Dense Coefficient Matrix Research Articles

An ACA-BM-SBM for 2D acoustic sensitivity analysis

<abstract> <p>In this paper, we present a novel computational approach (named ACA-BM-SBM) for the calculation of 2D acoustic sensitivity by combining the Burton-Miller-type singular boundary method (BM-SBM) with the adaptive cross-approximation (ACA) algorithm. The BM-SBM circumvents the source singularities of the fundamental solutions by introducing the origin intensity factors, and it eliminates the fictitious frequency problem in external acoustic fields by introducing the Burton-Miller formula. As a semi-analysis meshless method, the BM-SBM can accurately solve the external acoustic problem governed by the Helmholtz equation. Nevertheless, the computational inefficiency introduced by the dense coefficient matrix renders this method suboptimal, particularly for large-scale simulations. As the number of nodes increases, the computation time and store memory increase dramatically. ACA is a purely algebraic method based on hierarchical matrices which can be used to partition the coefficient matrix step by step. By employing ACA, the BM-SBM can be effectively accelerated, and this results in less computation time, as well as fewer memory requirements. Numerical experiments, including Dirichlet and Neumann boundary conditions, illustrate that the proposed approach is an accurate, efficient and fast numerical method for acoustic sensitivity analysis.</p> </abstract>

Composite Numerical Methods and Scalable Tile Algorithms

Numerical modeling is an important tool when studying various natural processes and phenomena. Fractional diffusion can be used for modeling many processes in biology, for example in silico experiments in molecular biology and medicine design, protein diffusion within cells, complex media geometry, etc. The problem is usually reduced to a system of linear algebraic equations and in many cases this system has a dense coefficient matrix. Numerically solving such problems with the traditional LU factorization is a computationally expensive endeavour - $O(n^3)$. In this paper we explore the use of a hierarchical compression method based on Hierarchical Semi-Separable compression and ULV-like factorization from the STRUctured Matrices PACKage (STRUMPACK) software library for a flow around airfoils problem discretized with Boundary Element Method and fractional diffusion problem discretized with the Finite Element Method. The HSS based method promises better overall computational complexity of $O(r^2n)$ for problems with suitable structure - low rank off-diagonal blocks. Here $r$ is the maximum rank of the off-diagonal blocks. We present analysis of the performance and accuracy of the HSS based method and compare it with the state of the art direct LU factorization solvers. This paper is based on the PhD thesis Composite Numerical Methods and Scalable Tile Algorithms of the author defended on 17.05.2022 in the Institute of Information and Communication Technologies at the Bulgarian Academy of Sciences.

A fast direct singular boundary method for three-dimensional potential problems

The main purpose of the study is to develop a fast singular boundary method for solving the three-dimensional (3D) potential problems. It is a difficult issue to simulate the large-scale problems by the traditional SBM, because of the fully populated coefficient matrix. A fast direct algorithm based on recursive skeletonization factorization (RSF) is combined in the SBM to reduce the CPU time and memory requirements in this study. For the dense coefficient matrix in the SBM, a hierarchically generalized LU decomposition is constructed to invert it fast and accurately. The numerical results demonstrate the great performance of the RSF-SBM for solving large-scale 3D potential problems.

A fast Fourier-Galerkin method solving boundary integral equations for the Helmholtz equation with exponential convergence

A boundary integral equation in general form will be considered, which can be used to solve Dirichlet problems for the Helmholtz equation. The goal of this paper is to develop a fast Fourier-Galerkin method solving these boundary integral equations. To this aim, a scheme for splitting integral operators is presented, which splits the corresponding integral operator into a convolution operator and a compact operator. A truncation strategy is presented, which can compress the dense coefficient matrix to a sparse one having only $\mathcal {O}(n)$ nonzero entries, where n is the order of the Fourier basis functions used in the method. The proposed fast method preserves the stability and optimal convergence order. Moreover, exponential convergence can be obtained under suitable assumptions. Numerical examples are presented to confirm the theoretical results for the approximation accuracy and computational complexity of the proposed method.

Simulation of two-dimensional steady-state heat conduction problems by a fast singular boundary method

The study presents a fast direct algorithm for solutions of systems arising from singular boundary method (SBM) in two-dimensional (2D) steady-state heat conduction problems. As a strong-form boundary discretization collocation technique, the SBM is mathematically simple, easy-to-implement, and free of mesh. Similar to boundary element method (BEM), the SBM generates a dense coefficient matrix, which requires O(N3) operations and O(N2) memory to solve the system with direct solvers. In this study, a fast direct solver based on recursive skeletonization factorization (RSF) is applied to solve the linear systems in the SBM. Due to the fact that the coefficient matrix is hierarchically block separable, we can construct a multilevel generalized LU decomposition that allows fast application of the inverse of the coefficient matrix. Combing the RSF and the SBM can dramatically reduce the operations and the memory to O(N). The accuracy and effectiveness of the RSF-SBM are tested through some 2D heat conduction problems including isotropic and anisotropic cases.

Induced Magnetic Field Computation for Ships Based on Fast Multipole Method and Integral Equation Method

The nonsymmetric and dense coefficient matrix of integral equation method results in low computation efficiency and large memory cost in induced magnetic field computation of large scale ships. In order to improve the computation efficiency, a new method called FMMIEM is developed based on fast multipole method and integral equation method. The basic formulas of the FMMIEM were deduced. The numerical implementations of the multipole expansion, multipole to local translation and local expansion were described in detail. The FMMIEM not only has the virtue of the vector integral equation method which needs only meshing source area and is easy to deal with the infinite domain problems, such as the induced magnetic field modeling, but also has the excellent performance of the fast multipole method in conjunction with iterative solver which can accelerate the matrix vector product. The analytical numerical experiment of the sphere magnetized in the uniform applied magnetic field and the magnetostatic field modeling of the main cabin of the simple submarine mockup were designed and the results showed that the FMMIEM is with the high calculation accuracy and reduces costing time obviously. The proposed method is very helpful in engineering.

Banded operational matrices for Bernstein polynomials and application to the fractional advection-dispersion equation

In the papers dealing with derivation and applications of operational matrices of Bernstein polynomials, a basis transformation, commonly a transformation to power basis, is used. The main disadvantage of this method is that the transformation may be ill-conditioned. Moreover, when applied to the numerical simulation of a functional differential equation, it leads to dense operational matrices and so a dense coefficient matrix is obtained. In this paper, we present a new property for Bernstein polynomials. Using this property, we build exact banded operational matrices for derivatives of Bernstein polynomials. Next, as an application, we propose a new numerical method based on a Petrov-Galerkin variational formulation and the new operational matrices utilizing the dual Bernstein basis for the time-fractional advection-dispersion equation. Finally, we show that the proposed method leads to a narrow-banded linear system and so less computational effort is required to obtain the desired accuracy for the approximate solution. Some numerical examples are provided to demonstrate the efficiency of the method.

Parallel Randomized and Matrix-Free Direct Solvers for Large Structured Dense Linear Systems

We design efficient and distributed-memory parallel randomized direct solvers for large structured dense linear systems, including a fully matrix-free version based on matrix-vector multiplications and a partially matrix-free one. The dense coefficient matrix $A$ has an off-diagonal low-rank structure, as often encountered in practical applications such as Toeplitz systems and discretized integral and partial differential equations. A distributed-memory parallel framework for randomized structured solution is shown. Scalable adaptive randomized sampling and hierarchical compression algorithms are designed to approximate $A$ by hierarchically semiseparable (HSS) matrices. Systematic process grid storage schemes are given for different HSS forms. Parallel hierarchical algorithms are proposed for the resulting HSS forms. As compared with existing work on parallel HSS methods, our algorithms have several remarkable advantages, including the matrix-free schemes that avoid directly using dense $A$, a synchroniz...

A rapid BEM-based method for cooling simulation of injection molding

Cooling simulation is significant for optimization of the cooling system of injection molds. The boundary element method (BEM) is thought to be one of the best approaches suiting the steady-state cooling simulation. In spite of the merits of the BEM, the long computational time and the exorbitant memory requirement are two bottleneck problems in the current BEM-based cooling simulation method for industrial applications. The problems are caused by two reasons. One is the coupled heat transfer between the mold and the part and another is attributed to the inherent drawback of the BEM. In this article, the outer iteration, which is traditionally used to achieve consistency of boundary conditions on the mold cavity surface, is eliminated by introducing analytical solutions of the part temperature into the BEM equations. Then the dense coefficient matrix is sparsified by a combination of coefficient items using geometric topology. Moreover, parallel computing has been employed to speed up the computation. The case study showed that the sparse ratio reaches 7% with a temperature error of ±1 oC and the total computational time is reduced by almost one order of magnitude simultaneously.

A fast semi-implicit difference method for a nonlinear two-sided space-fractional diffusion equation with variable diffusivity coefficients

In this paper, we derive a new nonlinear two-sided space-fractional diffusion equation with variable coefficients from the fractional Fick’s law. A semi-implicit difference method (SIDM) for this equation is proposed. The stability and convergence of the SIDM are discussed. For the implementation, we develop a fast accurate iterative method for the SIDM by decomposing the dense coefficient matrix into a combination of Toeplitz-like matrices. This fast iterative method significantly reduces the storage requirement of O(n2) and computational cost of O(n3) down to n and O(nlogn), where n is the number of grid points. The method retains the same accuracy as the underlying SIDM solved with Gaussian elimination. Finally, some numerical results are shown to verify the accuracy and efficiency of the new method.

Iterative addition of parallel temperature effects to finite-difference simulation of radio-frequency wave propagation in plasmas

Accurate simulations of how radio frequency (RF) power is launched, propagates, and absorbed in a magnetically confined plasma is a computationally challenging problem that for which no comprehensive approach presently exists. The underlying physics is governed by the Vlasov–Maxwell equations, and characteristic length scales can vary by three orders of magnitude. Present algorithms are, in general, based on finding the constituative relation between the induced RF current and the RF electric field and solving the resulting set of Maxwell’s equations. These linear equations use a Fourier basis set that is not amenable to multi-scale formulations and have a large dense coefficient matrix that requires a high-communications overhead factorization technique. Here the use of operator splitting to separate the current and field calculations, and a low-overhead iterative solver leads to an algorithm that avoids these issues and has the potential to solve presently intractable problems due to its data-parallel and favorable scaling characteristics. We verify the algorithm for the iterative addition of parallel temperature effects for a 1D electron Langmuir by reproducing the solution obtained with the existing Fourier kinetic RF code aorsa (Jaeger et al., 2008).

2차원 열전도 문제에 대한 Fast Multipole 경계요소법의 이론과 실행 알고리즘의 분석

This paper presents the fast multipole boundary element method (FM-BEM) as a new BEM solution methodology that overcomes many disadvantages of conventional BEM. In conventional BEM, large-scale problems cannot be treated easily because the computation time increases rapidly with an increase in the number of boundary elements owing to the dense coefficient matrix. Analysis results are obtained to compare FM-BEM with conventional BEM in terms of computation time and accuracy for a simple two-dimensional steady-state heat conduction problem. It is confirmed that the FM-BEM solution methodology greatly enhances the computation speed while maintaining solution accuracy similar to that of conventional BEM. As a result, the theory and implementation algorithm of FM-BEM are discussed in this study.

A Parallel implementation of Gram-Schmidt Algorithm for Dense Linear System of Equations

The linear system of equations with dense coefficient matrix is very common in science and engineering. In this paper, a parallel algorithm based on Gram-Schmidt QR factorization method for the exact solution of dense system of linear equations is presented. Although several parallel approaches have been proposed to solve the system of linear equations until now, the aim of this paper is to show the ability and limitation of this parallel algorithm in comparison with the sequential one. The suggested parallel algorithm is executed on M IM D architecture and distributed memory. In order to specify the efficiency of this algorithm, the amounts of speedup and FLOPs in executions with different size of matrix (from 2000 to 12000 equations) on up to 5 processors are compared together. The results show that the achieved speedup is significant, and also the performance of this practical parallel algorithm increases as the number of equations grows.

Global field interpolation for particle methods

A common problem in particle simulations is effective field interpolation. Field interpolation is any method for creating an accurate representation of a given continuous field by a linear combination of overlapping basis functions. This paper presents two techniques for field interpolation, based on a radial basis function (RBF) formulation using Gaussians. The application in mind is vortex methods, where one needs to determine the circulation (or strength) of individual vortex particles with known position and scale to represent a given vorticity field. This process is required both to initially discretize a given vorticity field, and to replace a configuration of particles with another for the purposes of maintaining spatial accuracy. The first technique presented is formulated as an RBF collocation problem, and obtains a solution accurately and with excellent algorithmic efficiency by means of a preconditioned iterative method. The preconditioner is a sparse approximation, based on localization, to the dense coefficient matrix of the RBF system. The second technique uses approximate solutions to the reverse heat equation, recognizing that the standard regularization used in vortex methods (estimating particle strengths using the local value of vorticity multiplied by particle area/volume) corresponds to a Gaussian blurring of the original field. A single time step is used, thus avoiding amplification of high frequencies, and accurate solutions are produced using explicit finite difference methods. Computational experiments were performed in two dimensions, demonstrating the accuracy and convergence of the proposed techniques. Application in three dimensions is straightforward, as radial basis function interpolation is neutral to dimension, but will require more computational effort.

Feasibility of simplified integral equation modeling of low-frequency marine CSEM with a resistive target

We have assessed the accuracy of a simplified integral equation (SIE) modeling approach for marine controlled-source electromagnetics (CSEM) with low applied frequencies and a resistive target. The most computationally intensive part of rigorous integral equation (IE) modeling is the computation of the anomalous electric field within the target itself. This leads to a matrix problem with a dense coefficient matrix. It is well known that, in general, the presence of many grid cells creates a computational disadvantage for dense-matrix methods compared to sparse-matrix methods. The SIE approach replaces the dense-matrix part of rigorous IE modeling by sparse-matrix calculations based on an approximation of Maxwell’s equations. The approximation is justified theoretically if a certain dimensionless parameter [Formula: see text] is small. As opposed to approximations of the Born type, the validity of the SIE approach does not rely on the anomalous field to be small in comparison with the background field in the target region. We have calculated [Formula: see text] for a range of parameter values typical for marine CSEM, and compared the SIE approach numerically to the rigorous IE method and to the quasi-linear (QL) and quasi-analytic (QA) approximate solutions. It is found that the SIE approach is very accurate for small [Formula: see text], corresponding to frequencies in the lower range of those typical for marine CSEM for petroleum exploration. In addition, the SIE approach is found to be significantly more accurate than the QL and QA approximations for small [Formula: see text].

A fast Petrov–Galerkin method for solving the generalized airfoil equation

In this paper we develop a fast Petrov–Galerkin method for solving the generalized airfoil equation using the Chebyshev polynomials. The conventional method for solving this equation leads to a linear system with a dense coefficient matrix. When the order of the linear system is large, the computational complexity for solving the corresponding linear system is huge. For this we propose the matrix truncation strategy, which compresses the dense coefficient matrix into a sparse matrix. We prove that the truncated method preserves the optimal order of the approximate solution for the conventional method. Moreover, we solve the truncated equation using the multilevel augmentation method. The computational complexity for solving this truncated linear system is estimated to be linear up to a logarithmic factor.

A fast solver for a hypersingular boundary integral equation

In this paper we develop a fast fully discrete Galerkin method for a hypersingular boundary integral equation based on trigonometric polynomials. Usually, the Galerkin method for this equation leads to a discrete linear system with a dense coefficient matrix. When the order of the linear system is large, the complexity for generating the fully discrete linear system and then solving the corresponding linear system is huge. For this purpose, we propose a truncation strategy to compress the dense coefficient matrix into a sparse matrix, and then we use a numerical integration method to generate the fully discrete truncated linear system, which will be solved by the multilevel augmentation method. An optimal order of the approximate solution is preserved. The computational complexity for generating and solving the fully discrete truncated linear system is estimated to be linear up to a logarithmic factor. Numerical examples complete the paper.

Accelerating the Multilevel Fast Multipole Algorithm with the Sparse-Approximate-Inverse (SAI) Preconditioning

With the help of the multilevel fast multipole algorithm, integral-equation methods can be used to solve real-life electromagnetics problems both accurately and efficiently. Increasing problem dimensions, on the other hand, necessitate effective parallel preconditioners with low setup costs. In this paper, we consider sparse approximate inverses generated from the sparse near-field part of the dense coefficient matrix. In particular, we analyze pattern selection strategies that can make efficient use of the block structure of the near-field matrix, and we propose a load-balancing method to obtain high scalability during the setup. We also present some implementation details, which reduce the computational cost of the setup phase. In conclusion, for the open-surface problems that are modeled by the electric-field integral equation, we have been able to solve ill-conditioned linear systems involving millions of unknowns with moderate computational requirements. For closed-surface problems that can be modeled by the combined-field integral equation, we reduce the solution times significantly compared to the commonly used block-diagonal preconditioner.

An Efficient Algorithm for Solving the Generalized Airfoil Equation

In this paper, we develop an efficient Petrov-Galerkin method for the generalized airfoil equation. In general, the Petrov-Galerkin method for this equation leads to a linear system with a dense coefficient matrix. When the order of the coefficient matrix is large, the complexity for solving the corresponding linear system is huge. For this purpose, we propose a matrix truncation strategy to compress the dense coefficient matrix into a sparse matrix. Subsequently, we use a numerical integration method to generate the fully discrete truncated linear system. At last we solve the corresponding linear system by the multilevel augmentation method. An optimal order of the approximate solution is preserved. The computational complexity for generating the fully discrete truncated linear system and solving it is estimated to be linear up to a logarithm. The spectral condition number of the truncated matrix is proved to be bounded. Numerical examples complete the paper.

Sparse transformations and preconditioners for 3-D capacitance extraction

Three-dimensional (3-D) capacitance-extraction algorithms are important due to their high accuracy. However, the current 3-D algorithms are slow and thus their application is limited. In this paper, we present a novel method to significantly speed up capacitance-extraction algorithms based on boundary element methods (BEMs), under uniform and multiple dielectrics. The n/spl times/n coefficient matrix in the BEM is dense, even when approximated with the fast multipole method or hierarchical-refinement method, where n is the number of panels needed to discretize the conductor surfaces and dielectric interfaces. As a result, effective preconditioners are hard to obtain and iterative solvers converge slowly. In this paper, we introduce a linear transformation to convert the n/spl times/n dense coefficient matrix into a sparse matrix with O(n) nonzero entries, and then use incomplete factorization to produce a very effective preconditioner. For the k/spl times/k bus-crossing benchmark, our method requires at most four iterations, whereas previous best methods such as FastCap and HiCap require 10-20 iterations. As a result, our algorithm is up to 70 times faster than FastCap and up to 2 times faster than HiCap on these benchmarks. Additional experiments illustrate that our method consistently outperforms previous best methods by a large magnitude on complex industrial problems with multiple dielectrics.