Restart Procedure Research Articles

Convergence of Symmetric Rank-One method based on Modified Quasi-Newton equation

In this paper we investigate on convergence rate of a modified symmetric rank-one (SR1) method for unconstrainedoptimization problems. In general, the modified SR1 method incorporates a modified secant equationinto the standard SR1 method. Also a restart procedure is applied to avoid the loss of positive definiteness and zero denominator.A remarkable feature of the modified SR1 method is that it possesses at most $n+1$-step $q$-superlinearly convergent and$2n$-step quadratic convergent without uniformly independent assumptions of steps.

A restarted Lanczos approximation to functions of a symmetric matrix

In this paper the error term in the 'first' method of Eiermann & Ernst (2006, SIAM J. Numer. Anal., 44, 2481-2504) for restarting the Lanczos approximation of the matrix-vector product f (A)b, where A ∈ ℝ n×n is symmetric, is re-derived and expressed as an explicit partial fraction expansion of the divided differences. The partial fraction representation makes the new variant slightly more stable (albeit still unstable) than the former method because it requires fewer finite-difference evaluations. We then present an error bound for the restarted Lanczos approximation of f (A)b for symmetric positive-definite A when f is in a particular class of completely monotone functions and illustrate for some important matrix function applications the usefulness of these bounds for terminating the restart process once the desired accuracy in the matrix function approximation has been achieved. Finally, in an attempt to overcome the inherent instability of our restart procedure, we propose a simple heuristic that identifies when to halt the iterations.

Constrained Globalized Nelder—Mead Method for Simultaneous Structural and Manufacturing Optimization of a Composite Bracket

The optimized design of composite structures is a difficult task. It requires optimizing simultaneously both structural and manufacturing objectives. The objectives do not have closed form solutions and have multiple local optima that calls for a global search. This paper improves the global search method called GBNM [1], which is based on several restarts of the Nelder—Mead method. Two issues are addressed here. First, the restart procedure is improved by using a one-dimensional probability function and a weighted selection procedure. Second, nonlinear constraints are included by projecting the infeasible points onto the nonlinear constraints. The improved procedure is applied to four mathematical test functions. Numerical results show the proposed approach is more efficient in terms of computational time and probability of finding the global minimum. The improved GBNM is then applied to the simultaneous structural and manufacturing design of a Z-shaped composite bracket. The results are compared to those obtained with the genetic algorithm.

Hydrate Remediation in Deepwater Gulf of Mexico Dry-Tree Wells: Lessons Learned

Summary Hydrate plugs were formed above the mudline in two dry tree oil wells in the Gulf of Mexico. The plugs were formed when trying to open the downhole safety valve with crude to return the wells to production after they were shut-in because of hurricane evacuation. Several unsuccessful attempts to melt the hydrate blockages included pumping methanol through the chemical injection lines below the plugs and lubricating in glycol above the plugs. As a last attempt, before using coiled tubing, injecting hot oil into the tubing-casing annulus was considered. Transient simulations were performed to determine the required injection temperature, rate, and time. Well integrity issues were mainly associated with the compatibility of the hot oil with the elastomers and possible as-phaltene or paraffin precipitation in the annulus. Sensitivity studies show that with a 1-bbl/min injection rate and 150°F injection temperature, the pressure-temperature condition inside the tubing located 3,000 ft below the sea level will come out of the hydrate formation region within 4 hours. However, as the section goes deeper, the warm up time increases and at some point the conditions will not warrant being out of the hydrate region even after several days of injection time. Hydrate plugs in two dry tree wells melted after 6 and 60 hours of injection time, respectively. A revised restart procedure has been implemented to eliminate the hydrate problem in future startups.

Impact of Micromolar Concentrations of Externally-Provided Chloride and Sulfide Contaminants on PEMFC Reversible Stability

In this paper, we evaluate the reversible performance stability of 3M membrane electrode assemblies containing either Nanostructured Thin Film Catalyst (NSTFC) or carbon-supported Pt (Pt/C) electrodes. Micromolar concentrations of chloride and sulfide ions, provided via the humidification DI water, were found to cause rapid but highly recoverable performance loss. The impact of chloride on the decay rate depends strongly on catalyst type, surface area, and cell temperature. The effect of these externally-provided species remains for several hours after they are removed from the humidification streams. The performance loss due to these species is largely recovered through a special shutdown and restart procedure, where liquid water is injected into the cell as it cools to room temperature.

Optimisation of start and restart procedure in TMF fatigue testing

Thermomechanical fatigue is a technique that is going to be widely used for the characterisation of gas turbine components. As the experimental technique is quite complex, it requires a defined procedure to initialise the experiments. In particular the evaluation of the systematic temperature-mechanical strain shift due to the equipment response and the thermal strain compensation during thermal cycling, the verification of zero stress before starting have to be defined in order to produce TMF tests comparable from different laboratories. This paper describes the experimental activity carried out in order to determine a procedure for the evaluation of thermal strain, for the optimisation of the phase between temperature cycle and mechanical strain and for the selection of the best point at which test should start. The zero stress test performed before TMF test is described and confirms its utility to ensure a correct start of TMF testing. TMF tests have been performed on Nimonic 90 nickel-base superalloy in the temperature range of 400–850 °C and 180° out of phase cycle. Some tests were stopped and restarted in order to determine the influence of such stops on TMF life and the best restart procedure. At the end few tests have been planned for the verification that the adopted procedure is generally correct.

Changes in the enzymatic hydrolysis rate of Avicel cellulose with conversion

The slow down in enzymatic hydrolysis of cellulose with conversion has often been attributed to declining reactivity of the substrate as the more easily reacted material is thought to be consumed preferentially. To better understand the cause of this phenomenon, the enzymatic reaction of the nearly pure cellulose in Avicel was interrupted over the course of nearly complete hydrolysis. Then, the solids were treated with proteinase to degrade the cellulase enzymes remaining on the solid surface, followed by proteinase inhibitors to inactive the proteinase and successive washing with water, 1.0 M NaCl solution, and water. Next, fresh cellulase and buffer were added to the solids to restart hydrolysis. The rate of cellulose hydrolysis, expressed as a percent of substrate remaining at that time, was approximately constant over a wide range of conversions for restart experiments but declined continually with conversion for uninterrupted hydrolysis. Furthermore, the cellulose hydrolysis rate per adsorbed enzyme was approximately constant for the restart procedure but declined with conversion when enzymes were left to react. Thus, the drop off in reaction rate for uninterrupted cellulose digestion by enzymes could not be attributed to changes in substrate reactivity, suggesting that other effects such as enzymes getting "stuck" or otherwise slowing down may be responsible.

The statistical dynamics of turbulent Rossby wave flow over topography

The statistical dynamics of Rossby wave turbulence is examined by comparing direct numerical simulation of the vorticity form of the 2-D Navier--Stokes equation with a non-Markovian statistical closure theory for inhomogeneous flow over mean topography. The quasi-diagonal direct interaction approximation closure theory is formulated for the interaction of mean fields, Rossby waves and inhomogeneous turbulence over topography on a generalized ?-plane. The competing effects of nonlinear waves at the large scales and fully developed turbulence at the small scales is examined by comparing closure theory with ensemble averaged results from direct numerical simulation at resolution k=48 for circularly truncated wavenumber space. This work builds on the low resolution ?-plane studies of Frederiksen and O'Kane (2005) and extends the high resolution f-plane studies of O'Kane and Frederiksen (2004) to incorporate waves. We also examine the performance of a computationally efficient restart or cumulant update procedure at moderate Reynolds number in the presence of waves.

Multistart Tabu Search Strategies for the Unconstrained Binary Quadratic Optimization Problem

This paper describes and experimentally compares five rather different multistart tabu search strategies for the unconstrained binary quadratic optimization problem: a random restart procedure, an application of a deterministic heuristic to specially constructed subproblems, an application of a randomized procedure to the full problem, a constructive procedure using tabu search adaptive memory, and an approach based on solving perturbed problems. In the solution improvement phase a modification of a standard tabu search implementation is used. A computational trick applied to this modification – mapping of the current solution to the zero vector – allowed to significantly reduce the time complexity of the search. Computational results are provided for the 25 largest problem instances from the OR-Library and, in addition, for the 18 randomly generated larger and more dense problems. For 9 instances from the OR-Library new best solutions were found.

Integro-differential closure equations for inhomogeneous turbulence

We present numerical and computational techniques to solve systems of integro-differential closure equations for inhomogeneous two-dimensional turbulent flow. The closure equations, representing the first tractable closure theory for inhomogeneous flow over mean (single realization) topography, are based on a quasi-diagonal direct interaction approximation derived via renormalization techniques. The equations are computationally challenging due to the potentially long time history integrals. In order to reduce the computational cost we have implemented a formal restart procedure for the two and three point cumulant terms. The restart procedure is shown to be in good agreement with the closure without restarts and results are compared to direct numerical simulation of the barotropic vorticity equation.

Improved Constructive Multistart Strategies for the Quadratic Assignment Problem Using Adaptive Memory

Multistart constructive approaches operate by applying a local search procedure to start from different initial solutions produced by a repeated (variable) constructive process. The classical Random Restart procedure and the more recent GRASP procedure are prominent examples of such approaches. Adaptive memory strategies that are the heart of tabu search methods give a foundation for alternative, enhanced, multistart approaches. We demonstrate this by showing that a simple implementation of adaptive memory search principles, even when restricted to the constructive phases, can provide more effective multistart methods. Computational experiments for the quadratic assignment problem disclose that these methods improve significantly over previous multistart methods that do not incorporate such memory based strategies.

Jacobi--Davidson Style QR and QZ Algorithms for the Reduction of Matrix Pencils

Recently the Jacobi--Davidson subspace iteration method has been introduced as a new powerful technique for solving a variety of eigenproblems. In this paper we will further exploit this method and enhance it with several techniques so that practical and accurate algorithms are obtained. We will present two algorithms, JDQZ for the generalized eigenproblem and JDQR for the standard eigenproblem, that are based on the iterative construction of a (generalized) partial Schur form. The algorithms are suitable for the efficient computation of several (even multiple) eigenvalues and the corresponding eigenvectors near a user-specified target value in the complex plane. An attractive property of our algorithms is that explicit inversion of operators is avoided, which makes them potentially attractive for very large sparse matrix problems. We will show how effective restarts can be incorporated in the Jacobi--Davidson methods, very similar to the implicit restart procedure for the Arnoldi process. Then we will discuss the use of preconditioning, and, finally, we will illustrate the behavior of our algorithms by a number of well-chosen numerical experiments.

A Conjugate Gradient Algorithm with a Trust Region for Molecular Geometry Optimization

An algorithm is presented for the optimization of molecular geometries and general nonquadratic functions using the nonlinear conjugate gradient method with a restricted step and restart procedure. The algorithm only requires the evaluation of the energy function and its gradient, therefor less memory storage is needed than for other conjugate gradient algorithms. Some numerical results are also presented and the efficiency and behaviour of the algorithm is compared with the standard conjugate gradient method. We also present comparisons of both conjugate gradient and variable metric methods with and without the trust region technique. One of the main conclusions of the present work is that a trust region always improves the converge of any optimization method. A sketch of the algorithm is also given.

Robust adaptive control of a time varying process using parallel recursive estimators

A parameter-adaptive controller to control the simulated plant of the “1993 IFAC World Congress Bench Mark Design Example” is presented. To cope with the rather fast time-varying process parameters a set of parallel estimators is used. Due to a cyclic restart procedure each of these estimators is working with another time window. For the controller itself an algorithm is used that realizes a controller with variable robustness. Simulation results are presented.

Robust Adaptive Control of a Time Varying Process Using Parallel Recursive Estimators

A parameter-adaptive controller to control the simulated plant of the "1993 IFAC World Congress Bench Mark Design Example" is presented. To cope with the rather fast time-varying process parameters a set of parallel estimators is used. Due to a cyclic restart procedure each of these estimators is working with another time window. For the controller itself an algorithm is used that realizes a controller with variable robustness. Simulation results are presented.

Fast transversal filters with data sequence weighting

A fast O(N) algorithm is presented for the adaptive recursive least-squares design of transversal filters. Data sequence weighting is introduced to allow for arbitrarily time-varying weighting strategies, facilitating the tracking of arbitrary nonstationary phenomena. A novel weighting adaptation mechanism is presented that has various desirable features and optimal performance under certain conditions. The algorithm is derived for the prewindowed given data case, and the underlying principles of the derivation are clearly exposed. Exact and soft-constraint initialization are discussed, and an improved restart procedure is proposed. >

Micromagnetic predictions of closure domain patterns in magnetic thin films

The magnetic domain structures of permalloy thin films have been calculated using a micromagnetic model. The films are assumed to be very thin (500 AA), so that the magnetization lies in the plane of the film. The model takes a small sample of the film, typically several microns in size, and discretizes it into several thousand elements. The total energy of the system is then minimized to obtain the equilibrium states. A restart procedure is designed to overcome the stiff convergence problem in the calculations for the soft magnetic materials. The closure domain patterns predicted are in agreement with experiments and are shown to have advantages compared with domain-level calculations. Disadvantages are that the computation takes an enormous amount of CPU time owing to the large number of unknowns, and that its application is limited to small samples to avoid having an excessive number of grid points.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Binary-Search SPOC: An Optimal Control Theory Version of ECHO

Abstract Two models for identifying the optimal harvest of old-growth forests—the supply potential optimal control model, SPOC, and the economic harvest optimization model, ECHO—are compared theoretically and numerically. Walker's ECHO restart procedure to solve convergence problems is examined and found to be theoretically sound. The primary difference between the models is that SPOC incorporates the effects of future rotations while ECHO does not. A new algorithm, binary-search SPOC, is identified which incorporates the effects of future rotations by including a difference equation for the shadow value of these rotations. This algorithm is the solution to a discrete time optimal control problem. The numerical analysis of Walker's original problem revealed that the ECHO solution was very close in the early years when the implicit land rental was low and rising, but generated the wrong long-run equilibrium. Forest Sci. 32:576-584.

Restart procedures for the conjugate gradient method

The conjugate gradient method is particularly useful for minimizing functions of very many variables because it does not require the storage of any matrices. However the rate of convergence of the algorithm is only linear unless the iterative procedure is "restarted" occasionally. At present it is usual to restart everyn or (n + 1) iterations, wheren is the number of variables, but it is known that the frequency of restarts should depend on the objective function. Therefore the main purpose of this paper is to provide an algorithm with a restart procedure that takes account of the objective function automatically. Another purpose is to study a multiplying factor that occurs in the definition of the search direction of each iteration. Various expressions for this factor have been proposed and often it does not matter which one is used. However now some reasons are given in favour of one of these expressions. Several numerical examples are reported in support of the conclusions of this paper.

Quadratic termination properties of Davidon's new variable metric algorithm

Davidon [1] provides a variable metric algorithm for minimization calculations that achieves quadratic termination without any line searches by a method that uses positive definite second derivative approximations. We give a new and elementary proof of these termination properties. It shows the great freedom allowed by the algorithm in the choice of search directions and it explains the effect of a restart procedure.