{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,14]],"date-time":"2026-07-14T21:58:20Z","timestamp":1784066300622,"version":"3.55.0"},"reference-count":32,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2018,3,16]],"date-time":"2018-03-16T00:00:00Z","timestamp":1521158400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The simulation of complex physics models may lead to enormous computer running times. Since the simulations are expensive it is necessary to exploit the computational budget in the best possible manner. If for a few input parameter settings an output data set has been acquired, one could be interested in taking these data as a basis for finding an extremum and possibly an input parameter set for further computer simulations to determine it\u2014a task which belongs to the realm of global optimization. Within the Bayesian framework we utilize Gaussian processes for the creation of a surrogate model function adjusted self-consistently via hyperparameters to represent the data. Although the probability distribution of the hyperparameters may be widely spread over phase space, we make the assumption that only the use of their expectation values is sufficient. While this shortcut facilitates a quickly accessible surrogate, it is somewhat justified by the fact that we are not interested in a full representation of the model by the surrogate but to reveal its maximum. To accomplish this the surrogate is fed to a utility function whose extremum determines the new parameter set for the next data point to obtain. Moreover, we propose to alternate between two utility functions\u2014expected improvement and maximum variance\u2014in order to avoid the drawbacks of each. Subsequent data points are drawn from the model function until the procedure either remains in the points found or the surrogate model does not change with the iteration. The procedure is applied to mock data in one and two dimensions in order to demonstrate proof of principle of the proposed approach.<\/jats:p>","DOI":"10.3390\/e20030201","type":"journal-article","created":{"date-parts":[[2018,3,20]],"date-time":"2018-03-20T06:57:11Z","timestamp":1521529031000},"page":"201","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":25,"title":["Global Optimization Employing Gaussian Process-Based Bayesian Surrogates"],"prefix":"10.3390","volume":"20","author":[{"given":"Roland","family":"Preuss","sequence":"first","affiliation":[{"name":"Max-Planck-Institute for Plasma Physics, EURATOM Association, 85748 Garching, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Udo","family":"Von Toussaint","sequence":"additional","affiliation":[{"name":"Max-Planck-Institute for Plasma Physics, EURATOM Association, 85748 Garching, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2018,3,16]]},"reference":[{"key":"ref_1","first-page":"409","article-title":"Design and Analysis of Computer Experiments","volume":"4","author":"Sacks","year":"1989","journal-title":"Stat. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1023\/A:1012771025575","article-title":"A Taxonomy of Global Optimization Methods Based on Response Surfaces","volume":"21","author":"Jones","year":"2001","journal-title":"J. Glob. Optim."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Barber, D. (2012). Bayesian Reasoning and Machine Learning, Cambridge University Press.","DOI":"10.1017\/CBO9780511804779"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Bishop, C. (1996). Neural Networks for Pattern Recognition, Oxford University Press.","DOI":"10.1201\/9781420050646.ptb6"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1071","DOI":"10.1016\/0893-6080(95)00137-9","article-title":"Neural Network Exploration Using Optimal Experiment Design","volume":"9","author":"Cohn","year":"1996","journal-title":"Neural Netw."},{"key":"ref_6","unstructured":"MacKay, D.J.C. (2013). Bayesian Approach to Global Optimization: Theory and Applications, Kluwer Academic."},{"key":"ref_7","unstructured":"Neal, R.M. (1997). Monte Carlo Implementation of Gaussian Process Models for Bayesian Regression and Classification. Technical Report 9702, University of Toronto."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Seo, S., Wallat, M., Graepel, T., and Obermayer, K. (2000, January 24\u201327). Gaussian process regression: active data selection and test point rejection. Proceedings of the International Joint Conference on Neural Networks, Como, Italy.","DOI":"10.1007\/978-3-642-59802-9_4"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1198\/TECH.2009.0015","article-title":"Adaptive Design and Analysis of Supercomputer Experiments","volume":"51","author":"Gramacy","year":"2009","journal-title":"Technometrics"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Mockus, J. (1989). Bayesian Approach to Global Optimization, Springer.","DOI":"10.1007\/978-94-009-0909-0"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1023\/A:1008294716304","article-title":"Bayesian Algorithms for One-Dimensional Global Optimization","volume":"10","author":"Locatelli","year":"1997","journal-title":"J. Glob. Optim."},{"key":"ref_12","unstructured":"Lafferty, J.D., Williams, C.K.I., Shawe-Taylor, J., Zemel, R.S., and Culotta, A. (2010). Batch Bayesian Optimization via Simulation Matching. Advances in Neural Information Processing Systems 23, Curran Associates, Inc."},{"key":"ref_13","unstructured":"Azimi, J., Jalali, A., and Fern, X. (July, January 26). Hybrid Batch Bayesian Optimization. Proceedings of the 29th International Conference on Machine Learning, Edinburgh, UK."},{"key":"ref_14","first-page":"790","article-title":"GLASSES: Relieving The Myopia of Bayesian Optimisation","volume":"51","author":"Gonzalez","year":"2016","journal-title":"J. Mach. Learn. Res."},{"key":"ref_15","first-page":"119","article-title":"A Statistical Approach to Some Basic Mine Valuation Problems on the Witwatersrand","volume":"52","author":"Krige","year":"1951","journal-title":"J. Chem. Metal. Min. Soc. S. Afr."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1246","DOI":"10.2113\/gsecongeo.58.8.1246","article-title":"Principles of geostatistics","volume":"58","author":"Matheron","year":"1963","journal-title":"Econ. Geol."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1007\/978-1-4471-0657-9_2","article-title":"Space and space-time modeling using process convolutions","volume":"Volume 3754","author":"Higdon","year":"2002","journal-title":"Quantitative Methods for Current Environmental Issues"},{"key":"ref_18","first-page":"217","article-title":"Dependent Gaussian processes","volume":"Volume 17","author":"Boyle","year":"2005","journal-title":"Advances in Neural Information Processing Systems"},{"key":"ref_19","unstructured":"Alvarez, M., Luengo, D., and Lawrence, N. (2009, January 16\u201318). Latent force models. Proceedings of the 12th International Conference on Artificial Intelligence and Statistics (AISTATS), Clearwater Beach, Florida."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2693","DOI":"10.1109\/TPAMI.2013.86","article-title":"Linear Latent Force Models Using Gaussian Processes","volume":"35","author":"Alvarez","year":"2013","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_21","first-page":"118","article-title":"Prediction of Plasma Simulation Data with the Gaussian Process Method","volume":"Volume 1636","author":"Niven","year":"2014","journal-title":"Bayesian Inference and Maximum Entropy Methods in Science and Engineering"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Rasmussen, C., and Williams, C. (2006). Gaussian Processes for Machine Learning, MIT Press.","DOI":"10.7551\/mitpress\/3206.001.0001"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Garnett, R., Osborne, M.A., and Roberts, S.J. (2010, January 12\u201316). Bayesian Optimization for Sensor Set Selection. Proceedings of the 9th ACM\/IEEE International Conference on Information Processing in Sensor Networks, Stockholm, Sweden.","DOI":"10.1145\/1791212.1791238"},{"key":"ref_24","unstructured":"Osborne, M.A., Garnett, R., and Roberts, S.J. (2018, March 15). Gaussian Processes for Global Optimization. Available online: http:\/\/www.robots.ox.ac.uk\/~parg\/pubs\/OsborneGarnettRobertsGPGO.pdf."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Gilks, W.R., Richardson, S., and Spiegelhalter, D.J. (1996). Markov Chain Monte Carlo in Practice, Chapman & Hall.","DOI":"10.1201\/b14835"},{"key":"ref_26","first-page":"11","article-title":"Global Versus Local Search in Constrained Optimization of Computer Models","volume":"Volume 34","author":"Flournoy","year":"1998","journal-title":"New Developments and Applications in Experimental Design"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"T\u00f6rn, A., and Zilinskas, A. (1989). Lecture Notes in Computer Science, Springer. Global Optimization.","DOI":"10.1007\/3-540-50871-6"},{"key":"ref_28","unstructured":"Press, W.H., Teukolsky, S.A., Vetterling, W.T., and Flannery, B.P. (2007). Numerical Recipes: The Art of Scientific Computing, Cambridge University Press. [3rd ed.]."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Preuss, R., and von Toussaint, U. (2017, January 9\u201314). Optimization employing Gaussian process-based surrogates. Proceedings of the 37th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Jarinu\/SP, Brazil.","DOI":"10.1007\/978-3-319-91143-4_26"},{"key":"ref_30","unstructured":"Dixon, L.C.W., and Szego, G.P. (1978). The global optimisation problem: An introduction. Towards Global Optimisation 2, North Holland."},{"key":"ref_31","first-page":"3227","article-title":"Robust Gaussian Process Regression with a Student-t Likelihood","volume":"12","author":"Vanhatalo","year":"2011","journal-title":"J. Mach. Learn. Res."},{"key":"ref_32","first-page":"877","article-title":"Studen-t Processes as Alternatives to Gaussian Processes","volume":"Volume 33","author":"Shah","year":"2014","journal-title":"Proceedings of the 17th International Conference on Artificial Intelligence and Statistics"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/3\/201\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T14:57:25Z","timestamp":1760194645000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/3\/201"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,3,16]]},"references-count":32,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2018,3]]}},"alternative-id":["e20030201"],"URL":"https:\/\/doi.org\/10.3390\/e20030201","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,3,16]]}}}