A mixed-precision algorithm for the solution of Lyapunov equations on hybrid CPU–GPU platforms

Benner, Peter; Ezzatti, Pablo; Kreßner, Daniel; Quintana–Ort́ı, Enrique S.; Remón, Alfredo

doi:10.1016/j.parco.2010.12.002

Cited by 35 publications

(20 citation statements)

References 25 publications

(27 reference statements)

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“…ComputeR k+1 via a matrix-matrix product (n 2 ×k c flops) Therefore, most of the computational effort is concentrated on the calculation of the matrix inverse A k −1 . This is reinforced from the numerical results reported in a previous work, where the same method is employed to solve a single Lyapunov equation (only steps 1 to 3 are required) with general dense coefficient matrices [5]. In that work, despite the use of a GPU to accelerate the computation of the inverse, this operation represented the 85% and 91% of the total computation time for two problems of dimension 5, 177 and 9, 699 respectively.…”

Section: The Sign Function Methodsmentioning

confidence: 78%

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Efficient Model Order Reduction of Large-Scale Systems on Multi-core Platforms

Ezzatti

Quintana–Ort́ı

Remón

2011

Computational Science and Its Applications - ICCSA 2011

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We propose an efficient implementation of the Balanced Truncation (BT) method for model order reduction when the state-space matrix is symmetric (positive definite). Most of the computational effort required by this method is due to the computation of matrix inverses. Two alternatives for the inversion of a symmetric positive definite matrix on multi-core platforms are studied and evaluated, the traditional approach based on the Cholesky factorization and the Gauss-Jordan elimination algorithm. Implementations of both methods have been developed and tested. Numerical experiments show the efficiency attained by the proposed implementations on the target architecture.

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Section: The Sign Function Methodsmentioning

confidence: 78%

“…In previous works we have addressed the cases where the state-space matrix A is sparse [4], a general dense matrix [5], and a band matrix [6]. In this work we focus in the case when −A is a dense symmetric positive definite (SPD) matrix.…”

Section: Introductionmentioning

confidence: 99%

Efficient Model Order Reduction of Large-Scale Systems on Multi-core Platforms

Ezzatti

Quintana–Ort́ı

Remón

2011

Computational Science and Its Applications - ICCSA 2011

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“…The necessary theory was derived in [80] and [76,Chapter 10]. We will provide this here in some detail, as the defect correction principle has, despite its importance for practice, received little attention since, and in particular new computing platforms including hardware accelerators may use this principle for obtaining fast and reliable algorithms, as has been already suggested for Lyapunov equation in [18].…”

Section: Defect Correctionmentioning

confidence: 97%

“…An interesting aspect of further research would be to derive a mixed precision CPU-GPU variant of Algorithm DC_CARE in the fashion of [18]. Also, in case of large-scale sparse solvers for CAREs as recently reviewed in [32], where low-rank approximations to the stabilizing solution are computed, it would be necessary to be able to represent Q F in low-rank format to use this concept.…”

Section: Theorem 8 Suppose the Invariant Subspace Of The Hamiltonian mentioning

confidence: 99%

Theory and Numerical Solution of Differential and Algebraic Riccati Equations

Benner

2015

Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory

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Since Kalman's seminal work on linear-quadratic control and estimation problems in the early 1960s, Riccati equations have been playing a central role in many computational methods for solving problems in systems and control theory, like controller design, Kalman filtering, model reduction, and many more. We will review some basic theoretical facts as well as computational methods to solve them, with a special emphasis on the many contributions Volker Mehrmann had regarding these subjects. IntroductionThe algebraic and differential Riccati equations (AREs/DREs) play a fundamental role in the solution of problems in systems and control theory. They have found widespread applications in applied mathematics and engineering, many of which can be found in the monographs [1,36,68,86]. In this chapter, we focus on Riccati equations associated to control problems, as these have always inspired Volker Mehrmann's work, and he has mainly focused on the resulting symmetric Riccati equations -symmetric in the sense that the associated Riccati operators map symmetric (Hermitian) matrices onto symmetric (Hermitian) matrices. Hence, also the solutions to the Riccati equations we will consider are expected to be symmetric (Hermitian). A class of nonsymmetric AREs that arises, e.g., in queuing theory, certain fluid flow problems, and transport theory (see, e.g., [36]) is of importance as well, but for conciseness, we will omit these AREs even though Volker has also contributed to this area [81].In most of the literature on AREs and DREs, the motivation is taken from the classical linear-quadratic regulator (LQR) problem. This was the topic of Volker's habilitation thesis [74], where, building upon earlier work by Bender and Laub [6,7], he extended the LQR theory to so-called descriptor systems, i.e., systems with

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“…In recent years, Graphics Processing Units (GPUs) have shown remarkable performance in the computation of large-scale matrix operations, and particularly, in the solution of matrix equations; see [4,5] among others. In addition to its performance, GPUs present other interesting properties, such as a low Watt-per-floating-point arithmetic operation ratio and an afforable price.…”

Section: Introductionmentioning

confidence: 99%

Extending lyapack for the solution of band Lyapunov equations on hybrid CPU–GPU platforms

Benner

Remón

Dufrechou³

et al. 2014

J Supercomput

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The solution of large-scale Lyapunov equations is an important tool for the solution of several engineering problems arising in optimal control and model order reduction. In this work we investigate the case when the coefficient matrix of the equations presents a band structure. Exploiting the structure of this matrix we can achive relevant reductions in the memory requirements and the number of floating-point operations. Additionally, the new solver efficiently leverages the parallelism of CPU-GPU platforms. Furthermore, it is integrated in the lyapack library to facilitate its use. The new codes are evaluated on the solution of several benchmarks, exposing significant runtime reductions with respect to the original CPU version in lyapack.

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A mixed-precision algorithm for the solution of Lyapunov equations on hybrid CPU–GPU platforms

Cited by 35 publications

References 25 publications

Efficient Model Order Reduction of Large-Scale Systems on Multi-core Platforms

Efficient Model Order Reduction of Large-Scale Systems on Multi-core Platforms

Theory and Numerical Solution of Differential and Algebraic Riccati Equations

Extending lyapack for the solution of band Lyapunov equations on hybrid CPU–GPU platforms

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