Parallel Numerical Methods Research Articles

A Review of High-Performance Computing Methods for Power Flow Analysis

Power flow analysis is critical for power systems due to the development of multiple energy supplies. For safety, stability, and real-time response in grid operation, grid planning, and analysis of power systems, it requires designing high-performance computing methods, accelerating power flow calculation, obtaining the voltage magnitude and phase angle of buses inside the power system, and coping with the increasingly complex large-scale power system. This paper provides an overview of the available parallel methods to fix the issues. Specifically, these methods can be classified into three categories from a hardware perspective: multi-cores, hybrid CPU-GPU architecture, and FPGA. In addition, from the perspective of numerical computation, the power flow algorithm is generally classified into iterative and direct methods. This review paper introduces models of power flow and hardware computing architectures and then compares their performance in parallel power flow calculations depending on parallel numerical methods on different computing platforms. Furthermore, this paper analyzes the challenges and pros and cons of these methods and provides guidance on how to exploit the parallelism of future power flow applications.

A novel MPI+MPI hybrid approach combining MPI-3 shared memory windows and C11/C++11 memory model

The increase of the number of cores in processors used in modern cluster architectures advocates hybrid parallel programming, combining Message Passing Interface (MPI) for internode operations and a shared memory treatment of intranode operations. We propose an MPI+MPI hybrid approach to parallel programming in which shared memory operations are managed by the combination of MPI shared memory windows introduced with MPI-3, C11/C++11 atomic operations and the associated multi-thread memory model. We illustrate the method on fundamental parallel operations (barrier, reduction) and on the ghost update, which is prevalent in many parallel numerical methods. The performance tests on Reedbush-U and Oakbridge-CX systems show that using the C11/C++11 memory model to manage shared memory windows can achieve levels of performance comparable to state of the art MPI implementations, while reducing the variance of execution times as well as increasing the level of synchronization between processes, especially in multiple nodes environments. It also reduces significantly the execution time of ghost updates compared to flat MPI, and the synchronization of shared data with the C++11 memory model is observed to be more efficient than other synchronization methods based on RMA utilities.

On Development of Parallel Algorithms for Solving Parabolic and Elliptic Equations

In this paper, we present results of the development of certain parallel numerical methods for solving three-dimensional evolutionary and stationary problems of diffusion and heat transfer. We present a detailed description of a special, explicit iteration scheme for parabolic equations and discuss a multigrid technology used for solving elliptic equations and implicit schemes for parabolic equations.

TO THE SOLUTION OF ONE-PARAMETRIC MATRIX EQUATIONS OF A(T)⋅X(T)+ X*(T) ⋅B(T) =C(T) TYPE

We consider the one-parameter conjugated analogs of Sylvester type matrix equations. On the basis of simple transformations we obtainthe equivalent matrix equation containing only the unknown matrix which should be determined. Next, using the apparatus of Kroneckerproducts of matrices, the analytical solution of the problem was obtained, which is limited in practical applications, but serves as a basis for the development of numerical-analytical methods for solving the original problem. Serial and parallel numerical and analytical solution methods are proposed based on the differential transformations G. E. Pukhov. With sequential numerical-analytical method we operate with numeric recursive procedures in the first stage of the calculation, and analytical relations – in the second stage. In parallel numerical-analytical method we operate with linear hypersystem of numerical equations in the first stage of the calculation, and analytical relations – in the second stage. For all the methods the relevant conditions for the unique solvability of the problem were obtained. A model example was considered for which using numerical and analytical methods an exact solution of the Taylor was obtained. The proposed numerical-analytical techniques can be efficiently implemented by means of modern information technology.

Parallel Multiprojection Preconditioned Methods Based on Subspace Compression

During the last decades, the continuous expansion of supercomputing infrastructures necessitates the design of scalable and robust parallel numerical methods for solving large sparse linear systems. A new approach for the additive projection parallel preconditioned iterative method based on semiaggregation and a subspace compression technique, for general sparse linear systems, is presented. The subspace compression technique utilizes a subdomain adjacency matrix and breadth first search to discover and aggregate subdomains to limit the average size of the local linear systems, resulting in reduced memory requirements. The depth of aggregation is controlled by a user defined parameter. The local coefficient matrices use the aggregates computed during the formation of the subdomain adjacency matrix in order to avoid recomputation and improve performance. Moreover, the rows and columns corresponding to the newly formed aggregates are ordered last to further reduce fill-in during the factorization of the local coefficient matrices. Furthermore, the method is based on nonoverlapping domain decomposition in conjunction with algebraic graph partitioning techniques for separating the subdomains. Finally, the applicability and implementation issues are discussed and numerical results along with comparative results are presented.

Parallel-Simulator Framework for Multipermeability Modeling With Discrete Fractures for Unconventional and Tight Gas Reservoirs

Summary Economic gas rate from ultralow-permeability shale reservoirs requires the creation of a complex fracture network in a large volume known as the stimulated reservoir volume (SRV). The fracture network connects a large surface area of the reservoir to the well. It is created by injecting low-viscosity fracturing fluid (slickwater) at very high rates in multiple stages along the horizontal wellbore. Numerical simulation is used to evaluate the stimulation designs and completion strategy. Microseismic (MS) -survey fracture mapping can provide a measurement of the overall SRV and an estimate of the fracture patterns. Special core analyses provide estimates of shale-matrix permeability. The extent of the fracture network indicates that there is insufficient proppant volume, and many stimulated fractures may be only partially propped or may be unpropped. Thus, fracture conductivity will vary spatially caused by uneven proppant distribution and temporally caused by stress sensitivity upon pressure decline during production. Because of the vast contrast in conductivity between stimulated/hydraulic fractures (darcy-ft) and shale matrix (nd-ft), the transient response in matrix/fracture flow cannot be captured accurately if the stimulated fractures are approximated with large dual-continuum (DC) gridblocks. The gridding requirement to achieve an accurate solution in fractured shale reservoirs is investigated and discussed. In this work, the stimulated and hydraulic fractures are discretized explicitly to form a discrete fracture network (DFN). This paper discusses the mathematical framework and parallel numerical methods for simulating unconventional reservoirs. The simulation methods incorporate known mechanisms and processes for shale, which include gas sorption in organic matter; combined Knudsen diffusion and viscous flow in nanopores; stress-sensitive fracture permeability; and velocity-dependent flow in the high-conductivity hydraulic fractures. The simulation system is based on a general finite-volume method that includes a multiconnected multicontinuum (MC) representation of the pore system with either a compositional or a black-oil fluid description. The MC model is used to represent the storage and intercommunication among the various porosities in shale (organic matter, inorganic matter, fine unstimulated natural fractures). Unconventional simulation involves many more nonlinearities, and the extreme contrast in permeabilities will make the problems harder to solve. We discuss numerical implementation of the methods for modeling the mechanisms and processes in fractured shale. In addition, we discuss the MC formulation, the discretization method, the unstructured parallel domain-decomposition method, and the solution method for the simulation system. Finally, we explain our efforts in numerical validation of the system with fine-grid single-porosity simulation. We show numerical examples to demonstrate the applications of the simulator and to study the transient flow behavior in shale reservoirs. The effects of the various mechanisms for gas production are also evaluated.

A Hybrid Algorithm Based on Optimal Quadratic Spline Collocation and Parareal Deferred Correction for Parabolic PDEs

Parareal is a kind of time parallel numerical methods for time-dependent systems. In this paper, we consider a general linear parabolic PDE, use optimal quadratic spline collocation (QSC) method for the space discretization, and proceed with the parareal technique on the time domain. Meanwhile, deferred correction technique is also used to improve the accuracy during the iterations. In fact, the optimal QSC method is a correction of general QSC method. Along the temporal direction we embed the iterations of deferred correction into parareal to construct a hybrid method, parareal deferred correction (PDC) method. The error estimation is presented and the stability is analyzed. To save computational cost, we find out a simple way to balance the two kinds of iterations as much as possible. We also argue that the hybrid algorithm has better system efficiency and costs less running time. Numerical experiments by multicore computers are attached to exhibit the effectiveness of the hybrid algorithm.

Co-design of Parallel Numerical Methods for Plasma Physics and Astrophysics

Physically meaningful simulations in plasma physics and astrophysics need powerful hybrid supercomputers equipped with computation accelerators. The development of parallel numerical codes for such supercomputers is a complex scientific problem. In order to solve it the concept of codesign is employed. The codesign is defined as considering the architecture of the supercomputer at all stages of the development of the code. The use of codesign is shown by the example of two physical problems: the interaction of an electron beam with plasma and the collision of galaxies. The efficiency is 92 % with 500 Tesla GPUs at the Lomonosov supercomputer. The test computation involved 160 million of model particles.

Parallel numerical methods for incompressible flows

The simulation of incompressible flows consists of an important part of computational fluid dynamics. Based on finite element discretization, this paper develops several parallel numerical methods for the incompressible flows governed by the Navier-Stokes (N-S) equations, which can be classified into two classes: one is based on two-grid discretization, where the fully nonlinear N-S equations are first solved on a coarse grid, and the correction is then calculated on a fine grid by solving a linear residual problem in a parallel manner; the other is based on a novel fully overlapping domain decomposition technique, where each processor computes a local finite element solution in its own sub-domain using a global multiscale mesh that is locally refined around its own sub-domain. These parallel numerical methods are easy to implement. They have low communication complexity, and can yield a finite element solution with the same order of convergence rate as the standard finite element methods. Theoretical analysis and numerical tests demonstrated the high efficiency of the studied methods.

An efficient graphics processing unit‐based parallel algorithm for pricing multi‐asset American options

SUMMARYWe develop highly efficient parallel PDE‐based pricing methods on graphics processing units (GPUs) for multi‐asset American options. Our pricing approach is built upon a combination of a discrete penalty approach for the linear complementarity problem arising because of the free boundary and a GPU‐based parallel alternating direction implicit approximate factorization technique with finite differences on uniform grids for the solution of the linear algebraic system arising from each penalty iteration. A timestep size selector implemented efficiently on GPUs is used to further increase the efficiency of the methods. We demonstrate the efficiency and accuracy of the parallel numerical methods by pricing American options written on three assets. Copyright © 2011 John Wiley & Sons, Ltd.

An Efficient GPU-Based Parallel Algorithm for Pricing Multi-Asset American Options

We develop highly-efficient parallel Partial Differential Equation (PDE) based pricing methods on Graphics Processing Units (GPUs) for multi-asset American options. Our pricing approach is built upon a combination of a discrete penalty approach for the linear complementarity problem arising due to the free boundary and a GPU-based parallel Alternating Direction Implicit Approximate Factorization technique with finite differences on uniform grids for the solution of the linear algebraic system arising from each penalty iteration. A timestep size selector implemented efficiently on GPUs is used to further increase the efficiency of the methods. We demonstrate the efficiency and accuracy of the parallel numerical methods by pricing American options written on three assets.

High performance parallel numerical methods for Volterra equations with weakly singular kernels

Non-stationary discrete time waveform relaxation methods for Abel systems of Volterra integral equations using fractional linear multistep formulae are introduced. Fully parallel discrete waveform relaxation methods having an optimal convergence rate are constructed. A significant expression of the error is proved, which allows us to estimate the number of iterations needed to satisfy a prescribed tolerance and allows us to identify the problems where the optimal methods offer the best performance. The numerical experiments confirm the theoretical expectations.

Special meshes and domain decomposition methods for evolutionary convection–diffusion–reaction problems

AbstractIn this work we propose new parallel numerical methods to solve certain evolutionary singularly perturbed problems in a robust and efficient way. To get this, we firstly consider a semidiscretization in time using a simple fractional step Runge–Kutta method, where the splittings for the convection–diffusion–reaction operator and the source term are subordinated to a decomposition of the spatial domain in many smaller subdomains. Such semidiscretization procedure reduces the original problem to a set of elliptic problems (in smaller subdomains), which can be solved in parallel, and it avoids the use of Schwarz iterations. To discretize in space such problems we have considered classical linear finite elements on certain piecewise uniform meshes which have been constructed with a very simple a priori criterion, similar to the one introduced by Shishkin for one‐dimensional stationary problems of this kind. We show that the use of these meshes permits to obtain uniformly convergent approximations even in the boundary layer regions. Copyright © 2005 John Wiley & Sons, Ltd.

Speedup in solving differential equations on clusters of workstations

A performance model is constructed for parallel iterative numerical methods under the assumption of a message-passing computing system. Arguments are given for the fact that the speedup of parallel iterative methods is mainly influenced by the speedup at one iterative step including computations and communications. Using the theoretical model, it is proved why explicit iterative methods for ordinary differential equations are inefficient in implementation on distributed memory multiprocessor systems. Numerical tests on a cluster of dedicated machines confirm the correctness of the theoretical model at least in the case of iterative methods for ordinary differential equations and time-dependent partial differential equations.

Study of parallel numerical methods for semiconductor device simulation

Simulators of semiconductor devices have to solve systems of equations generated by the discretization of partial differential equations, which are the most time-consuming part of the simulation process. Therefore, the use of an effective method to solve these linear systems is essential. In this work we have evaluated the efficiency of different parallel direct and iterative solvers used for the solution of the drift–diffusion equations in semiconductor device simulation. Several preconditioning techniques have been applied in order to minimize the execution times. We have found that FGMRES and BCGSTAB solvers preconditioned with Additive Schwarz are the most suitable for these types of problems. The results were obtained in an HP Superdome cluster with 128 Itanium2 1.5 GHz. Copyright © 2005 John Wiley & Sons, Ltd.

An Evaluation of Parallel Least Squares Methods for Multisatellite Orbit Determination

Previous studies of parallel processing methods for use with satellite related, many parameter estimation problems indicated the need for faster numerical integration methods. A study of parallel numerical integration methods for use in satellite orbit determination showed only slight improvement over the fastest serial methods, thus prompting the reexamination of the role of numerical integration in a parallel estimation scheme. A parallel orbit determination scheme, parallel in both observation processing and least squares solution, was implemented. The fastest parallel method, run on twenty-four processors, showed a 33% improvement over the fastest serial method for the case of estimating twenty-four GPS satellite orbits. To determine the relative time spent in numerical integration a detailed analysis of the time spent in each of the primary steps of the parallel method was completed. This analysis showed that numerical integration does not account for a significant fraction of the computing times. In addition, the analysis revleaed that delays due the the synchronization of processors were a limit on the performance.

Parallel Numeric Algorithms On Faster Computers

Parallel Numeric Algorithms On Faster Computers

Parallel dynamic simulation of multiple manipulator systems: temporal versus spatial methods

In this paper, parallel algorithms are developed for real-time dynamic simulation of a multiple manipulator system, cooperating to manipulate a large load. In an effort to achieve real-time computational rates on a general-purpose parallel system, temporal and spatial forms of parallelism are implemented to improve performance. Temporal parallelism is obtained with the use of parallel numerical integration methods. A speedup of 3.78 on four processors of a CRAY Y-MPS was achieved with a parallel four-point block predictor-corrector method for the simulation of a four manipulator system. To overcome a loss of efficiency because of a reduction in accuracy with the block integration methods, spatial parallelism is used in which the dynamics of each chain is computed simultaneously. With the same four manipulator system, this form of parallelism in conjunction with a serial integration method results in a speedup of 3.1 on four processors without the degradation in accuracy. In cases where there are more processors than chains, a new multipoint parallel integration method can still be advantageous despite the reduced accuracy. In this case, it is shown that greater effective speedups are achieved when both forms of parallelism are combined to generate more parallel tasks. >

PARALLELIZATION OF INVISCID AND VISCOUS FLOW SOLVERS

SUMMARY The objectives of this paper are to present the development and implementation of parallel numerical methods for inviscid and viscous compressible flows. A flux vector splitting scheme and a Riemann solver were used to discretize the Euler and Navier-Stokes equations. Furthermore, an implicit unfactored procedure with Gauss-Seidel relaxation was used to solve the equations in time dependent form. Acceleration of the convergence was achieved by the local solution method and the mesh sequencing technique. Grid partitioning was utilized for the parallelization of the methods. Calculations were performed on various parallel platforms and factors that can influence the parallel performance, such as numerical method, acceleration procedure, grid size, and interprocessor communication were investigated.

Parallel methods for initial value problems

As scientific technology becomes increasingly more sophisticated, the production of more data and/or the modelling of more complex systems requires computers with ever-increasing computational power. In order to cope with this situation numerical algorithms have to be developed which allow the distribution of both data and code segments over large numbers of processors with the hope that problems that were insoluble in a sequential environment because of either (or both) accuracy and size constraints can now be solved in a parallel environment. This paper will present a review of recently developed techniques in the area of parallel numerical methods for initial value problems. It will focus mainly on two different approaches—parallelism across time and parallelism across space—but will also consider special techniques developed for certain classes of problems.