Parallel Multigrid Method Research Articles

Patient Specific Simulation of Body Surface ECG using the Finite Element Method

Recent studies, supported by advances in computer science, have successfully simulated the excitation and repolarization processes of the heart, based on detailed cell models of electrophysiology and implemented with realistic morphology. In this study, we extend these approaches to simulate the body surface electrocardiogram (ECG) of specific individuals. Patient-specific finite element models of the heart and torso are created for four patients with various heart diseases, based on clinical data including computer tomography, while the parallel multi-grid method is used to solve the dynamic bi-domain problem. Personalization procedures include demarcation of nonexcitable tissue, allocation of the failing myocyte model of electrophysiology, and modification of the excitation sequence. In particular, the adjustment of QRS morphology requires iterative computations, facilitated by the simultaneous visualization of the propagation of excitation in the heart, average QRS vector in the torso, and 12-lead ECG. In all four cases we obtained reasonable agreement between the simulated and actual ECGs. Furthermore, we also simulated the ECGs of three of the patients under bi-ventricular pacing, and once again successfully reproduced the actual ECG morphologies. Since no further adjustments were made to the heart models in the pacing simulations, the good agreement provides strong support for the validity of the models. These results not only help us understand the cellular basis of the body surface ECG, but also open the possibility of heart simulation for clinical applications.

A parallel multigrid method of the Cahn–Hilliard equation

We present a parallel finite difference scheme and its implementation for solving the Cahn–Hilliard equation, which describes the phase separation process. Our numerical algorithm employs an unconditionally gradient stable splitting discretization method. The resulting discrete equations are solved using a parallel multigrid method. This parallel scheme facilitates the solution of large-scale problems. We provide numerical results related to the speed-up, efficiency, and scalability to demonstrate the high performance of our proposed method. We also propose a linearly stabilized splitting scheme for the Cahn–Hilliard equation with logarithmic free energy.

A Parallel Auxiliary Grid Algebraic Multigrid Method for Graphic Processing Units

In this paper, we develop a new parallel auxiliary grid algebraic multigrid (AMG) method to leverage the power of graphic processing units (GPUs). In the construction of the hierarchical coarse grid, we use a simple and fixed coarsening procedure based on a region quadtree generated from an auxiliary grid. This allows us to explicitly control the sparsity patterns and operator complexities of the AMG solver. This feature provides (nearly) optimal load balancing and predictable communication patterns on shape regular grids, which makes our new algorithm suitable for parallel computing, especially on GPUs. We also design a parallel smoother based on the special coloring of the quadtree to accelerate the convergence rate and improve the parallel performance of this solver. Based on the CUDA toolkit [NVIDIA CUDA Programming Guide, NVIDIA Corp., 2010], we implemented our new parallel auxiliary grid AMG method on GPUs and the numerical results of this implementation demonstrate the efficiency of our new method ...

Exposing Fine-Grained Parallelism in Algebraic Multigrid Methods

Algebraic multigrid methods for large, sparse linear systems are a necessity in many computational simulations, yet parallel algorithms for such solvers are generally decomposed into coarse-grained tasks suitable for distributed computers with traditional processing cores. However, accelerating multigrid methods on massively parallel throughput-oriented processors, such as graphics processing units, demands algorithms with abundant fine-grained parallelism. In this paper, we develop a parallel algebraic multigrid method which exposes substantial fine-grained parallelism in both the construction of the multigrid hierarchy as well as the cycling or solve stage. Our algorithms are expressed in terms of scalable parallel primitives that are efficiently implemented on the GPU. The resulting solver achieves an average speedup of $1.8\times$ in the setup phase and $5.7\times$ in the cycling phase when compared to a representative CPU implementation.

An implicit parallel multigrid computing scheme to solve coupled thermal-solute phase-field equations for dendrite evolution

An implicit, second-order space and time discretization scheme together with a parallel multigrid method involving a strip grid domain partitioning has been developed to solve fully coupled, nonlinear phase field equations involving solute and heat transport for multiple solidifying dendrites. The computational algorithm has been shown to be stable and monotonously convergent, and allowed time marching steps that were 3–4 orders of magnitude larger than those employed in similar explicit approaches, resulting in an increase of 3–4 orders of magnitude in computing efficiency. Full solute and thermal coupling was achieved for metallic alloys with a realistic, high Lewis number of >10 4. The parallel multigrid computing scheme is shown to provide a scalable methodology that allowed the efficient use of distributed supercomputing resource to simulate the evolution of tens of complex shaped 2D dendrites in a computational domain containing tens or even hundreds of millions of grid points. The simulations have provided insight into the dynamic interplay of many growing dendrites in a more realistic fully coupled thermal-solute condition, capturing for the first time fine scale features such as dendrite splitting.

Parallel multigrid method for finite element simulations of complex flow problems on locally refined meshes

We present a parallel multigrid solver on locally refined meshes for solving very complex three-dimensional flow problems. Besides describing the parallel implementation in detail, we prove the smoothing property of the suggested iteration for a simple model problem. For demonstration of the efficiency and feasibility of the solver, we show a chemically reactive flow simulation for a Methane burner using detailed chemical reaction modeling. Further, we give the results of an ocean flow simulation. All described methods are implemented in the finite element toolbox Gascoigne. Copyright © 2010 John Wiley & Sons, Ltd.

Reconsidering algorithms for iterative solvers in the multicore era

Efficient solution of computational problems require a match between the algorithm and the underlying architecture. New multicore processors feature low intra-chip communication cost and smaller per-thread caches compared to single-core implementations, indicating that data locality issues are more important than communication overheads. We investigate the impact of these changes on parallel multigrid methods. We present a temporally blocked, naturally ordered, smoother implementation that improves the data locality as much as ten times compared with the standard red-black algorithm. We present results of the performance of our new algorithm on an SMP system, an UltraSPARC T1 (Niagara) SMT/CMP, and a simulated CMP processor.

Parallel Unstructured-Solver Methods for Simulation of Complex Giant Reservoirs

Summary The major issues for parallel solvers in a modern reservoir simulator are robustness, scalability, efficiency, and flexibility. There is significant interest in running fast field-scale simulations for complex giant Middle Eastern reservoirs, which will require tens of millions to hundreds of millions of grid cells to give reasonable resolution. At the same time, significant geologic complexity will require the treatment of dual-permeability regions, faulting and fractures, and high variations of reservoir and fluid properties. Of course, the methods should also work well for extracted-sector simulation with local grid refinements in both the structured and unstructured discretization. The preconditioning methods considered in this work include both the single-stage and multistage frameworks. In the single-stage framework, a novel method is considered in addition to the well-known variants of incomplete lower-upper (ILU) factorizations [ILU0, ILU(k), and ILUT]. The new method is a highly parallel method, which, in this paper, will be referred to as the unstructured line-solve power-series (LSPS) method. The method will be discussed and contrasted in light of key issues for parallel linear solvers. The unstructured LSPS has certain interesting properties in the parallel construct, which make it a highly effective component. The multistage method researched in this work is of the constraint pressure residual (CPR) framework. The method uses approximate pressure solve as the first-stage preconditioning to the full-system preconditioning. A number of original adaptations based on this concept were researched. Here, the use of the parallel algebraic multigrid (PAMG) method and other single-level methods mentioned previously in combinations within the multistage CPR framework were explored. Certain methods constructed in this way are found to be highly efficient, scalable, and robust. The methods developed are discussed, and several test problems are included, in this paper. The largest simulation model tested to date using these solver methods is a 172-million-cell full-field model of a supergiant carbonate complex with more than 3,000 wells and 60 years of history simulation.

Modelling of signal processing in neurons

Event Abstract Back to Event Modelling of signal processing in neurons Gabriel Wittum1* 1 Ruprecht-Karls-University, Germany The crucial feature of neuronal ensembles is their high complexity and variability. This makes modelling and computation very difficult, in particular for detailed models based on first principles. The problem starts with modelling geometry, which has to extract the essential features from those highly complex and variable phenotypes and at the same time has to take in to account the stochastic variability. Moreover, models of the highly complex processes which are living on these geometries are far from being well established, since those are highly complex too and couple on a hierarchy of scales in space and time. Simulating such systems always puts the whole approach to test, including modeling, numerical methods and software implementations. In combination with validation based on experimental data, all components have to be enhanced to reach a reliable solving strategy. To handle problems of this complexity, new mathematical methods and software tools are required. In recent years, new approaches such as parallel adaptive multigrid methods and corresponding software tools have been developed allowing to treat problems of huge complexity. In the lecture we present a three dimensional model of signaling in neurons. First we show a method for the reconstruction of the geomety of cells and subcellular structures as three dimensional objects. With this tool, NeuRA, complex geometries of neuron nuclei were reconstructed. We present the results and discuss reasons for the complicated shapes. To that end, we present a model of calcium signaling to the nucleus and show simulation results on reconstructed nuclear geometries. We discuss the implications of these simulations. We further show reconstructed cell geometries and simulations with a three dimensional active model of signal transduction in the cell which is derived from the Maxwell equations and uses generalized Hodgkin-Huxley fluxes for the description of the ion channels. Conference: Neuroinformatics 2008, Stockholm, Sweden, 7 Sep - 9 Sep, 2008. Presentation Type: Oral Presentation Topic: Workshop Citation: Wittum G (2008). Modelling of signal processing in neurons. Front. Neuroinform. Conference Abstract: Neuroinformatics 2008. doi: 10.3389/conf.neuro.11.2008.01.148 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 28 Jul 2008; Published Online: 28 Jul 2008. * Correspondence: Gabriel Wittum, Ruprecht-Karls-University, Heidelberg, Germany, nemoABS01@frontiersin.org Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Gabriel Wittum Google Gabriel Wittum Google Scholar Gabriel Wittum PubMed Gabriel Wittum Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

Parallel multigrid summation for the [formula omitted]-body problem

An Θ ( n ) parallel multigrid summation method (MG) for the N-body problem is presented. The method was originally devised for vacuum boundary conditions. Here, it is extended to periodic boundary conditions and implemented in parallel using force decomposition and MPI. MG is based on a hierarchical decomposition of computational kernels on multiple grids. For low accuracy calculations, appropriate for molecular dynamics, a sequential implementation is as fast or faster than particle mesh Ewald (PME). Our parallel implementation is more scalable than PME. The method can be combined with multiple time stepping integrators to produce a powerful simulation protocol for simulation of biological molecules and other materials. The parallel implementation is tested on both a Linux cluster with Myrinet interconnect and a shared memory computer. It is available as open-source at http://protomol.sourceforge.net. An auxiliary tool allows the automatic selection of optimal parameters for MG, and is available at http://mdsimaid.cse.nd.edu.

Parallel computation of unsteady three-dimensional incompressible viscous flow using an unstructured multigrid method

The development and validation of a parallel unstructured tetrahedral non-nested multigrid (MG) method for simulation of unsteady 3D incompressible viscous flow is presented. The Navier–Stokes solver is based on the artificial compressibility method (ACM) and a higher-order characteristics-based finite-volume scheme on unstructured MG. Unsteady flow is calculated with an implicit dual time stepping scheme. The parallelization of the solver is achieved by a MG domain decomposition approach (MG-DD), using the Single Program Multiple Data (SPMD) programming paradigm. The Message-Passing Interface (MPI) Library is used for communication of data and loop arrays are decomposed using the OpenMP standard. The parallel codes using single grid and MG are used to simulate steady and unsteady incompressible viscous flows for a 3D lid-driven cavity flow for validation and performance evaluation purposes. The speedups and efficiencies obtained by both the parallel single grid and MG solvers are reasonably good for all test cases, using up to 32 processors on the SGI Origin 3400. The parallel results obtained agree well with those of serial solvers and with numerical solutions obtained by other researchers, as well as experimental measurements.

Parallel multigrid methods for the calculation of unsteady flows on unstructured grids: algorithmic aspects and parallel performances on clusters of PCs

We report on our efforts towards the design of efficient parallel hierarchical iterative methods for the solution of sparse and irregularly structured linear systems resulting from CFD applications. The solution strategies considered here share a central numerical kernel which consists in a linear multigrid by volume agglomeration method. Starting from this method, we study two parallel solution strategies. The first variant results from a direct intra-grid parallelization of multigrid operations on coarse grids. The second variant is based on an additive Schwarz domain decomposition algorithm which is formulated at the continuous level through the introduction of specific interface conditions. In this variant, the linear multigrid by volume agglomeration method is used to approximately solve the local systems obtained at each iteration of the Schwarz algorithm. As a result, the proposed hybrid domain decomposition/multigrid method can be viewed as a particular form of parallel multigrid in which multigrid acceleration is applied on a subdomain basis, these local calculations being coordinated by an appropriate domain decomposition iteration at the global level. The parallel performances of these two parallel multigrid methods are evaluated through numerical experiments that are performed on several clusters of PCs with different computational nodes and interconnection networks.

Parallel multigrid methods for the calculation of unsteady flows on unstructured grids: algorithmic aspects and parallel performances on clusters of PCs

Parallel multigrid methods for the calculation of unsteady flows on unstructured grids: algorithmic aspects and parallel performances on clusters of PCs

Parallel unsteady incompressible viscous flow computations using an unstructured multigrid method

The development and validation of a parallel unstructured non-nested multigrid method for simulation of unsteady incompressible viscous flow is presented. The Navier–Stokes solver is based on the artificial compressibility approach and a higher-order characteristics-based finite-volume scheme on an unstructured multigrid. Unsteady flow is calculated with an implicit dual time-stepping scheme. The parallelization of the solver is achieved by multigrid domain decomposition approach (MG-DD), using the single program multiple data (SPMD) programming paradigm and message-passing interface (MPI) for communication of data. The parallel codes using single grid and multigrid are used to simulate steady and unsteady incompressible viscous flows over a circular cylinder for validation and performance evaluation purposes. The speedups and parallel efficiencies obtained by both the parallel single grid and multigrid solvers are reasonably good for both test cases, using up to 32 processors on the SGI Origin 2000. A maximum speedup of 12 could be achieved on 16 processors for high-Reynolds number unsteady viscous flow. The parallel results obtained were compared with those using serial single grid and multigrid codes and it remains the same as those obtained by serial solvers and agrees well with numerical solutions obtained by other researchers as well as experimental measurements.

Comparison of models for finite plasticity: A numerical study

We introduce a general framework for the numerical approximation of finite multiplicative plasticity. The method is based on a fully implicit discretization in time which results in an iteratively evaluated stress response; the arising nonlinear problem is then solved by a Newton method where the linear subproblems are solved with a parallel multigrid method. The procedure is applied to models with different elastic free energy functionals and a plastic flow rule of von Mises type. In addition these models are compared to a recently derived frame indifferent approximation of finite multiplicative plasticity valid for small elastic strains which leads to linear balance equations. Rate independent and rate dependent realizations of the former models are considered. We demonstrate by various 3D simulations that the choice of the elastic free energy is not essential (for material parameters representative for metals) and that the new model gives the same response quantitatively and qualitatively as the standard models.

A parallel algebraic multigrid solver for finite element method based source localization in the human brain

Time plays an important role in medical and neuropsychological diagnosis and research. In the field of Electro- and MagnetoEncephaloGraphy (EEG/MEG) source localization, a current distribution in the human brain is reconstructed noninvasively by means of measured fields outside the head. High resolution finite element modeling for the field computation leads to a sparse, large scale, linear equation system with many different right hand sides to be solved. The presented solution process is based on a parallel algebraic multigrid method. It is shown that very short computation times can be achieved through the combination of the multigrid technique and the parallelization on distributed memory computers. A solver time comparison to a classical parallel Jacobi preconditioned conjugate gradient method is given.

Local block refinement with a multigrid flow solver

AbstractA local block refinement procedure for the efficient computation of transient incompressible flows with heat transfer is presented. The procedure uses patched structured grids for the blockwise refinement and a parallel multigrid finite volume method with colocated primitive variables to solve the Navier‐Stokes equations. No restriction is imposed on the value of the refinement rate and non‐integer rates may also be used. The procedure is analysed with respect to its sensitivity to the refinement rate and to the corresponding accuracy. Several applications exemplify the advantages of the method in comparison with a common block structured grid approach. The results show that it is possible to achieve an improvement in accuracy with simultaneous significant savings in computing time and memory requirements. Copyright © 2002 John Wiley & Sons, Ltd.

Convergence rate analysis of an asynchronous space decomposition method for convex Minimization

We analyze the convergence rate of an asynchronous space decomposition method for constrained convex minimization in a reflexive Banach space. This method includes as special cases parallel domain decomposition methods and multigrid methods for solving elliptic partial differential equations. In particular, the method generalizes the additive Schwarz domain decomposition methods to allow for asynchronous updates. It also generalizes the BPX multigrid method to allow for use as solvers instead of as preconditioners, possibly with asynchronous updates, and is applicable to nonlinear problems. Applications to an overlapping domain decomposition for obstacle problems are also studied. The method of this work is also closely related to relaxation methods for nonlinear network flow. Accordingly, we specialize our convergence rate results to the above methods. The asynchronous method is implementable in a multiprocessor system, allowing for communication and computation delays among the processors.

A parallel multigrid solver for 3D convection and convection–diffusion problems

Multigrid algorithms are known to be highly efficient in solving systems of elliptic equations. However, standard multigrid algorithms fail to achieve optimal grid-independent convergence rates in solving non-elliptic problems. In many practical cases, the non-elliptic part of a problem is represented by the convection operator. Downstream marching, when it is viable, is the simplest and most efficient way to solve this operator. However, in a parallel setting, the sequential nature of marching degrades the efficiency of the algorithm. The aim of this paper is to present, evaluate and analyze an alternative highly parallel multigrid method for 3D convection-dominated problems. This method employs semicoarsening, a four-color plane-implicit smoother, and discretization rules allowing the same cross-characteristic interactions on all the grids involved to be maintained. The load imbalance below the critical level is the main source of inefficiency in its parallel implementation. After comparing several strategies to relieve the load imbalance, a hybrid smoother that degrades the convergence properties of the method but improves its granularity has been found to be the best choice. The numerical and parallel properties of the multigrid algorithm with the four-color and hybrid smoothers are studied on SGI Origin 2000 and Cray T3E systems.

Multiplicative and additive parallel multigrid algorithms for the acceleration of compressible flow computations on unstructured meshes

In this paper we examine how parallel multigrid acceleration can be used to improve the efficiency of two-dimensional compressible steady flow calculations on unstructured meshes. We study two parallel multigrid formulations. The first one is based on the standard approach that relies on domain partitioning for the parallel treatment of the pre- and post-smoothing steps whereas the coarse grid levels are visited sequentially according to predefined cycles (V-cycle, F-cycle or W-cycle). Reducing communication overheads is of crucial importance for parallel multigrid methods. When adopting the standard parallelization technique (i.e., intra-level parallelism based on domain partitioning) the usual drawback is that, as the calculation in a given cycling strategy proceeds from the finest level to the coarsest ones, the ratio between communication and calculation becomes worse resulting in a notable degradation of the parallel efficiency. In order to improve this situation, the second formulation considered in this study makes use of residual and correction filtering techniques allowing a parallel treatment of the various grid levels. This leads to the notion of inter-level parallelism. We propose distributed memory parallel versions of these two multigrid formulations and evaluate them through numerical experiments that are performed on a cluster of Pentium Pro computers interconnected via a 100 Mbit/s FastEthernet switch.