An efficient solution of a partial differential equation with parameters or uncertainties in input data can be a difficult task. The stochastic Galerkin method solves a PDE with uncertainties via the discretization of both the physical and the stochastic space. The resulting basis is assumed as a product of the two bases: finite elements (FE) for the discretization of physical space, and polynomials for the discretization of stochastic space. This approach leads to an enormous system of equations with a number of degrees of freedom surpassing the capabilities of any standard method for the solution of linear systems. The solution of such a system requires specialized tools for a feasible solution.

A very efficient tool for the solution is the reduced basis (RB) approach. The RB approach aims at a compression of the FE basis. The compressed FE basis has orders of magnitude smaller size and the resulting reduced system is much easier to solve.

In this talk, we will discuss a Monte Carlo based approach to the construction of RB. We examine measures of error for stopping criteria and adaptive polynomial basis selection. Additionally, we briefly discuss an efficient solution for reduced systems using the Tensor Train approximation. We will use problems inspired by geotechnical applications for the demonstration of the discussed methods.

The discretisation of elliptic problems on fine grids leads to systems of linear equations with millions to billions of unknowns, which favour iterative solvers over direct solvers. However, the number of iterations of iterative solvers can grow with increasing system size. This issue may be overcome by the discrete Green's operator (DGO) preconditioning, which makes iteration count independent of the system/grid size. This approach uses the inverse of a linear system matrix of a reference elliptic problem discretised on the same grid as the preconditioner.

We studied the DGO preconditioning from a linear algebra viewpoint and showed that all individual eigenvalues of such preconditioned systems can be bounded purely from the knowledge of the material data of the problems, both original and reference. We developed a simple algorithm to compute these bounds.

In this lecture, we will discuss the theoretical aspects of these results and practical applications of the DGO preconditioning to periodic homogenisation problems discretised on regular grids.

In matrix computation, it is common to divide matrices into dense and sparse categories. Even though such categories are not precisely defined, we can think of a sparse matrix as one whose number of zero elements is large enough to be conveniently exploitable and a dense one as a matrix that is not sparse. It is important to note that the notion of sparsity does not consider the magnitude of the nonzero elements. This can be an issue since, in many applications, one has to deal with dense matrices in which most elements are so close to zero to being negligible. These matrices are close to being sparse in the sense that they are sparse upon truncation. Moreover, the nonnegligible elements are usually localized in some part of the matrix, and the magnitude of the other elements tends to decay to zero as we move away from them. As a simple example, consider the inverse of a tridiagonal matrix. Such an inverse is usually dense; however, under some conditions, the magnitude of its elements quickly decays to zero as we move away from the diagonal. Therefore, the inverse can be considered banded upon truncation.

In this lecture, we will introduce the basics of the decay phenomenon in the matrix function case. Then we will show how to predict the decay phenomenon and exploit it in matrix computation. In particular, we will present the results obtained with Valeria Simoncini on a-priori bounds for the approximation of matrix function by Krylov subspace methods, a relaxed strategy for the inexact Arnoldi method, and their extension to the rational case. We will also show how the decay phenomenon is fundamental for a new strategy for non-autonomous systems of linear ODEs that we have developed with Pierre-Louis Giscard and Niel Van Buggenhout.

The lecture deals with the modelling of flow and mechanics in porous media based on the Biot consolidation theory. In the first part, I will introduce the classical Biot system. I will give an overview of results on well-posedness and finite element approximation and present some solution techniques based on monolithic or splitting schemes and suitable preconditionings.

The second part will be devoted to applications in rock hydro-mechanics, which involves treating inhomogeneities and fractures. A brief survey of modelling approaches will be given. I will then focus on the so-called discrete fracture-matrix (DFM) approach which couples the rock matrix and discrete fracture network. Here, some nonlinear effects have to be taken into account such as non-penetration contact conditions and stress-permeability relations. I will explain the derivation of DFM models and their suitable discretizations and show examples of their use in geotechnical applications.

This contribution will present several results relating eigenvalues of matrices and spectra of self-adjoint operators, with connection to operator preconditioning. They were obtained jointly with Tomáš Gergelits, Kent-André Mardal and, in particular, Björn Fredrik Nielsen. But they are much in line with many discussions that we had together with Radim Blaheta over several decades and that were for me always enlightening and very pleasant. Due to involvement in many other projects and duties we have not transformed them into a real joint project that would end up in a joint paper. Our intention to change this will remain unfinished. But our interactions have definitely been for me very rewarding, professionally and even more personally.