720 research outputs found
Objective multiscale analysis of random heterogeneous materials
The multiscale framework presented in [1, 2] is assessed in this contribution for a study of random heterogeneous materials. Results are compared to direct numerical simulations (DNS) and the sensitivity to user-defined parameters such as the domain decomposition type and initial coarse scale resolution is reported. The parallel performance of the implementation is studied for different domain decompositions
Matching Multiple Rigid Domain Decompositions of Proteins
We describe efficient methods for consistently coloring and visualizing collections of rigid cluster decompositions obtained from variations of a protein structure, and lay the foundation for more complex setups, that may involve different computational and experimental methods. The focus here is on three biological applications: the conceptually simpler problems of visualizing results of dilution and mutation analyses, and the more complex task of matching decompositions of multiple Nucleic Magnetic Resonance (NMR) models of the same protein. Implemented into the KINematics And RIgidity (KINARI) web server application, the improved visualization techniques give useful information about protein folding cores, help examining the effect of mutations on protein flexibility and function, and provide insights into the structural motions of Protein Data Bank proteins solved with solution NMR. These tools have been developed with the goal of improving and validating rigidity analysis as a credible coarse-grained model capturing essential information about a protein\u27s slow motions near the native state
A Cache-Aware Approach to Domain Decomposition for Stencil-Based Codes
Partial Differential Equations (PDEs) lie at the heart of numerous scientific simulations depicting physical phenomena. The parallelization of such simulations introduces additional performance penalties in the form of local and global synchronization among cooperating processes. Domain decomposition partitions the largest shareable data structures into sub-domains and attempts to achieve perfect load balance and minimal communication. Up to now research efforts to optimize spatial and temporal cache reuse for stencil-based PDE discretizations (e.g. finite difference and finite element) have considered sub-domain operations after the domain decomposition has been determined. We derive a cache-oblivious heuristic that minimizes cache misses at the sub-domain level through a quasi-cache-directed analysis to predict families of high performance domain decompositions in structured 3-D grids. To the best of our knowledge this is the first work to optimize domain decompositions by analyzing cache misses - thus connecting single core parameters (i.e. cache-misses) to true multicore parameters (i.e. domain decomposition). We analyze the trade-offs in decreasing cache-misses through such decompositions and increasing the dynamic bandwidth-per-core. The limitation of our work is that currently, it is applicable only to structured 3-D grids with cuts parallel to the Cartesian Axes. We emphasize and conclude that there is an imperative need to re-think domain decompositions in this constantly evolving multicore era
A New Domain Decomposition Method for the Compressible Euler Equations
In this work we design a new domain decomposition method for the Euler
equations in 2 dimensions. The basis is the equivalence via the Smith
factorization with a third order scalar equation to whom we can apply an
algorithm inspired from the Robin-Robin preconditioner for the
convection-diffusion equation. Afterwards we translate it into an algorithm for
the initial system and prove that at the continuous level and for a
decomposition into 2 sub-domains, it converges in 2 iterations. This property
cannot be preserved strictly at discrete level and for arbitrary domain
decompositions but we still have numerical results which confirm a very good
stability with respect to the various parameters of the problem (mesh size,
Mach number, ....).Comment: Submitte
Additive domain decomposition operator splittings -- convergence analyses in a dissipative framework
We analyze temporal approximation schemes based on overlapping domain
decompositions. As such schemes enable computations on parallel and distributed
hardware, they are commonly used when integrating large-scale parabolic
systems. Our analysis is conducted by first casting the domain decomposition
procedure into a variational framework based on weighted Sobolev spaces. The
time integration of a parabolic system can then be interpreted as an operator
splitting scheme applied to an abstract evolution equation governed by a
maximal dissipative vector field. By utilizing this abstract setting, we derive
an optimal temporal error analysis for the two most common choices of domain
decomposition based integrators. Namely, alternating direction implicit schemes
and additive splitting schemes of first and second order. For the standard
first-order additive splitting scheme we also extend the error analysis to
semilinear evolution equations, which may only have mild solutions.Comment: Please refer to the published article for the final version which
also contains numerical experiments. Version 3 and 4: Only comments added.
Version 2, page 2: Clarified statement on stability issues for ADI schemes
with more than two operator
Parallel TREE code for two-component ultracold plasma analysis
The TREE method has been widely used for long-range interaction {\it N}-body
problems. We have developed a parallel TREE code for two-component classical
plasmas with open boundary conditions and highly non-uniform charge
distributions. The program efficiently handles millions of particles evolved
over long relaxation times requiring millions of time steps. Appropriate domain
decomposition and dynamic data management were employed, and large-scale
parallel processing was achieved using an intermediate level of granularity of
domain decomposition and ghost TREE communication. Even though the
computational load is not fully distributed in fine grains, high parallel
efficiency was achieved for ultracold plasma systems of charged particles. As
an application, we performed simulations of an ultracold neutral plasma with a
half million particles and a half million time steps. For the long temporal
trajectories of relaxation between heavy ions and light electrons, large
configurations of ultracold plasmas can now be investigated, which was not
possible in past studies
Laplacian Mixture Modeling for Network Analysis and Unsupervised Learning on Graphs
Laplacian mixture models identify overlapping regions of influence in
unlabeled graph and network data in a scalable and computationally efficient
way, yielding useful low-dimensional representations. By combining Laplacian
eigenspace and finite mixture modeling methods, they provide probabilistic or
fuzzy dimensionality reductions or domain decompositions for a variety of input
data types, including mixture distributions, feature vectors, and graphs or
networks. Provable optimal recovery using the algorithm is analytically shown
for a nontrivial class of cluster graphs. Heuristic approximations for scalable
high-performance implementations are described and empirically tested.
Connections to PageRank and community detection in network analysis demonstrate
the wide applicability of this approach. The origins of fuzzy spectral methods,
beginning with generalized heat or diffusion equations in physics, are reviewed
and summarized. Comparisons to other dimensionality reduction and clustering
methods for challenging unsupervised machine learning problems are also
discussed.Comment: 13 figures, 35 reference
Learning Adaptive FETI-DP Constraints for Irregular Domain Decompositions
Adaptive coarse spaces yield a robust convergence behavior for FETI-DP (Finite Element Tearing and Interconnecting - Dual Primal) and BDDC (Balancing Domain Decomposition by Constraints) methods for highly heterogeneous problems. However, the usage of such adaptive coarse spaces can be computationally expensive since, in general, it requires the setup and the solution of a relatively high amount of local eigenvalue problems on parts of the domain decomposition interface. In earlier works, see, e.g., [2], it has been shown that it is possible to train a neural network to make an automatic decision which of the eigenvalue problems in an adaptive FETI-DP method are actually necessary for robustness with a satisfactory accuracy. Moreover, these results have been extended in [6] by directly learning an approximation of the adaptive edge constraints themselves for regular, two-dimensional domain decompositions. In particular, this does not require the setup or the solution of any eigenvalue problems at all since the FETI-DP coarse space is, in this case, exclusively enhanced by the learned constraints obtained from the regression neural networks trained in an offline phase. Here, in contrast to [6], a regression neural network is trained with both, training data resulting from straight and irregular edges. Thus, it is possible to use the trained networks also for the approximation of adaptive constraints for irregular domain decompositions. Numerical results for a heterogeneous two-dimensional stationary diffusion problem are presented using both, a decomposition into regular and irregular subdomains
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