Efficient solution of liquid state integral equations using the Newton-GMRES algorithm

Booth, Michael J.; Schlijper, A. G.; Scales, L. E.; Haymet, A. D. J.

doi:10.1016/s0010-4655(99)00186-1

Cited by 22 publications

(29 citation statements)

References 38 publications

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“…The Newton-GMRES code NITSOL [17] was applied to the OZ equations in [2]. The algorithm performed well, which is not surprising in view of the mesh independence results for GMRES when applied to integral equations.…”

Section: Nested Newton-gmres the Newton-gmres Methods Is An Inexactmentioning

confidence: 79%

“…On the finer meshes we ask that the size of the residual be reduced by a factor of ten. This nesting is a step beyond the method in [2] and substantially improves performance, because the most of the matrix-vector products are done on coarse grids.…”

Section: End Whilementioning

confidence: 99%

“…In this paper we propose a fast multi-level method for the solution of a class of integral equations called the Ornstein-Zernike (OZ) equations, which are useful in calculating probability distributions of matter (atoms) in fluid states [10]. Our approach is faster than a Newton-Krylov approach, such as the one proposed in [2], because the linear solver is only used on a coarse mesh problem.The OZ equations were initially designed to model density fluctuations near the critical point via the equilibrium theory of liquids [8,15]. Since then the range of validity and usefulness has been extended to include the entire fluid range of states.…”

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confidence: 99%

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A fast solver for the Ornstein–Zernike equations

Kelley

Pettitt

2004

Journal of Computational Physics

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Abstract. In this paper we report on the design and analysis of a multilevel method for the solution of the OrnsteinZernike Equations and related systems of integro-algebraic equations. Our approach is based on an extension of the Atkinson-Brakhage method, with Newton-GMRES used as the coarse mesh solver. We report on several numerical experiments to illustrate the effectiveness of the method.Key words. Multilevel method, Newton-GMRES, Ornstein-Zernike equations, nonlinear equations AMS subject classifications. 65H10, 65N55, 65R20, 65T50, 92E10, 1. Introduction. In this paper we propose a fast multi-level method for the solution of a class of integral equations called the Ornstein-Zernike (OZ) equations, which are useful in calculating probability distributions of matter (atoms) in fluid states [10]. Our approach is faster than a Newton-Krylov approach, such as the one proposed in [2], because the linear solver is only used on a coarse mesh problem.The OZ equations were initially designed to model density fluctuations near the critical point via the equilibrium theory of liquids [8,15]. Since then the range of validity and usefulness has been extended to include the entire fluid range of states. This set of nonlinear coupled integral equations has been derived from the full partition function for atomic systems [20] and while essentially never solved without some approximations has proved a useful tool for understanding liquids at the atomic level for over 50 years. While the OZ equation has two unknowns it is usually closed with another often algebraic relation between the two unknown functions. Two useful approximate closure relations are the Perkus-Yevick equation [16] and the hyper netted chain equation [21]. These equations then provide essentially two equations and two unknowns and when convenient may be substituted into the OZ equation to provide a single nonlinear integral equation for the unknown probability distribution function.The equations, when the physical parameters are reasonably adjusted, have solutions which can be achieved by a variety of techniques [10]. Those methods include Picard iteration with or with out relaxation and basis set (variational) methods. In cases where the physical parameters make the equations stiff, iterative solutions are particularly tedious.The objectives of this paper are to describe an multilevel approach for solving the OZ equations and apply that new approach to two examples. The multilevel method is based on the enhanced version of the Atkinson-Brakhage [1, 3] method from [13]. We show that we can compute

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Section: Nested Newton-gmres the Newton-gmres Methods Is An Inexactmentioning

confidence: 79%

Section: End Whilementioning

confidence: 99%

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confidence: 99%

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A fast solver for the Ornstein–Zernike equations

Kelley

Pettitt

2004

Journal of Computational Physics

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“…23,24 These polynomials are constructed to be orthogonal with respect to one-body orientational distribution function f(cos θ ). In order to speed up the solution of integral equations we employ Newton-GRMES algorithm 25 as implemented in the public-domain solver NITSOL. 26 This algorithm solves a system of nonlinear algebraic equations F(x) = 0 where F(x) is a vector function in a space of dimension n (F : R n → R n ).…”

Section: Introductionmentioning

confidence: 99%

Reference hypernetted chain theory for ferrofluid bilayer: Distribution functions compared with Monte Carlo

Polyakov

Vorontsov-Vèlyaminov

2014

The Journal of Chemical Physics

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Properties of ferrofluid bilayer (modeled as a system of two planar layers separated by a distance h and each layer carrying a soft sphere dipolar liquid) are calculated in the framework of inhomogeneous Ornstein-Zernike equations with reference hypernetted chain closure (RHNC). The bridge functions are taken from a soft sphere (1/r 12 ) reference system in the pressure-consistent closure approximation. In order to make the RHNC problem tractable, the angular dependence of the correlation functions is expanded into special orthogonal polynomials according to Lado. The resulting equations are solved using the Newton-GRMES algorithm as implemented in the publicdomain solver NITSOL. Orientational densities and pair distribution functions of dipoles are compared with Monte Carlo simulation results. A numerical algorithm for the Fourier-Hankel transform of any positive integer order on a uniform grid is presented. © 2014 AIP Publishing LLC.

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Multigrid solver for the reference interaction site model of molecular liquids theory

2011

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In this article, we propose a new multigrid-based algorithm for solving integral equations of the reference interactions site model (RISM). We also investigate the relationship between the parameters of the algorithm and the numerical accuracy of the hydration free energy calculations by RISM. For this purpose, we analyzed the performance of the method for several numerical tests with polar and nonpolar compounds. The results of this analysis provide some guidelines for choosing an optimal set of parameters to minimize computational expenses. We compared the performance of the proposed multigrid-based method with the one-grid Picard iteration and nested Picard iteration methods. We show that the proposed method is over 30 times faster than the one-grid iteration method, and in the high accuracy regime, it is almost seven times faster than the nested Picard iteration method.

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Efficient solution of liquid state integral equations using the Newton-GMRES algorithm

Cited by 22 publications

References 38 publications

A fast solver for the Ornstein–Zernike equations

A fast solver for the Ornstein–Zernike equations

Reference hypernetted chain theory for ferrofluid bilayer: Distribution functions compared with Monte Carlo

Multigrid solver for the reference interaction site model of molecular liquids theory

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