A comparison of deterministic and Bayesian inverse with application in micromechanics. (English) Zbl 1499.82056

Blaheta, Radim; Béreš, Michal; Domesová, Simona; Pan, Pengzhi

Appl. Math., Praha 63, No. 6, 665-686 (2018).

Summary: 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.

MSC:

82M10	Finite element, Galerkin and related methods applied to problems in statistical mechanics
74S05	Finite element methods applied to problems in solid mechanics
74S60	Stochastic and other probabilistic methods applied to problems in solid mechanics
74M25	Micromechanics of solids
65N21	Numerical methods for inverse problems for boundary value problems involving PDEs
60-08	Computational methods for problems pertaining to probability theory

Keywords:

inverse problems; Bayesian approach; stochastic Galerkin method

Software:

Matlab; Optimization Toolbox