Jaroslav Vondřejc

Nachiketa Mishra, Jaroslav Vondřejc, Jan Zeman

In this paper, we assess the performance of four iterative algorithms for solving non-symmetric rank-deficient linear systems arising in the FFT-based homogenization of heterogeneous materials defined by digital images. Our framework is based on the Fourier-Galerkin method with exact and approximate integrations that has recently been shown to generalize the Lippmann-Schwinger setting of the original work by Moulinec and Suquet from 1994. It follows from this variational format that the ensuing system of linear equations can be solved by general-purpose iterative algorithms for symmetric positive-definite systems, such as the Richardson, the Conjugate gradient, and the Chebyshev algorithms, that are compared here to the Eyre-Milton scheme - the most efficient specialized method currently available. Our numerical experiments, carried out for two-dimensional elliptic problems, reveal that the Conjugate gradient algorithm is the most efficient option, while the Eyre-Milton method performs comparably to the Chebyshev semi-iteration. The Richardson algorithm, equivalent to the still widely used original Moulinec-Suquet solver, exhibits the slowest convergence. Besides this, we hope that our study highlights the potential of the well-established techniques of numerical linear algebra to further increase the efficiency of FFT-based homogenization methods.

Jaroslav Vondřejc, Hermann G. Matthies

This paper focuses on inverse problems to identify parameters by incorporating information from measurements. These generally ill-posed problems are formulated here in a probabilistic setting based on Bayes's theorem because it leads to a unique solution of the updated distribution of parameters. Many approaches build on Bayesian updating in terms of probability measures or their densities. However, the uncertainty propagation problems and their discretisation within the stochastic Galerkin or collocation method are naturally formulated for random vectors which calls for updating of random variables, i.e. a filter. Such filters typically build on some approximation to conditional expectation (CE). Specifically, the approximation of the CE with affine functions leads to the familiar Kalman filter which works best on linear or close to linear problems only. Our approach builds on a reformulation, which allows to localise the operator of the CE to the point of measured value. The resulting conditioned expectation (CdE) predicts correctly the quantities of interest, e.g. conditioned mean and covariance, even for general highly non-linear problems. The novel CdE allows straight-forward numerical integration; particularly, the approximated covariance matrix is always positive definite for integration rules with positive weights. The theoretical results are confirmed by numerical examples.

Jaroslav Vondřejc

This paper is focused on the double-grid integration with interpolation-projection (DoGIP), which is a novel matrix-free discretisation method of variational formulations introduced for Fourier--Galerkin approximation. Here, it is described as a more general approach with an application to the finite element method (FEM) on simplexes. The approach is based on treating the trial and a test function in variational formulation together, which leads to the decomposition of a linear system into interpolation and (block) diagonal matrices. It usually leads to reduced memory demands, especially for higher-order basis functions, but with higher computational requirements. The numerical examples are studied here for two variational formulations: weighted projection and scalar elliptic problem modelling, e.g. diffusion or stationary heat transfer. This paper also opens a room for further investigation, which is discussed in the conclusion.

Jaroslav Vondřejc, Jan Zeman, Ivo Marek

Guaranteed upper-lower bounds on homogenized coefficients, arising from the periodic cell problem, are calculated in a scalar elliptic setting. Our approach builds on the recent variational reformulation of the Moulinec-Suquet (1994) Fast Fourier Transform (FFT) homogenization scheme by Vondřejc et al. (2014), which is based on the conforming Galerkin approximation with trigonometric polynomials. Upper-lower bounds are obtained by adjusting the primal-dual finite element framework developed independently by Dvořák (1993) and Wieckowski (1995) to the FFT-based Galerkin setting. We show that the discretization procedure differs for odd and non-odd number of grid points. Thanks to the Helmholtz decomposition inherited from the continuous formulation, the duality structure is fully preserved for the odd discretizations. In the latter case, a more complex primal-dual structure is observed due to presence of the trigonometric polynomials associated with the Nyquist frequencies. These theoretical findings are confirmed with numerical examples. To conclude, the main advantage of the FFT-based approach over conventional finite-element schemes is that the primal and the dual problems are treated on the same basis, and this property can be extended beyond the scalar elliptic setting.

Jaroslav Vondřejc, Jan Zeman, Ivo Marek

In 1994, Moulinec and Suquet introduced an efficient technique for the numerical resolution of the cell problem arising in homogenization of periodic media. The scheme is based on a fixed-point iterative solution to an integral equation of the Lippmann-Schwinger type, with action of its kernel efficiently evaluated by the Fast Fourier Transform techniques. The aim of this work is to demonstrate that the Moulinec-Suquet setting is actually equivalent to a Galerkin discretization of the cell problem, based on approximation spaces spanned by trigonometric polynomials and a suitable numerical integration scheme. For the latter framework and scalar elliptic setting, we prove convergence of the approximate solution to the weak solution, including a-priori estimates for the rate of convergence for sufficiently regular data and the effects of numerical integration. Moreover, we also show that the variational structure implies that the resulting non-symmetric system of linear equations can be solved by the conjugate gradient method. Apart from providing a theoretical support to Fast Fourier Transform-based methods for numerical homogenization, these findings significantly improve on the performance of the original solver and pave the way to similar developments for its many generalizations proposed in the literature.