Nelder-Mead Downhill Simplex Method Research Articles

Asteroid absolute magnitudes and phase curve parameters from Gaia photometry

Aims. We perform light curve inversion for 491 asteroids to retrieve phase curve parameters, rotation periods, pole longitudes and latitudes, and convex and triaxial ellipsoid shapes by using the sparse photometric observations from Gaia Data Release 2 and the dense ground-based observations from the DAMIT database. We develop a method for the derivation of reference absolute magnitudes and phase curves from the Gaia data, allowing for comparative studies involving hundreds of asteroids. Methods. For both general convex shapes and ellipsoid shapes, we computed least-squares solutions using either the Levenberg-Marquardt optimization algorithm or the Nelder-Mead downhill simplex method. Virtual observations were generated by adding Gaussian random errors to the observations, and, later on, a Markov chain Monte Carlo method was applied to sample the spin, shape, and scattering parameters. Absolute magnitude and phase curve retrieval was developed for the reference geometry of equatorial illumination and observations based on model magnitudes averaged over rotational phase. Results. The derived photometric slope values showed wide variations within each assumed Tholen class. The computed Gaia G-band absolute magnitudes matched notably well with the V-band absolute magnitudes retrieved from the Jet Propulsion Laboratory Small-Body Database. Finally, the reference phase curves were well fitted with the H, G1, G2 phase function. The resulting G1, G2 distribution differed, in an intriguing way, from the G1, G2 distribution that is based on the phase curves corresponding to light curve brightness maxima.

Reliability modelling for multi-component systems subject to stochastic deterioration and generalized cumulative shock damages

For a complex system consisting of multiple components, it is often unrealistic that one type of environmental shocks affects all the components at the same time. Correspondingly, random shocks are categorized into several distinct sets according to their functions, attributes or sizes. This study develops generalized reliability models for multi-component systems, where each component is subject to two dependent competing failure processes, i.e., a soft failure process caused jointly by internal performance degradation and an incremental damage due to effective external shock sets, and a hard failure process caused by the same random shocks. A damage improvement coefficient and a damage aggravation coefficient are respectively introduced to extend the standard cumulative shock damage model into two more generalized shock cases. Analytical representations of system reliability for a series–parallel system and a parallel–series system are derived based on a gamma to normal distribution approximation approach. To quantitatively compare the effects of these two damage coefficients, a block replacement policy is further adopted by searching for the optimal replacement intervals with a Nelder–Mead downhill simplex method. Finally, an illustrative example of micro-electro-mechanical systems (MEMS) consisting of four silicon micro-mechanical resonators is provided to examine the effects of self-healing ability in the materials of polymer binder on system reliability and replacement period.

Reliability modeling and optimal random preventive maintenance policy for parallel systems with damage self-healing

Materials with intrinsic self-healing phenomenon possess the ability to heal in response to external random shocks. Introducing a recovery factor to quantitatively measure the damage self-recovery efficiency, this paper designs a self-healing mechanism corresponding to both damage load and shock arrival numbers for a parallel redundant system consisting of multiple non-identical components. From the actual engineering perspective, each shock arriving on the system selectively affects one component or more but not necessarily all units in parallel, and consequently, random shocks are categorized according to their sizes, attributes and affected components. This study investigates novel reliability models and schedules optimal preventive maintenance policies, in which the closed-form reliability quantities are derived analytically and the optimum preventive replacement interval is demonstrated theoretically. In addition, a Nelder-Mead downhill simplex method is introduced to seek the optimal replacement age in minimizing the long-run average maintenance cost rate for the condition system failure distribution is rather complex. A micro-electro-mechanical system (MEMS) whose constitutional materials are integrated by microcrystalline silicon, where polymer binders with self-healing capability are always synthesized, is designed to verify the results we obtained numerically, illustrating the significance of considering damage self-healing phenomena.

Economic design of exponentially weighted moving average chart with variable sampling interval at fixed times scheme incorporating Taguchi loss function

A control chart is a graph used to study how process changes over time. The exponentially weighted moving average (EWMA) chart is highly useful for not only detecting small changes in process parameters but also in inspecting individual unit observations. The objective of this paper is to develop an economic statistical design of the exponentially weighted moving average (EWMA) chart using variable sampling interval at fixed times (VSIFT) control scheme considering preventive maintenance and Taguchi loss function to determine the values of the seven test parameters (i.e., the sample size, the fixed sampling interval, the number of subintervals between two consecutive sampling times, the warning limit coefficient, the control limit coefficient, exponential weight constant and the time interval of preventive maintenance) such that the expected total cost per hour is minimised. A mathematical model is developed to analyse the cost of the integrated model. Nelder-Mead downhill simplex method and genetic algorithm approaches are applied to search for the optimal values of the seven test parameters for the economic statistical design of VSIFT EWMA chart and Nelder-Mead downhill simplex performed better than genetic algorithm. A hypothetical example and its solution are provided to have a better understanding about the demonstration of the proposed model.

Optimal warranty policies considering repair service and replacement service under the manufacturer’s perspective

In this paper, under the manufacturer’s point of view, a warranty cost model is proposed in consideration of both repair service and replacement service simultaneously upon the system failure to find the optimized warranty period in terms of an expected cost rate during the warranty cycle. If a failed system can’t be repaired by the threshold warranty servicing time in the warranty service center, the replacement service is provided for the failed system instead of continuing to provide the repair service. Under such circumstance, new warranty cost models are developed and the renewable warranty policy is investigated for free and pro-rata two-dimensional warranty dependent on both failure time and warranty servicing time. From the manufacturer’s point of view, the warranty period is newly defined starting from the system’s purchasing time and ending at the expiration time of the warranty. This paper determines the optimal warranty period by applying Nelder–Mead downhill simplex method to minimize the expected cost rate during the warranty period. The real application is discussed adopting the proposed approach based on the actual field data and numerical examples are presented to exemplify the applicability of the methodologies derived in this paper.

Application of the genetic algorithm and downhill simplex methods (Nelder–Mead methods) in the search for the optimum chiller configuration

Methods for searching for the optimum chiller configuration for a heat source plant were examined. In addition to the genetic algorithm (GA), a hybrid of the GA and the Nelder–Mead downhill simplex method was developed. By using the Nelder–Mead downhill simplex method, fewer calculations were required to determine the optimum chiller configuration. Due to applying the GA, the calculations for the developed system did not become trapped in a local optimum, and the algorithm could then determine the global optimum after a few iterations of the Nelder–Mead downhill simplex method.The performance of the developed system was examined for the heating/cooling energy plant of a building; the plant consisted of a centrifugal chiller, a gas-fired absorption chiller, air-source heat pumps, and boilers. It was confirmed that the ability of the GA to find the optimum configuration for the chiller was improved by combining the GA with the Nelder–Mead downhill simplex method.

Asteroid orbital inversion using a virtual-observation Markov-chain Monte Carlo method

A novel virtual-observation Markov-chain Monte Carlo method (MCMC) is presented for the asteroid orbital inverse problem posed by small to moderate numbers of astrometric observations. In the method, the orbital-element proposal probability density is chosen to mimic the convolution of the a posteriori density by itself: first, random errors are simulated for each observation, resulting in a set of virtual observations; second, least-squares orbital elements are derived for the virtual observations using the Nelder–Mead downhill simplex method; third, repeating the procedure gives a difference between two sets of what can be called virtual least-squares elements; and, fourth, the difference obtained constitutes a symmetric proposal in a random-walk Metropolis–Hastings algorithm, avoiding the explicit computation of the proposal density. In practice, the proposals are based on a large number of pre-computed sets of orbital elements. Virtual-observation MCMC is thus based on the characterization of the phase-space volume of solutions before the actual MCMC sampling. Virtual-observation MCMC is compared to MCMC orbital ranging, a random-walk Metropolis–Hastings algorithm based on sampling with the help of Cartesian positions at two observation dates, in the case of the near-Earth asteroid (85640) 1998 OX4. In the present preliminary comparison, the methods yield similar results for a 9.1-day observational time interval extracted from the full current astrometry of the asteroid. In the future, both of the methods are to be applied to the astrometric observations of the Gaia mission.

Issues of Nelder-Mead Simplex Optimisation with Constraints

The Nelder-Mead downhill simplex method is a very popular minimiser, often used in finance. However in practice, a lot of problems need to be solved according to some specific constraints, for example, boundaries on the possible values. We will look at various ways to add those constraints, and show how problematic it can be to minimize simple quadratics under constraints.

Multicomponent diffusion in garnets I: general theoretical considerations and experimental data for Fe–Mg systems

We have carried out a combined theoretical and experimental study of multicomponent diffusion in garnets to address some unresolved issues and to better constrain the diffusion behavior of Fe and Mg in almandine–pyrope-rich garnets. We have (1) improved the convolution correction of concentration profiles measured using electron microprobes, (2) studied the effect of thermodynamic non-ideality on diffusion and (3) explored the use of a mathematical error minimization routine (the Nelder-Mead downhill simplex method) compared to the visual fitting of concentration profiles used in earlier studies. We conclude that incorporation of thermodynamic non-ideality alters the shapes of calculated profiles, resulting in better fits to measured shapes, but retrieved diffusion coefficients do not differ from those retrieved using ideal models by more than a factor of 1.2 for most natural garnet compositions. Diffusion coefficients retrieved using the two kinds of models differ only significantly for some unusual Mg–Mn–Ca-rich garnets. We found that when one of the diffusion coefficients becomes much faster or slower than the rest, or when the diffusion couple has a composition that is dominated by one component (>75 %), then profile shapes become insensitive to one or more tracer diffusion coefficients. Visual fitting and numerical fitting using the Nelder-Mead algorithm give identical results for idealized profile shapes, but for data with strong analytical noise or asymmetric profile shapes, visual fitting returns values closer to the known inputs. Finally, we have carried out four additional diffusion couple experiments (25–35 kbar, 1,260–1,400 °C) in a piston-cylinder apparatus using natural pyrope- and almandine-rich garnets. We have combined our results with a reanalysis of the profiles from Ganguly et al. (1998) using the tools developed in this work to obtain the following Arrhenius parameters in D = D0 exp{–[Q1bar + (P–1)ΔV+]/RT} for DMg* and DFe*: Mg: Q1bar = 228.3 ± 20.3 kJ/mol, D0 = 2.72 (±4.52) × 10−10 m2/s, Fe: Q1bar = 226.9 ± 18.6 kJ/mol, D0 = 1.64 (±2.54) × 10−10 m2/s. ΔV+ values were assumed to be the same as those obtained by Chakraborty and Ganguly (1992).

Optimization of internal heat exchangers for hydrogen storage tanks utilizing metal hydrides

Two detailed, unit cell models, a transverse fin design and a longitudinal fin design, of a combined hydride bed and heat exchanger are developed in COMSOL ® Multiphysics incorporating and accounting for heat transfer and reaction kinetic limitations. MatLab ® scripts for autonomous model generation are developed and incorporated into (1) a grid-based and (2) a systematic optimization routine based on the Nelder-Mead downhill simplex method to determine the geometrical parameters that lead to the optimal structure for each fin design that maximizes the hydrogen stored within the hydride. The optimal designs for both the transverse and longitudinal fin designs point toward closely-spaced, small cooling fluid tubes. Under the hydrogen feed conditions studied (50 bar), a 25 times improvement or better in the hydrogen storage kinetics will be required to simultaneously meet the Department of Energy technical targets for gravimetric capacity and fill time. These models and methodology can be rapidly applied to other hydrogen storage materials, such as other metal hydrides or to cryoadsorbents, in future work.

An optimization-simulation model for a simple LNG process

A gradient free optimization-simulation method for processes modelled with the simulator Aspen HYSYS is developed. The tool is based on a Tabu Search (TS) and the Nelder-Mead Downhill Simplex (NMDS) method. The local optima that result from the TS are fine-tuned with NMDS to reduce the required number of simulations. The tool has been applied to find the total refrigerant flow rate, composition and the refrigerant suction and condenser pressures that minimize the energy requirements of a Prico process. The main strength of this method is that it has a high probability of obtaining a better solution with significantly fewer simulation runs than other metaheuristic methods. Also, by changing the TS step size it is possible to influence the initial search pattern, thereby taking advantage of already gained process knowledge to decrease the optimization time. The method is general and can be applied to other processes modelled in Aspen HYSYS.

Nonlinear regression-based method for pseudoenhancement correction in CT colonography

In CT colonography (CTC), orally administered positive-contrast tagging agents are often used for differentiating residual bowel contents from native colonic structures. However, tagged materials can sometimes hyperattenuate observed CT numbers of their adjacent untagged materials. Such pseudoenhancement complicates the differentiation of colonic soft-tissue structures from tagged materials, because pseudoenhanced colonic structures may have CT numbers that are similar to those of tagged materials. The authors developed a nonlinear regression-based (NLRB) method for performing a local image-based pseudoenhancement correction of CTC data. To calibrate the correction parameters, the CT data of an anthropomorphic reference phantom were correlated with those of partially tagged phantoms. The CTC data were registered spatially by use of an adaptive multiresolution method, and untagged and tagged partial-volume soft-tissue surfaces were correlated by use of a virtual tagging scheme. The NLRB method was then optimized to minimize the difference in the CT numbers of soft-tissue regions between the untagged and tagged phantom CTC data by use of the Nelder-Mead downhill simplex method. To validate the method, the CT numbers of untagged regions were compared with those of registered pseudoenhanced phantom regions before and after the correction. The CT numbers were significantly different before performing the correction (p<0.01), whereas, after the correction, the difference between the CT numbers was not significant. The effect of the correction was also tested on the size measurement of polyps that were covered by tagging in phantoms and in clinical cases. In phantom cases, before the correction, the diameters of 12 simulated polyps submerged in tagged fluids that were measured in a soft-tissue CT display were significantly different from those measured in an untagged phantom (p<0.01), whereas after the correction the difference was not significant. In clinical cases, before the correction, the diameters of 29 colonoscopy-confirmed 3-14 mm polyps affected by tagging that were measured in a soft-tissue CT display were significantly different from those measured in a lung CT display (p<0.0001) or in colonoscopy (p<0.05), whereas after the correction the difference was not significant. Finally, the effect of the correction was tested on automated detection of 25 polyps > or =6 mm affected by tagging in 56 clinical CTC cases. The application of the correction increased the detection accuracy from 60% with 5.0 FP detections per patient without correction to 96% with 2.9 FP detections with correction. This improvement in detection accuracy was statistically significant (p<0.05). The results indicate that the proposed NLRB method can yield an accurate pseudoenhancement correction with potentially significant benefits in clinical CTC examinations.

Optimal experiment design for time-lapse traveltime tomography

Geophysical monitoring techniques offer the only noninvasive approach capable of assessing the spatial and temporal dynamics of subsurface fluid processes. Increasingly, permanent sensor arrays in boreholes and on the ocean floor are being deployed to improve the repeatability and increase the temporal sampling of monitoring surveys. Because permanent arrays require a large up-front capital investment and are difficult (or impossible) to reconfigure once installed, a premium is placed on selecting a geometry capable of imaging the desired target at minimum cost. We have taken a simple approach to optimizing downhole sensor configurations for monitoring experiments making use of differential seismic traveltimes. We used a design quality metric based on the accuracy of tomographic reconstructions for a suite of imaging targets. By not requiring an explicit singular value decomposition of the forward operator, evaluation of this objective function scaled to problems with a large numberof unknowns. We restricted the design problem by recasting the array geometry into a low-dimensional form more suitable for optimization at a reasonable computational cost. We tested two search algorithms on the design problem: the Nelder-Mead downhill simplex method and the multilevel coordinate search algorithm. The algorithm was tested for four crosswell acquisition scenarios relevant to continuous seismic monitoring, a two-parameter array optimization, several scenarios involving four-parameter length/offset optimizations, and a comparison of optimal multisource designs. In the last case, we also examined trade-offs between source sparsity and the quality of tomographic reconstructions. Asymmetrical array lengths improved localized image quality in crosswell experiments with a small number of sources and a large number of receivers. Preliminary results also suggested that high-quality differential images could be generated using only a small number of optimally positioned sources in tandem with a more extensive receiver array.

Magnetoencephalography source localization using improved simplex method

Nelder–Mead downhill simplex method, a kind of deterministic optimization algorithms, has been used extensively for magnetoencephalography (MEG) dipolar source localization problems because it does not require any functional differentiation. Like many other deterministic algorithms, however, it can be easily trapped in local optima when being applied to complex inverse problems with multiple simultaneous sources. In the present study, some modifications have been made to improve its capability of finding global optima. Those include (1) constructing an initial simplex based upon sensitivity of variables and (2) introducing a shaking technique based on polynomial interpolation. The efficiency of the proposed method was tested using analytical test functions and simulated MEG data. The simulation results demonstrate that the improved downhill simplex method can result in more reliable inverse solutions than the conventional one.

Thin Film Optical Parameters Determination by the Dynamical Modelling and Stochastic Optimization Method

We report on a new method of experimental data processing to obtain optical parameters of thin films. Dynamical modelling of the spectral reflectance can be performed interactively in a graphical environment by the genetic search in wide interval of parameter space and then refined by the genetic algorithm method, by the Nelder-Mead downhill simplex method or Marquardt-Levenberg method. Optical parameters of hydrogenated amorphous silicon (a-Si:H) thin film used in solar cell technology are determined by this new method. The spectral reflectance is a function of optical properties and the thickness of the film. Optical parameters found by our approach do not depend on the initial spectral reflectance estimation.

An Inspection-Maintenance Model for Systems With Multiple Competing Processes

In some applications, the failure rate of the system depends not only on the time, but also upon the status of the system, such as vibration level, efficiency, number of random shocks on the system, etc., which causes degradation. In this paper, we develop a generalized condition-based maintenance model subject to multiple competing failure processes including two degradation processes, and random shocks. An average long-run maintenance cost rate function is derived based on the expressions for the degradation paths & cumulative shock damage, which are measurable. A geometric sequence is employed to develop the inter-inspection sequence. Upon inspection, one needs to decide whether to perform a maintenance, such as preventive or corrective, or to do nothing. The preventive maintenance thresholds for degradation processes & inspection sequences are the decision variables of the proposed model. We also present an algorithm based on the Nelder-Mead downhill simplex method to calculate the optimum policy that minimizes the average long-run maintenance cost rate. Numerical examples are given to illustrate the results using the optimization algorithm.

A probabilistic search method for sequential tolerance control

Tolerance control is the set of practices used to ensure that parts are made within their specifications. Sequential Tolerance Control (STC) is the practice of using the results of earlier operations in the execution of a process plan to locate appropriate set points for later operations. In a previously developed set-point adjustment method, the largest sphere that fits into the remaining feasible space is moved to its extreme positions in the coordinate direction of the next operation, x k , and the next set point is positioned at the mean of the k th coordinates of the centres in these two positions. Scaling the feasible space in proportion to the variability of each operation has also been investigated. This paper presents a probabilistic search method for STC. This method finds the new set point by using the Nelder-Mead downhill simplex method to optimize an estimation of the expected process yield. Probabilistic search is found to be about as effective as sphere-fitting methods for normally distributed process deviations, but for skewed distributions, the search method is shown to yield better set points than previous methods.

The Graded Screen Technique for Measuring the Diffusion Coefficient of Radon Decay Products

ABSTRACT The main focus of this paper is the graded screen array (GSA) sampler, a specialized diffusion battery developed modified at our Laboratory to measure the size distribution of unattached 218Po, a naturally occurring “atomic aerosol,” implicated in causing lung cancer. The GSA, as used in this study, consists of four stainless steel wire screens with increasing mesh count, each positioned on a circular mount. For sampling, the four mounted screens are stacked into a holder and air is drawn through them in the direction of increasing mesh count. A backup filter is placed behind the fourth screen. After sampling, the alpha radioactivity on each screen, and on the backup filter, is measured by standard methods. Size distributions are retrieved from these data using both the Nelder-Mead downhill simplex method and the expectation-maximization algorithm. To illustrate the usefulness of the GSA sampler, size distributions from 218Po samples taken from the radon chamber at Clarkson University, Potsdam, N...