Fast numerical techniques for FE simulations in electrical capacitance tomography

Abstract

Purpose Nonlinear solution approaches for inverse problems require fast simulation techniques for the underlying sensing problem. In this work, the authors investigate finite element (FE) based sensor simulations for the inverse problem of electrical capacitance tomography. Two known computational bottlenecks are the assembly of the FE equation system as well as the computation of the Jacobian. Here, existing computation techniques like adjoint field approaches require additional simulations. This paper aims to present fast numerical techniques for the sensor simulation and computations with the Jacobian matrix. Design/methodology/approach For the FE equation system, a solution strategy based on Green’s functions is derived. Its relation to the solution of a standard FE formulation is discussed. A fast stiffness matrix assembly based on an eigenvector decomposition is shown. Based on the properties of the Green’s functions, Jacobian operations are derived, which allow the computation of matrix vector products with the Jacobian for free, i.e. no additional solves are required. This is demonstrated by a Broyden–Fletcher–Goldfarb–Shanno-based image reconstruction algorithm. Findings MATLAB-based time measurements of the new methods show a significant acceleration for all calculation steps compared to reference implementations with standard methods. E.g. for the Jacobian operations, improvement factors of well over 100 could be found. Originality/value The paper shows new methods for solving known computational tasks for solving inverse problems. A particular advantage is the coherent derivation and elaboration of the results. The approaches can also be applicable to other inverse problems.

ECT in a large scale industrial pneumatic conveying system

“…Hereby J is the Jacobian matrix, which holds the derivatives of the simulated measurements with respect to the elements of x. Efficient techniques for the computation of J are presented in [46]. Σ meas is a covariance matrix incorporating knowledge about the measurement noise.…”

Efficient Jacobian Computations for Complex ECT/EIT Imaging

“…This section contains a summary of the fast numerical techniques presented in [38]. While the techniques were developed for the inverse problem of ECT, most aspects can be related to the complex-valued problems treated in this work.…”

“…In a previous study [38], our research group showed a set of fast numerical techniques for the solution of the inverse problem of ECT. This includes a fast technique for the assembly of the FE stiffness matrix K and, in particular, a set of techniques to evaluate matrix vector products with the Jacobian or its transpose without an explicit evaluation of J.…”

“…The rest of this paper is structured as follows. Section 2 presents a recap of the methods presented in [38], including an introduction to the considered sensor setup. In Section 3, the application of the Green's function approach to inverse problems with an underlying quasi-static electrical field problem is shown.…”

“…In Section 3, the application of the Green's function approach to inverse problems with an underlying quasi-static electrical field problem is shown. Here, the required computations with respect to the real-valued counterpart presented in [38] are derived. Afterward, the computational steps to solve inverse problems with the new methods are discussed.…”