Efficiency of pseudo-spectral algorithms with Anderson mixing for the SCFT of periodic block-copolymer phases

Stasiak, Pawel; Matsen, Mark W.

doi:10.1140/epje/i2011-11110-0

Cited by 96 publications

(123 citation statements)

References 20 publications

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“…A similar procedure was used by Thompson et al 42 Recently Stasiak and Matsen have carried out a comparative study of the various numerical algorithms to implement the SCFT calculations. 43 Let us consider an incompressible melt of the linear ternary ABC triblock copolymer chains consisting of the end blocks A and C and the middle block C. The free energy functional for the system under consideration reads:…”

Section: Model and The Scft Techniquementioning

confidence: 99%

Perpendicular lamellar-in-lamellar and other planar morphologies in A-b-(B-b-A)2-b-C and (B-b-A)2-b-C ternary multiblock copolymer melts

Марков

Kriksin

Erukhimovich

et al. 2013

The Journal of Chemical Physics

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Ordered planar morphologies in A-b-(B-b-A)2-b-C and (B-b-A)2-b-C terpolymer melts are studied within the framework of the self-consistent field theory for volume fractions of components A, B, and C in the ratio 1:1:2 and the Flory-Huggins interaction parameters satisfying χ(AB) = 2χ(AC). The stable phases turn out to be the disordered, hexagonal, parallel lamellar-in-lamellar L∥ (including the simple lamellar) as well as non-shifted and shifted (L⊥ and SL⊥) perpendicular lamellar-in-lamellar morphologies. Depending on the value of the ratio r = Θ(AB)/Θ(BC), where Θ is a characteristic temperature of the units involved, different sequences of phase transitions are shown to occur. The hexagonal phase is characteristic for r ≅ 1. The L⊥ and SL⊥ morphologies occur at weak and intermediate segregations whereas the L∥ morphology appears for stronger degrees of segregation. For (B-b-A)2-b-C a reduction in r favors the shifted SL⊥ phase over the non-shifted L⊥ one, whereas for A-b-(B-b-A)2-b-C we find re-entrant phase transitions SL⊥ - L⊥. The physics determining the particular phase behavior is discussed.

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Section: Model and The Scft Techniquementioning

confidence: 99%

Perpendicular lamellar-in-lamellar and other planar morphologies in A-b-(B-b-A)2-b-C and (B-b-A)2-b-C ternary multiblock copolymer melts

Марков

Kriksin

Erukhimovich

et al. 2013

The Journal of Chemical Physics

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“…Anderson in 1965 in the context of integral equations [2]. It has recently been used in many applications, including multisecant methods for fixed-point iterations in electronic structure computations [5], geometry optimization problems [12], various types of flow problems [11,13], radiation diffusion and nuclear physics [1,16], molecular interaction [14], machine learning [6], improving the alternating projections method for computing nearest correlation matrices [7], and on a wide range of nonlinear problems in the context of generalized minimal residual (GMRES) methods in [17]. We further refer readers to [8,10,11,17] and references therein for detailed discussions on both practical implementation and a history of the method and its applications.…”

Section: Introductionmentioning

confidence: 99%

A Proof That Anderson Acceleration Improves the Convergence Rate in Linearly Converging Fixed-Point Methods (But Not in Those Converging Quadratically)

Evans¹,

Pollock²,

Rebholz³

et al. 2020

SIAM J. Numer. Anal.

101

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This paper provides the first proof that Anderson acceleration (AA) improves the convergence rate of general fixed point iterations. AA has been used for decades to speed up nonlinear solvers in many applications, however a rigorous mathematical justification of the improved convergence rate has remained lacking. The key ideas of the analysis presented here are relating the difference of consecutive iterates to residuals based on performing the inner-optimization in a Hilbert space setting, and explicitly defining the gain in the optimization stage to be the ratio of improvement over a step of the unaccelerated fixed point iteration. The main result we prove is that AA improves the convergence rate of a fixed point iteration to first order by a factor of the gain at each step. In addition to improving the convergence rate, our results indicate that AA increases the radius of convergence. Lastly, our estimate shows that while the linear convergence rate is improved, additional quadratic terms arise in the estimate, which shows why AA does not typically improve convergence in quadratically converging fixed point iterations. Results of several numerical tests are given which illustrate the theory.

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“…A more detailed explanation of the self‐consistent mean field theory can be found in several reviews and books . The SCFT equations were solved in real space iteratively using the pseudospectral method coupled with Anderson mixing to accelerate the convergence …”

Section: Resultsmentioning

confidence: 99%

Effects of Blockiness and Polydispersity on the Phase Behavior of Random Block Copolymers

Vanderwoude

Shi

2016

Macro Theory & Simulations

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The effects of blockiness and compositional/sequential polydispersity on the order–disorder transition of random block copolymers are studied using a segment model of random block copolymers. Specifically, the copolymers are randomly assembled from a collection of equal‐length “segments,” while the characteristics of the segments are chosen according to the nature of the polydispersity. The phase behavior of the model system is examined using the random phase approximation. The critical points of the system are calculated over a range of blockiness and polydispersity for a variety of segment models. It is observed that the critical value of the Flory–Huggins parameter χ, above which the homogeneous phase becomes unstable, is inversely proportional to the blockiness. Furthermore, it is found that sequential and compositional dispersity decreases the critical value of χ, but that the effect of sequential dispersity is much weaker. At sufficiently large compositional dispersity, the random block copolymers can undergo macrophase separation. In this region of phase space, the critical point is entirely determined by the compositional polydispersity.

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Efficiency of pseudo-spectral algorithms with Anderson mixing for the SCFT of periodic block-copolymer phases

Cited by 96 publications

References 20 publications

Perpendicular lamellar-in-lamellar and other planar morphologies in A-b-(B-b-A)2-b-C and (B-b-A)2-b-C ternary multiblock copolymer melts

Perpendicular lamellar-in-lamellar and other planar morphologies in A-b-(B-b-A)2-b-C and (B-b-A)2-b-C ternary multiblock copolymer melts

A Proof That Anderson Acceleration Improves the Convergence Rate in Linearly Converging Fixed-Point Methods (But Not in Those Converging Quadratically)

Effects of Blockiness and Polydispersity on the Phase Behavior of Random Block Copolymers

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