Playa: High-Performance Programmable Linear Algebra

Howle, Victoria E.; Kirby, Robert C.; Long, Kevin R.; Brennan, Brian; Kennedy, Kimberly R

doi:10.1155/2012/606215

Cited by 7 publications

(11 citation statements)

References 14 publications

(16 reference statements)

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“…(This presents some practical complications, as the linear algebra has a nested block structure with the fluid block itself having subblocks. We address the software‐related issues in a separate publication . )…”

Section: Numerical Results — 2dmentioning

confidence: 99%

“…(This presents some practical complications, as the linear algebra has a nested block structure with the fluid block itself having subblocks. We address the software-related issues in a separate publication [26].) In Tables IV-VII, we apply the preconditioner (13) by using varying degrees of accuracy on the subsolves, starting with all tight tolerances and gradually loosening the tolerances on the subsolves.…”

Section: The 2 2 Block Preconditioner (2d Results)mentioning

confidence: 99%

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Block preconditioners for finite element discretization of incompressible flow with thermal convection

Howle

Kirby

2012

Numerical Linear Algebra App

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SUMMARY We derive block preconditioners for a finite element discretization of incompressible flow coupled to heat transport by the Boussinesq approximation. Our techniques rely on effectively approximating the Schur complement obtained by eliminating the fluid variables to obtain an equation for temperature alone. Additionally, the method utilizes existing block‐structured preconditioners and multilevel methods for the Navier–Stokes equations and scalar convection–diffusion. We find that the preconditioner remains robust and scalable even when the subsolves are applied quite inexactly. Copyright © 2012 John Wiley & Sons, Ltd.

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Section: Numerical Results — 2dmentioning

confidence: 99%

Section: The 2 2 Block Preconditioner (2d Results)mentioning

confidence: 99%

Block preconditioners for finite element discretization of incompressible flow with thermal convection

Howle

Kirby

2012

Numerical Linear Algebra App

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“…1. Centrally important to Sundance's interoperation with other Trilinos packages is one package, Playa [11], which provides Sundance with a single point of contact for the various vector, operator, and solver types available through Trilinos.…”

Section: Interoperation With Other Trilinos Packagesmentioning

confidence: 99%

“…The problem specification building blocks have subclasses. Sundance uses the reference-counted handle idiom ( [11] for discussion of this idiom in the context of Playa) to provide polymorphism and safe memory management along with value syntax. The actual relationship between a handle (Expr), a pointer to a base class (ExprBase), and a derived class (CoordExpr) is as shown in the left side of Fig.…”

Section: Overview Of Object Architecturementioning

confidence: 99%

Sundance: High-Level Software for PDE-Constrained Optimization

Long

Boggs

Waanders

2012

Scientific Programming

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Abstract. Sundance is a package in the Trilinos suite designed to provide high-level components for the development of highperformance PDE simulators with built-in capabilities for PDE-constrained optimization. We review the implications of PDEconstrained optimization on simulator design requirements, then survey the architecture of the Sundance problem specification components. These components allow immediate extension of a forward simulator for use in an optimization context. We show examples of the use of these components to develop full-space and reduced-space codes for linear and nonlinear PDE-constrained inverse problems.

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“…While a high-level PDE engine makes it possible to obtain these new operators at low user cost, additional care is required to develop a clean, extensible interface. For example, the PCD preconditioner has been implemented using Sundance and Playa [23], although the resulting code tightly fused the description of the problem with a highly specialised specification of the preconditioner. Similarly, in the FEniCS project, cbc.block [31] allows the model developer to write complex block preconditioners as a composition of high-level "symbolic" linear algebra operations; Trilinos provides similar functionality through Teko [12].…”

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confidence: 99%

Solver Composition Across the PDE/Linear Algebra Barrier

Kirby¹,

Mitchell²

2018

SIAM J. Sci. Comput.

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Abstract. The efficient solution of discretizations of coupled systems of partial differential equations (PDEs) is at the core of much of numerical simulation. Significant effort has been expended on scalable algorithms to precondition Krylov iterations for the linear systems that arise. With few exceptions, the reported numerical implementation of such solution strategies is specific to a particular model setup, and intimately ties the solver strategy to the discretization and PDE, especially when the preconditioner requires auxiliary operators. In this paper, we present recent improvements in the Firedrake finite element library that allow for straightforward development of the building blocks of extensible, composable preconditioners that decouple the solver from the model formulation. Our implementation extends the algebraic composability of linear solvers offered by the PETSc library by augmenting operators, and hence preconditioners, with the ability to provide any necessary auxiliary operators. Rather than specifying up front the full solver configuration tied to the model, solvers can be developed independently of model formulation and configured at runtime. We illustrate with examples from incompressible fluids and temperature-driven convection.Key words. iterative methods, preconditioning, composable solvers, multiphysics AMS subject classifications. 65N22, 65F08, 65F10 DOI. 10.1137/17M11332081. Introduction. For over a decade, domain-specific languages for numerical partial differential equations (henceforth PDEs) such as Sundance [30,29], FEniCS [28], and now Firedrake [40] have enabled users to efficiently generate algebraic systems from a high-level description of the variational problems. Both FEniCS and Firedrake make use of the Unified Form Language (UFL) [1] as a description language for the weak forms of PDEs, converting it into efficient low-level code for form evaluation. They also share a Python interface that, for the intersection of their feature sets, is nearly source-compatible. These high-level PDE codes succeed by connecting a rich description language for PDEs to effective lower-level solver libraries such as PETSc [4,5] or Trilinos [21] for the requisite, and performance-critical, numerical (non)linear algebra.These high-level PDE projects utilize the solver packages in an essentially unidirectional way: the residuals are evaluated, Jacobians formed, and are then handed off to mainly algebraic techniques. Hence, the codes work at their best when (compositions of) existing black-box matrix techniques effectively solve the algebraic systems. However, in many situations the best preconditioners require additional structure

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Playa: High-Performance Programmable Linear Algebra

Cited by 7 publications

References 14 publications

Block preconditioners for finite element discretization of incompressible flow with thermal convection

Block preconditioners for finite element discretization of incompressible flow with thermal convection

Sundance: High-Level Software for PDE-Constrained Optimization

Solver Composition Across the PDE/Linear Algebra Barrier

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