ABSTRACT. Multiple Hermite polynomials are an extension of the classical Hermite polynomials for which orthogonality conditions are imposed with respect to r>1 normal (Gaussian) weights with different means c_i, 1 \leq i \leq r. These polynomials have a number of properties, such as a Rodrigues formula, a recurrence relations (connecting polynomials with nearest neighbor multi-indices), a differential equation, etc. The asymptotic distribution of the (scaled) zeros is well understood and an interesting new feature happens: depending on the distance between the means c_i (1 \leq i \leq r), the zeros may accumulate on s disjoint intervals, wwhere 1 \leq s \leq r. We will use the zeros of these multiple Hermite polynomials to approximate integrals of the form \int_{-\infty}^{\infty} f(x) \exp(-x^2 + c_jx)\, dx$ simultaneously for $1 \leq j \leq r$. The behavior of the quadrature weights depends in an important way on whether or not the zeros are on disjoint intervals or on one interval.

ABSTRACT. Time parallel time integration methods have received renewed interest over the last decade because of the advent of massively parallel computers, due to the clock speed limit reached on today's processors. When solving time dependent partial differential equations, the time direction is usually not used for parallelization. But when parallelization in space saturates, the time direction offers itself as a further direction for parallelization. The time direction is however special, and for evolution problems there is a causality principle: the solution later in time is determined by the solution earlier in time, so the flow of information is just into the direction forward in time. Algorithms trying to use the time direction for parallelization must therefore be special, and take this very different property of the time dimension into account.

I will show in this talk how time parallel time integration methods were invented over the past five decades, and give a classification into four different groups: methods based on multiple shooting, space-time multigrid methods, methods based on domain decomposition and waveform relaxation, and direct time parallel methods. The performance of these methods depends on the nature of the underlying evolution problem, and it turns out that for the first two classes of methods, time parallelization is only really possible for parabolic problems, while the last two classes can also be used to parallelize hyperbolic problems in time. I will also explain in more detail one of the methods from each class: the parareal algorithm and a space-time multigrid method, which are currently among the most promising methods for parabolic problems, and a Schwarz waveform relaxation method related to tent-pitching and a direct time parallel method based on diagonalization of the time stepping matrix, which are very effective for hyperbolic problems.

ABSTRACT. Given a sequence $\left \{ P_{n}\right \}_{n\geq0}$ of monic orthogonal polynomials with respect to a linear functional $u$ and a fixed integer $k$, we establish necessary and sufficient conditions so that the quasi-orthogonal polynomials $Q_{n}$, defined by $Q_{n}(x) =P_{n}(x) + \sum _{i=1}^{k} b_{i,n}P_{n-i}(x), n\geq 0,$ with $b_{i,n} \in R$, and $b_{k,n}\neq 0$ for $n\geq k$, also constitute a sequence of orthogonal polynomials with respect to a linear functional $v$ which is called the Geronimus transformation of $u$ (\cite{Bracciali}). Indeed, there exists a polynomial $h(x)$ of degree $k$ such that $h(x) v=u$. This is an inverse problem of one analyzed in \cite{Elhay}. The characterization turns out to be equivalent to recurrence formulas for the coefficients $b_{i,n}$. An algorithm to find the polynomial $h(x)$ is presented. Some illustrative examples are provided.\\

These results are used to state explicit relations between various type quadrature rules. The classical situation, when the algebraic degree of precision is the highest possible, is well-known and the quadrature formulas are the Gaussian ones whose nodes coincide with the zeros of the corresponding orthogonal polynomials and the Christoffel numbers are expressed in terms of the so-called kernel polynomials \cite{Gautschi}. In our case, we can relax the requirement for the highest possible degree of precision in order to gain the possibility to either approximate integrals of more specific continuous functions containing a polynomial factor or including additional fixed nodes. The construction of such quadrature processes is related to quasi-orthogonal polynomials (see \cite{Brezinski}) and \cite{Bultheel}).\\

This is a joint work with C. F. Bracciali (UNESP, Brazil) and S. Varma (University of Ankara, Turkey).\\

ABSTRACT. Kronrod in 1964, trying to estimate economically the error of the $n$-point Gauss quadrature formula for the Legendre weight function, developed a new formula by adding to the $n$ Gauss nodes $n+1$ new ones, which are determined, together with all weights, such that the new formula has maximum degree of exactness. It turns out that the new nodes are zeros of a polynomial orthogonal with respect to a variable-sign weight function, considered by Stieltjes in 1894, without though making any reference to quadrature.

In recent years, Gauss-Kronrod quadrature formulae have attracted considerable attention from both the theoretical and the computational point of view, the former in view of the intriguing mathematical questions they pose and the latter on account of their use in packages of automatic integration; so, these formulae form an active area of research for over 50 years now.

We survey the recent advances on Gauss-Kronrod quadrature, paying particular attention to existence, nonexistence and error term results; at the same time, we point out the important questions that are still open in this area.

ABSTRACT. We discuss the application of Gauss-type quadrature rules to the approximation of certain matrix functional and to determine error estimates. Several generalizations of anti-Gauss rules will be discussed, including to matrix valued measures. Also anti-Gauss-type quadrature rules associated with multiple orthogonal polynomials will be described/