Sparse Approximate Inverse Preconditioner Research Articles

Novel and efficient implementation of multi-level fast multipole indirect BEM for industrial Helmholtz problems

Fast multipole boundary element methods (FMBEM) have been successfully applied to solve ultra-large acoustic problems. In practice, there are still a large number of industrial applications where model sizes are well below a million Degrees of freedom. Thus, it is of practical importance to further improve the solver’s efficiency for such applications. This paper presents a novel formulation of the multi-level fast multipole indirect BEM, which relies on redesigning a conventional FMBEM with considerations from several aspects. A new flexible partition algorithm is proposed to generate a flexible and efficient multi-level structure for fast multipole calculations, which also allows the explicit calculation of the far field contributions. A hybrid and efficient regularization approach for hypersingular integral is investigated and implemented, which significantly reduces the iterative solving time. The use of efficient sparse approximate inverse preconditioners is key for fast convergence. Two strategies for constructing the sparsity pattern are implemented and evaluated. Numerical results show that by bringing the aforementioned techniques all together in one package, the new flexible multi-level fast multipole indirect BEM (f-FMBEM) provides significant improvements on actual simulation time over the conventional BEM and FMBEM solvers for industrial size problems (with a non-dimensional Helmholtz number up to kA∼323) on a modern workstation. To efficiently address larger problems, a more efficient multipole translation should be considered.

Mixed Precision Iterative Refinement with Sparse Approximate Inverse Preconditioning

With the commercial availability of mixed precision hardware, mixed precision GMRES-based iterative refinement schemes have emerged as popular approaches for solving sparse linear systems. Existing analyses of these approaches, however, are based on using full LU factorizations to construct preconditioners for use within GMRES in each refinement step. In practical applications, inexact preconditioning techniques, such as incomplete LU or sparse approximate inverses, are often used for performance reasons. In this work, we investigate the use of sparse approximate inverse preconditioners based on Frobenius norm minimization within GMRES-based iterative refinement. We analyze the computation of sparse approximate inverses in finite precision and derive constraints under which user-specified stopping criteria will be satisfied. We then analyze the behavior of and convergence constraints for a five-precision GMRES-based iterative refinement scheme that uses sparse approximate inverse preconditioning, which we call SPAI-GMRES-IR. Our numerical experiments confirm the theoretical analysis and illustrate the resulting tradeoffs between preconditioner sparsity and GMRES-IR convergence rate.

Parallel Dynamic Sparse Approximate Inverse Preconditioning Algorithm on GPU

The dynamic sparse approximate inverse (SPAI) preconditioner has proven to be effective in accelerating the convergence of iterative methods for large linear systems. Recently, accelerating it on graphics processing unit (GPU) has attracted considerable attention due to the fact that the cost of constructing the preconditioner is high. However, the existing parallel dynamic SPAI preconditioning algorithms on GPU are usually ineffective because of the out-of-memory error for large matrices. This motivates us to investigate how to accelerate the construction of dynamic SPAI preconditioners on GPU. In this article, we propose an efficient dynamic SPAI preconditioning algorithm on GPU, called GDSPAI. For our proposed GDSPAI, there are the following novelties: (1) a well-known dynamic SPAI preconditioning algorithm is substantially modified to address the main challenges of parallelization on GPU, (2) a parallel framework of constructing the dynamic SPAI preconditioner on GPU is presented on the basis of the modified dynamic SPAI preconditioning algorithm; and (3) each component of the preconditioner is computed in parallel inside a group of threads. Experimental results show that the proposed GDSPAI is effective for large matrices, and outperforms the popular preconditioning algorithms in three public libraries, as well as a recent parallel static SPAI preconditioning algorithm.

Optimizing the sparse approximate inverse preconditioning algorithm on GPU

In this study, we present an optimization sparse approximate inverse (SPAI) preconditioning algorithm on GPU, called GSPAI-Opt. In GSPAI-Opt, it fuses the advantages of two popular SPAI preconditioning algorithms, and has the following novelties: (1) an optimization strategy is proposed to choose whether to use the constant or non-constant thread group for any sparse pattern of the preprocessor, and (2) a parallel framework of optimizing the SPAI preconditioner is proposed on GPU, and (3) for each component of the preconditioner, a decision tree is established to choose the optimal kernel of computing it. Experimental results validate the effectiveness of GSPAI-Opt.

An Efficient Discontinuous Galerkin Method for Cavity Design

Cavities are strong scattering parts for an aircraft. A discontinuous Galerkin (DG) method is presented to efficiently optimize cavity scattering. The DG method is shown to have ability of only remeshing the optimized parts and not remeshing the whole cavity in the computing process of scattering from the whole cavity. An effective and simple preconditioner, called distance sparse preconditioner (DSP), is applied to the DG method, different from the conventional block-diagonal preconditioner (BDP) and sparse approximate inverse preconditioner (SAI). The former one is usually recommended in the DG methods and the latter one is generally used in method of moment (MoM). Numerical results show that the DSP is more efficient than the BDP and SAI in the DG method for the cavity analysis. The effects of cavity opening shape, area, and transition shape are studied for scattering from cavities. Some interesting and useful numerical results are presented for cavity design.

A thread-adaptive sparse approximate inverse preconditioning algorithm on multi-GPUs

In this study, we present an efficient thread-adaptive sparse approximate inverse preconditioning algorithm on multiple GPUs, called GSPAI-Adaptive. For our proposed GSPAI-Adaptive, there are the following novelties: (1) a thread-adaptive allocation strategy is presented for each column of the preconditioner, and (2) a parallel framework of constructing the sparse approximate inverse preconditioner is proposed on multiple GPUs, and (3) each component of the preconditioner is computed in parallel inside a thread group of GPU. Experimental results show that GSPAI-Adaptive is effective, and is advantageous over the popular preconditioning algorithms in two public libraries, and a latest parallel sparse approximate inverse preconditioning algorithm.

Fast Preconditioner Computation for BICGSTAB-FFT Method of Moments with NURBS in Large Multilayer Structures

Fast computation of the coefficients of the reduced impedance matrix of the method of moment (MM) is proposed by expanding the basis functions (BFs) in pulses and solving an equivalent periodic problem (EPP) for analyzing large multilayer structures with non-uniform rational basis spline (NURBS) modeling of the embedded layout. These coefficients are required by the computation of sparse approximate inverse (SAI) preconditioner, which leads an efficient iterative version of the MM. This reduced coefficient matrix only considers the near field part of the MM matrix. Discrete functions of small sizes are required to implement the pulse expansion and EPP. These discrete functions of small size lead to discrete cyclic convolutions that are computed in a very fast way by fast Fourier transform (FFT)-accelerated matrix–vector multiplication. Results obtained using a conventional laptop show an analysis of very large multilayer structures with resonant layouts, as whole reflectarrays of electrical size 40 times the vacuum wavelengths, where the iterative MM with a SAI preconditioner can be 22.7 times faster than the pure iterative MM without any preconditioner.

A Parallel and Broadband Helmholtz FMBEM Model for Large-Scale Target Strength Modeling

The Fast Multipole Boundary Element Method (FMBEM) reduces the [Formula: see text] computational and memory complexity of the conventional BEM discretized with [Formula: see text] boundary unknowns, to [Formula: see text] and [Formula: see text], respectively. A number of massively parallel FMBEM models have been developed in the last decade or so for CPU, GPU and heterogeneous architectures, which are capable of utilizing hundreds of thousands of CPU cores to treat problems with billions of degrees of freedom (dof). On the opposite end of this spectrum, small-scale parallelization of the FMBEM to run on the typical workstation computers available to many researchers allows for a number of simplifications in the parallelization strategy. In this paper, a novel parallel broadband Helmholtz FMBEM model is presented, which utilizes a simple columnwise distribution scheme, element reordering and rowwise compression of data, to parallelize all stages of the fast multipole method (FMM) algorithm with a minimal communication overhead. The sparse BEM near-field and sparse approximate inverse preconditioner are also constructed and executed in parallel, while the flexible generalized minimum residual (fGMRES) solver has been modified to apply the FMBEM matrix-vector products and corresponding minimum residual convergence within the parallel environment. The algorithmic and memory complexities of the resulting parallel FMBEM model are shown to reaffirm the above estimates for both the serial and parallel configurations. The parallel efficiency (PE) of the FMBEM matrix-vector products and fGMRES solution for the present model is shown to be satisfactory; achieving PEs up to [Formula: see text] and [Formula: see text] in the fGMRES solution using 3 and 6 CPU cores respectively, when applied to models having [Formula: see text] dof per CPU core. The PE of the precalculation stages of the FMBEM — in particular the FMM precomputation stage which is largely unparallelized — reduces the overall PE of the FMBEM model; resulting in average efficiencies of [Formula: see text] and [Formula: see text] for the 3-core and 6-core models when treating problems with [Formula: see text] dof per CPU core. The present model is able to treat large-scale acoustic scattering problems involving up to [Formula: see text] dof on a workstation computer equipped with 128[Formula: see text]GB of RAM, while acoustic target strength (TS) results calculated up to 3[Formula: see text]kHz for the BeTSSi II submarine model demonstrate its capabilities for large-scale TS modeling.

Hybrid parallelisation scheme for the application of distributed near‐field sparse approximate inverse preconditioners on high‐performance computing clusters

Hybrid parallelisation scheme for the application of distributed near‐field sparse approximate inverse preconditioners on high‐performance computing clusters

An efficient sparse approximate inverse preconditioning algorithm on GPU

SummaryThe sparse approximate inverse (SPAI) preconditioner has proven to be effective in accelerating the convergence of iterative methods. Recently, accelerating it on the graphics processing unit (GPU) has attracted considerable attention due to the fact that the cost of constructing it is high. This motivates us to investigate how to accelerate the construction of SPAI preconditioners on GPU in this paper. We propose an efficient sparse approximate inverse algorithm on GPU, called SPAI‐Adaptive. For our proposed SPAI‐Adaptive, there are the following novelties: (1) an adaptive thread allocation strategy for SPAI‐Adaptive is proposed to assign the optimal thread number for each column of the preconditioner, and (2) Each component of the preconditioner, which includes finding indices I and J, constructing local submatrix, decomposing the local matrix into QR, and solving the upper triangular linear system, is computed in parallel inside a thread group of GPU. Experimental results show that the proposed SPAI‐Adaptive is effective, and has good performance and high parallelism.

Preconditioned linear solves for parametric model order reduction

ABSTRACTThe main computational cost of algorithms for computing reduced-order models of parametric dynamical systems is in solving sequences of very large and sparse linear systems of equations, which are predominantly dependent on slowly varying parameter values. We focus on efficiently solving these linear systems, specifically those arising in a set of algorithms for reducing linear dynamical systems with the parameters linearly embedded in the system matrices. We propose the use of the block variant of the problem-dependent underlying iterative method because often, all right hand sides are available together. Since Sparse Approximate Inverse (SPAI) preconditioner is a general preconditioner that can be naturally parallelized, we propose its use. Our most novel contribution is a technique to cheaply update the SPAI preconditioner, while solving parametrically changing linear systems. We support our proposed theory by numerical experiments where-in two different models are reduced by a commonly used parametric model order reduction algorithm called RPMOR. Experimentally, we demonstrate that using a block variant of the underlying iterative solver saves nearly 95% of the computation time over the non-block version. Further, and more importantly, block GCRO with SPAI update saves around 60% of the time over block GCRO with SPAI.

Efficient Analysis of Scattering by Multiple Moving Objects Using a Tailored MLFMA

In this communication, we present a tailored multilevel fast multipole algorithm (MLFMA) for efficient analysis of scattering by multiple moving objects. There are two major issues with the conventional MLFMA for multiple moving targets. One is that the relative positions and attitudes of the targets change with time and we have to remesh the solution domain and refill the matrix elements at each simulation moment. The other is that we have to solve the matrix equations repeatedly for the moving targets. To alleviate these two burdens, we adopt the stationary grouping scheme in using the MLFMA for individual object, which is unchanged as the object is moving. With the grouping scheme, the near interactions are invariant, and most far interactions are reusable when the objects are in motions. The number of levels of MLFMA remains the same even if the overall dimension of the multiple targets is changing. The interactions between objects are expressed via coordinate transforms at the highest levels of their octrees. To accelerate the iterative solutions, the sparse approximate inverse preconditioner is incorporated. Numerical results show that the saving of CPU time is substantial for continuous simulations of a maneuvering process of a group of targets.

A two–step radial basis function-based CFD mesh displacement tool

Mesh displacement based on Radial Basis Functions (RBF) interpolation is known for its ability to preserve the validity and quality of the mesh, even for large displacements, without being affected by mesh connectivity. However, in the case of large meshes, such as those used in real-world Computational Fluid Dynamics (CFD) applications, RBF interpolation, in its standard formulation, becomes excessively expensive. This paper proposes a cost reduction technique for mesh displacement based on RBF, by splitting the process into two steps. In the first step, named predictor, a data reduction algorithm that adaptively agglomerates mesh boundary nodes by reducing the RBF interpolation problem size is used. Upon completion of the first step, due to the agglomeration and the fact that the RBF interpolation is applied to the boundary nodes too, the so-displaced boundaries do not match the given displacements; thus, the position of the boundary nodes must be corrected during the second step, named corrector. The latter performs a local deformation based on RBF kernels with local support, to make the boundary conform to the known displacements of its nodes. The proposed method is accelerated by employing the Sparse Approximate Inverse preconditioner based on geometrical considerations and the Fast Multipole Method. The method and the programmed software are validated on three test cases related to the deformation of CFD meshes inside a duct and a turbine stator row as well as around a car model.

Mixed Inner–Outer Iteration Technique-Based Surface Integral Equations for Fast Solving EM Scattering From Penetrable Objects

In this paper, a mixed inner–outer iteration technique based on surface integral equation is proposed for solving scattering problems from penetrable and homogeneous dielectric structures. The most important novelty of the proposed technique is combining the tangential Poggio–Miller–Chang–Harrington–Wu–Tsai as outer iteration and the electric-magnetic current combined-field integral equation as inner iteration, respectively. In this case, the new method offers the guarantee of high accuracy with faster convergence, especially for electrically dielectric structures in large scale. Furthermore, it presents an excellent compatibility with sparse approximate inverse preconditioning. Numerical results are given to demonstrate the accuracy, efficiency, and compatibility of the proposed method.

A fast algorithm for calculating surface electric field of converter valve shield system by multipole boundary element method

The surface electric field analysis of the converter valve shield system is a large-scale electrostatic field problem, which is difficult to analyse. The fast multipole boundary element method (FMBEM), which is suitable for solving large-scale problems, can accelerate the computation speed and conserve memory. However, the coefficient matrix implicitly formed by using the FMBEM is sometimes ill-conditioned, especially for large-scale problems; thus, the convergence of iteration is poor. In this paper, a fast solver is proposed to improve efficiency. First, an adaptive GMRES( m ) with variant restart parameter is adjusted for the Galerkin FMBEM. In addition, the sparse approximate inverse preconditioner is improved, and a new sparsity pattern is proposed for the multiscale problem derived from the converter valve shield system. The numerical results show that the accuracy can meet the engineering requirements compared with the finite element method. Compared with other solvers and preconditioners, the algorithm can achieve a satisfactory convergence rate and reduce the computation time. In addition, a single bridge shield system of ±160 kV converter valve is successfully analysed using the proposed method.

Multilevel approaches for FSAI preconditioning

SummaryFactorized sparse approximate inverse (FSAI) preconditioners are robust algorithms for symmetric positive matrices, which are particularly attractive in a parallel computational environment because of their inherent and almost perfect scalability. Their parallel degree is even redundant with respect to the actual capabilities of the current computational architectures. In this work, we present two new approaches for FSAI preconditioners with the aim of improving the algorithm effectiveness by adding some sequentiality to the native formulation. The first one, denoted as block tridiagonal FSAI, is based on a block tridiagonal factorization strategy, whereas the second one, domain decomposition FSAI, is built by reordering the matrix graph according to a multilevel k‐way partitioning method followed by a bandwidth minimization algorithm. We test these preconditioners by solving a set of symmetric positive definite problems arising from different engineering applications. The results are evaluated in terms of performance, scalability, and robustness, showing that both strategies lead to faster convergent schemes regarding the number of iterations and total computational time in comparison with the native FSAI with no significant loss in the algorithmic parallel degree.

A Robust Multilevel Approximate Inverse Preconditioner for Symmetric Positive Definite Matrices

The use of factorized sparse approximate inverse (FSAI) preconditioners in a standard multilevel framework for symmetric positive definite (SPD) matrices may pose a number of issues as to the definiteness of the Schur complement at each level. The present work introduces a robust multilevel approach for SPD problems based on FSAI preconditioning, which eliminates the chance of algorithmic breakdowns independently of the preconditioner sparsity. The multilevel FSAI algorithm is further enhanced by introducing descending and ascending low-rank corrections, thus giving rise to the multilevel FSAI with low-rank corrections (MFLR) preconditioner. The proposed algorithm is investigated in a number of test problems. The numerical results show that the MFLR preconditioner is a robust approach that can significantly accelerate the solver convergence rate preserving a good degree of parallelism. The possibly large set-up cost, mainly due to the computation of the eigenpairs needed by low-rank corrections, makes its...

Incomplete Sparse Approximate Inverses for Parallel Preconditioning

Abstract In this paper, we propose a new preconditioning method that can be seen as a generalization of block-Jacobi methods, or as a simplification of the sparse approximate inverse (SAI) preconditioners. The “Incomplete Sparse Approximate Inverses” (ISAI) is in particular efficient in the solution of sparse triangular linear systems of equations. Those arise, for example, in the context of incomplete factorization preconditioning. ISAI preconditioners can be generated via an algorithm providing fine-grained parallelism, which makes them attractive for hardware with a high concurrency level. In a study covering a large number of matrices, we identify the ISAI preconditioner as an attractive alternative to exact triangular solves in the context of incomplete factorization preconditioning.

A supernodal block factorized sparse approximate inverse for non-symmetric linear systems

The concept of supernodes, originally developed to accelerate direct solution methods for linear systems, is generalized to block factorized sparse approximate inverse (Block FSAI) preconditioning of non-symmetric linear systems. It is shown that aggregating the unknowns in clusters that are processed together is particularly useful both to reduce the cost for the preconditioner setup and accelerate the convergence of the iterative solver. A set of numerical experiments performed on matrices arising from the meshfree discretization of 2D and 3D potential problems, where a very large number of nodal contacts is usually found, shows that the supernodal Block FSAI preconditioner outperforms the native algorithm and exhibits a much more stable behavior with respect to the variation of the user-specified parameters.

Change mechanism of solver performance: With a special focus on time history analysis of supertall buildings

SummaryThe correlation between the performance of some commonly used linear system solvers and the characteristics of system matrices, especially in the time history analysis of supertall buildings, is comprehensively identified. Relying on 5 well‐designed structural models with height varying from 47 to 660 m, the supernodal LLT solver and the conjugate gradient (CG) solver with the factorized sparse approximate inverse preconditioner are selected due to their actual efficiency. More importantly, basing on 90 fictitious supertall structural models generated from the parameterized method, the effects of structural characteristics on solver performance and the corresponding mechanism are studied by 3 mathematical indexes with parameters come from the equivalent dynamic stiffness matrices. Impacts of processors on the solvers are also clarified by associating the architectural characteristics of processors with the indexes. The performance of the 2 solvers is affected globally with increasing building size; different variation laws are shown for different structural systems. However, the accuracy of the CG solver does not exhibit distinct variation due to a slight change in the 2‐norm condition number of the matrices. The CG solver is found to be more applicable on GPU processor than the LLT solver due to its smaller gap between computational workload and the amount of data transferred.