Fast Iterative Solvers Research Articles

A preconditioned fast quadratic spline collocation method for two-sided space-fractional partial differential equations

Abstract Quadratic spline collocation methods for the one and two-dimensional fractional diffusion equations are proposed. By carefully exploring the mathematical structure of the coefficient matrix, we propose a matrix-free fast Krylov subspace iterative solver for the corresponding quadratic spline collocation scheme. And then, preconditioning technique is applied to further accelerate the convergence of the fast Krylov subspace iterative method. It shows that the fast quadratic spline collocation scheme has greatly reduced the computational cost from O ( N 2 ) to O ( N log N ) per iteration and memory requirement from O ( N 2 ) to O ( N ) , while still well approximates the fractional diffusion equations without any accuracy lost. Numerical experiments are given to demonstrate the efficiency and effectiveness of the fast method.

Noisy Accelerated Power Method for Eigenproblems With Applications

This paper introduces an efficient algorithm for finding the dominant generalized eigenvectors of a pair of symmetric matrices. Combining tools from approximation theory and convex optimization, we develop a simple scalable algorithm with strong theoretical performance guarantees. More precisely, the algorithm retains the simplicity of the well-known power method but enjoys the asymptotic iteration complexity of the powerful Lanczos method. Unlike these classic techniques, our algorithm is designed to decompose the overall problem into a series of subproblems that only need to be solved approximately. The combination of good initializations, fast iterative solvers, and appropriate error control in solving the subproblems lead to a linear running time in the input sizes compared to the superlinear time for the traditional methods. The improved running time immediately offers acceleration for several applications. As an example, we demonstrate how the proposed algorithm can be used to accelerate canonical correlation analysis, which is a fundamental statistical tool for learning of a low-dimensional representation of high-dimensional objects. Numerical experiments on real-world datasets confirm that our approach yields significant improvements over the current state of the art.

Fast iterative solvers and simulation for the space fractional Ginzburg–Landau equations

In this paper, we propose a new splitting iteration method for solving the Toeplitz-like complex linear system arising from the space fractional Ginzburg–Landau equations, of which the coefficient matrix is equal to the sum of a complex scaled Toeplitz-plus-diagonal matrix and a symmetric Toeplitz-plus-diagonal matrix. The new splitting method enjoys computational advantage since circulant preconditioner and fast Fourier transform(FFT) can be used for solving the involved linear subsystem. Under some conditions, the convergence properties of the corresponding iteration method are derived. Numerical examples are given to illustrate the effectiveness of the proposed method.

Preconditioners for Two-Phase Incompressible Navier--Stokes Flow

We consider iterative methods for solving the linearised Navier-Stokes equations arising from two-phase flow problems and the efficient preconditioning of such systems when using mixed finite element methods. Our target application is simulation within the Proteus toolkit; in particular, we will give results for a dynamic dam-break problem in 2D. We focus on a preconditioner motivated by approximate commutators which has proved effective, displaying mesh-independent convergence for the constant coefficient single-phase Navier-Stokes equations. This approach is known as the "pressure convection-diffusion" (PCD) preconditioner [H. C. Elman, D. J. Silvester and A. J. Wathen, Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics, second ed., Oxford University Press, 2014]. However, the original technique fails to give comparable performance in its given form when applied to variable coefficient Navier-Stokes systems such as those arising in two-phase flow models. Here we develop a generalisation of this preconditioner appropriate for two-phase flow, requiring a new form for PCD. We omit considerations of boundary conditions to focus on the key features of two-phase flow. Before considering our target application, we present numerical results within the controlled setting of a simplified problem using a variety of different mixed elements. We compare these results with those for a straightforward extension to another commutator-based method known as the "least-squares commutator" (LSC) preconditioner, a technique also discussed in the aforementioned reference. We demonstrate that favourable properties of the original PCD and LSC preconditioners (without boundary adjustments) are retained with the new preconditioners in the two-phase situation.

An Accurate Low-Order Discretization Scheme for the Identity Operator in the Magnetic Field and Combined Field Integral Equations

A new low-order discretization scheme for the identity operator in the magnetic field integral equation (MFIE) is discussed. Its concept is derived from the weak-form representation of combined sources which are discretized with Rao-Wilton-Glisson (RWG) functions. The resulting MFIE overcomes the accuracy problem of the classical MFIE while it maintains fast iterative solver convergence. The improvement in accuracy is verified with a mesh refinement analysis and with near- and far-field scattering results. Furthermore, simulation results for a combined field integral equation (CFIE) involving the new MFIE show that this CFIE is interior-resonance free and well-conditioned like the classical CFIE, but also accurate as the EFIE.

Isogeometric analysis for 2D and 3D curl–div problems: Spectral symbols and fast iterative solvers

Alfvén-like operators are of interest in magnetohydrodynamics, which is used in plasma physics to study the macroscopic behavior of plasma. Motivated by this important and complex application, we focus on a parameter-dependent curl–div problem that can be seen as a prototype of an Alfvén-like operator, and we discretize it using isogeometric analysis based on tensor-product B-splines. The involved coefficient matrices can be very ill-conditioned, so that standard numerical solution methods perform quite poorly here. In order to overcome the difficulties caused by such ill-conditioning, a two-step strategy is proposed. First, we conduct a detailed spectral study of the coefficient matrices, highlighting the critical dependence on the different physical and approximation parameters. Second, we exploit such spectral information to design fast iterative solvers for the corresponding linear systems. For the first goal we apply the theory of (multilevel block) Toeplitz and generalized locally Toeplitz sequences, while for the second we use a combination of multigrid techniques and preconditioned Krylov solvers. Several numerical tests are provided both for the study of the spectral problem and for the solution of the corresponding linear systems.

A Preconditioned Fast Parareal Finite Difference Method for Space-Time Fractional Partial Differential Equation

We develop a fast parareal finite difference method for space-time fractional partial differential equation. The method properly handles the heavy tail behavior in the numerical discretization, while retaining the numerical advantages of conventional parareal algorithm. At each time step, we explore the structure of the stiffness matrix to develop a matrix-free preconditioned fast Krylov subspace iterative solver for the finite difference method without resorting to any lossy compression. Consequently, the method has significantly reduced computational complexity and memory requirement. Numerical experiments show the strong potential of the method.

Fast iterative solvers for an optimal transport problem

Optimal transport problems pose many challenges when considering their numerical treatment. We investigate the solution of a PDE-constrained optimisation problem subject to a particular transport equation arising from the modelling of image metamorphosis. We present the nonlinear optimisation problem, and discuss the discretisation and treatment of the nonlinearity via a Gauss–Newton scheme. We then derive preconditioners that can be used to solve the linear systems at the heart of the (Gauss–)Newton method.

On segmentation model for vector valued images and fast iterative solvers

In this paper, we propose a new convex variational model for segmentation of vector valued images. The data term of the proposed model is based on the coefficient of variation, which works well in vector valued images having intensity inhomogeneity. Due to convexity of the model, it is independent of the placement of initial contour. Better performance of the proposed model can be seen from experimental results qualitatively and quantitatively. Images in practice are of large sizes, which makes numerical methods more important. In this paper, we also develop fast and stable numerical methods for solution of partial differential equation arisen from the minimization of the proposed model. We have developed a novel multigrid method based on a locally supported smoother. The proposed method is compared with the existing methods in terms of iterations and CPU time for vector valued images having large sizes.

On preconditioned BiCGSTAB solver for MLPG method applied to heat conduction in 3D complex geometry

Development of meshfree methods is in full swing in the last few decades as these methods overcome the shortcomings of mesh based methods. Development of meshfree methods for large scale problems requires fast iterative solvers. Preconditioner plays an important role in improving convergence rate of an iterative solver. Recently proposed SRJ method can be used as a stand-alone solver and as a preconditioner in CG type methods. Objective of the present work is to evaluate the performance of SRJ method as a preconditioner in BiCGSTAB method in connection with the MLPG method. Three dimensional Poisson equation and heat conduction equation are solved in regular and irregular domain respectively. The SRJ method is tested as a solver and as a preconditioner in BiCGSTAB method. Its performance as a preconditioner is compared with Jacobi and SOR preconditioners.

Second-kind integral equations for the Laplace-Beltrami problem on surfaces in three dimensions

The Laplace-Beltrami problem ΔΓψ = f has several applications in mathematical physics, differential geometry, machine learning, and topology. In this work, we present novel second-kind integral equations for its solution which obviate the need for constructing a suitable parametrix to approximate the in-surface Green’s function. The resulting integral equations are well-conditioned and compatible with standard fast multipole methods and iterative linear algebraic solvers, as well as more modern fast direct solvers. Using layer-potential identities known as Calderon projectors, the Laplace-Beltrami operator can be pre-conditioned from the left and/or right to obtain second-kind integral equations. We demonstrate the accuracy and stability of the scheme in several numerical examples along surfaces described by curvilinear triangles.

A Fast Discontinuous Galerkin Method for a Bond-Based Linear Peridynamic Model Discretized on a Locally Refined Composite Mesh

We develop a family of fast discontinuous Galerkin (DG) finite element methods for a bond-based linear peridynamic (PD) model in one space dimension. More precisely, we develop a preconditioned fast piecewise-constant DG scheme on a geometrically graded locally refined composite mesh which is suited for the scenario in which the jump discontinuity of the displacement field occurs at the one of the nodes in the original uniform partition. We also develop a preconditioned fast piecewise-linear DG scheme on a uniform mesh that has a second-order convergence rate when the jump discontinuity occurs at one of the computational nodes or has a convergence rate of one-half order otherwise. Motivated by these results, we develop a preconditioned fast hybrid DG scheme that is discretized on a locally uniformly refined composite mesh to numerically simulate the PD model where the jump discontinuity of the displacement field occurs inside a computational cell. We use a piecewise-constant DG scheme on a uniform mesh and a piecewise-linear DG scheme on a locally uniformly refined mesh of mesh size $$O(h^2)$$ , which has an overall convergence rate of O(h). Because of their nonlocal nature, numerical methods for PD models generate dense stiffness matrices which have $$O(N^2)$$ memory requirement and $$O(N^3)$$ computational complexity, where N is the number of computational nodes. In this paper, we explore the structure of the stiffness matrices to develop three preconditioned fast Krylov subspace iterative solvers for the DG method. Consequently, the methods have significantly reduced computational complexity and memory requirement. Numerical results show the utility of the numerical methods.

Spectral analysis and multigrid preconditioners for two-dimensional space-fractional diffusion equations

Fractional diffusion equations (FDEs) are a mathematical tool used for describing some special diffusion phenomena arising in many different applications like porous media and computational finance. In this paper, we focus on a two-dimensional space-FDE problem discretized by means of a second order finite difference scheme obtained as combination of the Crank–Nicolson scheme and the so-called weighted and shifted Grünwald formula.By fully exploiting the Toeplitz-like structure of the resulting linear system, we provide a detailed spectral analysis of the coefficient matrix at each time step, both in the case of constant and variable diffusion coefficients. Such a spectral analysis has a very crucial role, since it can be used for designing fast and robust iterative solvers. In particular, we employ the obtained spectral information to define a Galerkin multigrid method based on the classical linear interpolation as grid transfer operator and damped-Jacobi as smoother, and to prove the linear convergence rate of the corresponding two-grid method. The theoretical analysis suggests that the proposed grid transfer operator is strong enough for working also with the V-cycle method and the geometric multigrid. On this basis, we introduce two computationally favourable variants of the proposed multigrid method and we use them as preconditioners for Krylov methods. Several numerical results confirm that the resulting preconditioning strategies still keep a linear convergence rate.

On POD-based Deflation Vectors for DPCG applied to porous media problems

We study fast and robust iterative solvers for large systems of linear equations resulting from simulation of flow through strongly heterogeneous porous media. We propose the use of preconditioning and deflation techniques, based on information obtained from the system, to reduce the time spent in the solution of the linear system. An important question when using deflation techniques is how to find good deflation vectors, which lead to a decrease in the number of iterations and a small increase in the required computing time per iteration. In this paper, we propose the use of deflation vectors based on a POD-reduced set of snapshots. We investigate convergence and the properties of the resulting methods. Finally, we illustrate these theoretical results with numerical experiments. We consider compressible and incompressible single-phase flow in a layered model with variations in the permeability layers up to 1 0 3 and the SPE 10 benchmark model with a contrast in permeability coefficients of 1 0 7 . Using deflation for the incompressible problem, we reduce the number of iterations to 1 or 2 iterations. With deflation, for the compressible problem, we reduce up to ∼ 80 % the number of iterations when compared with the only-preconditioned solver.

Numerical Simulation of Shielding Current Density in HTS Film by Using Acceleration Technique With $\mathcal {H}$-Matrix Method

Numerical techniques are proposed for fast solving a linear system appearing in the shielding current analysis in a cracked high-temperature superconducting (HTS) film. After discretized with respect to time and space, an initial-boundary-value problem of the shielding current density reduces to a problem in which a linear system is solved at each iteration of the Newton method. Although the coefficient matrix of the system is dense, a fast iterative linear-system solver can be realized by implementing the H-matrix method to matrix-vector multiplications. On the other hand, the CPU time required for the iterative solver increases with the number of elements adjacent to cracks. To remove such unfavorable characteristics, the QR factorization is applied to a part of the linear system so that all unknowns other than correction of the current vector potential may be eliminated from the linear system. The performance of the techniques is evaluated and, as an application of the techniques, detectability of multiple cracks in an HTS film is investigated numerically.

A Modified Implementation of MINRES to Monitor Residual Subvector Norms for Block Systems

Saddle-point systems, i.e., structured linear systems with symmetric matrices are considered. A modified implementation of (preconditioned) MINRES is derived which allows subvectors of the residual to be monitored individually. Compared to the implementation from the textbook of [H. C. Elman, D. J. Silvester, and A. J. Wathen, Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, 2nd ed., Oxford University Press, Oxford, 2014], our method requires one extra vector of storage and no additional applications of the preconditioner. Numerical experiments are included.

A divide-and-conquer fast finite difference method for space–time fractional partial differential equation

Fractional partial differential equations (FPDEs) provide better modeling capabilities for challenging phenomena with long-range time memory and spatial interaction than integer-order PDEs do. A conventional numerical discretization of space–time FPDEs requires O(N2+MN) memory and O(MN3+M2N) computational work, where N is the number of spatial freedoms per time step and M is the number of time steps.We develop a fast finite difference method (FDM) for space–time FPDE: (i) We utilize the Toeplitz-like structure of the coefficient matrix to develop a matrix-free preconditioned fast Krylov subspace iterative solver to invert the coefficient matrix at each time step. (ii) We utilize a divide-and-conquer strategy, a recursive direct solver, to handle the temporal coupling of the numerical scheme. The fast method has an optimal memory requirement of O(MN) and an approximately linear computational complexity of O(NM(logN+log2M)), without resorting to any lossy compression. Numerical experiments show the utility of the method.

Fast iterative solvers for large matrix systems arising from time-dependent Stokes control problems

In this manuscript we consider the development of fast iterative solvers for Stokes control problems, an important class of PDE-constrained optimization problems. In particular we wish to develop effective preconditioners for the matrix systems arising from finite element discretizations of time-dependent variants of such problems. To do this we consider a suitable rearrangement of the matrix systems, and exploit the saddle point structure of many of the relevant sub-matrices involved – we may then use this to construct representations of these sub-matrices based on good approximations of their (1,1)-block and Schur complement. We test our recommended iterative methods on a distributed control problem with Dirichlet boundary conditions, and on a time-periodic problem.

Fast Multilevel Solvers for a Class of Discrete Fourth Order Parabolic Problems

In this paper, we study fast iterative solvers for the solution of fourth order parabolic equations discretized by mixed finite element methods. We propose to use consistent mass matrix in the discretization and use lumped mass matrix to construct efficient preconditioners. We provide eigenvalue analysis for the preconditioned system and estimate the convergence rate of the preconditioned GMRes method. Furthermore, we show that these preconditioners only need to be solved inexactly by optimal multigrid algorithms. Our numerical examples indicate that the proposed preconditioners are very efficient and robust with respect to both discretization parameters and diffusion coefficients. We also investigate the performance of multigrid algorithms with either collective smoothers or distributive smoothers when solving the preconditioner systems.

Indirect Coupling of the Cell Method and BEM for Solving 3-D Unbounded Magnetostatic Problems

A novel hybrid approach for solving magnetostatic problems with an unbounded air domain is presented. The basic idea is to use augmented dual grids for interfacing the cell method (CM) and Boundary Element Method by indirect coupling, introducing equivalent surface field sources. The field problem in bounded domains is discretized by the CM in terms of integral variables, i.e., line integrals of the magnetic vector potential. Boundary integral conditions, formulated with the reduced magnetic scalar potential, are applied to avoid the air region meshing. A mixed final symmetric algebraic system, which can be solved by fast iterative solvers, such as Minimal Residual Method or SYMMLQ, is finally obtained. The magnetic field in the air region is then easily reconstructed from the equivalent sources. Numerical tests show that the hybrid method is accurate even by using a collocation approach for discretizing the boundary integral conditions.