Optimization of Technical Systems

Qualification aims

This module equips students with the ability to apply numerical methods, develop mathematical models, and utilize data-driven optimization techniques to solve real-world problems, analyze large datasets, and enhance technical systems' performance through the use of advanced computational tools and optimization software.

Students can

• apply numerical methods
• develop mathematical models for technical systems
• data-driven optimization techniques

by

• translating real-world problems into computable form
• analyzing large datasets
• applying optimization algorithms (stochastic gradient descent)
• understanding bias in data
• utilizing programming and computational tools
• using “state of the art” optimization software and optimization algorithms
• using tools for visualizing model states

to

• solve optimization problems in technical systems, improving their efficiency and performance
• identify optimal solutions using various constraints and parameters
• predict system behavior and improve decision-making processes

Module Content

Numerical Methods

• Matrices
• Differences, Derivatives, and Boundary Conditions
• Inverses and Delta Functions
• Eigenvalues and Eigenvectors
• Positive Definite Matrices
• Numerical Linear Algebra: LU, QR, SVD
• Numerical integration of standard differential equation systems (linear, non-linear, formal procedures (Runge-Kutta etc.)
• Boundary value problems
• Differential Equations of Equilibrium
• Cubic Splines and Fourth Order Equations
• Gradient and Divergence
• Laplace's Equation
• Finite Differences and Fast Poisson Solvers
• The Finite Element Method
• Stochastic simulation
• Design and organisation of a Monte Carlo simulator

Optimization

• Optimization criteria
• Optimization basics (calculus of variation, Euler formula, Hamilton formula, maximum principle, etc.)
• Linear Programming (LP)
• Nonlinear Programming (NLP)
• Quadratic Programming (QP)
• Integer Programming (IP)
• Direct (extrapolation-free) searching procedures (pattern search)
• Stochastic procedures (simulated annealing, evolutionary algorithms)
• Application of optimization procedures to practical problems

Data-driven Optimization

• Data from real-world problems (industry, economy, science)
• Data preparation
• Linear regression, logistic regression
• Hypothesis testing
• Classification, Linear discriminant analysis
• Tree-based methods
• Sequential parameter optimization (SPO)
• Model selection
• Treatment of missing values and huge data sets
• Data visualization
• Data mining, CRISP-DM Process
• Learning, especially advanced modelling techniques: Bootstrap, bagging, meta learner (e.g. random forests), empirical learning problems
• Evaluation of modelling results (e.g., error measures, overfitting, cross valida-tion, precision and recall)

Bibliography

• Stoer, J., et.al.: Introduction to numerical analysis. ISBN 0-387-95452-X
• Kincaid, D., et.al.: Numerical analysis. ISBN 0-534-38905-8
• Gill, P.E., Murray, W., Wright, M.: Practical Optimization. Academic Press, London, 1989
• Edgar, T.F., Himmelblau, D.M.: Optimization of chemical processes. Mc Graw-Hill, 2001
• Gekeler, E.W.: Mathematical Methods for Mechanics with MATLAB Experiments. Springer, Berlin 2008
• Neumann, K. und Morlock, M: Operations Research. 2. Aufl. Hanser, München 2002
• Bartz-Beielstein, T.: Experimental Research in Evolutionary Computation. 1.Aufl., Springer, Berlin 2006
• Markon, S., Kita, H., Kise, H., Bartz-Beielstein, T.: Modern Supervisory and Optimal Control with Applications in the Control of Passenger Traffic Systems in Buildings. Springer, Berlin, Heidelberg, New York, 2006
• Witten, I. H., Frank, E.: Data Mining, Hanser, 2nd ed., 2005
• Hastie, T., Tibshirani, R., Friedeman, J.: The Elements of Statistical Learning. Springer, 2001
• James, G., Witten, D., Hastie, T., and Tibshirani, R.: An Introduction to Statistical Learning with Applications in R. Springer, 4th edition, 2014
• Law, A.M., Kelton, W.D.: Simulation Modeling and Analysis. McGraw-Hill, Boston, 2000
• Bartz-Beielstein, T. et al.: Experimental Methods for the Analysis of Optimization Algorithms. Springer, 2010
• Williams, G.: Data Mining with Rattle and R: The Art of Excavating Data for Knowledge Discovery. Springer, New York, 2011