Stochastic Optimization with High-Dimensional Uncertainties - dataset

Contact

Lennart Igel

Friedrich-Alexander-University Erlangen-Nürnberg (FAU), Department of Mathematics, Chair for Applied Mathematics (Continuous Optimization)

Email: lennart.g.igel@fau.de

Abstract

We consider a class of optimization problems, in which the objective function is, on the one hand, very expensive to evaluate and, on the other hand, exhibits discontinuous jumps over manifolds inside the domain, while remaining smooth in the rest of the domain. Problem classes of such type arise, for example, in mechanical contexts, where the objective depends on evaluating a state problem. Small variations of design parameters can lead to sudden changes in the system response, which in turn cause large jumps in the solution. Traditional methods to solve such problems can be categorized into sampling based, surrogate based and smoothing based strategies. Each of these approaches relies on some type of sampling of the domain, either locally or globally, to deal with the structure and location of the discontinuity. As a consequence, these methods scale poorly with increasing dimension sizes and are rendered unfeasible, due to the prohibitively expensive costs. 

We propose an Adaptive Smoothing Method for solving problems of this type, and avoiding scaling issues at very high dimension sizes. An approximation of a robust optimization problem is constructed by selectively smoothing the objective only in the region surrounding its discontinuities, exploiting the structure of the objective being continuously differentiable in the remaining domain. Additionally, smoothing is performed only along one-dimensional sets, which stand perpendicular to the discontinuous feature. This ensures that the relevant feature is regularized, while decoupling the dimension of smoothing from the dimension of the parameter space. In order to track both the distance and orientation of the discontinuity dynamically during the optimization process, the existence of a continuously differentiable level set function is assumed, which encodes the discontinuity set in its zero level set. 

Licence

Creative Commons Attribution 4.0 International

Context

Supplementary dataset to Dissertation "Stochastic Optimization with High-Dimensional Uncertainties " <DOI will come here > for the graphics and tables shown in the numerical examples. Data was generated by using the code provided in the github project https://github.com/lennartigel/StochOpt_Code and can also be found archived at https://doi.org/10.5281/zenodo.18806846 .

Content

All simulation results and data, which were used for figures and tables published in the dissertation are uploaded here. They are sorted by appearance of the numerical simulation within the dissertation. The code to read out the data and generate figures is given in the associated GitHub project. 

The data is structured into subfolders according to section to which they are relevant. Each folder includes subfolders for each numerical experiment of said section. Each of these subfolders includes a readme file that explains the filenaming scheme and data stored in it. It also explains what code was used to generate it. The folder structure is such that it corresponds to the folder structure of the github repository. 

Folder "Section 2" includes "Ex2.1", "Ex2.2" and "Ex2.3_dim_2_to_17" and "Ex2.3_dim_2_to_100000": 

• "Ex2.1" corresponds to first academic example shown in Section 2.5.1. It includes the generated landscape data, the example runs and the optimization runs of GSM and ASM.
• "Ex2.2" to the second academic example shown in Section 2.5.2.  It includes the generated landscape data, the example runs and the optimization runs of GSM and ASM. 
• "Ex2.3_dim_2_to_17" includes the data for comparing Adaptive Smoothing Method (ASM) to Global Smoothing Method (GSM) at dimensions 2 to 17 in Section 2.5.3. It includes the data for the optimization runs of GSM and ASM at those dimensions.
• "Ex2.3_dim_2_to_100000" the files for the optimization runs of the ASM for dimensions 2 to 100000 in Section 2.5.3. It includes the data for the optimization runs of ASM at those dimensions.

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Folder "Section 3" includes the files of "Ex3": 

• "Ex3" corresponds to the academic example for multiple discontintuity sets shown in Section 3.4. It includes the data for the optimization runs and example run of GSM and ASM and associated landscape data.  

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Folder "Section 4" includes "Ex4.1_spring" and "Ex4.2_QSAF": 

• "Ex4.1_spring"corresponds to the application example with the multiple spring model, based on [1], shown in Section 4.2.2 and 4.2.3. It includes the data for the optimization runs of GSM and ASM, the generated landscape data.
• "Ex4.2_QSAF" corresponds to the application of the ASM to a problem with the QSAF model by [2][3] as underlying state problem, as shown in Section 4.3.3 - 4.3.5.. It includes all the simulation, landscape and optimization data for the ASM application.

Funding

This research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 377472739/GRK 2423/1-2019 FRASCAL.

Software

The optimizations were performed in Python 3.9.12. using  NumPy version 1.22.2 and SciPy version 1.8.1.. The optimizations in Ex2.3_dim_2_to_17 in Section 2 and all optimizations in Section 4 were performed on the woody cluster. The author gratefully acknowledge the scientific support and HPC resources provided by the Erlangen National High Performance Computing Center (NHR@FAU) of the Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU). The hardware is funded by the German Research Foundation (DFG).

References

[1]  Singh, Sukhminder, Lukas Pflug, and Michael Stingl. "Material optimization to enhance delamination resistance of composite structures using viscous regularization." Computer Methods in Applied Mechanics and Engineering 382 (2021): 113881.

[2] Badal, Rufat, Manuel Friedrich, and Joscha Seutter. "Existence of quasi-static crack evolution for atomistic systems." Forces in Mechanics 9 (2022): 100138.

[3] Friedrich, Manuel, and Joscha Seutter. "Atomistic-to-continuum convergence for quasi-static crack growth in brittle materials." arXiv preprint arXiv:2402.02966 (2024).