Open-source optimization algorithms for optical design

Revisiting a classic lens design problem

Along with the increase in population and industry in many countries, the fuel oil demand also increases. Petroleum exploration on a large scale will accelerate the depletion of petroleum reserves. One alternative to meet fuel needs is the discovery of biodiesel which is renewable alternative energy. Synthesis biodiesel is carried out through an enzymatic reaction. In the enzymatic reaction model making biodiesel, there are parameters that must be estimated. The estimated parameters of the enzymatic reaction model will determine the success of the reaction. The parameter estimation of the enzymatic reaction model can be done using local optimization or global optimization algorithms, but the local optimization algorithm has a major disadvantage, the optimal value obtained is the local optimal value. Genetic algorithms are global optimization algorithms that are capable of working on high-dimensional problems. The success of genetic algorithms is determined by chromosome models, crossover operations, and mutation operations. The use of improper crossover operations often produces local optimum solutions. There are various types of crossover operation, each of which has weaknesses and advantages. This paper studies the parameters estimation of the enzymatic reaction model for biodiesel synthesis by using genetic algorithms with some crossover operation.

Geophysical inversion is usually carried out to quantitatively analyze the earth model and estimate its physical properties. Successful delineation of these properties such as layer boundaries, or other near-surface structures are crucial to understand the near-surface inhomogeneity. In this study, we focus on the use of joint inversion of seismic refraction and geoelectrical resistivity datasets using local and global optimization methods. The idea is to integrate the two optimization techniques to minimize the challenges faced by each algorithm when applied alone. This hybrid algorithm (local and global) is applied on synthetic data representing simple resistivity and velocity models. About 70% of the anomalies in both seismic and DC resistivity methods were reconstructed in terms of amplitude and geometry using the local optimization algorithm, while the global optimization algorithm shows improved results as it reconstructed about 80% of the amplitude and geometry of the anomalies in both geophysical methods. The result of the synthetic application shows that the hybrid algorithm provides promising outputs in terms of resolution, geometry and amplitude of the anomalies, and computation run time.

Dispersion curve inversion is a core component of Rayleigh wave surveys, primarily involving local linear optimization algorithms and global nonlinear optimization algorithms. Local linear optimization algorithms exhibit poor adaptability when addressing complex nonlinear features, making it challenging to identify the global optimum. The introduction of global nonlinear optimization algorithms enhances the accuracy of finding the global optimum and helps avoid the issue of converging to local minima. However, traditional global nonlinear optimization algorithms, such as genetic algorithms and particle swarm optimization (PSO), often demonstrate instability and slow convergence rates in the presence of noisy data. To address these issues, the snake optimization algorithm (SOA) is introduced first, which simulates the hunting paths of snakes to effectively search the parameter space and optimize the fitting of dispersion curves. Then, three theoretical models are constructed for inversion to verify the stability and noise resistance of the algorithm. Finally, SOA is applied to field data from a region in Jiangxi, China. The results indicate that, compared to PSO and whale optimization algorithm, the inversion results from SOA are closer to the actual borehole data, further demonstrating the algorithm's robustness against noise and its practicality in addressing engineering and geological applications.

The main geological structures in the Dammam Dome are defined by integrating geophysical measurements and applying new methodological approaches. Dammam Dome is characterized by a well-developed fracture/joints system; thus, high complexity of the subsurface is expected. Direct Current Resistivity (DCR) and Seismic Refraction (SR) geophysical survey aimed to map the Dammam Dome’s near-surface features. The geophysical data were acquired along two profiles in the northern part of Dammam Dome. To maximize the results from conducting DCR and SR measurements over a complex area, a combined local and global optimization algorithm was used to obtain high-resolution near-surface images in resistivity and velocity models. The local optimization technique involves individual and joint inversion of the DCR and SR data incorporating appropriate regularization parameters, while the global optimization uses single and multi-objective genetic algorithms in model parameter estimation. The combined algorithm uses the output from the local optimization method to define a search space for the global optimization algorithm. The results show that the local optimization produces satisfactory inverted models, and that the global optimization algorithm improves the local optimization results. The joint inversion and processing of the acquired data identified two major faults and a deformed zone with an almost N–S direction that corresponds with an outcrop were mapped in profile one, while profile two shows similar anomalies in both the resistivity and velocity models with the main E–W direction. This study not only demonstrates the capability of using the combined local and global optimization multi-objectives techniques to estimate model parameters of large datasets (i.e., 2D DCR and SR data), but also provides high-resolution subsurface images that can be used to study structural features of the Dammam Dome.

The purpose of this research was to compare the robustness and performance of a local and global optimization algorithm applied to the problem of fitting the parameters of a non-linear dose-response model utilized in the field of exercise physiology. Traditionally the parameters of dose-response models utilised in exercise physiology have been fit with non-linear least squares procedures in combination with local optimization algorithms. These algorithms have demonstrated limitations in their ability to converge on a globally optimal solution. This research purposes the use of an evolutionary computation based algorithm as an alternative method to fit a nonlinear dose-response model. The results of our comparison over 1000 experimental runs demonstrated the superior performance of the evolutionary computation based algorithm to consistently achieve a more consistent model fit and holdout evaluation performance in comparison to the local search algorithm. This initial research would suggest that global evolutionary computation based optimization algorithms are a fast and more robust alternative to local optimization algorithms when fitting the parameters of nonlinear dose-response models.

Many stellarator coil design problems are plagued by multiple minima, where the locally optimal coil sets can sometimes vary substantially in performance. As a result, solving a coil design problem a single time with a local optimization algorithm is usually insufficient and better optima likely do exist. To address this problem, we propose a global optimization algorithm for the design of stellarator coils and outline how to apply box constraints to the physical positions of the coils. The algorithm has a global exploration phase that searches for interesting regions of design space and is followed by three local optimization algorithms that search in these interesting regions (a ‘global-to-local’ approach). The first local algorithm (phase I), following the globalization phase, is based on near-axis expansions and finds stellarator coils that optimize for quasisymmetry in the neighbourhood of a magnetic axis. The second local algorithm (phase II) takes these coil sets and optimizes them for nested flux surfaces and quasisymmetry on a toroidal volume. The final local algorithm (phase III) polishes these configurations for an accurate approximation of quasisymmetry. Using our global algorithm, we study the trade-off between coil length, aspect ratio, rotational transform and quality of quasi-axisymmetry. The database of stellarators, which comprises approximately 200 000 coil sets, is available online and is called QUASR, for ‘quasi-symmetric stellarator repository’.

There are two types of optimization methods used in Water resources management: global optimization and local optimization. Global optimization algorithms, such as hydroPSO, MCMC, and genetic algorithms (GAs), have been used to optimize large-scale problems with many variables. Local optimization algorithms, such as PEST and UCODE, are best suited for fine-tuning and calibration of models to better fit specific observed data. Optimization techniques have become increasingly significant in water resource management over recent decades, aiming to efficiently allocate resources while minimizing environmental impacts. Two primary approaches dominate the field: global and local optimization. The chapter aims to compare global optimization techniques such as hydroPSO, MCMC, and others to local optimization techniques such as PEST, UCODE, and others in groundwater and surface water modeling. It is a flexible and powerful tool that can handle various types of hydrological models, such as MODFLOW, HYDRUS, and SWAT. The chapter will use different performance measures and case studies to provide a comprehensive comparison of these techniques in water resources management.

Calibration of computationally expensive watershed models is more feasible with algorithms that require fewer simulations. This paper compares the performance of seven global optimization algorithms on a 14-parameter and an 8-parameter watershed calibration problem. The optimization algorithms include Shuffled Complex Evolution (SCE), Differential Evolution (DE), an evolutionary algorithm that uses Radial Basis Function (RBF) approximation (ESGRBF), and four types of local optimization methods coupled with the Multi-Level Single Linkage (MLSL) multistart procedure. The four local optimization algorithms are: Sequential Quadratic Programming, which is a derivative-based method; Unconstrained Optimization by Quadratic Approximation (UOBYQA), which is a derivativefree trust-region method; Pattern Search; and Implicit Filtering. The results indicate that ESGRBF is the most effective algorithm on the two calibration problems, followed by Implicit Filtering coupled with the MLSL multistart approach. Hence, this study provides some promising alternatives to the currently most widely used methods in watershed calibration, which did not perform as well.

We developed an open source optical design program in C++ with a genetic and bisection optimization method. The genetic algorithm brings the system close to the global optimum using the survival of the fittest principle. After an abort criterion has been reached, a local optimization via the bisection method optimizes the system further until a new optimum is reached. We test our algorithm with different population numbers and search area sizes for the genetic selection and investigate how the calculation time or the accuracy of our algorithm behaves when we change those parameters. It is shown that our algorithm is able to optimize complex optical systems. The user can add different wavelengths, field positions, or a minimum surface gap. It is also possible to change the weight factor of specific parameters in the merit function. We explain the working principle of our algorithm and compare the results with commercial tools.

A hybrid Multi-Objective Optimization Algorithm based on the NSGA-II algorithm is presented and evaluated. The local optimization algorithm called SASS has been modified in order to be suitable for multi-objective optimization where the local optimization is intended towards non-dominated points. The modified local optimization algorithm has been incorporated into NSGA-II in order to improve performance.

A dynamic global and local combined particle swarm optimization algorithm

Today's optical design of imaging systems relies mostly on efficient ray tracing and (local or global) optimization algorithms. Such a traditional 'step-and-repeat' approach to optical design typically requires considerable experience, intuition, and sometimes trial-and-error guesswork. Such a time-consuming design process applies especially, but not only, to freeform optical systems. In particular, the identification of a suitable initial design to then adapt and further optimize has often proven to be a laborious process. We present our developed 'first time right' design method that allows a highly systematic generation and evaluation of directly calculated imaging optics design solutions and thus enables a rigorous, extensive, and real-time evaluation in solution space. The method is based on differential equations derived from Fermat's principle that can be solved effectively by using a power series method. This approach allows calculating all optical surface coefficients that ensure minimal image blurring for each individual order of aberrations. Such directly calculated optical design solutions can be readily used as starting point for further and final optimization. We demonstrate the deterministic and holistic nature of our method and the streamlined design process for various real-world examples ranging from spherical lens designs to freeform imaging systems. The method allows calculating all optical surface coefficients that ensure minimal image blurring for each individual order of aberrations. We demonstrate the systematic, deterministic, scalable, and holistic character of our method for various design examples ranging from spherical lens designs to freeform imaging systems.

Nowadays, sophisticated ray-tracing software packages are used for the design of optical systems, including local and global optimization algorithms. Nevertheless, the design process is still time-consuming with many manual steps, and it can take days or even weeks until an optical design is finished. To address this shortcoming, artificial intelligence, especially reinforcement learning, is employed to support the optical designer. In this work, different use cases are presented, in which reinforcement learning agents are trained to optimize a lens system. Besides the possibility of bending lenses to reduce spherical aberration, the movement of lenses to optimize the lens positions for a varifocal lens system is shown. Finally, the optimization of lens surface curvatures and distances between lenses are analyzed. For a predefined Cooke Triplet, an agent can choose the curvature of the different surfaces as optimization parameters. The chosen surfaces and the distances between the lenses will then be optimized with a least-squares optimizer1. It is shown, that for a Cooke Triplet, setting all surfaces as variables is a good suggestion for most systems if the runtime is not an issue. Taking the runtime into account, the selected number of variable surfaces decreases. For optical systems with a large number of degrees of freedom an intelligent selection of optimization variables can probably be a powerful tool for an efficient and time-saving optimization.

본 논문은 사장교의 주요부재인 경사 케이블에 대해 기존 장력추정기법들을 비교하고, FOA(Fast Optimization Algorithm)을 이용한 경사케이블의 장력추정기법을 소개한다. FOA는 기존의 LOA(Local Optimization Algorithm)의 단점과 GOA(Global Optimization Algorithm)의 단점을 극복한 기법으로 LOA의 국부수렴과 GOA의 조기 수렴의 문제점을 해결하였다. 기존 장력추정기법으로는 기계적인 방법과 고유진동수 기반의 동적방법으로 구분되며 최근 도입되고 있는 유한요소 모델과 시스템인식 기법을 이용한 방법이 제안되었다. 새그가 큰 케이블을 제외하고 기존 수학적 모델을 이용한 방법의 경우, 현이론과 Triantafyllou와 Gringfogel의 제안식은 5%이내의 오차를, 빔모델을 이용한 선형회귀식은 1%의 오차를 Shinke등의 제안식은 4%이내의 장력추정 오차가 확인되었고, 민감도방정식과 차분진화법(DE: Differential Evolutionary) 경우 0.5%이내의 오차를 GAs는 2%이내, <TEX>${\mu}GA$</TEX>는 3%이내 hGA는 5%이내의 오차를 보이며 장력이 추정되었다. 본 논문에서는 제안된 FOA를 이용하여 해석시간의 단축과 기존 기법들의 단점을 해결하였고, 모든 수치실험에서 0%에 가까운 오차를 보이며 정확한 장력이 추정되었다. This paper introduces the tension estimating technique using FOA(Fast Optimization Algorithm) for a major member of cable stayed bridge and compares the existing techniques. FOA is overcoming the disadvantages of LOA(Local Optimization) that is the local convergence and GOA(Global Optimization Algorithm) that is the problem of premature convergence. The existing methods are divided into the mechanical methods and the dynamics methods based on the natural frequency and they have been recently proposed the SI(System Identification) methods using the FE model and identification technique. Excepting that the sagged cable have a large sag, in the cases of conventional method using a mathematical model, the taut string theory and Triantafyllou & Gringfogel's proposed equation have the error less than 5%, the beam model using the linear regression equation hase the error less than 1%, Shinke et al.'s proposed equation is certificated the tension estimation errors less than 4%. SUA(Senstivity Updating Algorithm), DE(Differential Evolutionary) have the estimated tension value that have errors less than the 0.5%. GAs(Genetic Algorithm) has the errors less than 2%, <TEX>${\mu}GA$</TEX> less than 3%, hGA less than 5%. It is that the proposed technique using proposed FOA has resolved the weak of existing techniques and shorten analysis time. It seems that errors are less than 0% and the proposed technique has been estimated the exact tensions.