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AuthorTempone, Raul (6)Keyes, David E. (5)Ketcheson, David I. (4)Parsani, Matteo (4)Ltaief, Hatem (3)View MoreDepartmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division (24)Applied Mathematics and Computational Science Program (20)Extreme Computing Research Center (12)Computer Science Program (5)Physical Sciences and Engineering (PSE) Division (5)View MoreJournal

SIAM Journal on Scientific Computing (32)

KAUST Acknowledged Support UnitExtreme Computing Research Center (1)OSR (1)KAUST Grant NumberCRG3 Award ref. 2281 (1)OSR-2018-CARF-3666 (1)PublisherSociety for Industrial & Applied Mathematics (SIAM) (32)Subjectdomain decomposition (4)BDDC (2)Domain decomposition (2)hyperbolic PDEs (2)Monte Carlo methods (2)View MoreTypeArticle (32)Year (Issue Date)2019 (3)2018 (2)2017 (3)2016 (8)2015 (5)View MoreItem Availability
Open Access (32)

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Multilevel Balancing Domain Decomposition by Constraints Deluxe Algorithms with Adaptive Coarse Spaces for Flow in Porous Media

Zampini, Stefano; Tu, Xuemin (SIAM Journal on Scientific Computing, Society for Industrial & Applied Mathematics (SIAM), 2017-08-03) [Article]

Multilevel balancing domain decomposition by constraints (BDDC) deluxe algorithms are developed for the saddle point problems arising from mixed formulations of Darcy flow in porous media. In addition to the standard no-net-flux constraints on each face, adaptive primal constraints obtained from the solutions of local generalized eigenvalue problems are included to control the condition number. Special deluxe scaling and local generalized eigenvalue problems are designed in order to make sure that these additional primal variables lie in a benign subspace in which the preconditioned operator is positive definite. The current multilevel theory for BDDC methods for porous media flow is complemented with an efficient algorithm for the computation of the so-called malign part of the solution, which is needed to make sure the rest of the solution can be obtained using the conjugate gradient iterates lying in the benign subspace. We also propose a new technique, based on the Sherman--Morrison formula, that lets us preserve the complexity of the subdomain local solvers. Condition number estimates are provided under certain standard assumptions. Extensive numerical experiments confirm the theoretical estimates; additional numerical results prove the effectiveness of the method with higher order elements and high-contrast problems from real-world applications.

Toward a High Performance Tile Divide and Conquer Algorithm for the Dense Symmetric Eigenvalue Problem

Haidar, Azzam; Ltaief, Hatem; Dongarra, Jack (SIAM Journal on Scientific Computing, Society for Industrial & Applied Mathematics (SIAM), 2012-01) [Article]

Classical solvers for the dense symmetric eigenvalue problem suffer from the first step, which involves a reduction to tridiagonal form that is dominated by the cost of accessing memory during the panel factorization. The solution is to reduce the matrix to a banded form, which then requires the eigenvalues of the banded matrix to be computed. The standard divide and conquer algorithm can be modified for this purpose. The paper combines this insight with tile algorithms that can be scheduled via a dynamic runtime system to multicore architectures. A detailed analysis of performance and accuracy is included. Performance improvements of 14-fold and 4-fold speedups are reported relative to LAPACK and Intel's Math Kernel Library.

On NonAsymptotic Optimal Stopping Criteria in Monte Carlo Simulations

Bayer, Christian; Hoel, Hakon; Von Schwerin, Erik; Tempone, Raul (SIAM Journal on Scientific Computing, Society for Industrial & Applied Mathematics (SIAM), 2014-01) [Article]

We consider the setting of estimating the mean of a random variable by a sequential stopping rule Monte Carlo (MC) method. The performance of a typical second moment based sequential stopping rule MC method is shown to be unreliable in such settings both by numerical examples and through analysis. By analysis and approximations, we construct a higher moment based stopping rule which is shown in numerical examples to perform more reliably and only slightly less efficiently than the second moment based stopping rule.

Highly efficient strong stability preserving Runge-Kutta methods with Low-Storage Implementations

Ketcheson, David I. (SIAM Journal on Scientific Computing, Society for Industrial & Applied Mathematics (SIAM), 2008-05-16) [Article]

Strong stability-preserving (SSP) Runge–Kutta methods were developed for time integration of semidiscretizations of partial differential equations. SSP methods preserve stability properties satisfied by forward Euler time integration, under a modified time-step restriction. We consider the problem of finding explicit Runge–Kutta methods with optimal SSP time-step restrictions, first for the case of linear autonomous ordinary differential equations and then for nonlinear or nonautonomous equations. By using alternate formulations of the associated optimization problems and introducing a new, more general class of low-storage implementations of Runge–Kutta methods, new optimal low-storage methods and new low-storage implementations of known optimal methods are found. The results include families of low-storage second and third order methods that achieve the maximum theoretically achievable effective SSP coefficient (independent of stage number), as well as low-storage fourth order methods that are more efficient than current full-storage methods. The theoretical properties of these methods are confirmed by numerical experiment.

Active-Set Reduced-Space Methods with Nonlinear Elimination for Two-Phase Flow Problems in Porous Media

Yang, Haijian; Yang, Chao-he; Sun, Shuyu (SIAM Journal on Scientific Computing, Society for Industrial & Applied Mathematics (SIAM), 2016-07-26) [Article]

Fully implicit methods are drawing more attention in scientific and engineering applications due to the allowance of large time steps in extreme-scale simulations. When using a fully implicit method to solve two-phase flow problems in porous media, one major challenge is the solution of the resultant nonlinear system at each time step. To solve such nonlinear systems, traditional nonlinear iterative methods, such as the class of the Newton methods, often fail to achieve the desired convergent rate due to the high nonlinearity of the system and/or the violation of the boundedness requirement of the saturation. In the paper, we reformulate the two-phase model as a variational inequality that naturally ensures the physical feasibility of the saturation variable. The variational inequality is then solved by an active-set reduced-space method with a nonlinear elimination preconditioner to remove the high nonlinear components that often causes the failure of the nonlinear iteration for convergence. To validate the effectiveness of the proposed method, we compare it with the classical implicit pressure-explicit saturation method for two-phase flow problems with strong heterogeneity. The numerical results show that our nonlinear solver overcomes the often severe limits on the time step associated with existing methods, results in superior convergence performance, and achieves reduction in the total computing time by more than one order of magnitude.

A Componentwise Convex Splitting Scheme for Diffuse Interface Models with Van der Waals and Peng--Robinson Equations of State

Fan, Xiaolin; Kou, Jisheng; Qiao, Zhonghua; Sun, Shuyu (SIAM Journal on Scientific Computing, Society for Industrial & Applied Mathematics (SIAM), 2017-01-19) [Article]

This paper presents a componentwise convex splitting scheme for numerical simulation of multicomponent two-phase fluid mixtures in a closed system at constant temperature, which is modeled by a diffuse interface model equipped with the Van der Waals and the Peng-Robinson equations of state (EoS). The Van der Waals EoS has a rigorous foundation in physics, while the Peng-Robinson EoS is more accurate for hydrocarbon mixtures. First, the phase field theory of thermodynamics and variational calculus are applied to a functional minimization problem of the total Helmholtz free energy. Mass conservation constraints are enforced through Lagrange multipliers. A system of chemical equilibrium equations is obtained which is a set of second-order elliptic equations with extremely strong nonlinear source terms. The steady state equations are transformed into a transient system as a numerical strategy on which the scheme is based. The proposed numerical algorithm avoids the indefiniteness of the Hessian matrix arising from the second-order derivative of homogeneous contribution of total Helmholtz free energy; it is also very efficient. This scheme is unconditionally componentwise energy stable and naturally results in unconditional stability for the Van der Waals model. For the Peng-Robinson EoS, it is unconditionally stable through introducing a physics-preserving correction term, which is analogous to the attractive term in the Van der Waals EoS. An efficient numerical algorithm is provided to compute the coefficient in the correction term. Finally, some numerical examples are illustrated to verify the theoretical results and efficiency of the established algorithms. The numerical results match well with laboratory data.

Adaptive Selection of Primal Constraints for Isogeometric BDDC Deluxe Preconditioners

Beirão Da Veiga, L.; Pavarino, L. F.; Scacchi, S.; Widlund, O. B.; Zampini, Stefano (SIAM Journal on Scientific Computing, Society for Industrial & Applied Mathematics (SIAM), 2017-02-23) [Article]

Isogeometric analysis has been introduced as an alternative to finite element methods in order to simplify the integration of computer-aided design (CAD) software and the discretization of variational problems of continuum mechanics. In contrast with the finite element case, the basis functions of isogeometric analysis are often not nodal. As a consequence, there are fat interfaces which can easily lead to an increase in the number of interface variables after a decomposition of the parameter space into subdomains. Building on earlier work on the deluxe version of the BDDC (balancing domain decomposition by constraints) family of domain decomposition algorithms, several adaptive algorithms are developed in this paper for scalar elliptic problems in an effort to decrease the dimension of the global, coarse component of these preconditioners. Numerical experiments provide evidence that this work can be successful, yielding scalable and quasi-optimal adaptive BDDC algorithms for isogeometric discretizations.

Optimized Explicit Runge--Kutta Schemes for the Spectral Difference Method Applied to Wave Propagation Problems

Parsani, Matteo; Ketcheson, David I.; Deconinck, W. (SIAM Journal on Scientific Computing, Society for Industrial & Applied Mathematics (SIAM), 2013-04-10) [Article]

Explicit Runge--Kutta schemes with large stable step sizes are developed for integration of high-order spectral difference spatial discretizations on quadrilateral grids. The new schemes permit an effective time step that is substantially larger than the maximum admissible time step of standard explicit Runge--Kutta schemes available in the literature. Furthermore, they have a small principal error norm and admit a low-storage implementation. The advantages of the new schemes are demonstrated through application to the Euler equations and the linearized Euler equations.

Entropy Stable Staggered Grid Discontinuous Spectral Collocation Methods of any Order for the Compressible Navier--Stokes Equations

Parsani, Matteo; Carpenter, Mark H.; Fisher, Travis C.; Nielsen, Eric J. (SIAM Journal on Scientific Computing, Society for Industrial & Applied Mathematics (SIAM), 2016-10-04) [Article]

Staggered grid, entropy stable discontinuous spectral collocation operators of any order are developed for the compressible Euler and Navier--Stokes equations on unstructured hexahedral elements. This generalization of previous entropy stable spectral collocation work [M. H. Carpenter, T. C. Fisher, E. J. Nielsen, and S. H. Frankel, SIAM J. Sci. Comput., 36 (2014), pp. B835--B867, M. Parsani, M. H. Carpenter, and E. J. Nielsen, J. Comput. Phys., 292 (2015), pp. 88--113], extends the applicable set of points from tensor product, Legendre--Gauss--Lobatto (LGL), to a combination of tensor product Legendre--Gauss (LG) and LGL points. The new semidiscrete operators discretely conserve mass, momentum, energy, and satisfy a mathematical entropy inequality for the compressible Navier--Stokes equations in three spatial dimensions. They are valid for smooth as well as discontinuous flows. The staggered LG and conventional LGL point formulations are compared on several challenging test problems. The staggered LG operators are significantly more accurate, although more costly from a theoretical point of view. The LG and LGL operators exhibit similar robustness, as is demonstrated using test problems known to be problematic for operators that lack a nonlinear stability proof for the compressible Navier--Stokes equations (e.g., discontinuous Galerkin, spectral difference, or flux reconstruction operators).

PCBDDC: A Class of Robust Dual-Primal Methods in PETSc

Zampini, Stefano (SIAM Journal on Scientific Computing, Society for Industrial & Applied Mathematics (SIAM), 2016-10-27) [Article]

A class of preconditioners based on balancing domain decomposition by constraints methods is introduced in the Portable, Extensible Toolkit for Scientific Computation (PETSc). The algorithm and the underlying nonoverlapping domain decomposition framework are described with a specific focus on their current implementation in the library. Available user customizations are also presented, together with an experimental interface to the finite element tearing and interconnecting dual-primal methods within PETSc. Large-scale parallel numerical results are provided for the latest version of the code, which is able to tackle symmetric positive definite problems with highly heterogeneous distributions of the coefficients. Current limitations and future extensions of the preconditioner class are also discussed.

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