Full-Scale Simulation of Cardiac Electrophysiology on Parallel Computers

Dominant trends in parallel and distributed computing (parallel computers, parallel algorithms) aim to connect a number of powerful workstations (WS) based on single personal computers (PC) or symmetrical multiprocessor system (SMP) to solve complex problems in a parallel way. Typical representatives of these trends are network of workstation (NOW) and Grid as a high integrated network of NOW modules. In relation to it there is a bad need of unified modelling in traditionally evolved parallel computing and distributed computing as two separate research disciplines. Likewise current trends in high performance computing (HPC) aim to use NOW modules based on SMP parallel computers as cheaper alternatives to the traditionally used massive parallel multiprocessors or supercomputers and to profit from unifying modelling in parallel and distributed computing too. To exploit the parallel processing capability of such cluster, any application program must be paralleled. A way how to do it (decomposition model) belongs to a most important step in developing process of effective parallel algorithm. This article at first discusses the development of effective (optimised) iterative parallel algorithms (IPA) to solve partial differential equations (PDE). By carefully choosing an illustrative typical problem of PDE (Laplace equation) the proposed research has demonstrated the most important complexities in the process of complete modelling of substantial complexities and their optimisation in order to develop an effective PA. Such approach could be used as illustrative example for a complete modelling and effective solution of even other more complex PDE´s in a parallel way.

This paper examines the cost/performance of simulating a hypothetical target parallel computer using a commercial host parallel computer. We address the question of whether parallel simulation is simply faster than sequential simulation, or if it is also more cost-effective. To answer this, we develop a performance model of the Wisconsin Wind Tunnel (WWT), a system that simulates cache-coherent shared-memory machines on a message-passing Thinking Machines CM-5. The performance model uses Kruskal and Weiss's fork-join model to account for the effect of event processing time variability on WWT's conservative fixed-window simulation algorithm. A generalization of Thiebaut and Stone's footprint model accurately predicts the effect of cache interference on the CM-5. The model is calibrated using parameters extracted from a fully-parallel simulation (p=N), and validated by measuring the speedup as the number of processors (p) ranges from one to the number of target nodes (N. Together with simple cost models, the performance model indicates that for target system sizes of 32 nodes and larger, parallel simulation is more cost-effective than sequential simulation. The key intuition behind this result is that large simulations require large memories, which dominate the cost of a uniprocessor; parallel computers allow multiple processors to simultaneously access this large memory.

H.26L is an emerging video encoding standard proposed by ITU, which goal is for video communication and multicast in low bit rate conditions. We give detailed analysis and performance tests for the parallelism of the H.26L video encoder test model long-term (TML8.0). Based on these tests, several parallel schedule strategies are discussed in different aspects, and the optimal one is selected and implemented by using MPI. At last, experiment results on Dawning-2000 MPP machines with different processors are given. The analysis and simulation results show that the H.26L has a good parallel performance and scalability on slice layer, and parallel computing can greatly speed up the processing of H.26L encoder. The experiment methods and results can be applied to many parallel systems for H.26L video processing.

The application of parallel computers to reservoir simulation is studied with the aid of two recently introduced Multiple Instruction, Multiple Data (MIMD) computers. The problems posed by reservoir simulators are shown to be particularly well suited for such parallel computers* Fundamental parallel programming techniques using Fortran are presented for the Hetrogeneous Element Processor (HEP) produced by Denelcor and the iPSC Hypercube produced by Intel. These techniques are then applied to two of the most time consuming tasks in reservoir simulation: forming the matrix coefficients and producing a sparse matrix solution. The problem of parallel formation of sparse matrix coefficients is addressed for the single and multiphase cases using both black oil and compositional fluid models. Parallel sparse matrix solution is addressed by considering D4 ordered Gaussian elimination, and the multigrid, conjugate gradient, and Successive Over-Relaxation (SOR) methods. Three parallel algorithms based on SOR are developed to illustrate the possible diversity of parallel algorithms. The red-black and multicolored SOR methods which are often used on vector machines are shown to be easily adaptable to parallel MIMD machine. Matrix partitioning and a new method of iteration pipelining are also presented as parallel SOR methods. The various SOR algorithms, in parallel form, are mapped onto the HEP to illustrate the speedups currently possible on an MIMD machine. Projections are made on the future importance of these types of machines to reservoir simulation.

This is an introductory book on supercomputer applications written by a researcher who is working on solving scientific and engineering application problems on parallel computers. The book is intended to quickly bring researchers and graduate students working on numerical solutions of partial differential equations with various applications into the area of parallel processing.The book starts from the basic concepts of parallel processing, like speedup, efficiency and different parallel architectures, then introduces the most frequently used algorithms for solving PDEs on parallel computers, with practical examples. Finally, it discusses more advanced topics, including different scalability metrics, parallel time stepping algorithms and new architectures and heterogeneous computing networks which have emerged in the last few years of high performance computing. Hundreds of references are also included in the book to direct interested readers to more detailed and in-depth discussions of specific topics.

Parallel matrix computations in air pollution modelling

In the numerical computation of a partial differential equation, it is useful to employ a parallel computer, where a large number of processors are connected in a mesh form. Such a scheme is now realizable with the advent of modern VLSI technology. On the other hand, there arises a request for the fault tolerancy with the increase of the processing units. When a partial differential equation is to be solved by such a computer, the requirement is not severe for the completeness of the data, since the iterative method usually is employed. This paper considers a problem of this nature, and discusses the recovery from the fault within the execution. The method is to detect the fault, to modify the connections among the processing units, separating the processing unit in fault, and to continue the computation by modifying a part of the discrete algorithm. By the proposed method, the dynamic recovery from the fault is easily realized, and the degradation of the computation accuracy due to the fault is suppressed. The proposed scheme is implemented on the parallel computer PAX‐32 with 32 processing units. The actual result of solving a two‐dimensional Poisson equation is presented, and a theoretical evaluation is made for the error in the one‐dimensional problem.

Several distributed-memory-type massively parallel computers have been developed and have proved to be efficient for numerical simulation of Partial Differential Equation (PDE) problems [1, 2, 3]. Although performance is high, it is very time-consuming to develop programs by using languages that have a special facility to control parallelism [4]. (In this chapter, “massively parallel computer ” means the distributed memory type.) On the other hand, in the automatic parallelization of general languages like FORTRAN and C, it is technically very difficult to achieve sufficient parallelizing efficiency for programs that are developed without special care for parallelization. The PDEQSOL (Partial Differential Equation Solver Language) system for massively parallel computers is being developed to overcome these limitations. PDEQSOL is a high-level programming language specially designed to describe PDE problems in a natural way for numerical analysis. The parallelizing translator generates a highly parallelized program for massively parallel computers by using the implicit parallelism in the PDEQSOL program.

Solving partial or ordinary differential equation models in cardiac electrophysiology is a computationally demanding task, particularly when high-resolution meshes are required to capture the complex dynamics of the heart. Moreover, in clinical applications, it is essential to employ computational tools that provide only relevant information, ensuring clarity and ease of interpretation. In this work, we exploit two recently proposed operator learning approaches, namely Fourier Neural Operators (FNO) and Kernel Operator Learning (KOL), to learn the operator mapping the applied stimulus in the physical domain into the activation and repolarization time distributions. These data-driven methods are evaluated on synthetic 2D and 3D domains, as well as on a physiologically realistic left ventricle geometry. Notably, while the learned map between the applied current and activation time has its modeling counterpart in the Eikonal model, no equivalent partial differential equation (PDE) model is known for the map between the applied current and repolarization time. Our results demonstrate that both FNO and KOL approaches are robust to hyperparameter choices and computationally efficient compared to traditional PDE-based Monodomain models. These findings highlight the potential use of these surrogate operators to accelerate cardiac simulations and facilitate their clinical integration.

Partitioning strategies for parallel KIVA-4 engine simulations

Simulating the propagation of 3D ultrasonic waves in a nonlinear medium is a demanding task. It requires the solution of time-dependent and nonlinear partial differential equations (PDEs). For such nonlinear PDEs, we have to use an implicit numerical method that is very CPU-intensive. Parallel simulation is therefore essential for studying those ultrasonic waves with satisfactory accuracy. We present the parallelization of an ultrasonic wave simulator and report some numerical experiments that have been run on a 24-node Linux cluster. Our CPU-measurements indicate that Linux clusters can deliver satisfactory parallel computing power for the numerical solution of PDEs. For simulating 3D ultrasonic waves in particular; we have found that our Linux cluster; which consists of 48 Pentium-III 500 MHz processors inter-connected with a 100 Mbit/s Ethernet network, is fully comparable with an SGI Origin 2000 machine.

Simulating a supersonic wake flow field behind a conical body is a computing intensive task. It requires a large number of computational cells to capture the dominant flow physics and a robust numerical algorithm to obtain a reliable solution. High performance parallel computers with unique distributed processing and data storage capability can provide this need. They have larger computational memory and faster computing time than conventional vector computers. We apply the PINCA Navier-Stokes code to simulate a wind-tunnel supersonic wake experiment on Intel Gamma, Intel Paragon, and IBM SP2 parallel computers. These simulations are performed to study the mean flow in the near wake region of a sharp, 7-degree half-angle, adiabatic cone at Mach number 4.3 and freestream Reynolds number of 40,600. Overall the numerical solutions capture the general features of the hypersonic laminar wake flow and compare favorably with the wind tunnel data. With a refined and clustering grid distribution in the recirculation zone, the calculated location of the rear stagnation point is consistent with the 2D axisymmetric and 3D experiments. In this study, we also demonstrate the importance of having a large local memory capacity within a computer node and the effective utilization of the number of computermore » nodes to achieve good parallel performance when simulating a complex, large-scale wake flow problem.« less

The modeling of physical phenomena in computational fracture mechanics, computational fluid dynamics and other fields is based on solving systems of partial differential equations (PDEs). When PDEs are defined over geometrically complex domains, they often do not admit closed form solutions. In such cases, they are solved approximately using discretizations of domains into simple elements like triangles and quadrilaterals in two dimensions (2D), and tetrahedra and hexahedra in three dimensions (3D). These discretizations are called finite element meshes. Many applications, for example, real-time computer assisted surgery, or crack propagation from fracture mechanics, impose time and/or mesh size constraints that cannot be met on a single sequential machine. As a result, the development of parallel mesh generation algorithms is required. In this dissertation, we describe a complete solution for both sequential and parallel construction of guaranteed quality Delaunay meshes for 2D and 3D geometries. First, we generalize the existing 2D and 3D Delaunay refinement algorithms along with theoretical proofs of mesh quality in terms of element shape and mesh gradation. Existing algorithms are constrained by just one or two specific positions for the insertion of a Steiner point inside a circumscribed disk of a poorly shaped element. We derive an entire 2D or 3D region for the selection of a Steiner point (i.e., infinitely many choices) inside the circumscribed disk. Second, we develop a novel theory which extends both the 2D and the 3D Generalized Delaunay Refinement methods for the concurrent and mathematically guaranteed independent insertion of Steiner points. Previous parallel algorithms are either reactive relying on implementation heuristics to resolve dependencies in parallel mesh generation computations or require the solution of a very difficult geometric optimization problem (the domain decomposition problem) which is still open for general 3D geometries. Our theory solves both of these drawbacks. Third, using our generalization of both the sequential and the parallel algorithms we implemented prototypes of practical and efficient parallel generalized guaranteed quality Delaunay refinement codes for both 2D and 3D geometries using existing state-of-the-art sequential codes for traditional Delaunay refinement methods. On a heterogeneous cluster of more than 100 processors our implementation can generate a uniform mesh with about a billion elements in less than 5 minutes. Even on a workstation with a few cores, we achieve a significant performance improvement over the corresponding state-of-the-art sequential 3D code, for graded meshes.

Partial differential equations (PDEs) are core tools for characterizing physical laws and are widely used in fluid mechanics, power systems, additive manufacturing, and other fields. However, traditional numerical methods are limited by mesh partitioning, leading to a trade-off between accuracy and efficiency in complex geometric domains and high-dimensional problems. While Physically Informed Neural Networks (PINNs) achieve meshless PDE solutions by embedding prior physical knowledge, they still face challenges such as insufficient accuracy, low training efficiency, and poor stability. This study aims to address the core bottlenecks of PINN in solving PDEs by proposing a systematic optimization strategy to improve its numerical solution accuracy, efficiency, and stability. Methodologically, firstly, a composite loss function containing PDE residuals and boundary initial condition constraints is constructed based on automatic differentiation techniques to ensure that the network satisfies physical laws. Secondly, the PINN training system is optimized, including using Neural Architecture Search (NAS-PINN) to automatically match the optimal network structure, designing an adaptive data sampling strategy to improve data utilization, and introducing parallel computing and hardware acceleration techniques to reduce training time. Finally, the effectiveness of the strategy is verified through classical PDEs and engineering problems. The results show that the optimized PINN achieves high-precision numerical approximation in classical PDE solutions and can efficiently solve forward design and backward parameter inversion problems in engineering scenarios. Compared with traditional methods, it exhibits stronger adaptability in complex geometric domains and high-dimensional problems, while improving training efficiency by more than 30% and significantly enhancing stability. This research not only enriches the optimization theory of PINN and provides an efficient new path for PDE numerical solutions, but also promotes the practical application of PINN in engineering fields such as fluid mechanics, additive manufacturing, and power systems, possessing significant theoretical value and application significance.

Macro-elementwise preconditioning methods