New methods for parametric computations with non-symmetric or indefinite matrices on HPC architectures

M. Kim Liégeois will publicly defend his thesis entitled "New methods for parametric computations with non-symmetric or indefinite matrices on HPC architectures".

Summary

Parametric computations, and in particular, uncertainty quantification, are key components of predictive simulation.

Those computations in the context of multiphysics models typically require a substantial number of realizations of costly finite element models. 

In this work, we are particularly interested in problems with non-symmetric or indefinite matrices. 

As an illustration of this class of problems, we consider contact mechanics problems in saddle point formulation and thermomechanical simulations.

Embedded ensemble propagation was proposed by Phipps et al. to improve the efficiency of nonintrusive uncertainty quantification methods of computational models on emerging computational architectures.

It consists of simultaneously evaluating the model for a subset of samples together, instead of evaluating them individually. 

This method replaces scalars by vectors, vectors by matrices, and matrices by higher-order tensors. 

Having matrices instead of vectors raises questions such as the definition of an inner product.

A first approach introduced to solve parametric linear systems with ensemble propagation is ensemble reduction.

In Krylov methods for example, this reduction consists in coupling the samples together using an inner product that sums the sample contributions.

Ensemble reduction has the advantages of being able to use optimized implementations of BLAS functions and having a stopping criterion which involves only one scalar.

However, the reduction potentially decreases the rate of convergence of the iterative method due to the gathering of the spectra of the samples.

In the work of Phipps et al., they have investigated the effect of ensemble propagation to solve symmetric and positive definite linear problems using the conjugate gradient method.

In this work, we investigate ensemble propagation in the case of GMRES to be able to solve problems with non-symmetric or indefinite matrices.

In particular, we investigate GMRES without ensemble reduction to solve each sample simultaneously but independently to improve the convergence compared to ensemble reduction.

This raises two new issues which are solved in this thesis: the fact that optimized implementations of BLAS functions cannot be used anymore and that ensemble divergence, whereby individual samples within an ensemble must follow different code execution paths, can occur.  

We tackle those issues by implementing a high-performing ensemble dense matrix-vector product (GEMV) and by using masks.

The proposed ensemble GEMV leads to a similar cost per GMRES iteration for both approaches, i.e. with and without reduction.

For illustration, we study the performances of the new linear solver on four academic problems including one non-linear contact problem.

These examples demonstrate improved ensemble propagation speed-up without reduction. 

Finally, the method is applied to accelerate the uncertainty quantification study of a model problem relevant for the design of an optomechanical system for ITER, the fusion reactor, for which the measured final speed-up of using embedded ensemble propagation is about 2.

Practical information

The defense (in English) will take place on September 10th 2020 at 16:30 and is accessible to all via Lifesize meeting: https://call.lifesizecloud.com/5078575