Element-by-Element Preconditioners for Large Partially Separable Optimization Problems

Daydé, Michel; L’Excellent, Jean-Yves; Gould, Nicholas I. M.

doi:10.1137/s1064827594274796

Cited by 21 publications

(14 citation statements)

References 22 publications

(26 reference statements)

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“…The sparse matrix A is never assembled (computed). Typically, they compute an approximation to A −1 by computing the LU -factorizations of agglomerates of elements, and then combining them [17,30,26]. These preconditioners exhibit a high degree of parallelism and some can be applied to a vector using high-throughput dense matrixvector multiplications.…”

mentioning

confidence: 99%

IMF: An Incomplete Multifrontal $LU$-Factorization for Element-Structured Sparse Linear Systems

Vannieuwenhoven¹,

Meerbergen²

2013

SIAM J. Sci. Comput.

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We propose an incomplete multifrontal LU -factorization (IMF) that extends supernodal multifrontal methods to incomplete factorizations. IMF can be used as a preconditioner in a Krylov-subspace method to solve large-scale sparse linear systems with an underlying element structure. Such systems arise e.g. from a finite element discretization of a partial differential equation. The fact that the element matrices are dense is exploited to increase the computational performance and the robustness of the factorization (through partial pivoting within the dense matrices). We compare IMF with the multilevel ARMS, the level of fill-in ILU, the threshold-based ILUT and the SPAI preconditioner. IMF is demonstrated to be effective on linear systems derived from two incompressible flow simulation model problems, outperforming the aforementioned preconditioners by one order-of-magnitude in one instance. The preconditioner was also applied to solve general sparse systems, without an underlying element structure. It is shown to be effective and robust on some matrices from the University of Florida sparse matrix collection and the Matrix Market, provided that an artificial element structure can be extracted that is similar to a finite element discretization.Keywords : incomplete factorization, supernodal multifrontal method, multilevel preconditioner, block LU -factorization, finite element MSC : Primary : 65F08, 65N30 Secondary : 65Y05, 65F50, 65N22. IMF: AN INCOMPLETE MULTIFRONTAL LU-FACTORIZATION FOR ELEMENT-STRUCTURED SPARSE LINEAR SYSTEMS *NICK VANNIEUWENHOVEN ‡ AND KARL MEERBERGEN ‡ Abstract. We propose an incomplete multifrontal LU -factorization (IMF) that extends supernodal multifrontal methods to incomplete factorizations. IMF can be used as a preconditioner in a Krylov-subspace method to solve large-scale sparse linear systems with an underlying element structure. Such systems arise e.g. from a finite element discretization of a partial differential equation. The fact that the element matrices are dense is exploited to increase the computational performance and the robustness of the factorization (through partial pivoting within the dense matrices). We compare IMF with the multilevel ARMS, the level of fill-in ILU, the threshold-based ILUT and the SPAI preconditioner. IMF is demonstrated to be effective on linear systems derived from two incompressible flow simulation model problems, outperforming the aforementioned preconditioners by one order-of-magnitude in one instance. The preconditioner was also applied to solve general sparse systems, without an underlying element structure. It is shown to be effective and robust on some matrices from the University of Florida sparse matrix collection and the Matrix Market, provided that an artificial element structure can be extracted that is similar to a finite element discretization.

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mentioning

confidence: 99%

IMF: An Incomplete Multifrontal $LU$-Factorization for Element-Structured Sparse Linear Systems

Vannieuwenhoven¹,

Meerbergen²

2013

SIAM J. Sci. Comput.

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“…In [3], that structure was exploited in the design of efficient algorithms to compute the multilevel factorization in Equation (1). The factorization algorithm operates on the element structure of A k , rather than the sparse matrix A k .…”

Section: An Ebe-ilu Preconditionermentioning

confidence: 99%

“…They operate exclusively on the element matrices. A is typically approximated by combining the LU-factorizations of the element matrices [1]. They are presently out of favor due to their limited effectiveness.…”

Section: Introductionmentioning

confidence: 99%

An Element-by-Element Multilevel Block ILU Preconditioner Using GLAS

Vannieuwenhoven¹,

Meerbergen²

2010

AIP Conference Proceedings

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Abstract. We investigate a new element-by-element multilevel block-ILU preconditioner that combines the advantages of element-by-element (EBE) and incomplete LU-factorization (ILU) preconditioners. The preconditioner is constructed in EBE fashion with high-throughput dense operations. Numerical experiments demonstrate its effectiveness. Speedups as high as five were obtained over ILU(0) with a C++ implementation using GLAS.

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“…See e.g. [1,5,6,9,16,17,24,32,34] for approaches to partially separable optimization problems; and see [35,20] for smoothing techniques. To the best of our knowledge, none of the pre-existing work in portfolio optimization problems with transaction costs seems to have the versatility of our approach in terms of speed and accuracy.…”

Section: Background and Motivationmentioning

confidence: 99%

Large scale portfolio optimization with piecewise linear transaction costs

Potaptchik

Tunçel

Wolkowicz

2008

Optimization Methods and Software

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We consider the fundamental problem of computing an optimal portfolio based on a quadratic mean-variance model for the objective function and a given polyhedral representation of the constraints. The main departure from the classical quadratic programming formulation is the inclusion in the objective function of piecewise linear, separable functions representing the transaction costs. We handle the nonsmoothness in the objective function by using spline approximations. The problem is first solved approximately using a primal-dual interior-point method applied to the smoothed problem. Then, we crossover to an active set method applied to the original nonsmooth problem to attain a high accuracy solution. Our numerical tests show that we can solve large scale problems efficiently and accurately.

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Element-by-Element Preconditioners for Large Partially Separable Optimization Problems

Cited by 21 publications

References 22 publications

IMF: An Incomplete Multifrontal $LU$-Factorization for Element-Structured Sparse Linear Systems

IMF: An Incomplete Multifrontal $LU$-Factorization for Element-Structured Sparse Linear Systems

An Element-by-Element Multilevel Block ILU Preconditioner Using GLAS

Large scale portfolio optimization with piecewise linear transaction costs

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