Adjoint Hamiltonian Monte Carlo algorithm for the estimation of elastic modulus through the inversion of elastic wave propagation data

Abstract

SUMMARY Efficient inversion of noisy seismic waveform data produced due to elastic wave propagation for the estimation of a high‐dimensional elastic modulus vector is achieved. Estimation is carried out in a Bayesian framework using Hamiltonian Monte Carlo (HMC) that enables efficient statistical estimation over high‐dimensional parameters. The truncated Karhunen‐Loève (K‐L) expansion is introduced to reduce the dimensionality of the elastic modulus vector. Expensive computations of the gradient of the state vector with respect to the parameter vector at every step are also eliminated through the adjoint method, which is developed from a general one‐step discretization of the governing second‐order ordinary differential equations (ODEs). An Adjoint HMC algorithm that employs a truncated K‐L expansion of the elastic modulus vector is presented. The efficacy of the algorithm is investigated with respect to two representative problems with varying geometric complexity. Adjoint HMC offers a significant speed up in gradient calculation time over the direct differentiation counterpart as the number of terms in the K‐L expansion increases. The algorithm is able to estimate the true elastic modulus within the credible intervals for both cases.

Hamiltonian Monte Carlo for Simultaneous Interface and Spatial Field Detection (HMCSISFD) and its application to a piping zone interface detection problem

“…The computation of is straightforward and similar to the computations in Koch et al. 30 Considering the assembly , where is defined in (6), the derivatives at the elemental level can be given as Following the definition of in terms of the ‐term truncated KL expansion in Equation (8), it is easy to get the derivatives as …”

“…these large parameter vectors further adds to the computational demands. The adjoint based HMC method, 30 that uses dimensionality reduction and the adjoint method to promote computational efficiency solves these issues. Citing these advantages, this paper proposes a method to combine the explicit interface detection strategy of HMCID with the adjoint based HMC method for spatial field estimation to simultaneously estimate interface and spatial fields.…”

Numerical Gradient Computation for Simultaneous Detection of Geometry and Spatial Random Fields in a Statistical Framework

“…The interaction between the geometry and spatial parameters due to the domain of definition of the IEVP is also detailed. It should be noted that the procedure detailed above is applicable to both the direct differentiation and adjoint methods [9] of sensitivity analysis.…”

Simultaneous identification of multi-parameter for power hardening elastoplastic problems in three-dimensional geometries

“…Koch et al. developed Hamiltonian Monte Carlo (HMC) algorithm to efficiently estimate the elastic modulus through the inversion of elastic wave propagation data (Koch et al. , 2020a), and the solid-void interface through elastodynamic inversion (Koch et al.…”

“…Cooreman et al (2007) identified the metal's physical property parameters during plastic deformation by comparing an experimentally measured strain field with computed data from a finite element model, and an analytical method was used to calculate the sensitivity coefficients. Koch et al developed Hamiltonian Monte Carlo (HMC) algorithm to efficiently estimate the elastic modulus through the inversion of elastic wave propagation data (Koch et al, 2020a), and the solid-void interface through elastodynamic inversion (Koch et al, 2020b). Moreover, they also used the method to simultaneously identify the interface and spatial field properties through the inversion of hydraulic head and discharge rate data obtained from a steady seepage flow experiment on a domain containing a predefined piping zone (Koch et al, 2021).…”