Dipartimento di Matematica

Abstract

The advent of extreme scale computing platforms will require the use of parallel resources at an unprecedented scale. On the technological side, the continuous shrinking of transistor geometry and the increasing complexity of these devices affect dramatically their sensitivity to natural radiation leading to a high rate of hardware faults, and thus diminish their reliability. Handling fully these faults at the computer system level may have a prohibitive computational and energetic cost. High performance computing applications that aim at exploiting all these resources will thus need to be resilient. In this talk, we will first give an overview of the current trends towards exascale. We will discuss the new challenges to face in terms of platform reliability and associated variety of possible faults. We will then discuss some of the solutions that have been proposed to tackle these errors before discussing in more details some contributions in sparse numerical linear algebra. First, in the context of computing node crashes, we will discuss possible remedies in the framework of linear system or eigenproblem solutions, that are the inner most numerical kernels in many scientific and engineering applications and also ones of the most time consuming parts. Second, we will discuss a somehow more challenging problem related to silent transient soft-errors produced by natural radiation and consisting in a bit-flip in a memory cell producing unexpected results at the application level. In that context we will consider the conjugate gradient (CG) method that is the most widely used iterative scheme for the solution of large sparse systems of linear equations when the matrix is symmetric positive definite. We will investigate through extensive numerical experiments the sensitivity of of CG to bit-flips and further discuss possible numerical criteria to detect the occurrence of such faults.
The above mentioned research activities have been conducted in collaboration with many colleagues including E. Agullo (Inria), S. Cools (University of Antwerpen), E. Fatih-Yetkin (Kadir Has University), P. Salas (CERFACS), W. Vanroose (University of Antwerpen) and M. Zounon (NAG).

contact: luca.bonaventura@polimi.it

Abstract

The theory of poroelasticity models the interaction between the deformation and the fluid flow in a fluid-saturated porous medium. Poroelastic models are widely used nowadays in the modeling of many applications in different fields, ranging from geomechanics and petroleum engineer, to biomechanics. The poroelastic equations are often solved in a two-field formulation, where the unknowns are the displacement and the pressure, or in a three-field formulation where the velocity of the fluid is included as a primary variable as well. The numerical solution of these models is usually based on finite element methods. In this talk, we will study some stabilized finite element discretizations for both formulations of the poroelastic problem. Moreover, an important aspect in the numerical simulation of these problems is the efficient solution of the large systems of algebraic equations obtained after discretization. The resulting systems are of saddle point type and we will address their efficient solution by designing suitable geometric multigrid methods.

Contact: luca.formaggia@polimi.it

Abstract

The presentation will review recent advances in the area of designing and using polyconvex energy potentials to describe the behaviour of solids in the large strain regime in the presence of several physical phenomena such as thermal or electro-mechanical effects. The need for convexity will be justified from the viewpoint of ensuring the existence of real wave speeds in the material at any state of deformation. The convexity of energy potentials with respect to an extended set of variables describing deformation and other multi-physics effects enables the definition of conjugate stresses, thermal and electro-mechanical variables and the formulation of complementary energy functions via the application of the Legendre transforms. In the static case, this leads to the development of a variety of Hu-Washizu or Hellinger-Reissner mixed variation principles that permit the discretisation of different fields with specifically chosen order of accuracy. For instance, in the case of nearly and fully incompressible materials, discretisations that either meet the LBB condition or are appropriately stabilised can be constructed. In the dynamic case, first order conservation laws can be derived for each of the extended set of variables in the energy function. Moreover, convexity enables the definition of a convex entropy-like functional of the primary conserved variables, leading to a symmetrisation of the conservation laws in terms of conjugate stress-like variables and a robust implementation of a Petrov-Galerkin discretisation process. This can be done in a variety of ways, for instance in the case of thermoelasticity two approaches will be discussed, namely, using the Hamiltonian as “entropy-like” convex function or, alternatively, using the so-called “ballistic free energy” as convex entropy extension. Several examples demonstrating the theoretical developments will be provided.


contact: luca.formaggia@polimi.it