IFISS: A Computational Laboratory for Investigating Incompressible Flow Problems

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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“…Both systems are real, unsymmetric and not diagonally dominant. More details can be found in [20,22].…”

Section: Implementation Issuesmentioning

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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“…The problems are solved to a preconditioned residual tolerance of 10 −5 . In order to construct the relevant (stiffness, mass and divergence) matrices we make use of the Ifiss software package [4,5,26]. When approximating the inverse of a mass matrix we apply 20 steps of Table 2: Number of iterations and CPU times in seconds (emphasized) when applying Minres, with our block diagonal preconditioner, to the first example for Problem (P1).…”

Section: Numerical Resultsmentioning

confidence: 99%

Fast iterative solvers for large matrix systems arising from time-dependent Stokes control problems

Pearson

2016

Applied Numerical Mathematics

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In this manuscript we consider the development of fast iterative solvers for Stokes control problems, an important class of PDE-constrained optimization problems. In particular we wish to develop effective preconditioners for the matrix systems arising from finite element discretizations of time-dependent variants of such problems. To do this we consider a suitable rearrangement of the matrix systems, and exploit the saddle point structure of many of the relevant sub-matrices involved -we may then use this to construct representations of these sub-matrices based on good approximations of their (1, 1)-block and Schur complement. We test our recommended iterative methods on a distributed control problem with Dirichlet boundary conditions, and on a time-periodic problem.

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“…We utilize the IFISS software package [2,16] to construct the relevant matrices. We apply 20 steps of the Chebyshev semi-iteration method to approximate the inverse of mass matrices, and the AGMG software whenever a multigrid process is needed.…”

Section: Numerical Resultsmentioning

confidence: 99%

Block triangular preconditioning for time‐dependent Stokes control

Pearson

2015

Proc Appl Math and Mech

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We consider the numerical solution of time-dependent Stokes control problems, an important class of flow control problems within the field of PDE-constrained optimization. The problems we examine lead to large and sparse matrix systems which, with suitable rearrangement, can be written in block tridiagonal form, with the diagonal blocks given by saddle point systems.Using previous results for preconditioning PDE-constrained optimization and fluid dynamics problems, along with wellstudied saddle point theory, we construct a block triangular preconditioner for the matrix systems. Numerical experiments verify the effectiveness of our solver. Problem statementFlow control problems constitute a crucial application area of PDE-constrained optimization. The construction of fast and robust preconditioned iterative solvers for such problems is therefore an important research area. The particular formulation for which we consider such methods in this paper relates to a time-dependent Stokes control problem examined in [11], where block diagonal preconditioners are sought. We write this problem as follows:We solve this formulation for spatial coordinates x ∈ Ω ⊂ R d (d ∈ {2, 3}), with boundary ∂Ω, and time t ∈ [0, T ]. The state variables v and p denote the velocity and pressure of the flow, with the control variable given by u; further v d denotes the desired state, and β the Tikhonov regularization parameter.We apply a discretize-then-optimize approach, using the backward Euler method in time (with N t time-steps and time-step τ ) and the trapezoidal rule in space, with a Taylor-Hood finite element method (that is we discretize using Q2 nodes on the velocity space, and Q1 nodes on the pressure space). The discretized system, upon suitable rearrangement, is of the form [11]

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IFISS: A Computational Laboratory for Investigating Incompressible Flow Problems

Cited by 101 publications

References 35 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

Fast iterative solvers for large matrix systems arising from time-dependent Stokes control problems

Block triangular preconditioning for time‐dependent Stokes control

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