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Krylov subspace methods for computing hydrodynamic interactions in Brownian dynamics simulations
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10.1063/1.4742347
/content/aip/journal/jcp/137/6/10.1063/1.4742347
http://aip.metastore.ingenta.com/content/aip/journal/jcp/137/6/10.1063/1.4742347

Figures

Image of FIG. 1.
FIG. 1.

Convergence of (a) the Krylov subspace method and (b) the Chebyshev method measured by various error estimates, E k exact, E k , and E f . Diffusion matrices were constructed from five configurations of random polymer chains with length N = 1000 and the results represent the average over the five configurations. Standard deviations for all data points are so small that they are not displayed. E k with k = 1 was set to 1.

Image of FIG. 2.
FIG. 2.

(a) Effect of block size on convergence rate in the block Krylov subspace method. The errors are computed for the first vector of the block of vectors. Results are averages of the five random polymer chains with N = 1000. (b) Comparison of E k exact and E k in the block Krylov subspace method. Results for polymers with N = 1000 and block size = 50 are shown. All results are the average over the five independent configurations. Standard deviations for all data points are so small that they are not displayed. E k with k = 1 was set to 1.

Image of FIG. 3.
FIG. 3.

Effect of model, number of particles, and volume fraction on convergence of the Krylov subspace method. (a) Convergence of the random and collapsed polymer models with N = 200 and 1000. (b) Convergence of the monodisperse suspension model at various volume fractions Φ = 0.1, 0.2, 0.3, 0.4, and 0.5 with N = 200 and N = 1000. For monodisperse suspensions, convergence is slower for larger volume fractions and for larger numbers of particles.

Image of FIG. 4.
FIG. 4.

Effect of λ RPY on dynamic properties for the random and collapsed polymers, and monodisperse suspension model. (a) D cm for the random and collapsed polymers with various polymer lengths N obtained from BD simulation with various λ RPY. Lines are fit to the data for λ RPY = 1 with D cmN −ν and their exponents are shown. For both polymer models, results with different λ RPY are so close that their plots overlap and are almost indistinguishable in the figure. (b) D for the monodisperse suspension model with number of particles of 200 at various volume fractions Φ obtained from BD simulations with various λ RPY. The results with λ RPY = 1 are connected by a broken line to guide the eye. The Cholesky factorization method was used in the BD simulations.

Image of FIG. 5.
FIG. 5.

Values of E 1 for the covariance matrices constructed from sets of Brownian noise vectors generated by the Cholesky, Krylov subspace with E k = 0.1, and TEA methods. Results of the random polymer chain model (a, d), the collapsed polymer model (b, e), and the monodisperse suspension model at volume fraction of 0.3 (c, f) are shown. Left subfigures (a, b, c) are for N = 200 and right subfigures (d, e, f) are for N = 1000. For the monodisperse suspension model, results of the Cholesky (red lines) and Krylov subspace with E k = 0.1 (green lines) are so close that their lines overlap and are indistinguishable in the figure. All results are the average over five independent configurations. Standard deviations for all data points are so small that they are not displayed.

Image of FIG. 6.
FIG. 6.

Computational time required for generating 100 correlated Brownian noise vectors by the block-Krylov subspace method with different block sizes. Timings for (a) E k = 0.1 and (b) E k = 0.01 are shown. The random polymer model with length N = 1000–10 000 was used for this timing test. For block size = 1, the DSYMV BLAS routine was employed for matrix-vector multiplications. For other block sizes, the DSYMM BLAS routine was employed for matrix-matrix multiplications. The latter routine is not optimized for matrix-vector multiplications.

Image of FIG. 7.
FIG. 7.

Scaling of number of iterations and computational time of the block-Krylov subspace method with the number of particles. The random polymer model with length N = 1000–10 000 was used for this timing test. Number of iterations required with thresholds (a) E k = 0.1 and (b) E k = 0.01 for the block-Krylov subspace and Cholesky methods. Computational time required for generating 50 correlated Brownian noise vectors by the block-Krylov subspace and Cholesky methods with (c) E k = 0.1 and (d) E k = 0.01. A block size of 50 was used for the block-Krylov subspace method. Results for the Cholesky and TEA methods are also shown for comparison (also computed in block fashion). Dashed lines are fitted linear slopes for a range of N = 4000–10 000. The values of slopes for these methods are shown in inside the figures.

Tables

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Krylov subspace algorithm for computing

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Block-Krylov subspace algorithm for computing a block of s correlated vectors, , each vector with distribution N(0, D).

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Table I.

Errors (%) in D obtained with various diffusion matrix update intervals, λ RPY, relative to results with λ RPY = 1 for the monodisperse model at various volume fractions Φ with N = 200.

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Table II.

Errors (%) in D cm obtained from the simulations using the Krylov method with various levels of Brownian noise accuracies and the TEA method relative to results using the Cholesky method with λ RPY = 1 for various chain lengths for the random polymer model.

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Table III.

Errors (%) in D cm at equilibrated states obtained from the simulations using the Krylov method with various Brownian noise accuracies and the TEA method relative to results using the Cholesky method with λ RPY = 1 for various chain lengths for the collapsed polymer model.

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Table IV.

Errors (%) in D obtained from the simulations using the Krylov method with various Brownian noise accuracies and the TEA method relative to results using the Cholesky method with λ RPY = 1 for monodisperse model at various volume fractions Φ with number of particles N = 200.

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/content/aip/journal/jcp/137/6/10.1063/1.4742347
2012-08-10
2014-04-21
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Krylov subspace methods for computing hydrodynamic interactions in Brownian dynamics simulations
http://aip.metastore.ingenta.com/content/aip/journal/jcp/137/6/10.1063/1.4742347
10.1063/1.4742347
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