Performance of algebraic multigrid methods for non-symmetric matrices arising in particle methods

Large linear systems with sparse, non-symmetric matrices arise in the modeling of Markov chains or in the discretization of convection-diffusion problems. Due to their potential to solve sparse linear systems with an effort that is linear in the number of unknowns, algebraic multigrid (AMG) methods are of fundamental interest for such systems. For symmetric positive definite matrices, fundamental theoretical convergence results are established, and efficient AMG solvers have been developed. In contrast, for non-symmetric matrices, theoretical convergence results have been provided only recently. A property that is sufficient for convergence is that the matrix be an M-matrix. In this paper, we present how the simulation of incompressible fluid flows with particle methods leads to large linear systems with sparse, non-symmetric matrices. In each time step, the Poisson equation is approximated by meshfree finite differences. While traditional least squares approaches do not guarantee an M-matrix structure, an approach based on linear optimization yields optimally sparse M-matrices. For both types of discretization approaches, we investigate the performance of a classical AMG method, as well as an AMLI type method. While in the considered test problems, the M-matrix structure turns out not to be necessary for the convergence of AMG, problems can occur when it is violated. In addition, the matrices obtained by the linear optimization approach result in fast solution times due to their optimal sparsity.Comment: 16 pages, 7 figure

Multi-Architecture Monte-Carlo (MC) Simulation of Soft Coarse-Grained Polymeric Materials: SOft coarse grained Monte-carlo Acceleration (SOMA)

Multi-component polymer systems are important for the development of new materials because of their ability to phase-separate or self-assemble into nano-structures. The Single-Chain-in-Mean-Field (SCMF) algorithm in conjunction with a soft, coarse-grained polymer model is an established technique to investigate these soft-matter systems. Here we present an im- plementation of this method: SOft coarse grained Monte-carlo Accelera- tion (SOMA). It is suitable to simulate large system sizes with up to billions of particles, yet versatile enough to study properties of different kinds of molecular architectures and interactions. We achieve efficiency of the simulations commissioning accelerators like GPUs on both workstations as well as supercomputers. The implementa- tion remains flexible and maintainable because of the implementation of the scientific programming language enhanced by OpenACC pragmas for the accelerators. We present implementation details and features of the program package, investigate the scalability of our implementation SOMA, and discuss two applications, which cover system sizes that are difficult to reach with other, common particle-based simulation methods

Coupled coarse graining and Markov Chain Monte Carlo for lattice systems

We propose an efficient Markov Chain Monte Carlo method for sampling equilibrium distributions for stochastic lattice models, capable of handling correctly long and short-range particle interactions. The proposed method is a Metropolis-type algorithm with the proposal probability transition matrix based on the coarse-grained approximating measures introduced in a series of works of M. Katsoulakis, A. Majda, D. Vlachos and P. Plechac, L. Rey-Bellet and D.Tsagkarogiannis,. We prove that the proposed algorithm reduces the computational cost due to energy differences and has comparable mixing properties with the classical microscopic Metropolis algorithm, controlled by the level of coarsening and reconstruction procedure. The properties and effectiveness of the algorithm are demonstrated with an exactly solvable example of a one dimensional Ising-type model, comparing efficiency of the single spin-flip Metropolis dynamics and the proposed coupled Metropolis algorithm.Comment: 20 pages, 4 figure

Domain Growth, Budding, and Fission in Phase Separating Self-Assembled Fluid Bilayers

A systematic investigation of the phase separation dynamics in self-assembled multi-component bilayer fluid vesicles and open membranes is presented. We use large-scale dissipative particle dynamics to explicitly account for solvent, thereby allowing for numerical investigation of the effects of hydrodynamics and area-to-volume constraints. In the case of asymmetric lipid composition, we observed regimes corresponding to coalescence of flat patches, budding, vesiculation and coalescence of caps. The area-to-volume constraint and hydrodynamics have a strong influence on these regimes and the crossovers between them. In the case of symmetric mixtures, irrespective of the area-to-volume ratio, we observed a growth regime with an exponent of 1/2. The same exponent is also found in the case of open membranes with symmetric composition

Spontaneous Formation of Stable Capillary Bridges for Firming Compact Colloidal Microstructures in Phase Separating Liquids: A Computational Study

Computer modeling and simulations are performed to investigate capillary bridges spontaneously formed between closely packed colloidal particles in phase separating liquids. The simulations reveal a self-stabilization mechanism that operates through diffusive equilibrium of two-phase liquid morphologies. Such mechanism renders desired microstructural stability and uniformity to the capillary bridges that are spontaneously formed during liquid solution phase separation. This self-stabilization behavior is in contrast to conventional coarsening processes during phase separation. The volume fraction limit of the separated liquid phases as well as the adhesion strength and thermodynamic stability of the capillary bridges are discussed. Capillary bridge formations in various compact colloid assemblies are considered. The study sheds light on a promising route to in-situ (in-liquid) firming of fragile colloidal crystals and other compact colloidal microstructures via capillary bridges