Hierarchical Interpolative Factorization for Elliptic Operators: Integral Equations

This paper introduces the hierarchical interpolative factorization for integral equations (HIF‐IE) associated with elliptic problems in two and three dimensions. This factorization takes the form of an approximate generalized LU decomposition that permits the efficient application of the discretized operator and its inverse. HIF‐IE is based on the recursive skeletonization algorithm but incorporates a novel combination of two key features: (1) a matrix factorization framework for sparsifying structured dense matrices and (2) a recursive dimensional reduction strategy to decrease the cost. Thus, higher‐dimensional problems are effectively mapped to one dimension, and we conjecture that constructing, applying, and inverting the factorization all have linear or quasilinear complexity. Numerical experiments support this claim and further demonstrate the performance of our algorithm as a generalized fast multipole method, direct solver, and preconditioner. HIF‐IE is compatible with geometric adaptivity and can handle both boundary and volume problems.© 2016 Wiley Periodicals, Inc.     

COMPACT FOURTH-ORDER FINITE DIFFERENCE SCHEMES FOR HELMHOLTZ EQUATION WITH HIGH WAVE NUMBERS

In this paper, two fourth-order accurate compact difference schemes are presented for solving the Helmholtz equation in two space dimensions when the corresponding wave numbers are large. The main idea is to derive and to study a fourth-order accurate compact difference scheme whose leading truncation term, namely, the O（h^4） term, is independent of the wave number and the solution of the Helmholtz equation. The convergence property of the compact schemes are analyzed and the implementation of solving the resulting linear algebraic system based on a FFT approach is considered. Numerical results are presented, which support our theoretical predictions.     

A sweeping preconditioner for Yee’s finite difference approximation of time-harmonic Maxwell’s equations

This paper is concerned with the fast iterative solution of linear systems arising from finite difference discretizations in electromagnetics. The sweeping preconditioner with moving perfectly matched layers previously developed for the Helmholtz equation is adapted for the popular Yee grid scheme for wave propagation in inhomogeneous, anisotropic media. Preliminary numerical results are presented for typical examples.     

A Source Transfer Domain Decomposition Method for Helmholtz Equations in Unbounded Domain Part II: Extensions

In this paper we extend the source transfer domain decomposition method (STDDM) introduced by the authors to solve the Helmholtz problems in two-layered media, the Helmholtz scattering problems with bounded scatterer, and Helmholtz problems in 3D unbounded domains. The STDDM is based on the decomposition of the domain into non-overlapping layers and the idea of source transfer which transfers the sources equivalently layer by layer so that the solution in the final layer can be solved using a PML method defined locally outside the last two layers. The details of STDDM is given for each extension. Numerical results are presented to demonstrate the efficiency of STDDM as a preconditioner for solving the discretization problem of the Helmholtz problems considered in the paper.     

Incremental unknowns preconditioning for solving the Helmholtz equation

An efficient preconditioner is developed for solving the Helmholtz problem in both high and low frequency (wavenumber) regimes. The preconditioner is based on hierarchical unknowns on nested grids, known as incremental unknowns (IU). The motivation for the IU preconditioner is provided by an eigenvalue analysis of a simplified Helmholtz problem. The performance of our preconditioner is tested on the iterative solution of two‐dimensional electromagnetic scattering problems. When compared with other well‐known methods, our technique is shown to often provide a better numerical efficacy and, most importantly, to be more robust. Moreover, for the best performance, the number of IU levels used in the preconditioner should be designed for the coarsest grid to have roughly two points per linear wavelength. This result is consistent with the conventional sampling criteria for wave phenomena in contrast with existing IU applications for solving the Laplace/Poisson problem, where the coarsest grid comprises just one interior point. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

On a class of preconditioners for solving the Helmholtz equation

In 1983, a preconditioner was proposed [J. Comput. Phys. 49 (1983) 443] based on the Laplace operator for solving the discrete Helmholtz equation efficiently with CGNR. The preconditioner is especially effective for low wavenumber cases where the linear system is slightly indefinite. Laird [Preconditioned iterative solution of the 2D Helmholtz equation, First Year's Report, St. Hugh's College, Oxford, 2001] proposed a preconditioner where an extra term is added to the Laplace operator. This term is similar to the zeroth order term in the Helmholtz equation but with reversed sign. In this paper, both approaches are further generalized to a new class of preconditioners, the so-called “shifted Laplace” preconditioners of the form Δφ−k²φ with . Numerical experiments for various wavenumbers indicate the effectiveness of the preconditioner. The preconditioner is evaluated in combination with GMRES, Bi-CGSTAB, and CGNR.     

A multigrid-based preconditioned Krylov subspace method for the Helmholtz equation with PML

In this paper, we generalize the complex shifted Laplacian preconditioner to the complex shifted Laplacian-PML preconditioner for the Helmholtz equation with perfectly matched layer (Helmholtz-PML equation). The Helmholtz-PML equation is discretized by an optimal 9-point difference scheme, and the preconditioned linear system is solved by the Krylov subspace method, especially by the biconjugate gradient stabilized method (Bi-CGSTAB). The spectral analysis of the linear system is given, and a new matrix-based interpolation operator is proposed for the multigrid method, which is used to approximately invert the preconditioner. The numerical experiments are presented to illustrate the efficiency of the preconditioned Bi-CGSTAB method with the multigrid based on the new interpolation operator, also, numerical results are given for comparing the performance of the new interpolation operator with that of classic bilinear interpolation operator and the one suggested in Erlangga et al. (2006) [10].     

Improving the stability and robustness of incomplete symmetric indefinite factorization preconditioners

Sparse symmetric indefinite linear systems of equations arise in numerous practical applications. In many situations, an iterative method is the method of choice but a preconditioner is normally required for it to be effective. In this paper, the focus is on a class of incomplete factorization algorithms that can be used to compute preconditioners for symmetric indefinite systems. A limited memory approach is employed that incorporates a number of new ideas with the goal of improving the stability, robustness, and efficiency of the preconditioner. These include the monitoring of stability as the factorization proceeds and the incorporation of pivot modifications when potential instability is observed. Numerical experiments involving test problems arising from a range of real‐world applications demonstrate the effectiveness of our approach.     

Comparison between the shifted-Laplacian preconditioning and the controllability methods for computational acoustics

Processes that can be modelled with numerical calculations of acoustic pressure fields include medical and industrial ultrasound, echo sounding, and environmental noise. We present two methods for making these calculations based on Helmholtz equation. The first method is based directly on the complex-valued Helmholtz equation and an algebraic multigrid approximation of the discretized shifted-Laplacian operator; i.e. the damped Helmholtz operator as a preconditioner. The second approach returns to a transient wave equation, and finds the time-periodic solution using a controllability technique. We concentrate on acoustic problems, but our methods can be used for other types of Helmholtz problems as well. Numerical experiments show that the control method takes more CPU time, whereas the shifted-Laplacian method has larger memory requirement.     

Hierarchical Interpolative Factorization for Elliptic Operators: Differential Equations

This paper introduces the hierarchical interpolative factorization for elliptic partial differential equations (HIF‐DE) in two (2D) and three dimensions (3D). This factorization takes the form of an approximate generalized LU/LDL decomposition that facilitates the efficient inversion of the discretized operator. HIF‐DE is based on the nested dissection multifrontal method but uses skeletonization on the separator fronts to sparsify the dense frontal matrices and thus reduce the cost. We conjecture that this strategy yields linear complexity in 2D and quasilinear complexity in 3D. Estimated linear complexity in 3D can be achieved by skeletonizing the compressed fronts themselves, which amounts geometrically to a recursive dimensional reduction scheme. Numerical experiments support our claims and further demonstrate the performance of our algorithm as a fast direct solver and preconditioner. MATLAB® codes are freely available.© 2016 Wiley Periodicals, Inc.     

A multigrid‐based shifted Laplacian preconditioner for a fourth‐order Helmholtz discretization

In this paper, an iterative solution method for a fourth‐order accurate discretization of the Helmholtz equation is presented. The method is a generalization of that presented in (SIAM J. Sci. Comput. 2006; 27 :1471–1492), where multigrid was employed as a preconditioner for a Krylov subspace iterative method. The multigrid preconditioner is based on the solution of a second Helmholtz operator with a complex‐valued shift. In particular, we compare preconditioners based on a point‐wise Jacobi smoother with those using an ILU(0) smoother, we compare using the prolongation operator developed by de Zeeuw in (J. Comput. Appl. Math. 1990; 33 :1–27) with interpolation operators based on algebraic multigrid principles, and we compare the performance of the Krylov subspace method Bi‐conjugate gradient stabilized with the recently introduced induced dimension reduction method, IDR(s). These three improvements are combined to yield an efficient solver for heterogeneous problems. Copyright © 2009 John Wiley & Sons, Ltd.     

Flexible incomplete Cholesky factorization with multi‐parameters to control the number of nonzero elements in preconditioners

An incomplete Cholesky (IC) factorization with multi‐parameters is presented. The marked virtue of the proposed IC factorization algorithm is to dynamically control the number of nonzero elements in each column of the IC factorization preconditioner L with the help of these involved parameters. Parameter setting strategies are also given. Numerical results show that the total computing time for both computation of the preconditioner L and iterative solution is evidently reduced for almost all test matrices. In general, these parameters can obviously enhance the effectiveness and performance of the IC factorization. Copyright © 2011 John Wiley & Sons, Ltd.     

高波数波动问题的多水平方法

本文主要讨论求解高波数Helmholtz方程的多水平方法,主要回顾了一些具有代表性的多重网格方法.如Erlangga等人的shifted Laplacian预处理的多重网格法;Elman等提出的修正的多重网格方法;以及我们的基于连续内罚有限元(CIP-FEM)离散代数系统的多水平算法.最后还介绍了求解高波数时谐Maxwell方程的CIP-FEM离散代数系统的多水平算法.     

A comparison of preconditioned Krylov subspace methods for large‐scale nonsymmetric linear systems

Preconditioned Krylov subspace (KSP) methods are widely used for solving large‐scale sparse linear systems arising from numerical solutions of partial differential equations (PDEs). These linear systems are often nonsymmetric due to the nature of the PDEs, boundary or jump conditions, or discretization methods. While implementations of preconditioned KSP methods are usually readily available, it is unclear to users which methods are the best for different classes of problems. In this work, we present a comparison of some KSP methods, including GMRES, TFQMR, BiCGSTAB, and QMRCGSTAB, coupled with three classes of preconditioners, namely, Gauss–Seidel, incomplete LU factorization (including ILUT, ILUTP, and multilevel ILU), and algebraic multigrid (including BoomerAMG and ML). Theoretically, we compare the mathematical formulations and operation counts of these methods. Empirically, we compare the convergence and serial performance for a range of benchmark problems from numerical PDEs in two and three dimensions with up to millions of unknowns and also assess the asymptotic complexity of the methods as the number of unknowns increases. Our results show that GMRES tends to deliver better performance when coupled with an effective multigrid preconditioner, but it is less competitive with an ineffective preconditioner due to restarts. BoomerAMG with a proper choice of coarsening and interpolation techniques typically converges faster than ML, but both may fail for ill‐conditioned or saddle‐point problems, whereas multilevel ILU tends to succeed. We also show that right preconditioning is more desirable. This study helps establish some practical guidelines for choosing preconditioned KSP methods and motivates the development of more effective preconditioners.     

Combining a hybrid preconditioner and a optimal adjustment algorithm to accelerate the convergence of interior point methods

In this work, the optimal adjustment algorithm for p coordinates, which arose from a generalization of the optimal pair adjustment algorithm is used to accelerate the convergence of interior point methods using a hybrid iterative approach for solving the linear systems of the interior point method. Its main advantages are simplicity and fast initial convergence. At each interior point iteration, the preconditioned conjugate gradient method is used in order to solve the normal equation system. The controlled Cholesky factorization is adopted as the preconditioner in the first outer iterations and the splitting preconditioner is adopted in the final outer iterations. The optimal adjustment algorithm is applied in the preconditioner transition in order to improve both speed and robustness. Numerical experiments on a set of linear programming problems showed that this approach reduces the total number of interior point iterations and running time for some classes of problems. Furthermore, some problems were solved only when the optimal adjustment algorithm for p coordinates was used in the change of preconditioners.     

GPU implementation of a Helmholtz Krylov solver preconditioned by a shifted Laplace multigrid method

A Helmholtz equation in two dimensions discretized by a second order finite difference scheme is considered. Krylov methods such as Bi-CGSTAB and IDR(s) have been chosen as solvers. Since the convergence of the Krylov solvers deteriorates with increasing wave number, a shifted Laplace multigrid preconditioner is used to improve the convergence. The implementation of the preconditioned solver on CPU (Central Processing Unit) is compared to an implementation on GPU (Graphics Processing Units or graphics card) using CUDA (Compute Unified Device Architecture). The results show that preconditioned Bi-CGSTAB on GPU as well as preconditioned IDR(s) on GPU is about 30 times faster than on CPU for the same stopping criterion.     

An algebraic multifrontal preconditioner that exploits the low‐rank property

We present an algebraic structured preconditioner for the iterative solution of large sparse linear systems. The preconditioner is based on a multifrontal variant of sparse LU factorization used with nested dissection ordering. Multifrontal factorization amounts to a partial factorization of a sequence of logically dense frontal matrices, and the preconditioner is obtained if structured factorization is used instead. This latter exploits the presence of low numerical rank in some off‐diagonal blocks of the frontal matrices. An algebraic procedure is presented that allows to identify the hierarchy of the off‐diagonal blocks with low numerical rank based on the sparsity of the system matrix. This procedure is motivated by a model problem analysis, yet numerical experiments show that it is successful beyond the model problem scope. Further aspects relevant for the algebraic structured preconditioner are discussed and illustrated with numerical experiments. The preconditioner is also compared with other solvers, including the corresponding direct solver. Copyright © 2015 John Wiley & Sons, Ltd.     

On the Wiener–Hopf Method for Surface Plasmons: Diffraction from Semiinfinite Metamaterial Sheet

By formally invoking the Wiener–Hopf method, we explicitly solve a one‐dimensional, singular integral equation for the excitation of a slowly decaying electromagnetic wave, called surface plasmon‐polariton (SPP), of small wavelength on a semiinfinite, flat conducting sheet irradiated by a plane wave in two spatial dimensions. This setting is germane to wave diffraction by edges of large sheets of single‐layer graphene. Our analytical approach includes (i) formulation of a functional equation in the Fourier domain; (ii) evaluation of a split function, which is expressed by a contour integral and is a key ingredient of the Wiener–Hopf factorization; and (iii) extraction of the SPP as a simple‐pole residue of a Fourier integral. Our analytical solution is in good agreement with a finite‐element numerical computation.     

ARMS: an algebraic recursive multilevel solver for general sparse linear systems

This paper presents a general preconditioning method based on a multilevel partial elimination approach. The basic step in constructing the preconditioner is to separate the initial points into two parts. The first part consists of ‘block’ independent sets, or ‘aggregates’. Unknowns of two different aggregates have no coupling between them, but those in the same aggregate may be coupled. The nodes not in the first part constitute what might be called the ‘coarse’ set. It is natural to call the nodes in the first part ‘fine’ nodes. The idea of the methods is to form the Schur complement related to the coarse set. This leads to a natural block LU factorization which can be used as a preconditioner for the system. This system is then solved recursively using as preconditioner the factorization that could be obtained from the next level. Iterations between levels are allowed. One interesting aspect of the method is that it provides a common framework for many other techniques. Numerical experiments are reported which indicate that the method can be fairly robust. Copyright © 2002 John Wiley & Sons, Ltd.     

Hermite collocation solution of partial differential equations via preconditioned Krylov methods

We are concerned with the numerical solution of partial differential equations (PDEs) in two spatial dimensions discretized via Hermite collocation. To efficiently solve the resulting systems of linear algebraic equations, we choose a Krylov subspace method. We implement two such methods: Bi‐CGSTAB [1] and GMRES [2]. In addition, we utilize two different preconditioners: one based on the Gauss–Seidel method with a block red‐black ordering (RBGS); the other based upon a block incomplete LU factorization (ILU). Our results suggest that, at least in the context of Hermite collocation, the RBGS preconditioner is superior to the ILU preconditioner and that the Bi‐CGSTAB method is superior to GMRES. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:120–136, 2001