## Section: New Results

### Parallel numerical algorithms

#### High Performance Scientific Computing

Participant : Bernard Philippe.

This work was done in collaboration with several authors, from US, Greece, etc.

A book will appear on this subject in 2012 [39] and a chapter of this book is devoted to a historical perspective [38] .

This comprehensive text/reference, inspired by the visionary work of Prof. Ahmed H. Sameh, represents the state of the art in parallel numerical algorithms, applications, architectures, and system software. Articles in this collection address solutions to various challenges arising from concurrency, scale, energy efficiency, and programmability. These solutions are discussed in the context of diverse applications, ranging from scientific simulations to large-scale data analysis and mining.

As exascale computing is looming on the horizon while multicore and GPU’s are routinely used, we survey the achievements of Ahmed H. Sameh, a pioneer in parallel matrix algorithms [38] . Studying his contributions since the days of Illiac IV as well as the work that he directed and inspired in the building of the Cedar multiprocessor and his recent research, unfolds a useful historical perspective in the field of parallel scientific computing.

#### Updating the Diagonalization of a Symmetric Matrix

Participant : Bernard Philippe.

This work is done in the context of the DIAMS project.

Two methods are compared : Jacobi method and first order correction of the spectral projectors [25] ,[26] .

#### Counting eigenvalues in domains of the complex field

Participant : Bernard Philippe.

This work is done in collaboration with E. Kamgnia, from the University of Yaounde 1, Cameroon, in the context of the MOMAPLI project at LIRIMA.

It has been submitted to a journal [43] .

A procedure for counting the number of eigenvalues of a matrix in a region surrounded by a closed curve is presented. It is based on the application of the residual theorem. The quadrature is performed by evaluating the principal argument of the logarithm of a function. A strategy is proposed for selecting a path length that insures that the same branch of the logarithm is followed during the integration. Numerical tests are reported for matrices obtained from conventional matrix test sets.

#### Rescaling for time integration

Participant : Jocelyne Erhel.

This work is done in collaboration with N. Makhoul and N. Nassif, from the American University of Beirut, Lebanon.

It is published in a journal [17] .

This paper considers the mathematical framework of a sliced-time computation method for explosive solutions to systems of
ordinary differential equations: $Y\left(t\right)\in {\mathbb{R}}^{k}:\phantom{\rule{0.166667em}{0ex}}\frac{dY}{dt}=F\left(Y\right),\phantom{\rule{0.166667em}{0ex}}0<t,\phantom{\rule{0.166667em}{0ex}}Y\left(0\right)={Y}_{0}$,
that have **finite or infinite explosion time**. The method used generates automatically a sequence of non
uniform slices $\left\{[{T}_{n-1},{T}_{n}]\right|n\ge 1\}$ determined by an end-of-slice condition that controls the growth of
the solution within each slice. It also uses rescaling of the variables, whereas:
$t={T}_{n-1}+{\beta}_{n}s$ and $Y\left(t\right)=Y\left({T}_{n-1}\right)+{D}_{n}{Z}_{n}\left(s\right)$, ${D}_{n}{\mathbb{R}}^{k\times k}$ and ${\beta}_{n}$ being respectively
an invertible diagonal matrix and a rescaling time factor. Thus, the original system is transformed into a
sequence of slices-dependent initial-value shooting problems: $\frac{d{Z}_{n}}{ds}={G}_{n}\left({Z}_{n}\right),\phantom{\rule{0.166667em}{0ex}}0<s\le {s}_{n},\phantom{\rule{0.166667em}{0ex}}{Z}_{n}\left(0\right)=0$.
A suitable selection of ${\beta}_{n}$ and ${D}_{n}$ leads the rescaled systems to verify a concept of **uniform similarity**,
allowing to disable the extreme stiffness of the original ODE problem.
Then, on each time slice, the uniformly rescaled systems are locally solved using a 4^{th}
order explicit Runge-Kutta scheme, within a computational tolerance of ${\u03f5}_{loc}$.
After sequentially implementing the local solver on a total of $N$ slices,
a global tolerance ${\u03f5}_{glob}$ would result in approximating the solution $Y\left(t\right)$ of the original system.

The proper definition of Uniform Similarity leads to deriving, under a stability assumption, a relationship between ${\u03f5}_{loc}$, ${\u03f5}_{glob}$ and $N$. Numerical experiments are conducted for infinite and finite times explosive discrete reaction diffusion problems. These experiments validate the theoretical results and attest for the efficiency of the method in terms of stability and high accuracy.