Applications of Mathematics, Vol. 63, No. 6, pp. 665-686, 2018

A comparison of deterministic and Bayesian inverse with application in micromechanics

Radim Blaheta, Michal Béreš, Simona Domesová, Pengzhi Pan

Received July 16, 2018.   Published online November 19, 2018.

Abstract:  The paper deals with formulation and numerical solution of problems of identification of material parameters for continuum mechanics problems in domains with heterogeneous microstructure. Due to a restricted number of measurements of quantities related to physical processes, we assume additional information about the microstructure geometry provided by CT scan or similar analysis. The inverse problems use output least squares cost functionals with values obtained from averages of state problem quantities over parts of the boundary and Tikhonov regularization. To include uncertainties in observed values, Bayesian inversion is also considered in order to obtain a statistical description of unknown material parameters from sampling provided by the Metropolis-Hastings algorithm accelerated by using the stochastic Galerkin method. The connection between Bayesian inversion and Tikhonov regularization and advantages of each approach are also discussed.
Keywords:  inverse problems; Bayesian approach; stochastic Galerkin method
Classification MSC:  86-08, 82-08, 65N21, 65C60, 60-08
DOI:  10.21136/AM.2018.0195-18

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Affiliations:   Radim Blaheta, Institute of Geonics of the CAS, Studentská 1768, 708 00 Ostrava, Czech Republic, e-mail:; Michal Béreš, Simona Domesová, Institute of Geonics of the CAS, Studentská 1768, 708 00 Ostrava, Czech Republic; Department of Applied Mathematics, Faculty of Electrical Engineering and Computer Science, VŠB - Technical University of Ostrava, 17. listopadu 15, 708 33 Ostrava, Czech Republic; IT4Innovations National Supercomputing Center, VŠB - Technical University of Ostrava, Studentská 6231/1B, 708 33 Ostrava, Czech Republic, e-mail:,; Pengzhi Pan, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Xiaohongshan, Wuchang, Wuhan 430071, China, e-mail:

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