MP-RAS: A Multi-Parallel Restricted Additive Schwarz Preconditioner for Cardiac Electrophysiology Simulation

Atmospheric tomography, i.e., the reconstruction of the turbulence profile in the atmosphere, is a challenging task for adaptive optics (AO) systems for the next generation of extremely large telescopes. Within the AO community, the solver of first choice is the so-called Matrix Vector Multiplication (MVM) method, which directly applies the (regularized) generalized inverse of the system operator to the data. For small telescopes this approach is feasible, however, for larger systems such as the European Extremely Large Telescope (ELT), the atmospheric tomography problem is considerably more complex, and the computational efficiency becomes an issue. Iterative methods such as the Finite Element Wavelet Hybrid Algorithm (FEWHA) are a promising alternative. FEWHA is a wavelet-based reconstructor that uses the well-known iterative preconditioned conjugate gradient (PCG) method as a solver. The number of floating point operations and the memory usage are decreased significantly by using a matrix-free representation of the forward operator. A crucial indicator for the real-time performance are the number of PCG iterations. In this paper, we propose an augmented version of FEWHA, where the number of iterations is decreased by 50% using an augmented Krylov subspace method. We demonstrate that a parallel implementation of augmented FEWHA allows the fulfilment of the real-time requirements of the ELT.

We demonstrate that a small modification of the multiplicative, additive, and restricted additive Schwarz preconditioner at the algebraic level, motivated by optimized Schwarz methods defined at the continuous level, leads to a significant reduction in the iteration count of the iterative solver. Numerical experiments using finite difference and spectral element discretizations of the positive definite Helmholtz problem and an idealized climate simulation illustrate the effectiveness of the new approach.

The parallelization of the least-squares spectral element formulation of the Stokes problem has recently been discussed for incompressible flow problems on structured grids. In the present work, the extension to unstructured grids is discussed. It will be shown that, to obtain an efficient and scalable method, two different kinds of distribution of data are required involving a rather complicated parallel conversion between the data. Once the data conversion has been performed, a large symmetric positive definite algebraic system has to be solved iteratively. It is well known that the Conjugate Gradient method is a good choice to solve such systems. To improve the convergence rate of the Conjugate Gradient process, both Jacobi and Additive Schwarz preconditioners are applied. The Additive Schwarz preconditioner is based on domain decomposition and can be implemented such that a preconditioning step corresponds to a parallel matrix-by-vector product. The new results reveal that the Additive Schwarz preconditioner is very suitable for the p-refinement version of the least-squares spectral element method. To obtain good portable programs which may run on distributed-memory multiprocessors, networks of workstations as well as shared-memory machines we use MPI (Message Passing Interface). Numerical simulations have been performed to validate the scalability of the different parts of the proposed method. The experiments entailed simulating several large scale incompressible flows on a Cray T3E and on an SGI Origin 3800 with the number of processors varying from one to more than one hundred. The results indicate that the present method has very good parallel scaling properties making it a powerful method for numerical simulations of incompressible flows.

We present a parallel Newton-Krylov algorithm for solving the three-dimensional Euler equations on multi-block structured meshes. The Euler equations are discretized on each block independently using second-order accurate summation-by-parts operators and scalar numerical dissipation. Boundary conditions are imposed and block interfaces are coupled using simultaneous approximation terms (SATs). The resulting discrete equations are solved iteratively using an inexact Newton method. At each Newton iteration, the linear system is solved inexactly using a Krylov subspace iterative method, and both additive Schwarz and approximate Schur preconditioners are considered. The algorithm is tested on the ONERA M6 wing. The results show that a discretization based on SATs is well suited to a parallel Newton-Krylov solution strategy, and that the approximate Schur preconditioner is more efficient than the Schwarz preconditioner in terms of CPU time and Krylov iterations, for both flow and adjoint solves.

Groundwater flow simulation has become one of the top international issues in new generation of environmental applications. When managing large-scale groundwater flow problems, the intensive computational ability and large amounts of memory space required for modeling are the main bottlenecks for researchers. In order to solve three-dimensional large-scale groundwater flow problems more rapidly, the Open MP was adopted to parallelize the preconditioned conjugate gradient (PCG) algorithm in this paper. And this paper carried out a numerical experiment of the three-dimensional groundwater flow model on a computer with four cores. Based on the numerical experiment, it is found that the execution time of the original serial PCG program is about 1.74 to 2.86 times of the parallel PCG program executed with different number of threads. The experimental results also demonstrate that the PCG solver based on Open MP is an effective way for solving large-scale three-dimensional groundwater flow problem.

A novel sine transform based preconditioned conjugate gradient (PCG) method for signal restoration is presented. If the bandwidth 2? +1 in the model matrix is a constant, then the asymptotic complexity for the new algorithm is O(N) per iterative step, where Nis the order of the model. This complexity is reduced by an order of magnitude in comparison with the known fast Fourier transform (FFT) technique proposed recently in the literature. Moreover, if ? has a size similar to N, then the new PCG method is accomplished by the fast sine transform (FST) and the fast Hartley transform (FHT). The computational and storage cost per preconditioning step for the new PCG method is reduced by 50% as compared to the FFT approach. To stabilize the computation, a special Tikhonov regularization is introduced. Numerical experimentations show that the new PCG method is of a speedy convergence rate. The new PCG methods maintain less computational complexity per iterative step and have a better convergence rate than the other known PCG methods.

The $Q^N$- $Q^{N-2}$ spectral element discretization of the Stokes equation gives rise to an ill-conditioned, indefinite, symmetric linear system for the velocity and pressure degrees of freedom. We propose a domain decomposition method which involves the solution of a low-order global, and several local problems, related to the vertices, edges, and interiors of the subdomains. The original system is reduced to a symmetric equation for the velocity, which can be solved with the conjugate gradient method. We prove that the condition number of the iteration operator is bounded from above by $C(1+\log(N))^3/\beta_N$, where C is a positive constant independent of the degree N and the number of subdomains, and $\beta_N$ is the inf-sup condition of the pair $Q^N$- $Q^{N-2}$. We also consider the stationary Navier-Stokes equations; in each Newton step, a non-symmetric indefinite problem is solved using a Schwarz preconditioner. By using an especially designed low-order global space, and the solution of local problems analogous to those decribed above for the Stokes equation, we are able to present a complete theory for the method. We prove that the number of iterations of the GMRES method, at each Newton step, is bounded from above by $C(1+\log(N))^3/\beta_N$. The constant C does not depend on the number of subdomains or N, and it does not deteriorate as the Newton iteration proceeds.

In this work, we investigate devising efficient preconditioners for communication systems based on domain decomposition techniques. In particular, we focus on the uplink of a massive MIMO system and exploit the special structure of the channel cross-correlation matrix to derive improved preconditioners based on Schwarz preconditioning. Up to our knowledge, this is the first attempt to make use of domain decomposition techniques in the design of efficient preconditioners for communication systems. Two additive Schwarz preconditioners are investigated and tested within the context of cancelling inter-antenna interference in massive MIMO systems using two recently proposed parallel interference cancellation detectors. Simulation results reveal substantial improvement in convergence speed compared to the conventional Jacobi and block Jacobi preconditioning.

This paper presents a symmetric coupling between the finite element and boundary element methods for analysing open microwave devices. The proposed finite element boundary element (FEBE) method is based on the E-field vector Helmholtz equation. Second-order hierarchical Curl-conforming vector finite elements over tetrahedrons are used to discretize the interior region. The boundary element (BE) truncation surface is discretized with the second-order divergence-conforming surface elements with triangular support. The symmetry of the BE part is restored through the application of the Calderon-projector. Moreover, the BE computations are accelerated using a single level QR algorithm. This reduces both memory and computational time. The resulted symmetric system of equations is solved with a very efficient preconditioned conjugate gradient (PCCG) method with a p-multiplicative Schwarz (pMUS) preconditioner. Copyright © 2005 John Wiley & Sons, Ltd.

Gauss-Seidel preconditioned conjugate gradient (GS-PCG) method based on element-by-element (EBE) is derived and the calculation process of finite element method (FEM) is presented in this paper. The corresponding FEM program are developed and implemented on the magnetic field calculations which produced by straight conductor carrying current and 3-phases induction motor respectively. The validity and accuracy of the method have been verified by comparing the result with the analytical solution. Compared with Jacobi preconditioned conjugate gradient (J-PCG) method based on EBE-FEM, less computational time and cost are needed for the proposed method with same accuracy. The proposed method can be applied to electromagnetic field analysis of transformer and motor, which may refer large scale numerical computation.

Utility of the preconditioned conjugate gradient algorithm with a diagonal preconditioner for solving mixed-model equations in animal breeding applications was evaluated with 16 test problems. The problems included single- and multiple-trait analyses, with data on beef, dairy, and swine ranging from small examples to national data sets. Multiple-trait models considered low and high genetic correlations. Convergence was based on relative differences between left- and right-hand sides. The ordering of equations was fixed effects followed by random effects, with no special ordering within random effects. The preconditioned conjugate gradient program implemented with double precision converged for all models. However, when implemented in single precision, the preconditioned conjugate gradient algorithm did not converge for seven large models. The preconditioned conjugate gradient and successive overrelaxation algorithms were subsequently compared for 13 of the test problems. The preconditioned conjugate gradient algorithm was easy to implement with the iteration on data for general models. However, successive overrelaxation requires specific programming for each set of models. On average, the preconditioned conjugate gradient algorithm converged in three times fewer rounds of iteration than successive overrelaxation. With straightforward implementations, programs using the preconditioned conjugate gradient algorithm may be two or more times faster than those using successive overrelaxation. However, programs using the preconditioned conjugate gradient algorithm would use more memory than would comparable implementations using successive overrelaxation. Extensive optimization of either algorithm can influence rankings. The preconditioned conjugate gradient implemented with iteration on data, a diagonal preconditioner, and in double precision may be the algorithm of choice for solving mixed-model equations when sufficient memory is available and ease of implementation is essential.

In this study, we analyze the real-time solution of a series of augmented least squares problems, which are generated by adding information to an original least squares model repetitively. Instead of solving the least squares problems directly, we transform them into a batch of saddle point linear systems and subsequently solve the linear systems using restrictively preconditioned conjugate gradient (RPCG) methods. Approximation of the new Schur complement is generated effectively based on a previously approximated Schur complement. Owing to the variations of the preconditioned conjugate gradient method, the proposed methods generate convergence results similar to the conjugate gradient method and achieve a very fast convergent iterative sequence when the coefficient matrix is well preconditioned. Numerical tests show that the new methods are more effective than some standard Krylov subspace methods. Updated RPCG methods meet the requirement of real-time computing successfully for multifactor models.

A preconditioned Chebyshev basis communication-avoiding conjugate gradient method (P-CBCG) is applied to the pressure Poisson equation in a multiphase thermal-hydraulic CFD code JUPITER, and its computational performance and convergence properties are compared against a preconditioned conjugate gradient (P-CG) method and a preconditioned communication-avoiding conjugate gradient (P-CACG) method on the Oakforest-PACS, which consists of 8,208 KNLs. The P-CBCG method reduces the number of collective communications with keeping the robustness of convergence properties. Compared with the P-CACG method, an order of magnitude larger communication-avoiding steps are enabled by the improved robustness. It is shown that the P-CBCG method is 1.38 $$\times $$ and 1.17 $$\times $$ faster than the P-CG and P-CACG methods at 2,000 processors, respectively.

The preconditioned conjugate gradient (PCG) algorithm is a well-known iterative method for solving sparse linear systems in scientific computations. GPU-accelerated PCG algorithms for large-sized problems have attracted considerable attention recently. However, on a specific multi-GPU platform, producing a highly parallel PCG implementation for any large-sized problem requires significant time because several manual steps are involved in adjusting the related parameters and selecting an appropriate storage format for the matrix block that is assigned to each GPU. This motivates us to propose adaptive optimization modeling of PCG on multi-GPUs, which mainly involves the following parts: (1) an optimization multi-GPU parallel framework of PCG and (2) the profile-based optimization modeling for each one of the main components of the PCG algorithm, including vector operation, inner product, and sparse matrix-vector multiplication (SpMV). Our model does not construct a new storage format or kernel but automatically and rapidly generates an optimal parallel PCG algorithm for any problem on a specific multi-GPU platform by integrating existing storage formats and kernels. We take a vector operation kernel, an inner-product kernel, and five popular SpMV kernels for an example to present the idea of constructing the model. Given that our model is general, independent of the problems, and dependent on the resources of devices, this model is constructed only once for each type of GPU. The experiments validate the high efficiency of our proposed model.

In order to solve Toeplitz linear systems An(f)x=b generated by a nonnegative integrable function f, through use of the preconditioned conjugate gradient (PCG) method, several authors have proposed An(g) as preconditioner in the case where g is a trigonometric polynomial [10, 14, 27, 12, 28]. In preceding works, we studied the distribution and the extremal properties of the spectrum of the preconditioned matrix G=An−1(g) An(f). In this paper we prove that the union of the spectra of all the Gn is dense on the essential range of f/g, i.e.,ER(f/g) and we obtain asymptotic information about the rate of convergence of the smallest eigenvalue λln of Gn to r (and of λnn to R). As a consequence of this second order result, it is possible to handle the case where f has zeros of any order θ, through the PCG methods proposed in [10, 14]. This is a noteworthy extension since the techniques developed in [10, 14, 27, 12, 28] are shown to be effective only when f has zeros of even orders. The cost of this procedure is O(n1+c(θ) log n) arithmetic operations (ops) where the quantity c(θ) belongs to interval [0,2−1] and takes the maximum value 2−1 when f has a zero of odd order. Finally, for the special case of zeros of odd orders, we propose a further algorithm which makes use of the PCG techniques proposed in [10, 14, 27, 12, 28] for theeven order case, reducing the cost to O(n long n) ops.