The target of this chapter is the evaluation of gradients in inverse problems where spatial field parameters and geometry parameters are treated separately. Such an approach can be beneficial especially when the geometry needs to be detected accurately using L2-norm-based regularization. Emphasis is laid upon the computation of the gradients directly from the governing equations. Working in a statistical framework, the Karhunen-Loève (K-L) expansion is used for discretization of the spatial random field and inversion is done using the gradient-based Hamiltonian Monte Carlo (HMC) algorithm. The HMC gradients involve sensitivities w.r.t the random spatial field and geometry parameters. Building on a method developed by the authors, a procedure is developed which considers the gradients of the associated integral eigenvalue problem (IEVP) as well as the interaction between the gradients w.r.t random spatial field parameters and the gradients w.r.t the geometry parameters. The same mesh and linear shape functions are used in the finite element method employed to solve the forward problem, the artificial elastic deformation problem and the IEVP. Analysis of the rate of convergence using seven different meshes of increasing density indicates a linear rate of convergence of the gradients of the log posterior.
Part of the book: Inverse Problems
This chapter presents an application example of a nonlinear Kalman Filters (KFs), i.e., Particle Filter (PF), for state (or parameters) estimation and prediction of a dynamical system in geotechnical engineering. First key characteristics of dynamical systems in geotechnics, which need to be considered in filtering, are described by showing some figures, and why PF is necessary for geotechnical applications is explained. Then, a detailed algorithm and implementation of PF for geotechnical problems are presented with key equations. The PF is demonstrated through a case history focusing on deformation behavior of a ground due to embankment construction. The PF is applied to estimation of geotechnical parameters and predictions of future settlement behavior of the ground to discuss the effectiveness of the PF in geotechnical engineering. The results of the case history have shown that PF has presented great promise as an accurate parameter identification for a nonlinear dynamic model. The simulation with the identified parameters predicts the actual measurement data with high accuracy even though a limited amount of measurement data was used in identification stage. The PF provides more information on estimates than optimization methods because the estimates are obtained in the form of probability density functions (PDFs). This characteristic can contribute to risk analysis and reliability-based decision-making in geotechnical practice.
Part of the book: Applications and Optimizations of Kalman Filter and Their Variants