In this talk we consider the problem of solving large-scale linear inverse problems A x ≈ b contaminated by unknown noise. Such problems suffer from ill-posedness, meaning that their solution is extremely sensitive to small perturbations in the data.

Methods such as least squares produce here seriously noise-contaminated approximations. Thus, the problem must be regularized in order to obtain reasonable approximate solution. For large scale or even matrix-free problems (where A can not be constructed explicitly), iterative regularization by projection onto low-frequency dominated subspaces is advantageous. In this talk we give an overview of selected iterative regularization methods and discuss their properties. We concentrate mostly on regularization on Krylov subspaces. Applications will be discussed.

The traditional scheme for parallel iterative algorithms is a synchronous method where a new iteration is only started when all the data from the previous one has been received. Such algorithms meet serious scalability limitation due to the synchronization procedure occurring between the processors at the end of each iteration. Another kind of iterative scheme, called asynchronous iterations, can help solve these scalability problems, but lead to several convergence issues, as presented in the first talk.

Modifying an iterative scheme to make asynchronous iterations leads to several convergence difficulties, but the implementation of such methods is also a challenge. Indeed, two different parallel executions will lead to different numbers of iterations, and the asynchronous behavior will make difficult the computation of any stopping criteria. These aspects are discussed in the second talk and illustrated on numerical experiments performed in parallel on large scale engineering problems. Besides, the programming library developed is proved stable and powerful for the implementation of any asynchronous iterative methods.

A series of two 90 min lectures will present some difficulties and results in the numerical solution of algebraic systems stemming from the discretization of partial differential equations. We will see that an efficient procedure requires thorough understanding and interaction between all phases of the solution, such as discretization, preconditioning, algebraic solution, and error estimation. On simple examples, we will illustrate some risks of using inappropriate error measures or error estimates. In particular, we will see that the errors of a different origin, such as the algebraic and discretization errors, can have a very different spatial distribution over the computational domain. This motivates us to develop (and use) error bounds that can also estimate the local distribution of the errors. A construction of such bounds, using the so-called flux reconstruction, will be presented.

Stochastic Galerkin (finite element) method is a popular method for the numerical solution of differential equations with uncertain data. We present solution methods for the resulting large systems of linear equations. We also deal with preconditioning, adaptivity, and a posteriori error estimates.