Multiple Minima Research Articles

Reduce Distillation Column Cost by Hybrid Particle Swarm and Ant Colony Optimization Technique

A novel method for optimum design of plate type distillation column integrating the equilibrium, hydraulic and economic calculations is presented in the present paper. The present study explores the use of non-traditional optimization technique: called hybrid Particle swarm optimization (PSO) and Ant colony optimization (ACO), for design optimization of plate type distillation column from economic point of view. The optimization procedure involves the selection of the major plate geometric parameters such as hole diameters, ratio of downcomer area to tower area, weir height, fractional hole area in tray, tray spacing, tower diameter etc. and minimization of total annual cost is considered as design target subjected to operational constraints like flooding, weeping entrainment, quality specifications etc. The solution space of such type of problem is very complex due to presence of various nonlinear constraints and multiple minima. Hybrid Particle swarm optimization and Ant colony optimization (PSACO) technique is applied to deal with such complexity. The particle swarm optimization applies for global optimization and ant colony approach is employed to update positions of particles to attain rapidly the feasible solution space. Ant colony optimization works as a local search, wherein, ants apply pheromone-guided mechanism to update the positions found by the particles in the earlier stage. The presented hybrid Particle swarm optimization and Ant colony optimization (PSACO) technique is simple in concept, few in parameters and easy for implementations. Furthermore, the PSACO algorithm explores the good quality solutions quickly, giving the designer more degrees of freedom in the final choice with respect to traditional methods. One case study is presented to demonstrate the effectiveness and accuracy of proposed algorithm. The PSACO approach is able to reduce the total cost of distillation column as compare to cost obtained by commercial simulator.

Neural network exponential fitting of a potential energy surface with multiple minima: Application to HFCO

We have constructed a (semi)-global six-dimensional potential energy surface (PES) for HFCO, incorporating the equilibrium, cis- and trans- isomers (HOCF) as well as the transition states connecting them. The PES is based on a fit to 15000 CCSD(T)-F12/cc-pVTZ-F12 ab initio energies. The sum-of-products PES, obtained using neural network exponential fitting functions, was used to compute vibrational state frequencies using block improved relaxation with the multiconfiguration time dependent Hartree (MCTDH) approach. The PES is accurate (RMSE=130cm−1) up to 40,000cm−1 above the minimum energy. The equilibrium region of the PES was fit very well based on a comparison of the vibrational frequencies with those from a recent local HFCO PES (Pradhan and Brown, 2016) and experimental measurements (RMSE=10.9cm−1 compared to experiment). The vibrational frequencies for the trans- and cis-isomers are computed from the PES and compared to anharmonic MP2/aug-cc-pVTZ and CCSD(T)/aug-cc-pVTZ results; the trans- and cis-isomers have yet to be detected experimentally. Based on the accuracy of the vibrational energies at equilibrium, the present results for the cis- and trans-isomers could facilitate the identification of these species. The PES will also enable the study of intramolecular vibrational energy redistribution and its control leading to the elusive equilibrium HFCO to trans-HOCF conversion.

A fast Stokes inversion technique based on quadratic regression

Stokes inversion calculation is a key process in resolving polarization information on radiation from the Sun and obtaining the associated vector magnetic fields. Even in the cases of simple local thermodynamic equilibrium (LTE) and where the Milne-Eddington approximation is valid, the inversion problem may not be easy to solve. The initial values for the iterations are important in handling the case with multiple minima. In this paper, we develop a fast inversion technique without iterations. The time taken for computation is only 1/100 the time that the iterative algorithm takes. In addition, it can provide available initial values even in cases with lower spectral resolutions. This strategy is useful for a filter-type Stokes spectrograph, such as SDO/HMI and the developed two-dimensional real-time spectrograph (2DS).

Comparison of minimum-action and steepest-descent paths in gradient systems.

On high-dimensional and complex potential energy surfaces, the identification of the most likely mechanism for the transition between local minima is a challenging task. Usually the steepest-descent path is used interchangeably with the minimum-energy path and is associated with the most likely path. Here we compare the meaning of the steepest-descent path in complex energy landscapes to the path integral formulation of a trajectory that minimizes the action functional for Brownian dynamics. In particular, for energy landscapes with bifurcation points and multiple minima and saddle points, there can be several steepest-descent paths associated with specific saddles that connect two predetermined states but largely differ from the path of maximum likelihood. The minimum-action path, however, additionally takes into account the scalar work along the trajectory. Minimizing the scalar work can be less ambiguous in the identification of the most likely path in different gradient systems. It can also be used to distinguish between multiple steepest-descent paths that connect reactant and product states. We illustrate that in systems with complex energy landscapes a careful assessment of the steepest-descent path is thus advisable. Here the evaluation of the action can provide valuable information on the analysis and description of the most likely path.

Development of an Optimization Procedure for Magnetron-Sputtered Thin Films to Facilitate Combinatorial Materials Research

High-throughput experimentation (HTE) is an experimental paradigm that has shown potential for accelerating the evaluation of material systems by synthesizing a sample with a variation of material parameters and using parallel or fast serial characterization. Magnetron sputtering is one of the most widely used thin-film synthesis techniques in the HTE field. One major bottleneck for HTE magnetron sputtering is identifying appropriate sputtering conditions. Here, a Nelder–Mead optimization procedure with soft constraints is used to determine sputtering conditions of a desired deposition rate distribution using a continuum based sputter model. The optimization procedure uses a multiple stage multistart method to overcome issues with initialization and multiple minima. The procedure was validated against a simulated thin film. The method was able to determine a gun power, gun tilt, and substrate height within 1% of the simulation sputtering conditions in 29.8 s. This result is 3–4 orders of magnitude faster ...

Tabu search for critical non-circular slip surface in slope stability analysis

Stability analysis of slopes is of utmost importance because its failure may lead to loss of lives and great economic losses. Locating the critical circular slip surface is the key step of slope stability analysis. Location of the critical slip surface of a general soil slope is difficult because the objective function of the factor of safety is non-convex and multiple minima exist in general. In this paper, a new approach which combines tabu search and limited equilibrium method (Spencer method) was proposed and applied to some classic examples. The results show the proposed search approach is stable and highly correlated with the results of independent analyses. Furthermore, this paper demonstrates the successful application of tabu search to non-circular critical slip surface search problems.

Bayesian Uncertainty Quantification for Bond Energies and Mobilities Using Path Integral Analysis

Dynamic single-molecule force spectroscopy is often used to distort bonds. The resulting responses, in the form of rupture forces, work applied, and trajectories of displacements, are used to reconstruct bond potentials. Such approaches often rely on simple parameterizations of one-dimensional bond potentials, assumptions on equilibrium starting states, and/or large amounts of trajectory data. Parametric approaches typically fail at inferring complicated bond potentials with multiple minima, while piecewise estimation may not guarantee smooth results with the appropriate behavior at large distances. Existing techniques, particularly those based on work theorems, also do not address spatial variations in the diffusivity that may arise from spatially inhomogeneous coupling to other degrees of freedom in the macromolecule. To address these challenges, we develop a comprehensive empirical Bayesian approach that incorporates data and regularization terms directly into a path integral. All experimental and statistical parameters in our method are estimated directly from the data. Upon testing our method on simulated data, our regularized approach requires less data and allows simultaneous inference of both complex bond potentials and diffusivity profiles. Crucially, we show that the accuracy of the reconstructed bond potential is sensitive to the spatially varying diffusivity and accurate reconstruction can be expected only when both are simultaneously inferred. Moreover, after providing a means for self-consistently choosing regularization parameters from data, we derive posterior probability distributions, allowing for uncertainty quantification.

Spectral Modulation Effect in Teleseismic P-waves from North Korean Nuclear Tests Recorded in Broad Azimuthal Range and Possible Source Depth Estimation

Three underground nuclear explosions, conducted by North Korea in 2006, 2009 and 2013, are analyzed. The last two tests were recorded by the Israel Seismic Network. Pronounced coherent minima (spectral nulls) at 1.2–1.3 Hz were revealed in the spectra of teleseismic P-waves. For a ground-truth explosion with a shallow source depth, this phenomenon can be interpreted in terms of the interference between the down-going P-wave and the pP phase reflected from the Earth’s surface. This effect was also observed at ISN stations for a Pakistan nuclear explosion at a different frequency 1.7 Hz and the PNE Rubin-2 in West Siberia at 1 Hz, indicating a source-effect and not a site-effect. Similar spectral minima having essentially the same frequency, as at ISN, were observed in teleseismic P-waves for all the three North Korean explosions recorded at networks and arrays in Kazakhstan (KURK), Norway (NNSN), Australia (ASAR, WRA) and Canada (YKA), covering a broad azimuthal range. Data of 2009 and 2013 tests at WRA and KURK arrays showed harmonic spectral modulation with three multiple minima frequencies, evidencing the clear interference effect. These observations support the above-mentioned interpretation. Based on the null frequency dependency on the near-surface acoustic velocity and the source depth, the depth of the North Korean tests was estimated about 2.0–2.1 km. It was shown that the observed null frequencies and the obtained source depth estimates correspond to P-pP interference phenomena in both cases of a vertical shaft or a horizontal drift in a mountain. This unusual depth estimation needs additional validation based on more stations and verification by other methods.

New variants of glowworm swarm optimization based on step size

In 2005, Krishnanand and Ghose (Multimodal function optimization using a glowworm metaphor with applications to collective robotics, 2005a), presented the idea of glowworm metaphor to determine multiple minima in the optimization problem arising in robotics applications. That research paper highlights the glowworm swarm behavior for determining multiple local minima for multimodal functions with application to robotics. Since then, a number of research papers have appeared to improve the performance of glowworm swarm optimization (GSO). In this paper, two major contributions are made. Firstly, a mathematical result is proved which shows that the step size of GSO has a significant influence on the convergence of GSO. Secondly, three variants of GSO are proposed which depend on different step size. Based on the implementation of the proposed variants and the original GSO on 15 benchmark problems, it is concluded that one of the proposed variants is a definite improvement over the original GSO and the remaining variants.

Surface structure. Subatomic resolution force microscopy reveals internal structure and adsorption sites of small iron clusters.

Clusters built from individual iron atoms adsorbed on surfaces (adatoms) were investigated by atomic force microscopy (AFM) with subatomic resolution. Single copper and iron adatoms appeared as toroidal structures and multiatom clusters as connected structures, showing each individual atom as a torus. For single adatoms, the toroidal shape of the AFM image depends on the bonding symmetry of the adatom to the underlying structure [twofold for copper on copper(110) and threefold for iron on copper(111)]. Density functional theory calculations support the experimental data. The findings correct our previous work, in which multiple minima in the AFM signal were interpreted as a reflection of the orientation of a single front atom, and suggest that dual and triple minima in the force signal are caused by dimer and trimer tips, respectively.

An automated transition state search using classical trajectories initialized at multiple minima.

Very recently, we proposed an automated method for finding transition states of chemical reactions using dynamics simulations; the method has been termed Transition State Search using Chemical Dynamics Simulations (TSSCDS) (E. Martínez-Núñez, J. Comput. Chem., 2015, 36, 222-234). In the present work, an improved automated search procedure is developed, which consists of iteratively running different ensembles of trajectories initialized at different minima. The iterative TSSCDS method is applied to the complex C3H4O system, obtaining a total of 66 different minima and 276 transition states. With the obtained transition states and paths, statistical RRKM calculations and Kinetic Monte Carlo simulations are carried out to study the fragmentation dynamics of propenal, which is the global minimum of the system. The kinetic simulations provide a (three-body dissociation)/(CO elimination) ratio of 1.49 for an excitation energy of 148 kcal mol(-1), which agrees well with the corresponding value obtained in the photolysis of propenal at 193 nm (1.1), suggesting that at least these two channels: three-body dissociation (to give H2 + CO + C2H2) and CO elimination occur on the ground electronic state.

Solution of Economic Dispatch Problem usingBBO for Power system optimization

This paper presents the biogeography based optimization (BBO) algorithm to solve economic load dispatch problem. Economic load dispatch problem is basically an optimization problem where objective function may be highly nonlinear, non-convex, and no differentiable and may have multiple local minima. The models of biogeography explain how an organism arises, immigrate from an environment to another and gets eliminated. This method is based on two steps mutation and migration. This paper provides to show the minimum cost of the system constraint for each generating system also if the power demand increases the minimum cost also increases with the standard deviation.BBO technique is implemented in MATLAB environment.

Bayesian approach to the inverse problem in a light scattering application

In this article, static light scattering (SLS) measurements are processed to estimate the particle size distribution of particle systems incorporating prior information obtained from an alternative experimental technique: scanning electron microscopy (SEM). For this purpose we propose two Bayesian schemes (one parametric and another non-parametric) to solve the stated light scattering problem and take advantage of the obtained results to summarize some features of the Bayesian approach within the context of inverse problems. The features presented in this article include the improvement of the results when some useful prior information from an alternative experiment is considered instead of a non-informative prior as it occurs in a deterministic maximum likelihood estimation. This improvement will be shown in terms of accuracy and precision in the corresponding results and also in terms of minimizing the effect of multiple minima by including significant information in the optimization. Both Bayesian schemes are implemented using Markov Chain Monte Carlo methods. They have been developed on the basis of the Metropolis–Hastings (MH) algorithm using Matlab® and are tested with the analysis of simulated and experimental examples of concentrated and semi-concentrated particles. In the simulated examples, SLS measurements were generated using a rigorous model, while the inversion stage was solved using an approximate model in both schemes and also using the rigorous model in the parametric scheme. Priors from SEM micrographs were also simulated and experimented, where the simulated ones were obtained using a Monte Carlo routine. In addition to the presentation of these features of the Bayesian approach, some other topics will be discussed, such as regularization and some implementation issues of the proposed schemes, among which we remark the selection of the parameters used in the MH algorithm.

Inversion of Lattice Network Structure Subjected to Carbonate Mercury Intrusion Capillary Pressure: Hamiltonian Monte Carlo Posterior Sampling

The wide range of pore sizes in carbonates (differences of three to five orders of magnitude) aggravates the already difficult problem of multiple minima in pore network model (PNM) parameters. In this paper, we propose a method to estimate the PNM structural parameters. We have applied the method to four cases: one synthetic case and three carbonate rock samples. The PNM structural parameter estimates were conditioned to the mercury intrusion capillary pressure (MICP) via a stochastic inversion algorithm that uses molecular dynamics combined with stochastic steps using Hamiltonian mechanics. This algorithm, Hamiltonian Monte Carlo (HMC), allows for large moves in the phase space making use of the periodicity developed due to the Hamiltonian formulation, thus large modifications in the model parameters are possible. Additionally, we introduced a new estimator of the pore-size distribution using MICP. The statistics of the stochastic inversion found the multiple local minima and the global minimum of the misfit function in a difficult case. The results for the synthetic case showed that the volume and fraction of small pores in the pore-size distribution can lead to multiple local minima. For the three carbonate rock samples, the model parameters clearly showed that the volume exponents are greater than zero, which contradicts the usual assumption and inferences made by other methods. As the pore-size distribution becomes multi-modal, the volume exponent decreases but is greater than zero. These results suggest that when microporosity is ignored the volume exponent will systematically tend to be strongly underestimated (values close to zero). Our findings demonstrate that lattice-type PNM parameter estimation is a difficult problem because of multiple minima. The HMC method was able to escape from those multiple minima, resulting in better estimates of the PNM parameters.

Assimilation of vertical motion from simulated cloudy satellite imagery in an idealized single column model

Satellite infrared sounders are invaluable tools for making observations of the structure of the atmosphere. They provide much of the observational data used to initialize atmospheric models, especially in regions that do not have extensive surface‐based observing systems, such as oceans. However, information is lacking in the presence of cloud, as the cloud layer is opaque to infrared radiation. This means that where information is most desired (such as in a developing storm) it is often in the shortest supply.In order to explore the mathematics of assimilating data from cloudy radiances, a study has been performed using an idealized single‐column atmospheric model. The model simulates cloud development in an atmosphere with vertical motion, allowing the characteristics of a 2D‐Var data assimilation system using a single simulated infrared satellite observation taken multiple times to be studied. The strongly nonlinear nature of cloud formation poses a challenge for variational methods. The adjoint method produces an accurate gradient for the cost function and minimization is achieved using preconditioned conjugate gradients. The conditioning is poor and varies strongly with the atmospheric variables and the cost function has multiple minima, but acceptable results are achieved.The assimilation system is provided with a prior forecast simulated by adding random correlated Gaussian error to the truth. Assimilating observations comparable to those available from current geostationary satellites allows vertical motion to be retrieved with an error of less than a centimetre per second in most conditions. Moreover, evaluating the second derivative of the cost function at the minimum provides an estimate of the uncertainty in the retrieval. This allows atmospheric states that do not provide sufficient information for retrieval of vertical motion to be detected (such as a cloudless atmosphere or a non‐moving opaque cloud layer in the upper troposphere). Retrieval is most accurate with upwards motion.

Overview of nonlinear programming methods suitable for calibration of traffic flow models

The calibration of a macroscopic traffic flow model aims at enabling the model to reproduce, as accurately as possible, the real traffic conditions on a motorway network. Essentially, this procedure targets the best value for the parameter vector of the model and this can be achieved using appropriate optimization algorithms. The parameter calibration problem is formulated as a nonlinear, non-convex, least-squares optimization problem, which is known to attain multiple local minima; for this reason gradient-based solution algorithms are not considered to be an option. The methodologies that are more appropriate for application to this problem are mainly some meta-heuristic algorithms which use direct search approaches that allow them to avoid bad local minima. This paper presents an overview of the most suitable nonlinear programming methods for the calibration procedure of macroscopic traffic flow models. Furthermore, an application example, where two well-known macroscopic traffic flow models are evaluated through the calibration procedure, is presented.

Chiral bosonic Mott insulator on the frustrated triangular lattice

We study the superfluid and insulating phases of interacting bosons on the triangular lattice with an inverted dispersion, corresponding to frustrated hopping between sites. The resulting single-particle dispersion has multiple minima at nonzero wavevectors in momentum space, in contrast to the unique zero-wavevector minimum of the unfrustrated problem. As a consequence, the superfluid phase is unstable against developing additonal chiral order that breaks time reversal (T) and parity (P) symmetries by forming a condensate at nonzero wavevector. We demonstrate that the loss of superfluidity can lead to an even more exotic phase, the chiral Mott insulator, with nontrivial current order that breaks T, P. These results are obtained via variational estimates, as well as a combination of bosonization and DMRG of triangular ladders, which taken together permit a fairly complete characterization of the phase diagram. We discuss the relevance of these phases to optical lattice experiments, as well as signatures of chiral symmetry breaking in time-of-flight images.

Fractal free energy landscapes in structural glasses.

Glasses are amorphous solids whose constituent particles are caged by their neighbours and thus cannot flow. This sluggishness is often ascribed to the free energy landscape containing multiple minima (basins) separated by high barriers. Here we show, using theory and numerical simulation, that the landscape is much rougher than is classically assumed. Deep in the glass, it undergoes a 'roughness transition' to fractal basins, which brings about isostaticity and marginal stability on approaching jamming. Critical exponents for the basin width, the weak force distribution and the spatial spread of quasi-contacts near jamming can be analytically determined. Their value is found to be compatible with numerical observations. This advance incorporates the jamming transition of granular materials into the framework of glass theory. Because temperature and pressure control what features of the landscape are experienced, glass mechanics and transport are expected to reflect the features of the topology we discuss here.

Graphene ribbons for tunable coupling with plasmonic subwavelength cavities

Since graphene supports low loss plasmonic guided modes in the infrared range, we theoretically investigate the coupling of these modes in patterned sheets with nanocavities. We calculate cavity modes and (potentially critical) coupling in filter-type circuits, with resonances observed as multiple minima in the reflection spectrum. The origin and properties of the cavity modes are fully modeled by coupled mode theory, exploring for various positions of the cavity with respect to the access waveguide. A useful resonance frequency shift is examined by modifying the graphene doping (e.g., via voltage tuning). The deep subwavelength cavity modes reach quality factors up to 42 for ribbons of 30 nm width around 5 μm wavelength. These resonances provide opportunities for ultracompact optoelectronic circuits.

Application of the Firefly and Luus–Jaakola algorithms in the calculation of a double reactive azeotrope

The calculation of reactive azeotropes is an important task in the preliminary design and simulation of reactive distillation columns. Classically, homogeneous nonreactive azeotropes are vapor–liquid coexistence conditions where phase compositions are equal. For homogeneous reactive azeotropes, simultaneous phase and chemical equilibria occur concomitantly with equality of compositions (in the Ung–Doherty transformed space). The modeling of reactive azeotrope calculation is represented by a nonlinear algebraic system with phase equilibrium, chemical equilibrium and azeotropy equations. This nonlinear system can exhibit more than one solution, corresponding to a double reactive azeotrope. In a previous paper (Platt et al 2013 J. Phys.: Conf. Ser. 410 012020), we investigated some numerical aspects of the calculation of reactive azeotropes in the isobutene + methanol + methyl-tert-butyl-ether (with two reactive azeotropes) system using two metaheuristics: the Luus–Jaakola adaptive random search and the Firefly algorithm. Here, we use a hybrid structure (stochastic + deterministic) in order to produce accurate results for both azeotropes. After identifying the neighborhood of the reactive azeotrope, the nonlinear algebraic system is solved using Newton's method. The results indicate that using metaheuristics and some techniques devoted to the calculation of multiple minima allows both azeotropic coordinates in this reactive system to be obtains. In this sense, we provide a comprehensive analysis of a useful framework devoted to solving nonlinear systems, particularly in phase equilibrium problems.