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KRYLOV AFTERNOONS @ Maison de la simulation, Saclay, France


Maison de la simulation, Saclay, France

June 4-5, 2015

Krylov Subspace Methods  (KSMs) provide powerful and general-purpose ways to project a given problem into a much smaller one that yields good approximate solutions to the original problem.

The variety of applications of this general approach is phenomenal. They range from the well-known uses of KSMs for 
solving linear systems and eigenvalue problems, to less established ones such as computing approximate
spectral densities, solving systems of Ordinary Differential Equations, and dimension reduction in machine 

June 4: Tutorials

13-13:30 Registration, coffee and tea

13:30-16:30 Krylov subspace methods 

Yousef Saad, University of Minneapolis, MN, USA

This tutorial of two lectures will give an overview of Krylov subspace methods, and discuss the related algorithms, 
theory, and applications.

13:30-14:45 First lecture

14:45-15:15 Coffee and tea

15:15-16:00 Second lecture 

16:00-16:30 Discussions and Q&A 

June 5: Colloquium

13-13:30 Registration, coffee and tea

13:30-14:15 Divide and conquer algorithms for large Hermitian eigenvalue problems

Yousef Saad, University of Minnesota, MN, USA

Algorithms based on Divide and conquer paradigms can lead to complex, but efficient and flexible algorithms for    
solving large Hermitian eigenvalue problems.  This talk will discuss how various such strategies can be combined to 
exploit both `spectrum slicing', i.e., computing  slices of the spectrum independently, and domain decomposition. 
These strategies are independent of each other but both are essential if one has to compute very large parts of the 
spectrum, as is the case in approaches that deal with excited states in solid state physics.  The presentation will 
begin with an overview of polynomial filtering techniques. This general approach can be quite efficient in the 
situation where the matrix-vector product operation is inexpensive and when a large number of eigenvalues is  
sought. We present an algorithm (and package) that combines the Lanczos algorithm with partial 
reorthogonalization and polynomial filtering. n alternative to polynomial filtering that is generating a growing 
interest is a class of methods that exploit filtering by rational functions. Good representatives of this general 
approach are the FEAST eigensolver and the Sakurai-Sugiura algorithm.  Here we will argue that the standard way of 
selecting these rational filters, which is based on using the Cauchy integral, can  be substantially improved
-- especially when iterative solvers  are involved. Finally, the talk will discuss our ongoing work on Domain
Decomposition (DD) type methods that rely on spectral Schur complements combined with Newton's iteration.

14:15-15:00 Unite and Conquer Approach for High Scale Numerical Computation, Application to Krylov Methods

Nahid Emad, University of Versailles (Maison de la Simulation, PRiSM)

The ability to exploit emerging exascale computational systems will require a careful review and redesign of core 
numerical algorithms and their implementations to fully exploit multiple levels of concurrency, hierarchical memory 
structures and heterogeneous processing units available in these computational platforms. In this talk, we highlight 
the constraints imposed by these architectures allowing the conception of smart co-design algorithms. Following 
this insight, we present the "unite and conquer" approach to solve linear systems of equations and eigenvalue 
problems for extreme scale computing. Unite and conquer approach focuses on decreasing the number of restart 
cycles in restarted numerical methods by coupling either synchronously or asynchronously several restarted      
methods called co-methods. In the end of a restart cycle, each co-method locally gathers available results of all     
collaborating co-methods and selects the best one in order to create its restarting information. This can permit the       
global reduction of the number of cycles to convergence. We discuss the properties of these methods that make    
them well adapted to large-scale multi-level parallel architectures and highlight the generality of the approach. We 
present some experiments validating the approach for unite and conquer Krylov methods on several parallel 

15:00-15:30 Coffee and tea 

15:30-16:15 Parameter auto-tunings at runtime for Parallel and distributed KSM

Serge Petiton, University of Lille Sciences and Technologies and Maison de la Simulation

The convergence and the efficiencies of parallel and distributed KSM methods, such as GMRES and ERAM, greatly 
depend on the supercomputers used and on the choice of several parameters, such as the subspace size or the 
restarting strategies, which are difficult to set efficiently before the computation. Minimizing and avoiding 
communication strategies may also be developed to minimize several costs. Thus, these methods have a lot of 
correlated parameters which may be optimized using auto-tuning strategies to accelerate convergence, minimize 
storage space, data movements, and energy consumption. 
In this talk, we will survey some auto-tuning potential strategies for parallel and distributed KSM methods. We’ll 
discuss results obtained on clusters of accelerators concerning auto-tunings of subspace sizes, orthogonalisations 
and restarting strategies. As a conclusion, we will discuss auto-tuning strategies for KSM on future exascale 
supercomputers, leading to intelligent KSM methods capable of  “smart” decisions during computation.