scholarly journals Contaminant source localization via Bayesian global optimization

2017 ◽  
Author(s):  
Guillaume Pirot ◽  
Tipaluck Krityakierne ◽  
David Ginsbourger ◽  
Philippe Renard
Keyword(s):  
Global Optimization ◽  
Process Model ◽  
Bayesian Optimization ◽  
Expected Improvement ◽  
Optimization Approach ◽  
Objective Functions ◽  
Contaminant Source ◽  
Multiple Minima ◽  
Improvement Criterion ◽  
Linear Objective Functions

Abstract. A Bayesian optimization approach to localize a contaminant source is proposed. The localization problem is illustrated with two 2D synthetic cases which display sharp transmissivity contrasts and specific connectivity patterns. These cases generate highly non-linear objective functions that present multiple local minima. A derivative-free global optimization algorithm relying on a Gaussian Process model and on the Expected Improvement criterion is used to efficiently localize the minimum of the objective function which identifies the contaminant source. In addition, the generated objective functions are made available as a benchmark to further allow the comparison of optimization algorithms on functions characterized by multiple minima and inspired by concrete field applications.

scholarly journals Contaminant source localization via Bayesian global optimization

2019 ◽  
Vol 23 (1) ◽  
pp. 351-369 ◽  
Author(s):  
Guillaume Pirot ◽  
Tipaluck Krityakierne ◽  
David Ginsbourger ◽  
Philippe Renard
Keyword(s):  
Global Optimization ◽  
Objective Function ◽  
Source Localization ◽  
Source Location ◽  
Bayesian Optimization ◽  
Test Cases ◽  
Local Minima ◽  
Objective Functions ◽  
Contaminant Source ◽  
Highly Nonlinear

Abstract. Contaminant source localization problems require efficient and robust methods that can account for geological heterogeneities and accommodate relatively small data sets of noisy observations. As realism commands hi-fidelity simulations, computation costs call for global optimization algorithms under parsimonious evaluation budgets. Bayesian optimization approaches are well adapted to such settings as they allow the exploration of parameter spaces in a principled way so as to iteratively locate the point(s) of global optimum while maintaining an approximation of the objective function with an instrumental quantification of prediction uncertainty. Here, we adapt a Bayesian optimization approach to localize a contaminant source in a discretized spatial domain. We thus demonstrate the potential of such a method for hydrogeological applications and also provide test cases for the optimization community. The localization problem is illustrated for cases where the geology is assumed to be perfectly known. Two 2-D synthetic cases that display sharp hydraulic conductivity contrasts and specific connectivity patterns are investigated. These cases generate highly nonlinear objective functions that present multiple local minima. A derivative-free global optimization algorithm relying on a Gaussian process model and on the expected improvement criterion is used to efficiently localize the point of minimum of the objective functions, which corresponds to the contaminant source location. Even though concentration measurements contain a significant level of proportional noise, the algorithm efficiently localizes the contaminant source location. The variations of the objective function are essentially driven by the geology, followed by the design of the monitoring well network. The data and scripts used to generate objective functions are shared to favor reproducible research. This contribution is important because the functions present multiple local minima and are inspired from a practical field application. Sharing these complex objective functions provides a source of test cases for global optimization benchmarks and should help with designing new and efficient methods to solve this type of problem.

A MO-BAYESIAN OPTIMIZATION APPROACH USING THE WEIGHTED TCHEBYCHEFF METHOD

2021 ◽  
pp. 1-30
Author(s):  
Arpan Biswas ◽  
Claudio Fuentes ◽  
Christopher Hoyle
Keyword(s):  
Process Model ◽  
Low Cost ◽  
Black Box ◽  
Training Data ◽  
Bayesian Optimization ◽  
Optimization Approach ◽  
Objective Functions ◽  
Multi Objective ◽  
Input Variables ◽  
Future Evaluation

Abstract Bayesian optimization (BO) is a low-cost global optimization tool for expensive black-box objective functions, where we learn from prior evaluated designs, update a posterior surrogate Gaussian process model, and select new designs for future evaluation using an acquisition function. This research focuses upon developing a BO model with multiple black-box objective functions. In the standard Multi-Objective optimization (MOO) problem, the weighted Tchebycheff method is efficiently used to find both convex and non-convex Pareto frontiers. This approach requires knowledge of utopia values before we start optimization. However, in the BO framework, since the functions are expensive to evaluate, it is very expensive to obtain the utopia values as a prior knowledge. Therefore, in this paper, we develop a MO-BO framework where we calibrate with multiple linear regression (MLR) models to estimate the utopia value for each objective as a function of design input variables; the models are updated iteratively with sampled training data from the proposed multi-objective BO. This iteratively estimated mean utopia values is used to formulate the weighted Tchebycheff multi-objective acquisition function. The proposed approach is implemented in optimizing thin tube geometries under constant loading of temperature and pressure, with minimizing the risk of creep-fatigue failure and design cost, along with risk-based and manufacturing constraints. Finally, the model accuracy with frequentist, Bayesian and without MLR-based calibration are compared to true Pareto solutions.

scholarly journals A Comparative Study of Infill Sampling Criteria for Computationally Expensive Constrained Optimization Problems

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1631
Author(s):  
Kittisak Chaiyotha ◽  
Tipaluck Krityakierne
Keyword(s):  
Constrained Optimization ◽  
Process Model ◽  
Structural Engineering ◽  
Optimization Problems ◽  
Bayesian Optimization ◽  
Expected Improvement ◽  
Constrained Optimization Problems ◽  
Computationally Expensive ◽  
Physical Phenomena ◽  
Infill Sampling

Engineering optimization problems often involve computationally expensive black-box simulations of underlying physical phenomena. This paper compares the performance of four constrained optimization algorithms relying on a Gaussian process model and an infill sampling criterion under the framework of Bayesian optimization. The four infill sampling criteria include expected feasible improvement (EFI), constrained expected improvement (CEI), stepwise uncertainty reduction (SUR), and augmented Lagrangian (AL). Numerical tests were rigorously performed on a benchmark set consisting of nine constrained optimization problems with features commonly found in engineering, as well as a constrained structural engineering design optimization problem. Based upon several measures including statistical analysis, our results suggest that, overall, the EFI and CEI algorithms are significantly more efficient and robust than the other two methods, in the sense of providing the most improvement within a very limited number of objective and constraint function evaluations, and also in the number of trials for which a feasible solution could be located.

An Approach to Bayesian Optimization in Optimizing Weighted Tchebycheff Multi-Objective Black-Box Functions

2020 ◽  
Author(s):  
Arpan Biswas ◽  
Claudio Fuentes ◽  
Christopher Hoyle
Keyword(s):  
Process Model ◽  
Large Scale ◽  
Low Cost ◽  
Pareto Frontier ◽  
Black Box ◽  
Training Data ◽  
Bayesian Optimization ◽  
Future Research ◽  
Objective Functions ◽  
Multi Objective

Abstract Bayesian optimization (BO) is a low-cost global optimization tool for expensive black-box objective functions, where we learn from prior evaluated designs, update a posterior surrogate Gaussian process model, and select new designs for future evaluation using an acquisition function. This research focuses upon developing a BO model with multiple black-box objective functions. In the standard multi-objective optimization problem, the weighted Tchebycheff method is efficiently used to find both convex and non-convex Pareto frontier. This approach requires knowledge of utopia values before we start optimization. However, in the BO framework, since the functions are expensive to evaluate, it is very expensive to obtain the utopia values as a priori knowledge. Therefore, in this paper, we develop a Multi-Objective Bayesian Optimization (MO-BO) framework where we calibrate with Multiple Linear Regression (MLR) models to estimate the utopia value for each objective as a function of design input variables; the models are updated iteratively with sampled training data from the proposed multi-objective BO. The iteratively estimated mean utopia values are used to formulate the weighted Tchebycheff multi-objective acquisition function. The proposed approach is implemented in optimizing a thin tube design under constant loading of temperature and pressure, with multiple objectives such as minimizing the risk of creep-fatigue failure and design cost along with risk-based and manufacturing constraints. Finally, the model accuracy with and without MLR-based calibration is compared to the true Pareto solutions. The results show potential broader impacts, future research directions for further improving the proposed MO-BO model, and potential extensions to the application of large-scale design problems.

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