Block LU Decomposition Research Articles

Robust fast direct integral equation solver for three-dimensional doubly periodic scattering problems with a large number of layers

Frequency-domain wave scattering problems that arise in acoustics and electromagnetism can be often described by the Helmholtz equation. This work presents a boundary integral equation method for the Helmholtz equation in 3-D multilayered media with many doubly periodic smooth layers. Compared with conventional quasi-periodic Green's function method, the new method is robust at all scattering parameters. A periodizing scheme is used to decompose the solution into near- and far-field contributions. The near-field contribution uses the free-space Green's function in an integral equation on the interface in the unit cell and its immediate eight neighbors; the far-field contribution uses proxy point sources that enclose the unit cell. A specialized high-order quadrature is developed to discretize the underlying surface integral operators to keep the number of unknowns per layer small. We achieve overall linear computational complexity in the number of layers by reducing the linear system into block tridiagonal form and then solving the system directly via block LU decomposition. The new solver is capable of handling a 100-interface structure with 961.3k unknowns to 10−5 accuracy in less than 2 hours on a desktop workstation.

Substructure exploitation of a nonsmooth Newton method for large-scale optimal control problems with full discretization

We investigate the application of full discretization and a nonsmooth Newton method to large-scale optimal control problems. Based on a first discretize, then optimize approach, we discretize the state and control variables in time following a collocation method. Then, a nonsmooth Newton method combined with a line search globalization strategy is used to find a solution to the resulting finite-dimensional nonlinear optimization problem. In order to reduce the computational effort of solving the linear systems that arise from the application of the nonsmooth Newton method, we propose a structure exploitation strategy that results in a sparse banded matrix. We propose as well a substructure exploitation strategy based on a block LU decomposition. The different exploitation strategies combined with the use of appropriate linear solvers are demonstrated and compared for a quadratic 2D heat equation control problem discretized with the method of lines, and the approach that proved to be the most efficient is applied to a nonlinear version of the problem.

A target-fixed immersed-boundary formulation for rigid bodies interacting with fluid flow

We present an immersed boundary projection method for flow-structure interaction problems involving rigid bodies with complex geometries. Dynamics of a rigid body interacting with fluid flow and kinematics of other rigid bodies undergoing prescribed motions are coupled implicitly with the incompressible vorticity equations. In particular, the method is formulated in a frame of reference fixed on the rigid body under flow-structure interaction (the target) so that the coupled system can be solved non-iteratively, accurately, and efficiently. The implicit coupling of the fluid solver and dynamics and kinematics of rigid bodies ensures the method being stable for low solid-to-fluid mass ratios. The influence of fictitious fluid inside the rigid body is considered and the spurious oscillations in surface stresses are filtered to impose physically correct rigid body dynamics. Similar to many predecessors of the immersed boundary projection method, the resulting discrete system is solved efficiently using a block-LU decomposition. We then validate the method with two-dimensional test problems of a neutrally buoyant cylinder migrating in a planar Couette flow and a freely falling or rising cylindrical rigid body.

A fast method for solving a block tridiagonal quasi-Toeplitz linear system

This paper addresses the problem of solving block tridiagonal quasi-Toeplitz linear systems. Inspired by [10], we propose a more general algorithm for such systems. The algorithm is based on a block decomposition for block tridiagonal quasi-Toeplitz matrices and the Sherman–Morrison–Woodbury inversion formula. We also compare the proposed approach to the standard block LU decomposition method and the Gauss algorithm. A theoretical error analysis is also presented. All algorithms have been implemented in Matlab. Numerical experiments performed on a wide variety of test problems show the effectiveness of our algorithm in terms of efficiency, stability and robustness.

An immersed boundary projection method for simulating the inextensible vesicle dynamics

We develop an immersed boundary projection method (IBPM) based on an unconditionally energy stable scheme to simulate the vesicle dynamics in a viscous fluid. Utilizing the block LU decomposition of the algebraic system, a novel fractional step algorithm is introduced by decoupling all solution variables, including the fluid velocity, fluid pressure, and the elastic tension. In contrast to previous works, the present method preserves both the fluid incompressibility and the interface inextensibility at a discrete level simultaneously. In conjunction with an implicit discretization of the bending force, the present method alleviates the time-step restriction, so the numerical stability is assured by non-increasing total discrete energy during the simulation. The numerical algorithm takes a linearithmic complexity by using preconditioned GMRES and FFT-based solvers. The grid convergence studies confirm the solution variables exhibit first-order convergence rate in L2-norm. We demonstrate the numerical results of the vesicle dynamics in a quiescent fluid, Poiseuille flow, and shear flow, which are congruent with the results in the literature.

Recursive Block LU Decomposition based ELM in Apache Spark

Due to the massive memory and computational resources required to build complex machine learning models on large datasets, many researchers are employing distributed environments for training the models on large datasets. The parallel implementations of Extreme Learning Machine (ELM) with many variants have been developed using MapReduce and Spark frameworks in the recent years. However, these approaches have severe limitations in terms of Input-Output (I/O) cost, memory, etc. From the literature, it is known that the complexity of ELM is directly propositional to the computation of Moore-Penrose pseudo inverse of hidden layer matrix in ELM. Most of the ELM variants developed on Spark framework have employed Singular Value Decomposition (SVD) to compute the Moore-Penrose pseudo inverse. But, SVD has severe memory limitations when experimenting with large datasets. In this paper, a method that uses Recursive Block LU Decomposition to compute the Moore-Penrose generalized inverse over the Spark cluster has been proposed to reduce the computational complexity. This method enhances the ELM algorithm to be efficient in handling the scalability and also having faster execution of the model. The experimental results have shown that the proposed method is efficient than the existing algorithms available in the literature.

Key pre-distribution approach using block LU decomposition in wireless sensor network

At present, many types of applications rely on wireless sensor networks because of their availability and advantages. The working environments of the wireless sensor networks are usually hostile and difficult to access because the sensor nodes are distributed randomly to their environments. Due to the random distribution of the sensor nodes, none of these nodes can authenticate the other sensor nodes within their range. Therefore, these sensor nodes need a key agreement scheme that allows them to establish secure communication with one another, hence protecting the communicating information before being conveyed to the base station. This paper proposes a new and efficient key management scheme for wireless sensor network based on the polynomial pool-based key pre-distribution scheme and the block LU decomposition algorithm. The proposed scheme ensures that any two communicating nodes must share a common key before communication starts. The performance of the proposed scheme is evaluated based on network connectivity, resilience against node capture, memory overhead, energy consumption and the number of Sybil nodes and the messages needed by the adversary to break the entire network. Therefore, a simulation-based evaluation was conducted, and the simulation results were compared with related works. The simulation results show that the proposed scheme performs better compared to the existing approaches. The BLU-decomposition algorithm allows the proposed approach to use more than one space to obtain the shared keys and decrease the amount of pre-distributed information to the sensor nodes.

A Class of Iterative Solvers for the Helmholtz Equation: Factorizations, Sweeping Preconditioners, Source Transfer, Single Layer Potentials, Polarized Traces, and Optimized Schwarz Methods

Solving time-harmonic wave propagation problems by iterative methods is a difficult task, and over the last two decades an important research effort has gone into developing preconditioners for the simplest representative of such wave propagation problems, the Helmholtz equation. A specific class of these new preconditioners is considered here. They were developed by researchers with various backgrounds using formulations and notations that are very different, and all are among the most promising preconditioners for the Helmholtz equation. \indent The goal of the present article is to show that this class of preconditioners is based on a common mathematical principle, and that they can all be formulated in the context of domain decomposition methods known as optimized Schwarz methods. This common formulation allows us to explain in detail how and why all these methods work. The domain decomposition formulation also allows us to avoid technicalities in the implementation description we give of these recent methods. \indent The equivalence of these methods with optimized Schwarz methods translates at the discrete level into equivalence with approximate block LU decomposition preconditioners, and in each case we give the algebraic version, including a detailed description of the approximations used. While we choose to use the Helmholtz equation for which these methods were developed, our notation is completely general and the algorithms we give are written for an arbitrary second-order elliptic operator. The algebraic versions are even more general, assuming only a connectivity pattern in the discretization matrix. \indent All the new methods studied here are based on sequential decomposition of the problem in space into a sequence of subproblems, and they have in their optimal form the property to lead to nilpotent iterations, like an exact block LU factorization. Using our domain decomposition formulation, we finally present an algorithm for two-dimensional decompositions, i.e., decompositions that contain cross points, which is still nilpotent in its optimal form. Its approximation is currently an active area of research, and it would have been difficult to discover such an algorithm without the domain decomposition framework.

A pre-conditioned implicit direct forcing based immersed boundary method for incompressible viscous flows

A novel immersed boundary (IB) method based on an implicit direct forcing (IDF) scheme is developed for incompressible viscous flows. The key idea for the present IDF method is to use a block LU decomposition technique in momentum equations with Taylor series expansion to construct the implicit IB forcing in a recurrence form, which imposes more accurate no-slip boundary conditions on the IB surface. To accelerate the IB forcing convergence during the iterative procedure, a pre-conditioner matrix is introduced in the recurrence formulation of the IB forcing. A Jacobi-type parameter is determined in the pre-conditioner matrix by minimizing the Frobenius norm of the matrix function representing the difference between the IB forcing solution matrix and the pre-conditioner matrix. In addition, the pre-conditioning parameter is restricted due to the numerical stability in the recurrence formulation. Consequently, the present pre-conditioned IDF (PIDF) enables accurate calculation of the IB forcing within a few iterations. We perform numerical simulations of two-dimensional flows around a circular cylinder and three-dimensional flows around a sphere for low and moderate Reynolds numbers. The result shows that PIDF yields a better imposition of no-slip boundary conditions on the IB surfaces for low Reynolds number with a fairly larger time step than IB methods with different direct forcing schemes due to the implicit treatment of the diffusion term for determining the IB forcing. Finally, we demonstrate the robustness of the present PIDF scheme by numerical simulations of flow around a circular array of cylinders, flows around a falling sphere, and two sedimenting spheres in gravity.

Сравнение вычислительных и аналитических оценок ускорения многократного решения СЛАУ блочным LU-разложением

Сравнение вычислительных и аналитических оценок ускорения многократного решения СЛАУ блочным LU-разложением

Многократное вычисление емкостной матрицы системы проводников и диэлектриков с изменяющимися параметрами с помощью блочного LU-разложения при решении СЛАУ

Многократное вычисление емкостной матрицы системы проводников и диэлектриков с изменяющимися параметрами с помощью блочного LU-разложения при решении СЛАУ

A stable fluid–structure-interaction solver for low-density rigid bodies using the immersed boundary projection method

Dispersion of low-density rigid particles with complex geometries is ubiquitous in both natural and industrial environments. We show that while explicit methods for coupling the incompressible Navier-Stokes equations and Newton's equations of motion are often sufficient to solve for the motion of cylindrical particles with low density ratios, for more complex particles - such as a body with a protrusion - they become unstable. We present an implicit formulation of the coupling between rigid body dynamics and fluid dynamics within the framework of the immersed boundary projection method. Similarly to previous work on this method, the resulting matrix equation in the present approach is solved using a block-LU decomposition. Each step of the block-LU decomposition is modified to incorporate the rigid body dynamics. We show that our method achieves second-order accuracy in space and first-order in time (third-order for practical settings), only with a small additional computational cost to the original method. Our implicit coupling yields stable solution for density ratios as low as $10^{-4}$. We also consider the influence of fictitious fluid located inside the rigid bodies on the accuracy and stability of our method.

High-Performance Method for Analyzing Shielding Current Density in HTS Films: Application to the Scanning Permanent-Magnet Method

A fast and stable method is proposed for calculating the shielding current density in a high-temperature superconducting film containing cracks. After discretized with the finite-element method, the initial-boundary-value problem of the shielding current density reduces to semi-explicit differential algebraic equations (DAEs). Although the DAEs can be solved with standard ordinary-differential-equation (ODE) solvers, much CPU time is required for its numerical solution. In order to shorten the CPU time, a high-speed algorithm is proposed. In the algorithm, the block LU decomposition is incorporated into function evaluations in ODE solvers. A numerical code is developed on the basis of the proposed method, and detectability of cracks by the scanning permanent-magnet method (SPM) is numerically investigated. The results of computations show that the SPM can realize high-speed crack detection. In addition, it is also found that, for multiple cracks, resolution of the SPM will be remarkably degraded.

Analytic Evaluation of the Computational Costs for Solving Systems of Linear Algebraic Equations in Multiple Computing of the Capacitance Matrix in a Range of the Dielectric Permittivity of Dielectrics

The efficiency of using block LU decomposition in multiple solution of systems of linear algebraic equations is analytically evaluated. By numerical experiments it is demonstrated that a significant speed up (up to 134) can be achieved. Bibliography: 8 titles.

GP-GPU의 캐시메모리를 활용하기 위한 병렬 블록 LU 분해 프로그램의 구현

GP-GPU는 그래픽 처리를 위한 GPU의 다중쓰레드를 일반 수치 계산에 활용하여 초고속으로 계산하는 장치이다. GP-GPU에서는 CPU의 캐시메모리와는 달리 다중쓰레드가 공유하는 공유메모리의 형태로 캐시메모리를 제공하며, 공유메모리는 사용자 프로그램에서 직접 제어할 수 있다. 본 연구에서는 GP-GPU의 캐시메모리를 사용하여 계산 성능을 향상시키기 위한 블록 구조의 병렬 LU 분해 프로그램을 구현하였다. Nvidia CUDA C로 구현된 병렬 블록 LU 분해 프로그램은 동일한 GP-GPU 상에서 일반 LU 분해 프로그램에 비교하여 7~8배 이상의 속도 개선을 보였다. GP-GPUs are general purposed GPUs for numerical computation based on multiple threads which are originally for graphic processing. GP-GPUs provide cache memory in a form of shared memory which user programs can access directly, unlikely typical cache memory. In this research, we implemented the parallel block LU decomposition program to utilize cache memory in GP-GPUs. The parallel blocked LU decomposition program designed with Nvidia CUDA C run 7~8 times faster than nun-blocked LU decomposition program in the same GP-GPU computation environment.

A High Performance and Memory Efficient LU Decomposer on FPGAs

LU decomposition for dense matrices is an important linear algebra kernel that is widely used in both scientific and engineering applications. To efficiently perform large matrix LU decomposition on FPGAs with limited local memory, a block LU decomposition algorithm on FPGAs applicable to arbitrary matrix size is proposed. Our algorithm applies a series of transformations, including loop blocking and space-time mapping, onto sequential nonblocking LU decomposition. We also introduce a high performance and memory efficient hardware architecture, which mainly consists of a linear array of processing elements (PEs), to implement our block LU decomposition algorithm. Our design can achieve optimum performance under various hardware resource constraints. Furthermore, our algorithm and design can be easily extended to the multi-FPGA platform by using a block-cyclic data distribution and inter-FPGA communication scheme. A total of 36 PEs can be integrated into a Xilinx Virtex-5 XC5VLX330 FPGA on our self-designed PCI-Express card, reaching a sustained performance of 8.50 GFLOPS at 133 MHz for a matrix size of 16,384, which outperforms several general-purpose processors. For a Xilinx Virtex-6 XC6VLX760, a newer FPGA, we predict that a total of 180 PEs can be integrated, reaching 70.66 GFLOPS at 200 MHz. Compared to the previous work, our design can integrate twice the number of PEs into the same FPGA and has significantly higher performance.

FPGA-Based High-Performance and Scalable Block LU Decomposition Architecture

Decomposition of a matrix into lower and upper triangular matrices (LU decomposition) is a vital part of many scientific and engineering applications, and the block LU decomposition algorithm is an approach well suited to parallel hardware implementation. This paper presents an approach to speed up implementation of the block LU decomposition algorithm using FPGA hardware. Unlike most previous approaches reported in the literature, the approach does not assume the matrix can be stored entirely on chip. The memory accesses are studied for various FPGA configurations, and a schedule of operations for scaling well is shown. The design has been synthesized for FPGA targets and can be easily retargeted. The design outperforms previous hardware implementations, as well as tuned software implementations including the ATLAS and MKL libraries on workstations.

Preconditioning the bidomain model with almost linear complexity

The bidomain model is widely used in electro-cardiology to simulate spreading of excitation in the myocardium and electrocardiograms. It consists of a system of two parabolic reaction diffusion equations coupled with an ODE system. Its discretisation displays an ill-conditioned system matrix to be inverted at each time step: simulations based on the bidomain model therefore are associated with high computational costs. In this paper we propose a preconditioning for the bidomain model either for an isolated heart or in an extended framework including a coupling with the surrounding tissues (the torso). The preconditioning is based on a formulation of the discrete problem that is shown to be symmetric positive semi-definite. A block LU decomposition of the system together with a heuristic approximation (referred to as the monodomain approximation) are the key ingredients for the preconditioning definition. Numerical results are provided for two test cases: a 2D test case on a realistic slice of the thorax based on a segmented heart medical image geometry, a 3D test case involving a small cubic slab of tissue with orthotropic anisotropy. The analysis of the resulting computational cost (both in terms of CPU time and of iteration number) shows an almost linear complexity with the problem size, i.e. of type nlog α ( n) (for some constant α) which is optimal complexity for such problems.

A parallel block LU decomposition method for distributed finite element matrices

In this work we present a new parallel direct linear solver for matrices resulting from finite element problems. The algorithm follows the nested dissection approach, where the resulting Schur complements are also distributed in parallel. The sparsity structure of the finite element matrices is used to pre-compute an efficient block structure for the LU factors. We demonstrate the performance and the parallel scaling behavior by several test examples.

Reflectivity method for geomechanical equilibria

SUMMARY It is shown that the block LU decomposition of the transfer and scattering matrix convert these matrices into each other. This allows to introduce a generalization of the Kennett reflectivity method, which is applicable to arbitrary systems of linear differential equations. The introduced method is convenient to analyse equilibria, where the governing matrix is degenerate. The resulting algorithm is compact and numerically stable. To illustrate the concept, we consider elastic equilibrium of a layered medium. We also derive closed-form expressions for a quasi-stationary poroelastic case taking into account solid–fluid and electrokinetic coupling.