Improvements to Closed-Loop Ascent Guidance

This paper presents numerical enhancements for optimal closed-loop ascent guidance through atmospheric. For 3-dimensional ascent formulation, optimal endo-atmospheric ascent trajectory is numerically obtained by the relaxation approach, and the exo-atmospheric ascent trajectory is generated by an analytical multiple-shooting method. A new root-finding method based on double dogleg method and More’s Levenberg-Marquardt method with Gaussian elimination is presented. The simulation results indicate that our new algorithm has remarkable computation and convergence performances.

The aerodynamic shape optimization of airfoils using unconstrained trust region methods

In this article, unconstrained trust region methods are studied and applied in aerodynamic shape optimization. Trust region methods are high-performance gradient-based optimization methods. The theory of unconstrained trust region methods is explained, and a trust region subproblem is solved using the dogleg method and the Moré–Sorensen method. After verifying the trust region methods via analytical test problems, they are applied to the aerodynamic shape design optimization for transonic and subsonic flows.

Inexact Newton Dogleg Methods

The dogleg method is a classical trust-region technique for globalizing Newton's method. While it is widely used in optimization, including large-scale optimization via truncated-Newton approaches, its implementation in general inexact Newton methods for systems of nonlinear equations can be problematic. In this paper, we first outline a very general dogleg method suitable for the general inexact Newton context and provide a global convergence analysis for it. We then discuss certain issues that may arise with the standard dogleg implementational strategy and propose modified strategies that address them. Newton-Krylov methods have provided important motivation for this work, and we conclude with a report on numerical experiments involving a Newton-GMRES dogleg method applied to benchmark CFD problems.

Finite Element Model Updating of a Steel–Concrete Composite Moment - Resisting Structure with Partial Strength Joints

Dynamic and static identification of a full scale moment-resisting steel-concrete composite structure with partial strength joints that was tested by means of the pseudo-dynamic testing technique at the ELSA laboratory of the Joint Research Centre at Ispra, Italy, is the subject of this paper. The structure was subjected to pseudo-dynamic and dynamic tests at different damage and peak ground acceleration levels; and the results were used for identifying the behaviour of the structure. Two and three-dimensional refined finite element models of the structure accompanied by a robust nonlinear optimization method, the Powell’s Dog Leg method, were updated in order to reproduce in an optimal fashion the experimental static and dynamic behaviour of the structure.

Modeling and simulation of methane steam reforming in a thermally coupled membrane reactor

A distributed mathematical model for thermally coupled membrane reactor that is composed of three channels is developed for methane steam reforming. Methane combustion takes place in the first channel on a Pt / δ – Al 2 O 3 catalyst layer that supplies the necessary heat for the endothermic steam reforming reaction. In the second channel, catalytic steam reforming reactions take place in the presence of Ni/MgO–Al 2O 3 catalyst. The combustion catalyst forms a thin layer next to the reactor wall to minimize the heat transfer resistance. Selective permeation of hydrogen through the palladium membrane is achieved either by co-current or counter-current flow of sweep gas through the third channel. The burner is modeled as a monolith reactor and the reformer is assumed to behave as a pseudo-homogenous reactor. The mass and energy balance equations for the thermally coupled membrane reactor form a set of 22 coupled ordinary differential equations. With the application of appropriate boundary conditions, the distributed reactor model for steady-state operation is solved as a boundary value problem. The model equations are discretized using spline collocation on finite elements. The discretized nonlinear modeling equations, along with the boundary conditions, form a system of algebraic equations that are solved using the trust region dogleg method. The performance of the reactor is numerically investigated for various key operating variables such as inlet fuel concentration, inlet steam/methane ratio, inlet reformer gas temperature and inlet reformer gas velocity. Simulations for both the co-current and the countercurrent flow modes are also performed using different sweep gas flow rates. For each case, the reactor performance is analyzed based on methane conversion and hydrogen recovery yield.

Dogleg trust-region application in electrical impedance tomography

This paper presents a trust-region implementation for image reconstruction of conductivity changes in electrical impedance tomography. A dogleg trust-region algorithm is applied in different cases to detect abnormalities. The dogleg algorithm approximates a Levenberg–Marquardt step within the trust region of the model function with a quadratic model. The comparison of Levenberg–Marquardt and dogleg algorithms is presented using the reconstructed images. This comparison of two techniques suggests the implementation of the dogleg method could result in the reduction of the execution time to less than 50% of that of the Levenberg–Marquardt algorithm without any quantifiable loss of quality of reconstructed images.

On a trust region method without exact Jacobian matrices

AbstractAutomatic differentiation (AD) provides a possibility to evaluate exact derivative information within working accuracy. Here, we present an approach for equality constrained optimization that is based essentially on AD by computing only direct sensitivities and adjoints of first and second order. Employing this information, we generate approximations of the required derivative matrices using the STR1 update instead of computing the full constraint Jacobian or the full Lagrangian Hessian at each iteration explicitly. Hence, this approach avoids the forming and factoring of the exact constraint Jacobian that is often required in each iteration step.In order to globalize this provable local convergent method, the algorithm was embedded in a trust‐region framework. We apply a composite‐step method similar to the Byrd‐Omojokun approach that is well suited for the available information from the approximated matrices. For that purpose, the normal step is given by the dogleg method whereas a generalized CG‐iteration is applied to compute the tangential step. Here, the fact that only inexact information of the constraint Jacobian is available forms the main difference to other existing algorithms, where often a factorization of the exact Jacobian is used.Numerical results are shown for an equality constrained problem of the CUTE set and a PDE‐based optimization problem. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A new trust region dogleg method for unconstrained optimization

This paper presents a new trust region dogleg method for unconstrained optimization. The method can deal with the case when the Hessian B of quadratic models is indefinite. It is proved that the method is globally convergent and has a quadratic convergence rate if B (k) = ▿2 f(x (k)). Under certain conditions, the solution obtained by the method is even a second order stationary point. Numerical results also declare effectiveness of the method.

An Acceleration Method for Dogleg Methods in Simple Singular Regions

The behavior of dogleg methods in singular regions that have a one-dimensional null space is studied. A two-tier approach of identifying singular regions and accelerating convergence to a singular point is proposed. It is shown that singular regions are easily identified using a ratio of the two-norm of the Newton step to the two-norm of the Cauchy step since Newton steps tend to infinity and Cauchy steps tend to zero as a singular point is approached. Convergence acceleration is accomplished by bracketing the singular point using a projection of the gradient of the two-norm of the process model functions onto the normalized Newton direction in conjunction with bisection, thus preserving the global convergence properties of the dogleg method. Numerical examples for a continuous-stirred tank reactor and vapor-liquid equilibrium flash are used to illustrate the reliability and effectiveness of the proposed approach. Several geometric illustrations are presented.

A modular package for simultaneous calculation of complex interlinked separation processes

A compressor, an expander and an arbitrary separator model have been added to the BELSIM-COL70 steady state simulation package for multi-stage separation processes. The performance of the dogleg method applied to the global equation system of an extended version of Cavett's problem is compared to the classical Wegstein method and the dogleg method applied to the torn variables only. The results of the simulation of a C3-splitter are compared to industrial measurements. Finally, the algorithm is applied to an azeotrope separation process with demixion in the liquid phase.

On solving large sparse nonlinear equation systems

An approach toward solving large sparse nonlinear equation system using a modified Powell's dogleg method is described. Schubert's update is used either to update the Jacobian, or to update the U factor of the Jacobian while holding the L factor constant. This use of Schubert's update differs from others since it is used in connection with a dogleg method rather than the usual quasi-Newton method. Results on a set of standard test problems indicate that this leads to an efficient and reliable method for sparse nonlinear problems.

A modification of Powell's dogleg method for solving systems of nonlinear equations

We present in this paper an algorithm for solving nonlinear equation systems that is a modification of Powell's dogleg method. The modifications are designed to make the technique more efficient and reliable and to reduce storage requirements. The performance of the new algorithm on a set of standard test problems demonstrates its effectiveness.