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. (see 8.3.1 and 8.2.1 ).

A book appeared on this subject in 2012 [45] and a chapter of this book is devoted to a historical perspective [44] .

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. 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.

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 (see 8.3.5 ).

It is accepted for publication in a journal [15] , and was presented in conferences [31] , [32] , [38] , [40] .

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.

The procedure is now combined with the PPAT methodology (see 5.10 ). A list of triangles is built for overlapping the boundary of the pseudo-spectra. From the list of vertices, a closed polygonal line is defined and the number of enclosed eigenvalues is determined.

Ratio-Based Parallel Time Integration

Participant : Jocelyne Erhel.

This work is done in in the context of the MODNUM project (see 8.3.2 ), in collaboration with American University of Beirut (AUB), Lebanon.

It was presented at a conference [41] and is submitted to the proceedings. It was also presented at a seminar of Inria Rennes.

Because time-integration of time-dependent problems is inherently sequential, time parallelism aims mainly at reducing the computational time of some real-time evolutionary problems and may be done through predictor-corrector schemes.

We apply the rescaling method onto initial value problems having an explosive or oscillatory solution, in infinite time. We show how a relevant choice of the end-of-slice condition and the time-rescaling factor might lead to rescaled systems having a uniform convergence to a limit problem. This property provides much better predictions and enhances the relevance of RaPTI that consists mainly of (i) the little sequential computations it involves (predictions and corrections are done in parallel), (ii) the relatively low communication cost it induces and (iii) the similarity of the computation on all slices yielding similar computational times on all processors. Hence, significant speed-ups are achieved. This is illustrated on two problems: a non-linear diffusion-reaction problem having an explosive solution, and a membrane problem having an oscillatory and explosive solution.