In this presentation, we will first present the main goals and principles of reservoir simulation. Then we will focus on linear systems that arise in such simulation. The main HPC challenge is to solve those systems efficiently on massively parallel computers. The specificity of those systems is that their convergence is mostly governed by the elliptic part of the equations and the linear solver needs to take advantage of it to be efficient. The reference method in reservoir simulation is CPR-AMG which usually relies on AMG to solve the quasi elliptic part of the system. We will present some works on improving AMG scalability for the reservoir linear systems (work done in collaboration with CERFACS). We will then introduce an on-going work with INRIA to take advantage of their enlarged Krylov method (EGMRES) in the CPR method.[-]

Both multigrid and domain decomposition methods are so called optimal solvers for Laplace type problems, but how do they compare? I will start by showing in what sense these methods are optimal for the Laplace equation, which will reveal that while both multigrid and domain decomposition are iterative solvers, there are fundamental differences between them. Multigrid for Laplace's equation is a standalone solver, while classical domain decomposition methods like the additive Schwarz method or Neumann-Neumann and FETI methods need Krylov acceleration to work. I will explain in detail for each case why this is so, and then also present modifications so that Krylov acceleration is not necessary any more. For overlapping methods, this leads to the use of partitions of unity, while for non-overlapping methods, the coarse space can be a remedy. Good coarse spaces in domain decomposition methods are very different from coarse spaces in multigrid, due to the very aggressive coarsening in domain decomposition. I will introduce the concept of optimal coarse spaces for domain decomposition in a sense very different from the optimal above, and then present approximations of this coarse space. Together with optimized transmission conditions, this leads to a two level domain decomposition method of Schwarz type which is competitive with multigrid for Laplace's equation in wallclock time.[-]

Iterative methods for linear systems were invented for the same reasons as they are used today,namely to reduce computational cost. Gauss states in a letter to his friend Gerling in 1823: 'you will in the future hardly eliminate directly, at least not when you have more than two unknowns'.
Richardson's paper from 1910 was then very influential, and is a model of a modern numerical analysis paper: modeling, discretization, approximate solution of the discrete problem,and a real application. Richardson's method is much more sophisticated that how it is usually presented today, and his dream became reality in the PhD thesis of Gene Golub.
The work of Stiefel, Hestenes and Lanczos in the early 1950 sparked the success story of Krylov methods, and these methods can also be understood in the context of extrapolation, pioneered by Brezinski and Sidi, based on seminal work by Wynn.
This brings us to the modern iterative methods for solving partial differential equations,which come in two main classes: domain decomposition methods and multigrid methods. Domain decomposition methods go back to the alternating Schwarz method invented by Herman Amandus Schwarz in 1869 to close a gap in the proof of Riemann's famous Mapping Theorem. Multigrid goes back to the seminal work by Fedorenko in 1961, with main contributions by Brandt and Hackbusch in the Seventies.
I will show in my presentation how these methods function on the same model problem ofthe temperature distribution in a simple room. All these methods are today used as preconditioners for Krylov methods, which leads to the most powerful iterative solvers currently knownfor linear systems.[-]

There are more and more computing elements in modern supercomputers. This increases the probability of computer errors. Errors that do not stop the computation are called soft errors or silent errors. Of course, they could have a negative impact on the output of the code. So, it is of interest to be able to detect these silent errors and to correct them.
In this talk we are concerned with the detection and correction of silent errors in the conjugate gradient (CG) algorithm to solve linear systems Ax = b with a symmetric positive definite matrix A. Silent errors in CG may affect or even prevent the convergence of the algorithm. We propose a new way to detect silent errors using a scalar relation that must be satisfied by CG variables,
$\alpha_{2 k-1}\tfrac{\left(A p_{k-1}, A p_{k-1}\right)}{\left(r_{k-1}, r_{k-1}\right)}=1+\beta_{k},(1)$
where rj's are the residual vectors, pj's the descent directions and
$\alpha_{k-1}=\tfrac{\left(r_{k-1}, r_{k-1}\right)}{\left(\mathrm{p}_{\mathrm{k}-1}, \mathrm{Ap}_{\mathrm{k}-1}\right)}$, $\beta_{\mathrm{k}}=\frac{\left(\mathrm{r}_{\mathrm{k}}, \mathrm{r}_{\mathrm{k}}\right)}{\left(r_{k-1}, r_{k-1}\right)}$
are the coefficients computed in $\mathrm{CG}$.
We study how relation (1) is modified in finite precision arithmetic and define a criterion to detect when this relation is not satisfied.
Checking relation (1) involves computing an additional dot product, but, as it was shown some time ago in [1] and more recently in [2], relation (1) can be used to introduce more parallelism in the algorithm.
Assuming that the input data $(A, b)$ is not corrupted, we model silent errors by bit flips in the output of some CG steps. When an error is detected in some iteration $\mathrm{k}$, we could restore the CG data from iteration $k-2$ to be able to continue the computation safely.
Numerical experiments will show the efficiency of this approach.[-]

In domain decomposition methods, most of the computational cost lies in the successive solutions of the local problems in subdomains via forward-backward substitutions and in the orthogonalization of interface search directions. All these operations are performed, in the best case, via BLAS-1 or BLAS-2 routines which are inefficient on multicore systems with hierarchical memory. A way to improve the parallel efficiency of the method consists in working with several search directions, since multiple forward-backward substitutions and reorthogonalizations involve BLAS-3 routines. In the case of a problem with several right-hand-sides, using a block Krylov method is a straightforward way to work with multiple search directions. This will be illustrated with an application in electromagnetism using FETI-2LM method. For problems with a single right-hand-side, deriving several search directions that make sense from the optimal one constructed by the Krylov method is not so easy. The recently developed S-FETI method gives a very good approach that does not only improve parallel efficiency but can also reduce the global computational cost in the case of very heterogeneous problems.[-]

Multigrid is an iterative method for solving large linear systems of equations whose Toeplitz system matrix is positive definite. One of the crucial steps of any Multigrid method is based on multivariate subdivision. We derive sufficient conditions for convergence and optimality of Multigrid in terms of trigonometric polynomials associated with the corresponding subdivision schemes.
(This is a joint work with Marco Donatelli, Lucia Romani and Valentina Turati).[-]

The performance of numerical algorithms, both regarding stability and complexity, can be understood in a unified way in terms of condition numbers. This requires to identify the appropriate geometric settings and to characterize condition in geometric ways.
A probabilistic analysis of numerical algorithms can be reduced to a corresponding analysis of condition numbers, which leads to fascinating problems of geometric probability and integral geometry. The most well known example is Smale's 17th problem, which asks to find a solution of a given system of n complex homogeneous polynomial equations in $n$ + 1 unknowns. This problem can be solved in average (and even smoothed) polynomial time.
In the course we will explain the concepts necessary to state and solve Smale's 17th problem. We also show how these ideas lead to new numerical algorithms for computing eigenpairs of matrices that provably run in average polynomial time. Making these algorithms more efficient or adapting them to structured settings are challenging and rewarding research problems. We intend to address some of these issues at the end of the course.[-]

The performance of numerical algorithms, both regarding stability and complexity, can be understood in a unified way in terms of condition numbers. This requires to identify the appropriate geometric settings and to characterize condition in geometric ways.
A probabilistic analysis of numerical algorithms can be reduced to a corresponding analysis of condition numbers, which leads to fascinating problems of geometric probability and integral geometry. The most well known example is Smale's 17th problem, which asks to find a solution of a given system of n complex homogeneous polynomial equations in $n$ + 1 unknowns. This problem can be solved in average (and even smoothed) polynomial time.
In the course we will explain the concepts necessary to state and solve Smale's 17th problem. We also show how these ideas lead to new numerical algorithms for computing eigenpairs of matrices that provably run in average polynomial time. Making these algorithms more efficient or adapting them to structured settings are challenging and rewarding research problems. We intend to address some of these issues at the end of the course.[-]