Time schedule is valid for Central European Time (UTC+01:00 Prague).

Winter school lectures

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.

Rules for the online conference

The conference will be held online. The registered participants can access the
conference remotely via ZOOM service.
Details for a remote connection are sent on their email. The participants are kindly
asked for a compliance of the following simple rules:

State your real first and family names during log-in to the ZOOM service.

Mute the microphone and camera upon entering the meeting.
It should be set automatically.

Turn on the microphone (or camera eventually) only when giving a lecture,
asking a question, or making comments or remarks.

Share the screen only when giving a lecture.

Use Raise hand if you want to ask a question or make comments.

Important:
Each conference block is managed by a chairman, who watches pre-defined
lengths of lectures and controls a discussion.