Misfit landforms imposed by ill-conditioned inverse parametric problems

Marcin Łoś, Maciej Smołka, Robert Schaefer, Jakub Sawicki

Abstract


In the paper, we put forward a new topological taxonomy which allows to distinguish and separate multiple solutions to the ill-conditioned parametric inverse problems appearing in engineering, geophysics, medicine, etc. This taxonomy distinguishes the areas of insensitivity to parameters, called landforms of the misfit landscape, be it around minima (lowlands), maxima (uplands), or stationary points (shelves). We proved 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 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 the ill-conditioned inverse problems and for selecting and profiling complex optimization strategies used to solve these problems.


Keywords


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

Full Text:

PDF

References


Addis B.: Global Optimization Using Local Searches. Ph.D. thesis, Universitá Degli Studi di Firenze, 2004.

Barabasz B., Gajda-Zagórska E., Migórski S., Paszyński 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.

Barabasz B., Migórski S., Schaefer R., Paszyński M.: Multi-deme, twin adaptive strategy hp-HGS. In: Inverse Problems in Science and Engineering, vol. 19(1), pp. 3–16, 2011.

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.

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.

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.

Dixon L.C.W., Szegö G.P., eds.: Toward Global Optimization. North Holland, 1975.

Dixon L.C.W., Szegö G.P., eds.: Towards Global Optimisation 2. North-Holland, Amsterdam, 1978.

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.

Gajda-Zagórska E., Schaefer R., Smołka M., Paszyński 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.

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.

Kabanikhin S.I.: Definitions and examples of inverse and ill-posed problems. In: Journal of Inverse Ill-Posed Problems, vol. 16, pp. 317–337, 2008.

Koper K., Wysession M., Wiens D.: Multimodal function optimization with a niching genetic algorithm: A seismological example. In: Bulletin of the Seismological Society of America, vol. 89(4), pp. 978–988, 1999.

Łoś 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.

Łoś 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.

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.

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.

Preuss M.: Multimodal Optimization by Means of Evolutionary Algorithms. Natural Computing. Springer, 2015.

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.

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.

Smołka M., Gajda-Zagórska E., Schaefer R., Paszyński M., Pardo D.: A hybrid method for inversion of 3D AC logging measurements. In: Applied Soft Computing, vol. 36, pp. 422–456, 2015.

Smołka M., Schaefer R., Paszyński 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.

Tikhonov A., Goncharsky A., Stepanov V., Yagola A.: Numerical Methods for the Solution of Ill-Posed Problems. Kluwer, 1995.

Tikhonov A.N., Goncharskii A., Stepanov, V.V., Yagola A.G.: Numerical Methods for the Solution of Ill-Posed Problems. Springer-Verlag, 1995.

Törn A.: A Search Clustering Approach to Global Optimization. In: Dixon and Szegö, pp. 49–62.

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.

Wolny A., Schaefer R.: Improving Population-Based Algorithms with Fitness Deterioration. In: Journal of Telecommunications and Information Technology, (4), pp. 31–44, 2011.

Zeidler E.: Nonlinear Functional Analysis and its Application. II/A: Linear Monotone Operators. Springer, 2000.




DOI: http://dx.doi.org/10.7494/csci.2018.19.2.2781

Refbacks

  • There are currently no refbacks.