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The Induced Dimension Reduction method applied to convection-diffusion-reaction problems

Astudillo, R. (author), Van Gijzen, M.B. (author)

Discretization of (linearized) convection-diffusion-reaction problems yields a large and sparse non symmetric linear system of equations, Ax = b. (1) In this work, we compare the computational behavior of the Induced Dimension Reduction method (IDR(s)) [10], with other short-recurrences Krylov methods, specifically the Bi-Conjugate Gradient...

report 2016

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Preconditioning optimal in-domain control of navier-stokes equation using multilevel sequentially semiseparable matrix computations

Qiu, Y. (author), Van Gijzen, M.B. (author), Van Wingerden, J.W. (author), Verhaegen, M.H.G. (author), Vuik, C. (author)

In this manuscript, we study preconditioning techniques for optimal in-domain control of the Navier-Stokes equation, where the control only acts on a few parts of the domain. Optimization and linearization of the optimal in-domain control problem results in a generalized linear saddle-point system. The Schur complement for the generalized saddle...

report 2015

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Induced Dimension Reduction method for solving linear matrix equations

Astudillo, R. (author), Van Gijzen, M.B. (author)

This paper discusses the solution of large-scale linear matrix equations using the Induced Dimension reduction method (IDR(s)). IDR(s) was originally presented to solve system of linear equations, and is based on the IDR(s) theorem. We generalize the IDR(s) theorem to solve linear problems in any finite-dimensional space. This generalization...

report 2015

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Preconditioning Optimal In-Domain Control of Navier-Stokes Equation Using Multilevel Sequentially Semiseparable Matrix Computations

Qiu, Y. (author), Van Gijzen, M.B. (author), Van Wingerden, J.W. (author), Verhaegen, M. (author), Vuik, C. (author)

In this manuscript, we study preconditioning techniques for optimal in-domain control of the Navier-Stokes equation, where the control only acts on a few parts of the domain. Optimization and linearization of the optimal in-domain control problem results in a generalized linear saddle-point system. The Schur complement for the generalized saddle...

report 2015

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A Restarted Induced Dimension Reduction method to approximate eigenpairs of large unsymmetric matrices

Astudillo, R. (author), Van Gijzen, M.B. (author)

A new algorithm to compute eigenpairs of large unsymmetric matrices is presented. Using the Induced Dimension Reduction method (IDR), which was originally proposed for solving linear systems, we obtain a Hessenberg decomposition from which we approximate the eigen-values and eigenvectors of a matrix. This decomposition has two main advantages....

report 2014

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Efficient Preconditioners for PDE-Constrained Optimization Problems with a Multi-level Sequentially Semi-Separable Matrix Structure

Qiu, Y. (author), Van Gijzen, M.B. (author), Van Wingerden, J.W. (author), Verhaegen, M. (author), Vuik, C. (author)

In this paper, we consider preconditioning for PDE-constrained optimization problems. The underlying problems yield a linear saddle-point system. We study a class of preconditioners based on multilevel sequentially semiseparable (MSSS) matrix computations. The novel global preconditioner is to make use of the global structure of the saddle-point...

report 2014

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Nested Krylov methods for shifted linear systems

Baumann, M. (author), Van Gijzen, M.B. (author)

report 2014

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The eigenvectors corresponding to the second eigenvalue of the google matrix and their relation to link spamming

Sangers, A. (author), Van Gijzen, M.B. (author)

Google uses the PageRank algorithm to determine the relative importance of a website. Link spamming is the name for putting links between websites with no other purpose than to increase the PageRank value of a website. To give a fair result to a search query it is important to detect whether a website is link spammed so that it can be filtered...

report 2013

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Evaluation of Multilevel Sequentially Semiseparable Preconditioners on CFD Benchmark Problems Using IFISS

Qiu, Y. (author), Van Gijzen, M.B. (author), Van Wingerden, J. (author), Verhaegen, M. (author), Vuik, C. (author)

This paper studies a new preconditioning technique for sparse systems arising from discretized partial differential equations (PDEs) in computational fluid dynamics (CFD), which exploit the multilevel sequentially semiseparable (MSSS) structure of the system matrix. MSSS matrix computations give a data-sparse way to approximate the LU...

report 2013

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An induced dimension reduction algorithm to approximate eigenpairs of large nonsymmetric matrices

Astudillo, R. (author), Van Gijzen, M.B. (author)

This work presents an algorithm to approximate eigenpairs of large, sparse and nonsymmetric matrices based on the Induced Dimension Reduction method (IDR(s)) introduced in [1]. We obtain a Hessenberg relation from IDR(s) computations and in conjunction with Implicitly Restarting and shift-and-invert techniques [2] we created a short recurrence...

conference paper 2013

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A parallel linear solver exploiting the physical properties of the underlying mechanical problem

Lingen, F.J. (author), Bonnier, P.G. (author), Brinkgreve, R.B.J. (author), Van Gijzen, M.B. (author), Vuik, C. (author)

report 2012

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Comparison of the deflated preconditioned conjugate gradient method and parallel direct solver for composite materials

Jönsthövel, T.B. (author), Van Gijzen, M.B. (author), MacLachlan, S. (author), Vuik, C. (author), Scarpas, A. (author)

The demand for large FE meshes increases as parallel computing becomes the standard in FE simulations. Direct and iterative solution methods are used to solve the resulting linear systems. Many applications concern composite materials, which are characterized by large discontinuities in the material properties. An example of such a material is...

report 2011

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Efficient two-level preconditionined conjugate gradient method on the GPU

Gupta, R. (author), Van Gijzen, M.B. (author), Vuik, K. (author)

We present an implementation of Two-Level Preconditioned Conjugate Gradient Method for the GPU. We investigate a Truncated Neumann Series based preconditioner in combination with deflation and compare it with Block Incomplete Cholesky schemes. This combination exhibits fine-grain parallelism and hence we gain considerably in execution time. It’s...

report 2011

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On the use of rigid body modes in the deflated preconditioned conjugate gradient method

Jönsthövel, T.B. (author), Van Gijzen, M.B. (author), Vuik, C. (author), Scarpas, A. (author)

Large discontinuities in material properties, such as encountered in composite materials, lead to ill-conditioned systems of linear equations. These discontinuities give rise to small eigenvalues that may negatively affect the convergence of iterative solution methods such as the Preconditioned Conjugate Gradient (PCG) method. This paper...

report 2011

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Flexible and multi-shift induced dimension reduction algorithms for solving large sparse linear systems

Van Gijzen, M.B. (author), Sleijpen, G.L.G. (author), Zemke, J.P. (author)

We give two important generalizations of the Induced Dimension Reduction (IDR) approach for the solution of linear systems. We derive a flexible and a multi-shift Quasi-Minimal Residual IDR (QMRIDR) variant. Numerical examples are presented to show the effectiveness of these new IDR variants compared to existing ones and to other Krylov subspace...

report 2011

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Interpreting IDR(s) as a deflation method

Collignon, T.P. (author), Sleijpen, G.L.G. (author), Van Gijzen, M.B. (author)

In this paper the IDR(s) method is interpreted in the context of deflation methods. It is shown that IDR(s) can be seen as a Richardson iteration preconditioned by a variable deflation–type preconditioner. The main result of this paper is the IDR projection theorem, which relates the spectrum of the deflated system in each IDR(s) cycle to all...

report 2010

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Exploiting BiCGstab(?) strategies to induce dimension reduction

Sleijpen, G.L.G. (author), Van Gijzen, M.B. (author)

IDR(s) [P. Sonneveld and M. B. van Gijzen, SIAM J. Sci. Comput., 31 (2008), pp. 1035–1062] and BiCGstab(?) [G. L. G. Sleijpen and D. R. Fokkema, Electron. Trans. Numer. Anal., 1 (1993), pp. 11–32] are two of the most efficient short-recurrence iterative methods for solving large nonsymmetric linear systems of equations. Which of the two is best...

journal article 2010

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An elegant IDR(s) variant that efficiently exploits bi-orthogonality properties

Van Gijzen, M.B. (author), Sonneveld, P. (author)

The IDR(s) method that is proposed in [18] is a very efficient limited memory method for solving large nonsymmetric systems of linear equations. IDR(s) is based on the induced dimension reduction theorem, that provides a way to construct subsequent residuals that lie in a sequence of shrinking subspaces. The IDR(s) algorithm that is given in [18...

report 2010

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The Deflated Relaxed Incomplete Cholesky CG method for use in a real-time ship simulator

Van 't Wout, E. (author), Van Gijzen, M.B. (author), Ditzel, A. (author), Van der Ploeg, A. (author), Vuik, C. (author)

Ship simulators are used for training purposes and therefore have to calculate realistic wave patterns around the moving ship in real time. We consider a wave model that is based on the variational Boussinesq formulation, which results in a set of partial differential equations. Discretization of these equations gives a large system of linear...

journal article 2010

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Exploiting the flexibility of IDR(s) for grid computing

Van Gijzen, M.B. (author), Collignon, T.P. (author)

The IDR(s) method that is proposed in [26] is an efficient limited memory method for solving large nonsymmetric systems of linear equations. In [11] an IDR(s) variant is described that has a single synchronisation point per iteration step, which makes this variant well-suited for parallel and grid computing. In this paper, we combine this IDR(s)...

report 2010