Best-conditioned circulant preconditioners

Chan, Raymond H.; Wong, Charles

doi:10.1016/0024-3795(93)00171-u

Cited by 7 publications

(7 citation statements)

References 8 publications

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“…In our method we set D a = D and solve the matrix equation Dx = b for vector x using the conjugate gradients method 30 for each conformation (no preaveraging was used). As the preconditioner we use the inverse of the nearest circulant matrix to D. 31 Choosing this form of a preconditioner allows for efficient calculations using the discrete fast Fourier transform and results in a rapid convergence of the conjugate gradients method. For the diffusion tensor we use the usual Rotne−Prager−Yamakawa form.…”

Section: ■ Experimental Sectionmentioning

confidence: 99%

Intrinsic Viscosity of Cyclic Polystyrene

et al. 2017

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The intrinsic viscosity of cyclic polystyrenes (C-PS) and linear polystyrenes (L-PS) with molecular weight (MW) ranging from 16K to 370K was measured in THF using size exclusion chromatography coupled with a triple detection system. The C-PS samples were prepared by a ring-closure reaction of telechelic linear precursor synthesized by anionic polymerization. As-synthesized C-PS samples after the ringclosure reaction contain various byproducts, and they were fractionated by liquid chromatography at the critical condition to obtain highly pure C-PS. While the intrinsic viscosity of L-PS agrees well with the literature data, C-PS shows significantly lower value than the literature values. The Mark−Houwink exponent of C-PS is 0.67, somewhat lower than 0.70 for L-PS in the good solvent over the MW range examined. Therefore, the ratio of the intrinsic viscosity of C-PS to L-PS (g′ =[ η] C /[η] L) in THF is not MW independent but decreases as MW increases (0.63 ≥ [η] C /[η] L ≥ 0.57). The trend of intrinsic viscosity agrees well with the computer simulation result. The discrepancy in [η] C from the literature values can largely be attributed to the contamination of linear byproducts in the earlier studies.

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Section: ■ Experimental Sectionmentioning

confidence: 99%

Intrinsic Viscosity of Cyclic Polystyrene

et al. 2017

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“…Although the proof is similar to that of Theorem 1 in Chart and Wong [3], we give it here for completeness.…”

Section: Lemma 2 (Huckle [7]) Let a Be An N-by-n Matrix And Mebe Thmentioning

confidence: 86%

A note on best conditioned preconditioners

Jin

1994

BIT

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Abstract.We discuss the solution of Hermitian positive definite systems Ax = b by the preconditioned conjugate gradient method with a preconditioner M, In general, the smaller the condition numberis, the faster the convergence rate will be. For a given unitary matrix Q, let ~t a = {Q*A.Q 1A. is an n-by-n complex diagonal matrix} and 914~ = {Q*A.Q 1 An is an n-by-n positive definite diagonal matrix}. The preconditioner Mb that minimizes K(M-'hAM -~) over 5Mff is called the best conditioned preconditioner for the matrix A over Yctff. We prove that if QAQ* has Young's Property A, then Mb is nothing new but the minimizer of IIM -A lie over Yvt a. Here II" lit denotes the Frobenius norm. Some applications are also given here.

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“…While fast direct approaches for solving such systems have been proposed in the literature (see for example [5,37]), they tend to be either unstable or problem specific. 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 [13,12,36,35,34,8,11,9,7,49,31,10]).…”

Section: Toeplitz Matrices -Circulant Preconditionersmentioning

confidence: 99%

“…Other approximations are possible, however, we focus on the previously presented one. Nevertheless, the reader is referred to [11,49,10], for alternative level-1 circulant approximations. In the following theorem, we summarize some of the wellknown properties of C 1 (•).…”

Section: Toeplitz Matrices -Circulant Preconditionersmentioning

confidence: 99%

Fast Solution Methods for Convex Quadratic Optimization of Fractional Differential Equations

Pougkakiotis,

Pearson,

Leveque

et al. 2019

Preprint

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In this paper, we present numerical methods suitable for solving convex Fractional Differential Equation (FDE) optimization problems, with potential box constraints on the state and control variables. First we derive powerful multilevel circulant preconditioners, which may be embedded within Krylov subspace methods, for solving equality constrained FDE optimization problems. We then develop an Alternating Direction Method of Multipliers (ADMM) framework, which uses preconditioned Krylov solvers for the resulting subproblems. The latter allows us to tackle FDE optimization problems with box constraints, posed on space-time domains, that were previously out of the reach of state-of-the-art preconditioners. 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), and our proposed multilevel circulant preconditioners are suitable for approximating discretized FDEs of arbitrary dimension. Focusing on timedependent 2-dimensional FDEs, 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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Best-conditioned circulant preconditioners

Cited by 7 publications

References 8 publications

Intrinsic Viscosity of Cyclic Polystyrene

Intrinsic Viscosity of Cyclic Polystyrene

A note on best conditioned preconditioners

Fast Solution Methods for Convex Quadratic Optimization of Fractional Differential Equations

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