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Open AccessArticle
Statistics of Global Stochastic Optimisation: How Many Steps to Hit the Target?
by
Godehard Sutmann
Godehard Sutmann
Godehard Sutmann has studied theoretical physics and has been working as a computational scientist a [...]
Godehard Sutmann has studied theoretical physics and has been working as a computational scientist in the field of high performance parallel computing, applied mathematics, and method development. He has studied in Göttingen, Montpellier, and Heidelberg and received his PhD in 1999 at the Technical University of Munich. He has spent a postdoc at the University of Trento and had research stays at NIH Washington, Santa Barbara, and Mexico City. One of his main interests is in computational methods for complex many-particle physics and statistical physics. He is working at the Jülich Supercomputing Centre at Forschungszentrum Jülich, Germany, where he is the leader of the “Data and Simulation Laboratory Molecular Systems”. He is the director of the CECAM Node Jülich and professor for High Performance Computing at the Ruhr-University Bochum, where he is a member of the Interdisciplinary Center for Advanced Materials Simulation (ICAMS).
1,2
1
Jülich Supercomputing Centre (JSC), Forschungszentrum Jülich (FZJ), D-52425 Jülich, Germany
2
Interdisciplinary Center for Advanced Materials Simulation (ICAMS), Ruhr-University Bochum, D-44801 Bochum, Germany
Mathematics 2025, 13(20), 3269; https://doi.org/10.3390/math13203269 (registering DOI)
Submission received: 28 August 2025
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Revised: 30 September 2025
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Accepted: 7 October 2025
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Published: 13 October 2025
Abstract
Random walks are considered in a one-dimensional monotonously decreasing energy landscape. To reach the minimum within a region , a number of downhill steps have to be performed. A stochastic model is proposed which captures this random downhill walk and to make a prediction for the average number of steps, which are needed to hit the target. Explicit expressions in terms of a recurrence relation are derived for the density distribution of a downhill random walk as well as probability distribution functions to hit a target region within a given number of steps. For the case of stochastic optimisation, the number of rejected steps between two successive downhill steps is also derived, providing a measure for the average total number of trial steps. Analytical results are obtained for generalised random processes with underlying polynomial distribution functions. Finally the more general case of non-monotonously decreasing energy landscapes is considered for which results of the monotonous case are transferred by applying the technique of decreasing rearrangement. It is shown that the global stochastic optimisation can be fully described analytically, which is verified by numerical experiments for a number of different distribution and objective functions. Finally we discuss the transition to higher dimensional objective functions and discuss the change in computational complexity for the stochastic process.
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MDPI and ACS Style
Sutmann, G.
Statistics of Global Stochastic Optimisation: How Many Steps to Hit the Target? Mathematics 2025, 13, 3269.
https://doi.org/10.3390/math13203269
AMA Style
Sutmann G.
Statistics of Global Stochastic Optimisation: How Many Steps to Hit the Target? Mathematics. 2025; 13(20):3269.
https://doi.org/10.3390/math13203269
Chicago/Turabian Style
Sutmann, Godehard.
2025. "Statistics of Global Stochastic Optimisation: How Many Steps to Hit the Target?" Mathematics 13, no. 20: 3269.
https://doi.org/10.3390/math13203269
APA Style
Sutmann, G.
(2025). Statistics of Global Stochastic Optimisation: How Many Steps to Hit the Target? Mathematics, 13(20), 3269.
https://doi.org/10.3390/math13203269
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