Explicit preconditioned iterative methods for solving large unsymmetric finite element systems

A new class of explicit preconditioning methods based on the concept of sparse approximate factorization procedures and inverse matrix techniques is introduced for solving biharmonic equations. Isomorphic methods in conunction with explicit preconditioned schemes based on approximate inverse matrix techniques are presented for the efficient solution of biharmonic equations. Application of the proposed method on linear systems is discussed and numerical results are given.

A new class of normalized explicit approximate inverse matrix techniques, based on normalized approximate factorization procedures, for solving sparse linear systems resulting from the finite difference discretization of partial differential equations in three space variables are introduced. A new parallel normalized explicit preconditioned conjugate gradient square method in conjunction with normalized approximate inverse matrix techniques for solving efficiently sparse linear systems on distributed memory systems, using Message Passing Interface (MPI) communication library, is also presented along with theoretical estimates on speedups and efficiency. The implementation and performance on a distributed memory MIMD machine, using Message Passing Interface (MPI) is also investigated. Applications on characteristic initial/boundary value problems in three dimensions are discussed and numerical results are given.

A new class of approximate inverse matrix techniques based on the concept of sparse LU-type factorization procedures is introduced for computing explicitly inverses of arrowhead matrices without inverting the decomposition factors. Explicit preconditioned iterative schemes in conjunction with AIM techniques are presented for the efficient solution of linear systems. Applications of the method on a linear system are discussed and numerical results are given.

This paper focuses on the uplink (UL) of a single-cell massive multiple-input multiple-output (MIMO) systems in which the base station (BS) equipped with M antennas communicates with K single-antenna user equipments (UEs). Due to the finer spatial resolution achieved by a huge number of antennas at the base stations, these systems can substantially reduce intra-cell interference with simple signal processing. However, most precoding schemes exhibit prohibitively high computation complexity as the system dimensions increase. For example, the near-optimal minimum mean squared error (MMSE) precoding requires the inversion of a large matrix. To reduce further the complexity, the truncated polynomial expansion (TPE) is applied to approximate the inverse of large matrices that appear on the expressions of the optimal linear transceivers. A key requirement to achieve good detection performance is to find good polynomial coefficients (J), where the polynomial coefficients (J) are optimized to maximize the system performance. The simulation results show that the proposed scheme provides very close to optimal performance while reducing significantly the computational complexity.

In this paper, we advocate the use of the uncompressed form of i-vector and depend on subspace modeling using probabilistic linear discriminant analysis (PLDA) in handling the speaker and session (or channel) variability. An i-vector is a low-dimensional vector containing both speaker and channel information acquired from a speech segment. When PLDA is used on an i-vector, dimension reduction is performed twice: first in the i-vector extraction process and second in the PLDA model. Keeping the full dimensionality of the i-vector in the i-supervector space for PLDA modeling and scoring would avoid unnecessary loss of information. We refer to the uncompressed i-vector as the i-supervector. The drawback in using the i-supervector with PLDA is the inversion of large matrices in the estimation of the full posterior distribution, which we show can be solved rather efficiently by portioning large matrices into smaller blocks. We also introduce the Gaussianized rank-norm, as an alternative to whitening, for feature normalization prior to PLDA modeling. We found that the i-supervector performs better during normalization. A better performance is obtained by combining the i-supervector and i-vector at the score level. Furthermore, we also analyze the computational complexity of the i-supervector system, compared with that of the i-vector, at four different stages of loading matrix estimation, posterior extraction, PLDA modeling, and PLDA scoring.

We evaluate imaginary time density-density correlation functions for two-dimensional homogeneous electron gases of up to 42 particles in the continuum using the phaseless auxiliary field quantum Monte Carlo method. We use periodic boundary conditions and up to 300 plane waves as basis set elements. We show that such methodology, once equipped with suitable numerical stabilization techniques necessary to deal with exponentials, products, and inversions of large matrices, gives access to the calculation of imaginary time correlation functions for medium-sized systems. We discuss the numerical stabilization techniques and the computational complexity of the methodology and we present the limitations related to the size of the systems on a quantitative basis. We perform the inverse Laplace transform of the obtained density-density correlation functions, assessing the ability of the phaseless auxiliary field quantum Monte Carlo method to evaluate dynamical properties of medium-sized homogeneous fermion systems.

In this paper, a novel reconstruction of the gradient and Hessian tensors on an arbitrary unstructured grid, developed for implementation in a cell‐centered finite volume framework, is presented. The reconstruction, based on the application of Gauss' theorem, provides a fully second‐order accurate estimate of the gradient, along with a first‐order estimate of the Hessian tensor. The reconstruction is implemented through the construction of coefficient matrices for the gradient components and independent components of the Hessian tensor, resulting in a linear system for the gradient and Hessian fields, which may be solved to an arbitrary precision by employing one of the many methods available for the efficient inversion of large sparse matrices. Numerical experiments are conducted to demonstrate the accuracy, robustness, and computational efficiency of the reconstruction by comparison with other common methods. Copyright © 2009 John Wiley & Sons, Ltd.

The linear wave theory and the nonlinear-unsteady porous flow model are applied to analyze the energy dissipation and the bed pore water pressure induced by the interaction of wave, uniform current and porous bottom without considering the nonlinear waves and the viscosity effect inside the boundary layer. In this model, the linear, inertial and turbulent resistances are combined into a linearized resistance coefficient and the present system can be analyzed by a linear boundary value problem. The numerical result is quite agreement with the existing experimental data. It shows that the energy dissipation is reduced by the Doppler shift and the distribution of energy loss moves to the lower relative water depth region in the wave-following current. On the other hand, the bed pore water pressure in the wave-following current is always lager than that in the pure wave and the wave-opposing current.

PurposeTo propose novel parallel/distributed normalized explicit finite element (FE) approximate inverse preconditioning for solving sparse FE linear systems.Design/methodology/approachThe design of suitable methods was the main objective for which several families of the normalized approximate inverse, based on sparse normalized approximate factorization, are produced. The main motive for the derivation of the new normalized approximate inverse FE matrix algorithmic techniques is that they can be efficiently used in conjunction with normalized explicit preconditioned conjugate gradient (NEPCG) – type schemes on parallel and distributed systems. Theoretical estimates on the rate of convergence and computational complexity of the NEPCG method are also derived.FindingsApplication of the proposed method on a three‐dimensional boundary value problem is discussed and numerical results for uniprocessor systems along with speed‐ups and efficiency for multicomputer systems are given. These results tend to become optimum, which are in qualitative agreement with the theoretical results presented for uniprocessor and distributed memory systems, using message passing interface (MPI) communication library.Research limitations/implicationsFurther parallel algorithmic techniques will be investigated in order to improve the speed‐ups and the computational complexity of the parallel normalized explicit approximate inverse preconditioning.Originality/valueThe proposed parallel/distributed normalized explicit approximate inverse preconditioning, using approximate factorization and approximate inverse algorithms, is an efficient computational method that is valuable for computer scientists and for scientists and engineers in engineering computations.

In this paper we present new parallel inverse arrow-type matrix algorithms based on the concept of sparse factorization procedures, for computing explicitly exact and approximate inverses, on symmetric multiprocessor systems. The parallel implementation of the new inversion algorithms is discussed and numerical results are presented, using the simulation tool of Multi-Pascal.

Efficient ad hoc preconditioners are a key factor for a successful implementation of linear solvers in a parallel computing environment. The class of Factorized Sparse Approximate Inverses (FSAI), although originally developed for scalar machines, has proven extremely promising in multicore hardware. A recent evolution of FSAI is Block FSAI (BFSAI) which clusters the largest coefficients of the preconditionedmatrix in a number of diagonal blocks defined in advance. A further improvement of BFSAI is the adaptive BFSAI (labelled ABF) where the non zero pattern of the BFSAI preconditioner is not prescribed a priori but computed automatically and adaptively by a suitable algorithm. Numerical results from large finite element (FE) geomechanical models show that ABF coupled with an incomplete Cholesky factorization of each individual diagonal block, i.e. ABF-IC, may outperform BFSAI-IC by up to a factor 4 while exhibiting an excellent degree of parallelization on any multiprocessor computer.KeywordsFrobenius NormDiagonal BlockApproximate InverseGeomechanical ModelPreconditioned MatrixThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper studies efficient complex valued matrix manipulations for multi-user STBC-MIMO decoding. A novel method called Alamouti blockwise analytical matrix inversion (ABAMI) is proposed for the inversion of large complex matrices that are based on Alamouti sub-blocks. Another method using a variant of Givens rotation is proposed for fast QR decomposition of this kind of matrices. Our solutions significantly reduce the number of operations which makes them more than 4 times faster than several other solutions in the literature. Furthermore, compared to fixed function VLSI implementations, our solution is more flexible and consumes less silicon area because the hardware is programmable and it can be reused for many other operations such as filtering, correlation and FFT/IFFT. Besides the analysis of the general computational complexity based on the number of basic operations, the computational latency is also measured in clock cycles based on the conceptual hardware for real-time matrix manipulations.

We consider construction of surrogate models based on variable fidelity samples generated by a high fidelity function an exact representation of some physical phenomenon and by a low fidelity function a coarse approximation of the exact representation. A surrogate model is constructed to replace the computationally expensive high fidelity function. For such tasks Gaussian processes are generally used. However, if the sample size reaches a few thousands points, a direct application of Gaussian process regression becomes impractical due to high computational costs. We propose two approaches to circumvent this difficulty. The first approach uses approximation of sample covariance matrices based on the Nystrom method. The second approach relies on the fact that engineers often can evaluate a low fidelity function on the fly at any point using some blackbox; thus each time calculating prediction of a high fidelity function at some point, we can update the surrogate model with the low fidelity function value at this point. So, we avoid issues related to the inversion of large covariance matrices -- as we can construct model using only a moderate low fidelity sample size. We applied developed methods to a real problem, dealing with an optimization of the shape of a rotating disk.

A forward-advancing wave expansion method for numerical solution of large-scale sound propagation problems

An algorithm is presented to invert a (large and full) positive matrix approximatively, based on a (generalized) band of data centered around the diagonal. The approximative inverse is also positive with its off-band elements zero. As a result, the original matrix is approximated in the maximum entropy sense with another positive matrix which coincides in the band with the original but has (implicit) off-band elements. The algorithm is an extension of the classical Schur algorithm for maximum entropy estimation, has minimal complexity and is ideally suited for pipelined implementation in a concurrent processor.