215 research outputs found
DERIVING A NEW DOMAIN DECOMPOSITION METHOD FOR THE STOKES EQUATIONS USING THE SMITH FACTORIZATION
No full textIn this paper the Smith factorization is used systematically to derive a new domain decomposition method for the Stokes problem. In two dimensions the key idea is the transformation of the Stokes problem into a scalar bi-harmonic problem. We show, how a proposed domain decomposition method for the bi-harmonic problem leads to a domain decomposition method for the Stokes equations which inherits the convergence behavior of the scalar problem. Thus, it is sufficient to study the convergence of the scalar algorithm. The same procedure can also be applied to the three-dimensional Stokes problem. As transmission conditions for the resulting domain decomposition method of the Stokes problem we obtain natural boundary conditions. Therefore it can be implemented easily. A Fourier analysis and some numerical experiments show very fast convergence of the proposed algorithm. Our algorithm shows a more robust behavior than Neumann-Neumann or FETI type methods
A domain decomposition method for solving the three-dimensional time-harmonic Maxwell equations discretized by discontinuous Galerkin methods
Full text linkWe present here a domain decomposition method for solving the three-dimensional time-harmonic Maxwell equations discretized by a discontinuous Galerkin method. In order to allow the treatment of irregularly shaped geometries, the discontinuous Galerkin method is formulated on unstructured tetrahedral meshes. The domain decomposition strategy takes the form of a Schwarz-type algorithm where a continuity condition on the incoming characteristic variables is imposed at the interfaces between neighboring subdomains. A multifrontal sparse direct solver is used at the subdomain level. The resulting domain decomposition strategy can be viewed as a hybrid iterative/direct solution method for the large, sparse and complex coefficients algebraic system resulting from the discretization of the time-harmonic Maxwell equations by a discontinuous Galerkin method
A course space construction based on local Dirichlet-to-Neumann maps
Full text linkCoarse-grid correction is a key ingredient of scalable domain decomposition methods. In this work we construct coarse-grid space using the low-frequency modes of the subdomain Dirichlet-to-Neumann maps and apply the obtained two-level preconditioners to the extended or the original linear system arising from an overlapping domain decomposition. Our method is suitable for parallel implementation, and its efficiency is demonstrated by numerical examples on problems with large heterogeneities for both manual and automatic partitionings
Optimized Schwarz methods for Maxwell's equations
Full text linkOver the last two decades, classical Schwarz methods have been extended to systems of hyperbolic partial differential equations, using characteristic transmission conditions, and it has been observed that the classical Schwarz method can be convergent even without overlap in certain cases. This is in strong contrast to the behavior of classical Schwarz methods applied to elliptic problems, for which overlap is essential for convergence. More recently, optimized Schwarz methods have been developed for elliptic partial differential equations. These methods use more effective transmission conditions between subdomains than the classical Dirichlet conditions, and optimized Schwarz methods can be used both with and without overlap for elliptic problems. We show here why the classical Schwarz method applied to both the time harmonic and time discretized Maxwell's equations converges without overlap: the method has the same convergence factor as a simple optimized Schwarz method for a scalar elliptic equation. Based on this insight, we develop an entire new hierarchy of optimized overlapping and nonoverlapping Schwarz methods for Maxwell's equations with greatly enhanced performance compared to the classical Schwarz method. We also derive for each algorithm asymptotic formulas for the optimized transmission conditions, which can easily be used in implementations of the algorithms for problems with variable coefficients. We illustrate our findings with numerical experiments
Two-level domain decomposition methods for highly heterogeneous Darcy equations. Connections with multiscale methods
No full textMultiphase, compositional porous media flow models lead to the solution of highly heterogeneous systems of Partial Differential Equations (PDE). We focus on overlapping Schwarz type methods on parallel computers and on multiscale methods. We present a coarse space [Nataf F., Xiang H., Dolean V., Spillane N. (2011) SIAM J. Sci. Comput. 33, 4, 1623-1642] that is robust even when there are such heterogeneities. The two-level domain decomposition approach is compared to multiscale methods
Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps
Full text linkCoarse spaces are instrumental in obtaining scalability for domain decomposition methods for partial differential equations (PDEs). However, it is known that most popular choices of coarse spaces perform rather weakly in the presence of heterogeneities in the PDE coefficients, especially for systems of PDEs. Here, we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities. We achieve this by solving local generalized eigenvalue problems in the overlaps of subdomains that isolate the terms responsible for slow convergence. We prove a general theoretical result that rigorously establishes the robustness of the new coarse space and give some numerical examples on two and three dimensional heterogeneous PDEs and systems of PDEs that confirm this property
Scalable Domain Decomposition Methods for Nonlinear and Time-Dependent Stochastic Systems
No full textComputational modelling is one of the most important tools to understand and predict real-life physical processes. However, the accuracy of their predictions becomes questionable when the uncertainties associated with model parameters, assumptions to the mathematical models and the noise in experimental data are not properly accounted for. Sampling-based approaches to handle these uncertainties become overwhelmingly expensive for large-scale models with high resolution discretizations in space/time. This thesis proposes a sampling-free intrusive stochastic Galerkin-based approach to handle the uncertainties associated with model parameters for time-dependent and nonlinear problems. The increased cost of solving high resolution models using this sampling-free approach is handled using domain decomposition (DD)-based solvers by efficiently distributing the workload to many processes. Developing parallel scalable iterative solvers for uncertainty quantification of these high-resolution models in high performance computing (HPC) environments is the main objective of this thesis. An acoustic wave propagation model with a random field representation of wave speed is handled using a non-overlapping DD method. The symmetric and positive-definite coefficient matrix of the system can be solved using a conjugate-gradient iterative method and associated Neumann-Neumann vertex-based preconditioner in two dimensions. However, the complex spatial coupling and the coupling among the stochastic expansion coefficients can affect the scalabilities of the solver in three dimensions. Hence, a wirebasket-based preconditioner is utilized to enrich the coarse grid allowing better global error propagation and improved scalability for the elastic wave propagation model. For nonlinear stochastic partial differential equations (PDEs), the coefficient matrix of the associated linearized algebraic system is non-symmetric which requires the use of generalized minimum residual (GMRES) method-based iterative solvers. A multilevel Schwarz preconditioner combining DD and algebraic multigrid method is proposed for efficient error reduction for large-scale models. The scalabilities of the proposed solver are demonstrated for the PDE-based compartmental model of the geospatial spread of COVID-19 considering uncertain population movement in a large geographical domain of Southern Ontario
Numerical solution for a portable medical scanner
No full textLes déchirures de la coiffe des rotateurs (RCTs) représentent l'une des blessures les plus fréquentes de l'épaule et évoluent souvent vers des conditions plus graves au fil du temps. Bien que l'IRM soit la modalité d'imagerie standard pour détecter les RCTs, son utilisation est limitée aux centres d'imagerie et elle n'est pas toujours précise pour représenter la présence et l'étendue des déchirures. Un outil de diagnostic portable, non invasif et rentable pour le diagnostic sur site des RCTs est en demande. Cette thèse présente les contributions apportées au développement d'un modèle numérique pour ce scanner médical, qui repose sur la résolution répétée du problème Maxwell. La discrétisation par éléments finis de ce problème aboutit à un système linéaire de grande taille et mal conditionné, difficile à résoudre. Notre première contribution est le développement d'un préconditionneur de type Schwarz basé sur le PML qui améliore l'efficacité de la résolution de ce problème, en termes de taux de convergence et de temps de calcul. Ensuite, nous utilisons une modélisation numérique de pointe pour introduire un système d'imagerie portable pour la reconstruction tridimensionnelle de l'épaule. Cette tâche est difficile en raison de la grande taille électrique de l'épaule, de son anatomie complexe et de la nature hétérogène des tissus, caractérisée par des pertes élevées. L'étude de faisabilité montre des résultats prometteurs dans la détection des RCTs. Cependant, cette méthode peut être limitée par de forts niveaux de bruit ou les habitudes corporelles du patient. Pour remédier à cela, nous générons un grand ensemble de données, en utilisant une version optimisée du système d'imagerie et nous employons un algorithme d'apprentissage automatique pour la détection automatique et en temps réel des RCTs.Rotator cuff tears (RCTs) represent one of the most frequent shoulder injuries and often progress to more severe conditions over time. Although MRI is the standard imaging modality for detecting RCTs, it is limited to use in imaging centers, and it is not always accurate in depicting the presence and extent of the tears. A portable, non-invasive, and cost-effective diagnostic tool for on-site diagnosis of RCTs is in demand. This thesis presents the contributions made to the development of a numerical model for this medical scanner which relies on repeated solves of Maxwell's equations. The finite element discretization of this problem results in a large-scale, ill-conditioned linear system that is challenging to solve. Our first contribution is the development of a PML-based Schwarz-type preconditioner that improves the efficiency of the solution method, in terms of convergence rate and computing time. Next, we utilize state-of-the-art numerical modeling to design a wearable imaging system for three-dimensional image reconstruction of the shoulder. This task is challenging due to the electrically large size of the shoulder, its complex anatomy, and the heterogeneous nature of the tissues, which are characterized by high losses. The feasibility study shows promising results in the detection of RCTs. However, this method can be limited by high noise levels or the patient's body habits. To address this, we generate a large dataset, using an optimized version of the numerical imaging system and employ a machine learning algorithm, for automatic and real-time detection of RCTs
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