A preconditioner for constrained and weighted least squares problems with Toeplitz structure

Jin, Xiaoqing

doi:10.1007/bf01740547

Cited by 9 publications

(5 citation statements)

References 11 publications

(16 reference statements)

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“…Some of the algorithms considered in this paper are also applicable if one or more of the blocks in A happen to be dense, as long as matrixvector products with A can be performed efficiently, typically in O(n + m) time. This means that if a dense block is present, it must have a special structure (e.g., Toeplitz, as in Benzi and Ng (2004) and Jin (1996)) or it must be possible to approximate its action on a vector with (nearly) linear complexity, as in the fast multipole method (Mahawar and Sarin 2003).…”

Section: Sparsity Structure and Sizementioning

confidence: 99%

Numerical solution of saddle point problems

Benzi¹,

Golub²,

Liesen³

2005

Acta Numerica

1,977

1,774

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Abstract. Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for solving this type of systems. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.

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Section: Sparsity Structure and Sizementioning

confidence: 99%

Numerical solution of saddle point problems

Benzi¹,

Golub²,

Liesen³

2005

Acta Numerica

1,977

1,774

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show abstract

“…Thus, this preconditioner is well defined even when some of the entries on the diagonal of W are zero. We note that this case does indeed arise in many situations of practical interest [9,20]. The scalar 1/(2α) in (3.3) has no impact on the preconditioned system…”

Section: Hermitian and Skew-hermitian Preconditioningmentioning

confidence: 86%

“…However, such a preconditioner is not easy to obtain, and it is too expensive. One possible choice is to approximate D by a circulant matrix C D ; see [20]. It is clear that the 1 For any square matrix A, consider the matrix operator…”

Section: Introductionmentioning

confidence: 99%

Preconditioned Iterative Methods for Weighted Toeplitz Least Squares Problems

Benzi

2006

SIAM J. Matrix Anal. & Appl.

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Abstract. We consider the iterative solution of weighted Toeplitz least squares problems. Our approach is based on an augmented system formulation. We focus our attention on two types of preconditioners: a variant of constraint preconditioning, and the Hermitian/skew-Hermitian splitting (HSS) preconditioner. Bounds on the eigenvalues of the preconditioned matrices are given in terms of problem and algorithmic parameters, and numerical experiments are used to illustrate the performance of the preconditioners.

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“…One alternative is to solve them directly by exploiting the matrix structure (see for example [6,16,47,73]). A more popular approach is to employ some iterative method to solve the system, assisted by an appropriately designed preconditioner, to ensure that the iterative method achieves fast convergence (as in [8,9,10,11,12,13,14,15,42,44,45,46,56,71]). An equally rich literature exists for preconditioning Toeplitz-like linear systems arising specifically from the discretization of fractional diffusion equations (see [17,24,25,29,32,45,46,48,53], among others).…”

Section: The Fde-constrained Optimization Modelmentioning

confidence: 99%

Fast Solution Methods for Convex Quadratic Optimization of Fractional Differential Equations

Pougkakiotis¹,

Pearson²,

Leveque³

et al. 2020

SIAM J. Matrix Anal. Appl.

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In this paper, we present numerical methods suitable for solving convex quadratic Fractional Differential Equation (FDE) constrained optimization problems, with box constraints on the state and/or control variables. We develop an Alternating Direction Method of Multipliers (ADMM) framework, which uses preconditioned Krylov subspace solvers for the resulting subproblems. The latter allows us to tackle a range of Partial Differential Equation (PDE) optimization problems with box constraints, posed on space-time domains, that were previously out of the reach of state-of-the-art preconditioners. In particular, by making use of the powerful Generalized Locally Toeplitz (GLT) sequences theory, we show that any existing GLT structure present in the problem matrices is preserved by ADMM, and we propose some preconditioning methodologies that could be used within the solver, to demonstrate the generality of the approach. Focussing on convex quadratic programs with time-dependent 2-dimensional FDE constraints, we derive multilevel circulant preconditioners, which may be embedded within Krylov subspace methods, for solving the ADMM sub-problems. Discretized versions of FDEs involve large dense linear systems. In order to overcome this difficulty, we design a recursive linear algebra, which is based on the Fast Fourier Transform (FFT). We manage to keep the storage requirements linear, with respect to the grid size N , while ensuring an order N log N computational complexity per iteration of the Krylov solver. We implement the proposed method, and demonstrate its scalability, generality, and efficiency, through a series of experiments over different setups of the FDE optimization problem.

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A preconditioner for constrained and weighted least squares problems with Toeplitz structure

Cited by 9 publications

References 11 publications

Numerical solution of saddle point problems

Numerical solution of saddle point problems

Preconditioned Iterative Methods for Weighted Toeplitz Least Squares Problems

Fast Solution Methods for Convex Quadratic Optimization of Fractional Differential Equations

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