Fast Numerical Techniques for FE Simulations in Electrical Capacitance Tomography

Markus Neumayer*, Thomas Suppan, Thomas Bretterklieber, Hannes Wegleiter, Colin Fox

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review



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.

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.

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.

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.
Original languageEnglish
Pages (from-to)1101-1112
Number of pages12
JournalCOMPEL - The International Journal for Computation and Mathematics in Electrical and Electronic Engineering
Issue number5
Early online dateJul 2023
Publication statusPublished - 21 Nov 2023
Event20th International IGTE Symposium on Computational Methods in Electromagnetics and Multiphysics: IGTE 2022 - TU Graz, Hybrider Event, Graz, Austria
Duration: 18 Sept 202221 Sept 2022


  • Electrical capacitance tomography
  • pneumatic conveying
  • flow measurement
  • mass concentration
  • uncertainty
  • Green’s function
  • FE simulation
  • ECT
  • Jacobian operations

ASJC Scopus subject areas

  • Applied Mathematics
  • Electrical and Electronic Engineering
  • Computer Science Applications
  • Computational Theory and Mathematics

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