Misfit landforms imposed by ill-conditioned inverse parametric problems

• Autorzy: Łoś M. , Smołka M. , Schaefer R. , Sawicki J.

Identyfikatory

DOI

10.7494/csci.2018.19.2.2781

• Języki publikacji: EN

Abstrakty

EN

In this paper, we put forward a new topological taxonomy that allows us to distinguish and separate multiple solutions to ill-conditioned parametric inverse problems appearing in engineering, geophysics, medicine, etc. This taxonomy distinguishes the areas of insensitivity to parameters called the landforms of the misfit landscape, be it around minima (lowlands), maxima (uplands), or stationary points (shelves). We have proven their important separability and completeness conditions. In particular, lowlands, uplands, and shelves are pairwise disjoint, and there are no other subsets of the positive measure in the admissible domain on which the misfit function takes a constant value. The topological taxonomy is related to the second, “local” one, which characterizes the types of ill-conditioning of the particular solutions. We hope that the proposed results will be helpful for a better and more precise formulation of ill-conditioned inverse problems and for selecting and profiling complex optimization strategies used in solving these problems.

Słowa kluczowe

EN

taxonomy of ill-conditioned problems; ill-posed global optimization problems; fitness landscapes

• Wydawca: Wydawnictwa AGH
• Czasopismo: Computer Science
• Rocznik: 2018
• Tom: Vol. 19 (2)
• Strony: 157--178
• Opis fizyczny: Bibliogr. 28 poz., rys.

Twórcy

autor

Łoś M.

• AGH University of Science and Technology, Krakow, Poland

autor

Smołka M.

• AGH University of Science and Technology, Krakow, Poland

autor

Schaefer R.

• AGH University of Science and Technology, Krakow, Poland

autor

Sawicki J.

• AGH University of Science and Technology, Krakow, Poland

Bibliografia

• [1] Addis B.: Global Optimization Using Local Searches. Ph.D. thesis, Universitá Degli Studi di Firenze, 2004.
• [2] Barabasz B., Gajda-Zagórska E., Migórski S., Paszynski M., Schaefer R., Smołka M.: A hybrid algorithm for solving inverse problems in elasticity. In: International Journal of Applied Mathematics and Computer Science, vol. 24(4), pp. 865–886, 2014.
• [3] Barabasz B., Migórski S., Schaefer R., Paszynski M.: Multi-deme, twin adaptive strategy hp-HGS. In: Inverse Problems in Science and Engineering, vol. 19(1), pp. 3–16, 2011.
• [4] Beilina L., Klibanov M.V.: Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems. Springer, 2012. URL http://dx.doi.org/10.1007/978-1-4419-7805-9.
• [5] Boender C.G.E., Rinnoy-Kan A.H.G., Stougie L., Timmer G.T.: A Stochastic Method for Global Optimization. In: Mathematical Programming, vol. 22, pp. 125–140, 1982.
• [6] Cabib E., Davini C., Chong-Quing R.: A problem in the optimal design of networks under transverse loading. In: Quarterly of Appl. Math., vol. 48(2), pp. 251–263, 1990.
• [7] Cotta C., Schaefer R.: Complex Metaheuristics. In: Journal of Computational Sciences, vol. 17, pp. 171–173, 2016. URL http://dx.doi.org/doi:10.1016/j.jocs.2016.06.001.
• [8] Dixon L.C.W., Szegö G.P., eds.: Toward Global Optimization. North Holland, 1975.
• [9] Dixon L.C.W., Szegö G.P., eds.: Towards Global Optimisation 2. North-Holland, Amsterdam, 1978.
• [10] Duan Q., Sorooshian S., Gupta V.: Effective and Efficient Global Optimization for Conceptual Rainfall-Runoff Models. In: Water Resource Research, vol. 28(4), pp. 1015–1031, 1992.
• [11] Gajda-Zagórska E., Schaefer R., Smołka M., Paszynski M., Pardo D.: A hybrid method for inversion of 3D DC logging measurements. In: Natural Computing, (3), pp. 355–374, 2014. URL http://dx.doi.org/10.1007/s11047-014-9440-y.
• [12] Isshiki M., Sinclair D., Kaneko S.: Lens Design: Global Optimization of Both Performance and Tolerance Sensitivity. In: International Optical Design, p. TuA5. Optical Society of America, 2006. URL http://dx.doi.org/10.1364/IODC.2006.TuA5.
• [13] Kabanikhin S.I.: Definitions and examples of inverse ill-posed problems. In: Journal of Inverse and Ill-Posed Problems, vol. 16, pp. 317–357, 2008. URL http://dx.doi.org/10.1515/JIIP.2008.069.
• [14] Koper K., Wysession M., Wiens D.: Multimodal function optimization with aniching genetic algorithm: A seismological example. In: Bulletin of the Seismological Society of America, vol. 89(4), pp. 978–988, 1999.
• [15] Łos M., Sawicki J., Smołka M., Schaefer R.: Memetic approach for irremediable ill-conditioned parametric inverse problems. In: Procedia Computer Science, vol. 108C, pp. 867–876. Elsevier, 2017. URL http://dx.doi.org/10.1016/j.procs.2017.05.007.
• [16] Łos M., Schaefer R., Sawicki J., Smołka M.: Local Misfit Approximation in Memetic Solving of Ill-posed Inverse Problems. In: Lecture Notes in Computer Science, vol. 10199, pp. 297–309. Springer, 2017.
• [17] Pardalos P., Romeijn H., eds.: Handbook of Global Optimization, vol. 2. Springer US, 2002. URL http://dx.doi.org/10.1007/978-1-4757-5362-2.
• [18] Paruch M., Majchrzak E.: Identification of tumor region parameters using evolutionary algorithm and multiple reciprocity boundary element method. In: Engineering Applications of Artificial Intelligence, vol. 20(5), pp. 647–655, 2007.
• [19] Preuss M.: Multimodal Optimization by Means of Evolutionary Algorithms. Natural Computing. Springer, 2015.
• [20] Rinnoy-Kan A.H.G., Timmer G.T.: Stochastic Global Optimization Methods. Part 1: Clustering Methods. In: Mathematical Programming, vol. 39, pp. 27–56, 1987.
• [21] Schaefer R., Adamska K., Telega H.: Genetic Clustering in Continuous Landscape Exploration. In: Engineering Applications of Artificial Intelligence (EAAI), vol. 17, pp. 407–416, 2004.
• [22] Smołka M., Gajda-Zagórska E., Schaefer R., Paszynski M., Pardo D.: A hybrid method for inversion of 3D AC logging measurements. In: Applied Soft Computing, vol. 36, pp. 422–456, 2015.
• [23] Smołka M., Schaefer R., Paszynski M., Pardo D., Álvarez-Aramberri J.: An Agent-oriented Hierarchic Strategy for Solving Inverse Problems. In: International Journal of Applied Mathematics and Computer Science, vol. 25(3), pp. 483–498, 2015. URL http://dx.doi.org/10.1515/amcs-2015-0036.
• [24] Tikhonov A., Goncharsky A., Stepanov V., Yagola A.: Numerical Methods for the Solution of Ill-Posed Problems. Kluwer, 1995.
• [25] Törn A.: A Search Clustering Approach to Global Optimization. In: Dixon and Szegö G.P. (eds.) Towards Global Optimization 2, North-Holland, Amsterdam, pp. 49–62, 1978.
• [26] Törn A., Ali M.M., Viitanen S.: Stochastic Global Optimization: Problem Classes and Solution Techniques. In: Journal of Global Optimization, vol. 14, pp. 437–447, 1999.
• [27] Wolny A., Schaefer R.: Improving Population-Based Algorithms with Fitness Deterioration. In: Journal of Telecommunications and Information Technology, (4), pp. 31–44, 2011.
• [28] Zeidler E.: Nonlinear Functional Analysis and its Application. II/A: Linear Monotone Operators. Springer, 2000.

Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).