Robust algebraic Schur complement preconditioners based on low rank corrections

In this paper we introduce LORASC, a robust algebraic preconditioner for solving sparse linear systems of equations involving symmetric and positive definite matrices. The graph of the input matrix is partitioned by using k-way partitioning with vertex separators into N disjoint domains and a separator formed by the vertices connecting the N domains. The obtained permuted matrix has a block arrow structure. The preconditioner relies on the Cholesky factorization of the first N diagonal blocks and on approximating the Schur complement corresponding to the separator block. The approximation of the Schur complement involves the factorization of the last diagonal block and a low rank correction obtained by solving a generalized eigenvalue problem or a randomized algorithm. The preconditioner can be build and applied in parallel. Numerical results on a set of matrices arising from the discretization by the finite element method of linear elasticity models illustrate the robusteness and the efficiency of our preconditioner

HMOE: Hypernetwork-based Mixture of Experts for Domain Generalization

Due to domain shift, machine learning systems typically fail to generalize well to domains different from those of training data, which is what domain generalization (DG) aims to address. Although various DG methods have been developed, most of them lack interpretability and require domain labels that are not available in many real-world scenarios. This paper presents a novel DG method, called HMOE: Hypernetwork-based Mixture of Experts (MoE), which does not rely on domain labels and is more interpretable. MoE proves effective in identifying heterogeneous patterns in data. For the DG problem, heterogeneity arises exactly from domain shift. HMOE uses hypernetworks taking vectors as input to generate experts' weights, which allows experts to share useful meta-knowledge and enables exploring experts' similarities in a low-dimensional vector space. We compare HMOE with other DG algorithms under a fair and unified benchmark-DomainBed. Our extensive experiments show that HMOE can divide mixed-domain data into distinct clusters that are surprisingly more consistent with human intuition than original domain labels. Compared to other DG methods, HMOE shows competitive performance and achieves SOTA results in some cases

Adaptive linear solution process for single-phase Darcy flow

International audienceThis article presents an adaptive approach for solving linear systems arising from self-adjoint partial differential equations (PDE) problems discretized by cell-centered finite volume method and stemming from single-phase flow simulations. This approach aims at reducing the algebraic error in targeted parts of the domain using a posteriori error estimates. Numerical results of a reservoir simulation example for heterogeneous porous media in two dimensions are discussed. Using the adaptive solve procedure, we obtain a significant gain in terms of the number of time steps and iterations compared to a standard solve.Cet article présente une approche adaptative pour la résolution de systèmes linéaires qui découlent de problèmes d’équations aux dérivées partielles (EDP) avec opérateurs auto-adjoints discrétisés par la méthode des volumes finis centrés en cellules et issus de simulations d’écoulement mono-phasique. Cette approche vise à réduire l’erreur algébrique dans des parties ciblées du domaine à l’aide d’estimateurs d’erreur a posteriori. Des résultats numériques d’un exemple de simulation de réservoir en milieux poreux hétérogènes en deux dimensions sont présentés. En utilisant la procédure adaptative de résolution, nous obtenons un gain significatif en termes de nombre de pas de temps et d’itérations par rapport à une résolution classique

An a posteriori-based adaptive preconditioner for controlling a local algebraic error norm

International audienceThis paper introduces an adaptive preconditioner for iterative solution of sparse linear systems arising from partial differential equations with self-adjoint operators. This preconditioner allows to control the growth rate of a dominant part of the algebraic error within a fixed point iteration scheme. Several numerical results that illustrate the efficiency of this adaptive preconditioner with a PCG solver are presented and the preconditioner is also compared with a previous variant in the literature

A posteriori error estimates, stopping criteria, and adaptivity for multiphase compositional Darcy flows in porous media

ABSTRACT In this paper we derive a posteriori error estimates for the compositional model of multiphase Darcy flow in porous media REFERENCE

Adaptive inexact smoothing Newton method for a nonconforming discretization of a variational inequality

International audienceWe develop in this work an adaptive inexact smoothing Newton method for a nonconforming discretization of a variational inequality. As a model problem, we consider the contact problem between two membranes. Discretized with the finite volume method, this leads to a nonlinear algebraic system with complementarity constraints. The non-differentiability of the arising nonlinear discrete problem a priori requests the use of an iterative linearization algorithm in the semismooth class like, e.g., the Newton-min. In this work, we rather approximate the inequality constraints by a smooth nonlinear equality, involving a positive smoothing parameter that should be drawn down to zero. This makes it possible to directly apply any standard linearization like the Newton method. The solution of the ensuing linear system is then approximated by any iterative linear algebraic solver. In our approach, we carry out an a posteriori error analysis where we introduce potential reconstructions in discrete subspaces included in H1 (Ω), as well as H (div, Ω)-conforming discrete equilibrated flux reconstructions. With these elements, we design an a posteriori estimate that provides guaranteed upper bound on the energy error between the unavailable exact solution of the continuous level and a postprocessed, discrete, and available approximation, and this at any resolution step. It also offers a separation of the different error components, namely, discretization, smoothing, linearization, and algebraic. Moreover, we propose stopping criteria and design an adaptive algorithm where all the iterative procedures (smoothing, linearization, algebraic) are adaptively stopped; this is in particular our way to fix the smoothing parameter. Finally, we numerically assess the estimate and confirm the performance of the proposed adaptive algorithm, in particular in comparison with the semismooth Newton method

A posteriori error estimates, stopping criteria, and adaptivity for multiphase compositional Darcy flows in porous media

International audienceIn this paper we derive a posteriori error estimates for the compositional model of multiphase Darcy flow in porous media, consisting of a system of strongly coupled nonlinear unsteady partial differential and algebraic equations. We show how to control the dual norm of the residual augmented by a nonconformity evaluation term by fully computable estimators. We then decompose the estimators into the space, time, linearization, and algebraic error components. This allows to formulate criteria for stopping the iterative algebraic solver and the iterative linearization solver when the corresponding error components do not affect significantly the overall error. Moreover, the spatial and temporal error components can be balanced by time step and space mesh adaptation. Our analysis applies to a broad class of standard numerical methods, and is independent of the linearization and of the iterative algebraic solvers employed. We exemplify it for the two-point finite volume method with fully implicit Euler time stepping, the Newton linearization, and the GMRes algebraic solver. Numerical results on two real-life reservoir engineering examples confirm that significant computational gains can be achieved thanks to our adaptive stopping criteria, already on fixed meshes, without any noticeable loss of precision

The unfinished agenda of communicable diseases among children and adolescents before the COVID-19 pandemic, 1990-2019: a systematic analysis of the Global Burden of Disease Study 2019

BACKGROUND: Communicable disease control has long been a focus of global health policy. There have been substantial reductions in the burden and mortality of communicable diseases among children younger than 5 years, but we know less about this burden in older children and adolescents, and it is unclear whether current programmes and policies remain aligned with targets for intervention. This knowledge is especially important for policy and programmes in the context of the COVID-19 pandemic. We aimed to use the Global Burden of Disease (GBD) Study 2019 to systematically characterise the burden of communicable diseases across childhood and adolescence. METHODS: In this systematic analysis of the GBD study from 1990 to 2019, all communicable diseases and their manifestations as modelled within GBD 2019 were included, categorised as 16 subgroups of common diseases or presentations. Data were reported for absolute count, prevalence, and incidence across measures of cause-specific mortality (deaths and years of life lost), disability (years lived with disability [YLDs]), and disease burden (disability-adjusted life-years [DALYs]) for children and adolescents aged 0-24 years. Data were reported across the Socio-demographic Index (SDI) and across time (1990-2019), and for 204 countries and territories. For HIV, we reported the mortality-to-incidence ratio (MIR) as a measure of health system performance. FINDINGS: In 2019, there were 3·0 million deaths and 30·0 million years of healthy life lost to disability (as measured by YLDs), corresponding to 288·4 million DALYs from communicable diseases among children and adolescents globally (57·3% of total communicable disease burden across all ages). Over time, there has been a shift in communicable disease burden from young children to older children and adolescents (largely driven by the considerable reductions in children younger than 5 years and slower progress elsewhere), although children younger than 5 years still accounted for most of the communicable disease burden in 2019. Disease burden and mortality were predominantly in low-SDI settings, with high and high-middle SDI settings also having an appreciable burden of communicable disease morbidity (4·0 million YLDs in 2019 alone). Three cause groups (enteric infections, lower-respiratory-tract infections, and malaria) accounted for 59·8% of the global communicable disease burden in children and adolescents, with tuberculosis and HIV both emerging as important causes during adolescence. HIV was the only cause for which disease burden increased over time, particularly in children and adolescents older than 5 years, and especially in females. Excess MIRs for HIV were observed for males aged 15-19 years in low-SDI settings. INTERPRETATION: Our analysis supports continued policy focus on enteric infections and lower-respiratory-tract infections, with orientation to children younger than 5 years in settings of low socioeconomic development. However, efforts should also be targeted to other conditions, particularly HIV, given its increased burden in older children and adolescents. Older children and adolescents also experience a large burden of communicable disease, further highlighting the need for efforts to extend beyond the first 5 years of life. Our analysis also identified substantial morbidity caused by communicable diseases affecting child and adolescent health across the world. FUNDING: The Australian National Health and Medical Research Council Centre for Research Excellence for Driving Investment in Global Adolescent Health and the Bill & Melinda Gates Foundation

The global burden of cancer attributable to risk factors, 2010-19 : a systematic analysis for the Global Burden of Disease Study 2019

Background Understanding the magnitude of cancer burden attributable to potentially modifiable risk factors is crucial for development of effective prevention and mitigation strategies. We analysed results from the Global Burden of Diseases, Injuries, and Risk Factors Study (GBD) 2019 to inform cancer control planning efforts globally. Methods The GBD 2019 comparative risk assessment framework was used to estimate cancer burden attributable to behavioural, environmental and occupational, and metabolic risk factors. A total of 82 risk-outcome pairs were included on the basis of the World Cancer Research Fund criteria. Estimated cancer deaths and disability-adjusted life-years (DALYs) in 2019 and change in these measures between 2010 and 2019 are presented. Findings Globally, in 2019, the risk factors included in this analysis accounted for 4.45 million (95% uncertainty interval 4.01-4.94) deaths and 105 million (95.0-116) DALYs for both sexes combined, representing 44.4% (41.3-48.4) of all cancer deaths and 42.0% (39.1-45.6) of all DALYs. There were 2.88 million (2.60-3.18) risk-attributable cancer deaths in males (50.6% [47.8-54.1] of all male cancer deaths) and 1.58 million (1.36-1.84) risk-attributable cancer deaths in females (36.3% [32.5-41.3] of all female cancer deaths). The leading risk factors at the most detailed level globally for risk-attributable cancer deaths and DALYs in 2019 for both sexes combined were smoking, followed by alcohol use and high BMI. Risk-attributable cancer burden varied by world region and Socio-demographic Index (SDI), with smoking, unsafe sex, and alcohol use being the three leading risk factors for risk-attributable cancer DALYs in low SDI locations in 2019, whereas DALYs in high SDI locations mirrored the top three global risk factor rankings. From 2010 to 2019, global risk-attributable cancer deaths increased by 20.4% (12.6-28.4) and DALYs by 16.8% (8.8-25.0), with the greatest percentage increase in metabolic risks (34.7% [27.9-42.8] and 33.3% [25.8-42.0]). Interpretation The leading risk factors contributing to global cancer burden in 2019 were behavioural, whereas metabolic risk factors saw the largest increases between 2010 and 2019. Reducing exposure to these modifiable risk factors would decrease cancer mortality and DALY rates worldwide, and policies should be tailored appropriately to local cancer risk factor burden. Copyright (C) 2022 The Author(s). Published by Elsevier Ltd. This is an Open Access article under the CC BY 4.0 license.Peer reviewe

Etude d'estimations d'erreur a posteriori et d'adaptivité basée sur des critères d'arrêt et raffinement de maillages pour des problèmes d'écoulements multiphasiques et thermiques. Application aux procédés de récupération assistée d'huile

The goal of this thesis is the a posteriori error analysis and the conception of adaptive strategies based on stopping criteria and local mesh refinement. We treat a class of multidimensional degenerate parabolic equations which represent typical examples of industrial interest. In Chapter 1 we consider the time-dependent two-phase Stefan problem that models a phase change process governed by the Fourier law. We regularize the relation between the enthalpy and the temperature and we discretize the problem by the backward Euler temporal stepping method with a conforming spatial discretization, such as the finite element or the vertex-centered finite volume one. We prove un upper bound for the dual norm of the residual, the L2(0; T;H-1) error in the enthalpy, and L2(0; T;L2) error in the temperature, by fully computable error estimators. These estimators include: an estimator associated to the regularization error, an estimator associated to the linearization error, an estimator associated to the temporal error, and an estimator associated to the spatial error. Consequently, these estimators allow to formulate an adaptive resolution algorithm where the corresponding errors can be equilibrated. We also propose a strategy of local mesh refinement. Finally, we prove the efficiency of our a posteriori estimates. A numerical test illustrates the efficiency of our estimates and the performance of the adaptive algorithm. In particular, effectivity indices close to the optimal value of 1 are obtained. In Chapter 2 we derive a posteriori error estimates for the isothermal compositional model of the multiphase Darcy ow in porous media, consisting of a system of strongly coupled nonlinear unsteady partial differential and nonlinear algebraic equations. This model is discretized by a cell-centered finite volume scheme in space with the backward Euler temporal stepping. We establish an upper bound for a dual norm of the residual augmented by a nonconformity evaluation term by fully computable estimators. We focus in this chapter on the formulation of criteria for the iterative linearization (such as the Newton method) and iterative algebraic solvers (such as the GMRes method) that stop the iterations when the corresponding error components no longer affect the overall estimate significantly. We apply our analysis to several real-life reservoir engineering examples to confirm that significant computational gains (up to an order of magnitude in terms of the total number of algebraic solver iterations) can be achieved thanks to our adaptive stopping criteria, already on fixed meshes, and this without any noticeable loss of precision. In Chapter 3 we complete the model described in Chapter 2 by considering a nonisothermal condition for the flow in order to treat the general thermal multiphase compositional foow in porous media. For this problem, we derive fully computable a posteriori error estimates analogous to Chapter 2 for a dual norm of the residual supplemented by a nonconformity evaluation term. We then show how to estimate separately the space, time, linearization, and algebraic errors, giving the possibility to formulate adaptive stopping and balancing criteria. Specification of the abstract theory to the so-called dead oil model closes the chapter. The proof of efficiency of our a posteriori estimate is also provided. Finally, in Chapter 4 we consider the Steam-Assisted Gravity Drainage (SAGD) process, more precisely a thermal oil-recovery technique of the deal oil type with steam injection designed to increase the oil mobility. The main subjects of this chapter are to apply the a posteriori error analysis of Chapters 2 and 3, propose a simplification and a quadrature formula for an easy evaluation of the estimators, propose a space-time adaptive mesh reffinement algorithm, and illustrate by numerical results on real-life examples its performance. In particular, a signi cant gain in terms of the number of mesh cells is achieved on examples in 3 space dimensions.L'objectif de cette thèse est l'analyse d'erreur a posteriori et la proposition de stratégies d'adaptivité basées sur des critères d'arrêt et de raffinement local de maillage. Nous traitons une classe d'équations paraboliques dégénér ées multidimensionnelles modélisant des problèmes importants pour l'industrie. Au chapitre 1 nous considérons le problème de Stefan instationaire a deux phases qui modélise un processus de changement de phase régi par la loi de Fourier. Nous régularisons la relation entre l'enthalpie et la température et nous discrétisons le problème par la méthode d'Euler implicite en temps et un schéma numérique conforme en espace tel que les élément finis conformes, ou les volumes finis centrés aux sommets du maillage. Nous démontrons une borne supérieure de la norme duale du résidu, de l'erreur sur l'enthalpie dans L2(0; T;H-1) et de l'erreur sur la température dans L2(0; T;L2), par des estimateurs d'erreur entièrement calculables. Ces estimateurs comprennent : un estimateur associé à l'erreur de régularisation, un estimateur associé à l'erreur d'une méthode de linéarisation (par exemple, la méthode de Newton), un estimateur associé à l'erreur en temps et un estimateur associé à l'erreur du schéma en espace. Par conséquent, ces estimateurs permettent de formuler un algorithme adaptatif de résolution où les erreurs associées peuvent être équilibrées. Nous proposons également une stratégie de raffinement local de maillages. En fin, nous prouvons l'efficacité de nos estimations d'erreur a posteriori. Un test numérique illustre l'efficacité de nos estimateurs et la performance de l'algorithme adaptatif. En particulier, des indices d'efficacité proches de la valeur optimale de 1 sont obtenus. Au chapitre 2 nous développons des estimations d'erreur a posteriori pour l'écoulement de Darcy polyphasique et isothermique, décrit par un système couplé d'équations aux dérivées partielles non linéaires et d'équations algébriques non linéaires. Ce système est discrétisé en espace par une méthode de volume finis centrés par maille et la méthode d'Euler implicite en temps. Nous etablissons une borne supérieure d'une norme duale du résidu augmentée d'un terme qui tiens compte de la non-conformité des volumes finis par des estimateurs d'erreur a posteriori entièrement calculables. Dans ce chapitre, nous nous concentrons sur la formulation d'un critère d'arrêt de l'algorithme de linéarisation du problème discrète (tel que la méthode de Newton) avec un critère d'arrêt du solveur algébrique de résolution du système linéarité (par exemple la méthode GMRes), de sort que les contributions des estimateurs d'erreur correspondant n'affectent plus la somme globale des estimateurs d'erreur de manière significative. Nous appliquons notre analyse sur des exemples réalistes d'ingénierie de réservoir pour confirmer qu'en général notre ajustement des critères d'arrêt apporte une économie significative (jusqu'au un ordre de magnitude en termes du nombre total des itérations du solveur algébrique), déjà sur des maillages fixes, et ceci sans perte notable de précision. Au chapitre 3 nous complétons le modèle décrit au chapitre 2 en considérant une condition non-isothermique pour l'écoulement a fin de traiter le modèle général d'écoulement polyphasique thermique dans les milieux poreux. Pour ce problème, nous développons des estimateurs d'erreur analogues a ceux du chapitre 2 pour lesquels nous établissons une borne supérieure d'erreur entièrement calculable, pour une norme duale du résidu complétée par un terme d'évaluation de la non-conformité. Nous montrons ensuite comment estimer séparément chaque composante d'erreur, ce qui nous permet d'ajuster les critères d'arrêt et d'équilibrer les contributions des différents estimateurs d'erreur : erreur d'approximation en temps, erreur d'approximation en espace, erreur de linéarisation et erreur du solveur algébrique. Ce chapitre se termine par une application des estimateurs au modèle d'huile morte. La preuve de l'efficacité de notre estimation a postiriori est egalement fournie. Finalement, au chapitre 4 nous considérons les procédés de récupération assistée d'huile. Plus précisément, nous étudions une technique de récupération thermique d'huile de type huile morte par injection de vapeur destinée a augmenter la mobilité des hydrocarbures. Dans ce chapitre, nous appliquons l'analyse a posteriori des chapitres 2 et 3, nous proposons une formule de quadrature pour simplifier l'évaluation des estimateurs, nous proposons un algorithme adaptatif de raffinement de maillages en espace et en temps basé sur les estimateurs et nous illustrons pas des essais numériques sur des exemples réalistes la performance de cette stratégie de raffinement. Notamment, des gains significatifs sont réalisés en terme du nombre de mailles nécessaires pour la simulation sur des exemples en dimension trois