We will review the basic building blocks of iterative solvers, i.e. sparse matrix-vector multiplication, in the context of GPU devices such 
as the cards by NVIDIA; we will then discuss some techniques in preconditioning by approximate inverses, and we will conclude with an 
application to an image processing problem from the biomedical field.

Quadrature is the term for the numerical evaluation of integrals.  It's a beautiful subject because it's so accessible, yet full of conceptual surprises and challenges.  This talk will review ten of these, with plenty of history and numerical demonstrations.  Some are old if not well known, some are new, and two are subjects of my current research.

Partial differential equations with more than three coordinates arise naturally if the model features certain kinds of stochasticity. Typical examples are the Schroedinger, Fokker-Planck and Master equations in quantum mechanics or cell biology, as well as quantification of uncertainty.
The principal difficulty of a straightforward numerical solution of such equations is the `curse of dimensionality': the storage cost of the discrete solution grows exponentially with the number of coordinates (dimensions).

One way to reduce the complexity is the low-rank separation of variables. One can see all discrete data (such as the solution) as multi-index arrays, or tensors. These large tensors are never stored directly.
We approximate them by a sum of products of smaller factors, each carrying only one of the original variables. I will present one of the simplest but powerful of such representations, the Tensor Train (TT) decomposition. The TT decomposition generalizes the approximation of a given matrix by a low-rank matrix to the tensor case. It was found that many interesting models allow such approximations with a significant reduction of storage demands.

A workhorse approach to computations with the TT and other tensor product decompositions is the alternating optimization of factors. The simple realization is however prone to convergence issues.
I will show some of the recent improvements that are indispensable for really many dimensions, or solution of linear systems with non-symmetric or indefinite matrices.

Functions are usually approximated numerically in a basis, a non-redundant and complete set of functions that span a certain space. In this talk we highlight a number of benefits of using overcomplete sets, in particular using the more general notion of a "frame". The main 
benefit is that frames are easily constructed even for functions of several variables on domains with irregular shapes. On the other hand, allowing for possible linear depencies naturally leads to ill-conditioning of approximation algorithms. The ill-conditioning is 
potentially severe. We give some useful examples of frames and we first address the numerical stability of best approximations in a frame. Next, we briefly describe special point sets in which interpolation turns out to be stable. Finally, we review so-called Fourier extensions and an efficient algorithm to approximate functions with spectral accuracy on domains without structure.