Numerical Investigation of Some Potential Problems of Univariate Minimization Methods

1979 ◽  
Vol 101 (4) ◽  
pp. 663-666 ◽  
Author(s):  
G. E. Johnson ◽  
M. A. Townsend
Keyword(s):  
Golden Section ◽  
Step Length ◽  
Companion Paper ◽  
Test Functions ◽  
Convergence Criteria ◽  
One Dimensional ◽  
Exact Arithmetic ◽  
Golden Section Search ◽  
Minimization Methods ◽  
Programming Algorithms

Many nonlinear programming algorithms employ a univariate subprocedure to determine the step length at each multivariate iteration. In this note, a popular polynomial approximation-interpolation univariate algorithm (DSC-P) is compared to two versions of the golden section search. One-dimensional test functions which model the behavior of barrier and penalty functions are used for the comparison. In general, the polynominal method indicates convergence in fewer function evaluations than the golden section search. However, it is significantly less reliable. Tight convergence criteria do not necessarily lead to accurate results with the polynomial-based univariate strategy. A companion paper provides a theoretical basis for the observations and gives conditions underwhich DSC-P will fail—even for strictly convex, unimodal functions and exact arithmetic.

scholarly journals Generalized Fibonacci Search Method in One-Dimensional Unconstrained Non-Linear Optimization

2021 ◽  
Vol 29 (2) ◽  
Author(s):  
Chin Yoon Chong ◽  
Soo Kar Leow ◽  
Hong Seng Sim
Keyword(s):  
Golden Section ◽  
Linear Optimization ◽  
Fibonacci Numbers ◽  
Search Method ◽  
Initial Ratio ◽  
Microsoft Excel ◽  
One Dimensional ◽  
Golden Section Search ◽  
Non Linear ◽  
Number Of Iterations

In this paper, we develop a generalized Fibonacci search method for one-dimensional unconstrained non-linear optimization of unimodal functions. This method uses the idea of the “ratio length of 1” from the golden section search. Our method takes successive lower Fibonacci numbers as the initial ratio and does not specify beforehand, the number of iterations to be used. We evaluated the method using Microsoft Excel with nine one-dimensional benchmark functions. We found that our generalized Fibonacci search method out-performed the golden section and other Fibonacci-type search methods such as the Fibonacci, Lucas and Pell approaches.

Quantification of thermal energy generation in annular hyperbolic porous-finned heat sinks using inverse optimization

2021 ◽  
pp. 095440892110243
Author(s):  
Sagnik Pal ◽  
Ranjan Das
Keyword(s):  
Heat Transfer ◽  
Surface Temperature ◽  
Heat Generation ◽  
Thermal Energy ◽  
Inverse Method ◽  
Golden Section ◽  
Energy Generation ◽  
Heat Sinks ◽  
Search Technique ◽  
Golden Section Search

The present paper introduces an accurate numerical procedure to assess the internal thermal energy generation in an annular porous-finned heat sink from the sole assessment of surface temperature profile using the golden section search technique. All possible heat transfer modes and temperature dependence of all thermal parameters are accounted for in the present nonlinear model. At first, the direct problem is numerically solved using the Runge–Kutta method, whereas for predicting the prevailing heat generation within a given generalized fin domain an inverse method is used with the aid of the golden section search technique. After simplifications, the proposed scheme is credibly verified with other methodologies reported in the existing literature. Numerical predictions are performed under different levels of Gaussian noise from which accurate reconstructions are observed for measurement error up to 20%. The sensitivity study deciphers that the surface temperature field in itself is a strong function of the surface porosity, and the same is controlled through a joint trade-off among heat generation and other thermo-geometrical parameters. The present results acquired from the golden section search technique-assisted inverse method are proposed to be suitable for designing effective and robust porous fin heat sinks in order to deliver safe and enhanced heat transfer along with significant weight reduction with respect to the conventionally used systems. The present inverse estimation technique is proposed to be robust as it can be easily tailored to analyse all possible geometries manufactured from any material in a more accurate manner by taking into account all feasible heat transfer modes.

Comparison of iterative desmearing procedures for one-dimensional small-angle scattering data

2011 ◽  
Vol 44 (1) ◽  
pp. 32-42 ◽  
Author(s):  
Thomas Vad ◽  
Wiebke F. C. Sager
Keyword(s):  
Small Angle ◽  
Incident Beam ◽  
Scattering Data ◽  
Data Sets ◽  
Convergence Criteria ◽  
Small Momentum Transfer ◽  
One Dimensional ◽  
Noise Filter ◽  
Large Numbers ◽  
Angle Scattering

Two simple iterative desmearing procedures – the Lake algorithm and the Van Cittert method – have been investigated by introducing different convergence criteria using both synthetic and experimental small-angle neutron scattering data. Implementing appropriate convergence criteria resulted in stable and reliable solutions in correcting resolution errors originating from instrumental smearing,i.e.finite collimation and polychromaticity of the incident beam. Deviations at small momentum transfer for concentrated ensembles of spheres encountered in earlier studies are not observed. Amplification of statistical errors can be reduced by applying a noise filter after desmearing. In most cases investigated, the modified Lake algorithm yields better results with a significantly smaller number of iterations and is, therefore, suitable for automated desmearing of large numbers of data sets.

scholarly journals The faint young Sun problem revisited with a 3-D climate–carbon model – Part 1

2014 ◽  
Vol 10 (2) ◽  
pp. 697-713 ◽  
Author(s):  
G. Le Hir ◽  
Y. Teitler ◽  
F. Fluteau ◽  
Y. Donnadieu ◽  
P. Philippot
Keyword(s):  
Radiative Forcing ◽  
General Circulation ◽  
Freezing Point ◽  
Climatic Factors ◽  
General Circulation Models ◽  
Companion Paper ◽  
One Dimensional ◽  
Carbon Dioxide Concentrations ◽  
Wide Range ◽  
High Concentrations

Abstract. During the Archaean, the Sun's luminosity was 18 to 25% lower than the present day. One-dimensional radiative convective models (RCM) generally infer that high concentrations of greenhouse gases (CO2, CH4) are required to prevent the early Earth's surface temperature from dropping below the freezing point of liquid water and satisfying the faint young Sun paradox (FYSP, an Earth temperature at least as warm as today). Using a one-dimensional (1-D) model, it was proposed in 2010 that the association of a reduced albedo and less reflective clouds may have been responsible for the maintenance of a warm climate during the Archaean without requiring high concentrations of atmospheric CO2 (pCO2). More recently, 3-D climate simulations have been performed using atmospheric general circulation models (AGCM) and Earth system models of intermediate complexity (EMIC). These studies were able to solve the FYSP through a large range of carbon dioxide concentrations, from 0.6 bar with an EMIC to several millibars with AGCMs. To better understand this wide range in pCO2, we investigated the early Earth climate using an atmospheric GCM coupled to a slab ocean. Our simulations include the ice-albedo feedback and specific Archaean climatic factors such as a faster Earth rotation rate, high atmospheric concentrations of CO2 and/or CH4, a reduced continental surface, a saltier ocean, and different cloudiness. We estimated full glaciation thresholds for the early Archaean and quantified positive radiative forcing required to solve the FYSP. We also demonstrated why RCM and EMIC tend to overestimate greenhouse gas concentrations required to avoid full glaciations or solve the FYSP. Carbon cycle–climate interplays and conditions for sustaining pCO2 will be discussed in a companion paper.

scholarly journals GOLDEN EXPONENTIAL SMOOTHING: A SELF-ADJUSTED METHOD FOR IDENTIFYING OPTIMUM ALPHA

2020 ◽  
Vol 5 (2) ◽  
pp. 587
Author(s):  
Fong Yeng Foo ◽  
Azrina Suhaimi ◽  
Soo Kum Yoke
Keyword(s):  
Golden Section ◽  
Exponential Smoothing ◽  
Series Data ◽  
Training Process ◽  
Times Series ◽  
Golden Section Search ◽  
Training Stage ◽  
Forecasting Method ◽  
Double Exponential Smoothing ◽  
Accuracy Of Prediction

The conventional double exponential smoothing is a forecasting method that troubles the forecaster with a tremendous choice of its parameter, alpha. The choice of alpha would greatly influence the accuracy of prediction. In this paper, an integrated forecasting method named Golden Exponential Smoothing (GES) was proposed to solve the problem. The conventional method was reformed and interposed with golden section search such that an optimum alpha which minimizes the errors of forecasting could be identified in the algorithm training process.  Numerical simulations of four sets of times series data were employed to test the efficiency of GES model. The findings show that the GES model was self-adjusted according to the situation and converged fast in the algorithm training process. The optimum alpha, which was identified from the algorithm training stage, demonstrated good performance in the stage of Model Testing and Usage.

scholarly journals Idealized walking and running gaits minimize work

2007 ◽  
Vol 463 (2086) ◽  
pp. 2429-2446 ◽  
Author(s):  
Manoj Srinivasan ◽  
Andy Ruina
Keyword(s):  
Numerical Optimization ◽  
Phase Plane ◽  
Step Length ◽  
Metabolic Energy ◽  
One Dimensional ◽  
High Speeds ◽  
Locomotor Patterns ◽  
Positive Work ◽  
Selection Of ◽  
Trajectory Problem

Even though human legs allow a wide repertoire of movements, when people travel by foot they mostly use one of two locomotor patterns, namely, walking and running. The selection of these two gaits from the plethora of options might be because walking and running require less metabolic energy than other more unusual gaits. We addressed this possibility previously using numerical optimization of a minimal mathematical model of a biped. We had found that, for a given step-length, the two classical descriptions of walking and running, ‘inverted pendulum walking’ and ‘impulsive running’, do indeed minimize the amount of positive work required at low and high speeds respectively. Here, for the case of small step-lengths, we establish the previous results analytically. First, we simplify the two-dimensional particle trajectory problem to a one-dimensional ‘elevator’ problem. Then we use elementary geometric arguments on the resulting phase plane to show optimality of the two gaits: walking at low speeds and running at high speeds.

Sign in / Sign up

Export Citation Format

Share Document