Towards Parallel Large-Scale Genomic Prediction by Coupling Sparse and Dense Matrix Algebra

Genomic prediction for plant breeding requires taking into account environmental effects and variations of genetic effects across environments. The latter can be modelled by estimating the effect of each genetic marker in every possible environmental condition, which leads to a huge amount of effects to be estimated. Nonetheless, the information about these effects is only sparsely present, due to the fact that plants are only tested in a limited number of environmental conditions. In contrast, the genotypes of the plants are a dense source of information and thus the estimation of both types of effects in one single step would require as well dense as sparse matrix formalisms. This paper presents a way to efficiently apply a high performance computing infrastructure for dealing with large-scale genomic prediction settings, relying on the coupling of dense and sparse matrix algebra

The Parallelism Motifs of Genomic Data Analysis

Genomic data sets are growing dramatically as the cost of sequencing continues to decline and small sequencing devices become available. Enormous community databases store and share this data with the research community, but some of these genomic data analysis problems require large scale computational platforms to meet both the memory and computational requirements. These applications differ from scientific simulations that dominate the workload on high end parallel systems today and place different requirements on programming support, software libraries, and parallel architectural design. For example, they involve irregular communication patterns such as asynchronous updates to shared data structures. We consider several problems in high performance genomics analysis, including alignment, profiling, clustering, and assembly for both single genomes and metagenomes. We identify some of the common computational patterns or motifs that help inform parallelization strategies and compare our motifs to some of the established lists, arguing that at least two key patterns, sorting and hashing, are missing

High performance computing for large-scale genomic prediction

In the past decades genetics was studied intensively leading to the knowledge that DNA is the molecule behind genetic inheritance and starting from the new millennium next-generation sequencing methods made it possible to sample this DNA with an ever decreasing cost. Animal and plant breeders have always made use of genetic information to predict agronomic performance of new breeds. While this genetic information previously was gathered from the pedigree of the population under study, genomic information of the DNA makes it possible to also deduce correlations between individuals that do not share any known ancestors leading to so-called genomic prediction of agronomic performance. Nowadays, the number of informative samples that can be taken from a genome ranges from one thousand to one million. Using all this information in a breeding context where agronomic performance is predicted and optimized for different environmental conditions is not a straightforward task. Moreover, the number of individuals for which this information is available keeps on growing and thus sophisticated computational methods are required for analyzing these large scale genomic data sets. This thesis introduces some concepts of high performance computing in a genomic prediction context and shows that analyzing phenotypic records of large numbers of genotyped individuals leads to a better prediction accuracy of the agronomic performance in different environments. Finally, it is even shown that the parts of the DNA that influence the agronomic performance under certain environmental conditions can be pinpointed, and this knowledge can thus be used by breeders to select individuals that thrive better in the targeted environment

High performance selected inversion methods for sparse matrices: direct and stochastic approaches to selected inversion

The explicit evaluation of selected entries of the inverse of a given sparse matrix is an important process in various application fields and is gaining visibility in recent years. While a standard inversion process would require the computation of the whole inverse who is, in general, a dense matrix, state-of-the-art solvers perform a selected inversion process instead. Such approach allows to extract specific entries of the inverse, e.g., the diagonal, avoiding the standard inversion steps, reducing therefore time and memory requirements. Despite the complexity reduction already achieved, the natural direction for the development of the selected inversion software is the parallelization and distribution of the computation, exploiting multinode implementations of the algorithms. In this work we introduce parallel, high performance selected inversion algorithms suitable for both the computation and estimation of the diagonal of the inverse of large, sparse matrices. The first approach is built on top of a sparse factorization method and a distributed computation of the Schur-complement, and is specifically designed for the parallel treatment of large, dense matrices including a sparse block. The second is based on the stochastic estimation of the matrix diagonal using a stencil-based, matrix-free Krylov subspace iteration. We implement the two solvers and prove their excellent performance on Cray supercomputers, focusing on both the multinode scalability and the numerical accuracy. Finally, we include the solvers into two distinct frameworks designed for the solution of selected inversion problems in real-life applications. First, we present a parallel, scalable framework for the log-likelihood maximization in genomic prediction problems including marker by environment effects. Then, we apply the matrix-free estimator to the treatment of large-scale three-dimensional nanoelectronic device simulations with open boundary conditions

Application de la compression de rang faible à la solution directe de grands systèmes linéaires couplés creux et denses sur un nœud de calcul multi-cœur dans un contexte industriel

While hierarchically low-rank compression methods are now commonly available in both dense and sparse direct solvers, their usage for the direct solution of coupled sparse/dense linear systems has been little investigated. The solution of such systems is though central for the simulation of many important physics problems such as the simulation of the propagation of acoustic waves around aircrafts. Indeed, the heterogeneity of the jet flow created by reactors often requires a Finite Element Method (FEM) discretization, leading to a sparse linear system, while it may be reasonable to assume as homogeneous the rest of the space and hence model it with a Boundary Element Method (BEM) discretization, leading to a dense system. In an industrial context, these simulations are often operated on modern multicore workstations with fully-featured linear solvers. Exploiting their low-rank compression techniques is thus very appealing for solving larger coupled sparse/dense systems (hence ensuring a finer solution) on a given multicore workstation, and – of course – possibly do it fast. The standard method performing an efficient coupling of sparse and dense direct solvers is to rely on the Schur complement functionality of the sparse direct solver. However, to the best of our knowledge, modern fully-featured sparse direct solvers offering this functionality return the Schur complement as a non compressed matrix. In this paper, we study the opportunity to process larger systems in spite of this constraint. For that we propose two classes of algorithms, namely multi-solve and multi-factorization, consisting in composing existing parallel sparse and dense methods on well chosen submatrices. An experimental study conducted on a 24 cores machine equipped with 128 GiB of RAM shows that these algorithms, implemented on top of state-of-the-art sparse and dense direct solvers, together with proper low-rank assembly schemes, can respectively process systems of 9 million and 2.5 million total unknowns instead of 1.3 million unknowns with a standard coupling of compressed sparse and dense solvers.Bien que des méthodes basées sur la compression de rang faible hiérarchique soient de nos jours généralement fournies dans des solveurs direct denses et creux, leur utilisation pour la solution directe des systèmes linéaires couplés creux/denses n'a été que peu explorée. Résoudre ce type de systèmes est pourtant une étape centrale dans la simulation de nombreux problèmes en physique tels que la propagation des ondes acoustiques autour des avions. En effet, la hétérogénéité du flux d'air généré par des réacteurs nécessite souvent une discrétisation avec la méthode des éléments finis (FEM) conduisant à un système linéaire creux tandis que le reste de l'espace peut être raisonnablement considéré comme homogène et donc modélisé avec la méthode des éléments finis de frontière (BEM) conduisant à un système dense. Dans un contexte industriel, ces simulations sont souvent effectuées sur des machines modernes multi-cœurs en utilisant des solveurs avancés. Il y a donc une forte motivation pour exploiter leurs techniques de compression de rang faible pour la solution des systèmes couplés creux/denses plus grands (conduisant à des modèles plus précis) sur une machine multi-cœur donnée et - bien sûr - le faire de façon efficace. La méthode standard pour effectuer un couplage d'un solveur direct creux avec un solveur direct dense est de se baser sur la fonctionnalité de complément de Schur du solveur direct creux. Cependant, à notre connaissance, les solveurs modernes avancés proposant cette fonctionnalité retournent le complément de Schur dans une matrice dense non compressée. Dans cet article, nous étudions la possibilité de traiter des systèmes plus grands en dépit de cette contrainte. Pour cela, nous proposons deux classes d'algorithmes, c'est-à-dire « multi-solve » et « multi-factorization », qui consistent en la combinaison des méthodes parallèles creuses et denses existantes sur des matrices bien choisies. Une étude expérimentale, conduite sur une machine à 24 cœurs équipée de 128 Go de RAM, montre que ces algorithmes, implémentés par-dessus des solveurs directs creux et denses de l'état de l'art et grâce à un bon assemblage de schémas de compression de rang faible, peuvent traiter des systèmes avec respectivement 9 millions et 2,5 millions d'inconnues au total au lieu de 1,3 millions d'inconnues avec un couplage standard de solveurs creux et denses compressés

Comparison of coupled solvers for FEM/BEM linear systems arising from discretization of aeroacoustic problems

National audienceWhen discretization of an aeroacoustic physical model is based on the application of both the Finite Elements Method (FEM) and the Boundary Elements Method (BEM), this leads to coupled FEM/BEM linear systems combining sparse and dense parts. In this work, we propose and compare a set of implementation schemes relying on the coupling of the open-source sparse direct solver MUMPS with the proprietary direct solvers from Airbus Central R&T, i.e. the scalapack-like dense solver SPIDO and the hierarchical H-matrix compressed solver HMAT. For this preliminary study, we limit ourselves to a single 24-core computational node

Fast cross-validation for multi-penalty ridge regression

High-dimensional prediction with multiple data types needs to account for potentially strong differences in predictive signal. Ridge regression is a simple model for high-dimensional data that has challenged the predictive performance of many more complex models and learners, and that allows inclusion of data type specific penalties. The largest challenge for multi-penalty ridge is to optimize these penalties efficiently in a cross-validation (CV) setting, in particular for GLM and Cox ridge regression, which require an additional estimation loop by iterative weighted least squares (IWLS). Our main contribution is a computationally very efficient formula for the multi-penalty, sample-weighted hat-matrix, as used in the IWLS algorithm. As a result, nearly all computations are in low-dimensional space, rendering a speed-up of several orders of magnitude. We developed a flexible framework that facilitates multiple types of response, unpenalized covariates, several performance criteria and repeated CV. Extensions to paired and preferential data types are included and illustrated on several cancer genomics survival prediction problems. Moreover, we present similar computational shortcuts for maximum marginal likelihood and Bayesian probit regression. The corresponding R-package, multiridge, serves as a versatile standalone tool, but also as a fast benchmark for other more complex models and multi-view learners

Une comparaison de solveurs choisis pour la résolution de systèmes linéaires couplés FEM/BEM résultant de la discrétisation de problèmes aéroacoustiques

When discretization of an aeroacoustic physical model is based on the application of both the Finite Elements Method (FEM) and the Boundary Elements Method (BEM), this leads to coupled FEM/BEM linear systems combining sparse and dense parts. In this preliminary study, we compare a set of sparse and dense solvers applied on the solution of such type of linear systems with the aim to identify the best performing configurations of existing solvers.Lorsque la discrétisation d'un modèle aéroacoustique repose sur l'application d'à la fois la méthodes des éléments fini (FEM) et de la méthode des éléments finis de frontière (BEM), celle-ci conduit à des systèmes linéaires couplés FEM/BEM ayant des parties creuses ainsi que des parties denses. Dans cette étude préliminaire, nous faisons la comparaison d'un ensemble de solveurs creux et denses appliqués à la résolution de ce type de systèmes linéaires dans le but d'identifier les configurations les plus performantes des solveurs existants