Local Relaxation Method Research Articles

Local Structure-Preserving Relaxation Method for Equilibrium of Charged Systems on Unstructured Meshes

Local Structure-Preserving Relaxation Method for Equilibrium of Charged Systems on Unstructured Meshes

SGO: A fast engine for ab initio atomic structure global optimization by differential evolution

As the high throughout calculations and material genome approaches become more and more popular in material science, the search for optimal ways to predict atomic global minimum structure is a high research priority. This paper presents a fast method for global search of atomic structures at ab initio level. The structures global optimization (SGO) engine consists of a high-efficiency differential evolution algorithm, accelerated local relaxation methods and a plane-wave density functional theory code running on GPU machines. The purpose is to show what can be achieved by combining the superior algorithms at the different levels of the searching scheme. SGO can search the global-minimum configurations of crystals, two-dimensional materials and quantum clusters without prior symmetry restriction in a relatively short time (half or several hours for systems with less than 25 atoms), thus making such a task a routine calculation. Comparisons with other existing methods such as minima hopping and genetic algorithm are provided. One motivation of our study is to investigate the properties of magnetic systems in different phases. The SGO engine is capable of surveying the local minima surrounding the global minimum, which provides the information for the overall energy landscape of a given system. Using this capability we have found several new configurations for testing systems, explored their energy landscape, and demonstrated that the magnetic moment of metal clusters fluctuates strongly in different local minima.

Emergence of a few distinct structures from a single formal structure type during high-throughput screening for stable compounds: The case of RbCuS and RbCuSe

Theoretical sorting of stable and synthesizable ``missing compounds'' from those that are unstable is a crucial step in the discovery of previously unknown functional materials. This active research area often involves high-throughput (HT) examination of the total energy of a given compound in a list of candidate formal structure types (FSTs), searching for those with the lowest energy within that list. While it is well appreciated that local relaxation methods based on a fixed list of structure types can lead to inaccurate geometries, this approach is widely used in HT studies because it produces answers faster than global optimization methods (that vary lattice vectors and atomic positions without local restrictions). We find, however, a different failure mode of the HT protocol: specific crystallographic classes of formal structure types each correspond to a series of chemically distinct ``daughter structure types'' (DSTs) that have the same space group but possess totally different local bonding configurations, including coordination types. Failure to include such DSTs in the fixed list of examined candidate structures used in contemporary high-throughput approaches can lead to qualitative misidentification of the stable bonding pattern, not just quantitative inaccuracies. In this work, we (i) clarify the understanding of the general DST-FST relationship, thus improving current discovery HT approaches, (ii) illustrate this failure mode for RbCuS and RbCuSe (the latter being a yet unreported compound and is predicted here) by developing a synthesis method and accelerated crystal-structure determination, and (iii) apply the genetic-algorithm-based global space-group optimization (GSGO) approach which is not vulnerable to the failure mode of HT searches of fixed lists, demonstrating a correct identification of the stable DST. The broad impact of items (i)--(iii) lies in the demonstrated predictive ability of a more comprehensive search strategy than what is currently used---use HT calculations as the preliminary broad screening followed by unbiased GSGO of the final candidates.

A local relaxation method for the cardinality constrained portfolio optimization problem

The NP-hard nature of cardinality constrained mean-variance portfolio optimization problems has led to a number of different algorithms with varying degrees of success in reaching optimality given limited computational resources and under the presence of strict time constraints present in practice. The proposed local relaxation algorithm explores the inherent structure of the objective function. It solves a sequence of small, local, quadratic-programs by first projecting asset returns onto a reduced metric space, followed by clustering in this space to identify sub-groups of assets that best accentuate a suitable measure of similarity amongst different assets. The algorithm can either be cold started using a suitable heuristic method such as the centroids of initial clusters or be warm started based on the last output. Results, using a basket of up to 3,000 stocks and with different cardinality constraints, indicates that the proposed algorithm can lead to significant performance gain over popular branch-and-cut methods. One key application of this algorithm is in dealing with large scale cardinality constrained portfolio optimization under tight time constraint, such as for the purpose of index tracking or index arbitrage at high frequency.

A Local Relaxation Method for Cardinality Constrained Portfolio Optimization Problem

The NP-hard nature of cardinality constrained mean-variance portfolio optimization problems has led to a variety of different algorithms with varying degrees of success in reaching optimality given limited computational resources and under the presence of strict time constraints in practice. The proposed local relaxation algorithm exploits the inherent structure of the objective function. It solves a sequence of small, local, quadratic-programs by first projecting asset returns onto a reduced metric space, followed by clustering in this space to identify sub-groups of assets that best accentuate a suitable measure of similarity amongst different assets. The algorithm can either be cold started using the centroids of initial clusters or be warm started based on the output of a previous result. Empirical result, using baskets of up to 3,000 stocks and with different cardinality constraints, indicates that the proposed algorithm is able to achieve significant performance gain over a sophisticated branch-and-cut method. One key application of this algorithm is in dealing with large scale cardinality constrained portfolio optimization under tight time constraint, such as for the purpose of index tracking or index arbitrage at high frequency

Local Block Relaxation Method for the Solution of Equations of Gasdynamics

A new explicit pseudo-time-integration algorithm, termed local block relaxation, for the solution of the equations of gasdynamics in block-structured domains is proposed. The aim of the method is to reduce the computational costs in obtaining the numerical solution of complex flowfields. Attention is given to the definition of proper interface boundary conditions that ensure stability even if the smoothing of the solution is not synchronized in time among the blocks. Once stability is guaranteed, oversmoothings can be performed in a restricted number of blocks, namely, those that exhibit a slower rate of convergence. The method proposed is closely coupled with a standard multigrid cycle. A theoretical analysis of the savings in computational work is presented so that the convergence speed up can be estimated a priori. The strategy is shown to be effective in both two-dimensional and three-dimensional Navier-Stokes simulations. The present algorithm can be extended to a broad variety of numerical schemes based on explicit time integrations

Local block relaxation method for the solution of equations of gasdynamics

Local block relaxation method for the solution of equations of gasdynamics

A local relaxation method for optical flow estimation

Abstract The application of the local relaxation (LR) method to the optical flow estimation has been shown to provide much better performance than the Gauss-Seidel method. However, the direct application of the existing LR method results in a relatively slow convergence because of the coupling of the Poisson equations in which optical flow estimation problem is framed. This paper presents modification of the LR method based on the decoupling of the Poisson equations resulting in a fast convergence. This speed up in convergence is more noticeable when the proposed method is combined with registered gradients such as displaced frame difference and motion compensated spatial gradients. The paper includes simulation results on artificial and real image sequences verifying the superiority of the proposed modification.

Local relaxation method for boundary layer equations with separation

In the context of the laminar steady two-dimensional flow of an incompressible Newtonian fluid in a hilly channel for large values of the Reynolds number, we propose a numerical model of the boundary layer equations, when separation appears. Therefore, when the separation of the flow occurs, we consider the mathematical problem, justified by a formal asymptotic analysis, as elliptic.

A parallel multigrid algorithm for percolation clusters

A new parallel cluster-finding algorithm is formulated by using multigrid relaxation methods very similar to those used for differential equation solvers. For percolation clusters, this approach drastically reduces critical slowing down relative to local or scan relaxation methods. Numerical studies of scaling properties with system size are presented in the case of the 2D percolation clusters of the Swendsen-Wang Ising dynamics running on the Connection Machine.

Inelastic processes in extended x-ray-absorption fine structure.

Inelastic processes in extended x-ray-absorption fine structure (EXAFS) are studied using an extension of the semiclassical model of dynamical screening of a core hole and a photoelectron. This treatment effectively includes both extrinsic and intrinsic inelastic losses, as well as interference between them. The local-density approximation is used to derive a complex, energy-dependent final-state potential which includes dynamical corrections to the static exchange-correlation potential. A local relaxation method is developed to calculate the core-hole Green's function, from which the EXAFS amplitude-reduction factor can be determined. Applications to EXAFS amplitudes for the diatomic molecule ${\mathrm{Br}}_{2}$ and for metallic Cu yield results in reasonable agreement with experiment.

A Local Relaxation Method for Solving Elliptic PDEs on Mesh-Connected Arrays

A local relaxation method for solving linear elliptic PDEs with $O(N)$ processors and $O(\sqrt N )$ computation time is proposed. We first examine the implementation of traditional relaxation algorithms for solving elliptic PDEs on mesh-connected processor arrays, which require $O(N)$ processors and $O(N)$ computation time. The disadvantage of these implementations is that the determination of the acceleration factors requires some global communication at each iteration. The high communication cost increases the computation time per iteration significantly. Therefore, a local relaxation scheme is proposed to achieve the acceleration effect with very little global communication in the loading stage. We use a Fourier analysis approach to analyze the local relaxation method and also show its convergence. The convergence rate of the local relaxation method is studied by computer simulation.

SEMICLASSICAL TREATMENT FOR INELASTIC PROCESSES IN EXAFS

Inelastic processes in EXAFS (extended x-ray absorption fine structure) are studied using an extension of the semiclassical model of dynamical screening of a core hole and a photoelectron. This treatment effectively includes both extrinsic and intrinsic inelastic losses as well as interference between them. A complex, energy-dependent potential which accounts for these inelastic effects is derived. This potential is compared with static final state potentials and with empirically determined mean free paths. An local relaxation method is developed to calculate the core hole Green's function in an inhomogeneous system. Application is made to the EXAFS amplitude of the diatomic molecule Br2.

Determination of relaxation factors for high cell peclet number flow simulation

For the numerical simulation of transport problems with high cell Peclet number, the coefficient matrix of finite difference equations may lose the diagonal dominance if a scheme more accurate than the first-order upwind is used to approximate the convection terms. Hence, in many cases it is difficult to choose a suitable relaxation factor a priori for these schemes when the commonly used iterative method is applied to obtain the convergent solution. However, it is found that after appropriate normalization, an easier determined relaxation factor useful to all three schemes studied here, i.e., second-order central differencing, second-order upwinding, and QUICK, exists. Two model problems are used to compare the performances among the three schemes. The second-order central differencing is found to be less efficient in the cases investigated. Without the aid of the normalization, the second-order upwind scheme has the widest range of allowable values for the relaxation factor. Yet QUICK may be computationally more efficient after normalization. This is mainly because the number of iterations needed for QUICK is less sensitive to the choice of the relaxation factor when the optimum value isn't known. For a flow problem with non-constant velocities, the present method becomes the iterative algorithm with different relaxation parameters for different points, i.e., a local relaxation method. It is demonstrated that, in addition to helping choose the relaxation factors, this idea may also substantially reduce the computing time compared with the iterative algorithm with the uniform relaxation factor for all the grid points.

Local Relaxation Algorithms for Event-Driven Simulation of MOS Networks Including Assignable Delay Modeling

In this paper a new algorithm for switch level simulation is presented. The algorithm is compatible with classical time wheel mechanisms for logic simulation and therefore allows for assignable delay modeling also at the switch level. The key to this algorithm is a local relaxation method which takes care of localized bilateral effects. The algorithm is shown to be 0(n) and proven to converge to a unique solution.

On local relaxation methods and their application to convection-diffusion equations

This paper discusses local relaxation (LR) methods which can be regarded as generalizations of the successive overrelaxation (SOR) method. The difference is that within an LR method the relaxation factor is allowed to vary from equation to equation. A number of existing methods are found to be in fact special LR methods. Moreover, based on SOR theory, a new LR method is developed. The performance of LR methods is illustrated by applying them to central difference approximations of convection-diffusion equations. It is found that equations with small diffusion coefficients can be handled without difficulty. For equations with strongly varying coefficients, and for nonlinear equations, a properly selected LR method can be significantly more efficient than the optimum SOR method. As a special example, a 16 × 16 driven cavity problem for a Reynolds number of 10 6 can be solved in just a few seconds on a modern computer.