Preconditioned Krylov Subspace Methods Research Articles

A preconditioning technique in conjunction with Krylov subspace methods for solving multilinear systems

In this paper, a framework is proposed for left/right preconditioning of multi-linear system with Einstein product. More precisely, the inverse of preconditioned tensor is derived analytically. Some numerical results are disclosed to experimentally illustrate the feasibility of preconditioned Krylov subspace methods based on Hessenberg process and compare their performances with those based on the Arnoldi process.

A 6M digital twin for modeling and simulation in subsurface reservoirs

A 6M digital twin for modeling and simulation in subsurface reservoirs

Some uses of the field of values in numerical analysis

In this expository paper we illustrate the role that the field of values (or numerical range) of a matrix plays in connection with certain problems of numerical analysis. These include the approximation of matrix functions and the convergence of preconditioned Krylov subspace methods for solving large systems of equations arising from the discretization of partial differential equations.

On the regularization matrix of the regularized DPSS preconditioner for non-Hermitian saddle-point problems

In this work, with respect to the regularization matrix of the new regularized deteriorated positive and skew-Hermitian splitting (RDPSS) preconditioner for non-Hermitian saddle-point problems, we provide a class of Hermitian positive semidefinite matrices, which depend on certain parameters, for practical computations. A precise description about the eigenvalue distribution of the corresponding preconditioned matrix is given. The condition number of the eigenvector matrix, which partly determines the convergence rate of the related preconditioned Krylov subspace method, is also discussed in this work. Finally, some numerical experiments are carried out to identify the effectiveness of the presented special choices for the regularization matrix to solve the non-Hermitian saddle-point problems.

A Class of Preconditioners Based on Positive-Definite Operator Splitting Iteration Methods for Variable-Coefficient Space-Fractional Diffusion Equations

After discretization by the finite volume method, the numerical solution of fractional diffusion equations leads to a linear system with the Toeplitz-like structure. The theoretical analysis gives sufficient conditions to guarantee the positive-definite property of the discretized matrix. Moreover, we develop a class of positive-definite operator splitting iteration methods for the numerical solution of fractional diffusion equations, which is unconditionally convergent for any positive constant. Meanwhile, the iteration methods introduce a new preconditioner for Krylov subspace methods. Numerical experiments verify the convergence of the positive-definite operator splitting iteration methods and show the efficiency of the proposed preconditioner, compared with the existing approaches.

Variational Density Fitting with a Krylov Subspace Method.

In this work, we present the implementation of a variational density fitting methodology that uses iterative linear algebra for solving the associated system of linear equations. It is well known that most difficulties with this system arise from the fact that the coefficient matrix is in general ill-conditioned and, due to finite precision round-off errors, it may not be positive definite. The dimensionality, given by the number of auxiliary functions, also poses a challenge in terms of memory and time demand since the coefficient matrix is dense. The methodology presented is based on a preconditioned Krylov subspace method able to deal with indefinite ill-conditioned equation systems. To assess its potential, it has been combined with double asymptotic electron repulsion integral expansions as implemented in the deMon2k package. A numerical study on a set of problems with up to 130,000 auxiliary functions shows its effectiveness to alleviate the abovementioned problematic. A comparison with the default methodology used in deMon2k based on a truncated eigenvalue decomposition of the coefficient matrix indicates that the proposed method exhibits excellent robustness and scalability when implemented in a parallel setting.

Solution Optimization of RBF Interpolation for Implicit Modeling of Orebody

To interpolate large drillhole data sets efficiently in implicit modeling of orebody, we optimize the solution of the RBF equation based on the two-level domain decomposition method. The solution performance is improved by balancing the two goals of the convergence rate and the iteration efficiency. The solution method converts the large domain into small subdomains which can be solved directly. The optimum division of subdomains can improve both the convergence of iteration and the efficiency of a single iteration, whereas the subdivision of a trivial subdomain leads the iteration failed to converge. For the efficiency of a single iteration, the kernel independent fast multipole method is used to calculate the matrix-vector product efficiently. Moreover, the initial solution of iteration and center reduction strategy are optimized for the dynamic updating of implicit surface. Combined with the preconditioned Krylov subspace method, the optimized solution method was implemented. The experimental results of several drillhole data sets show that the optimized solution method converges rapidly within a small number of iterations.

Performance of Krylov subspace method with SOR preconditioner supported by Eisenstat’s technique for linear system derived from time-periodic FEM

PurposeThis paper aims to present the affinity of BiCGStab and BiCGStab2 with successive over-relaxation (SOR) preconditioner supported by Eisenstat’s technique for a linear system derived from the time-periodic finite element method (TP-FEM). To solve the time domain electromagnetic field problem with long transient state, TP-FEM is very useful from the perspective of rapidly achieving a steady state. Because TP-FEM solves all of the state variables at once, the linear system derived from TP-FEM becomes the large scale and nonsymmetric, whereas the detailed performance of some preconditioned Krylov subspace method is not reported.Design/methodology/approachIn this paper, BiCGStab and BiCGStab2 are used as the linear solver for a large-sparse nonsymmetric linear system derived from TP-FEM. In addition, incomplete LU (ILU) factorization is applied as a preconditioner to compare SOR supported by Eisenstat’s technique. As examples, the pot-type reactor and three-phase transformer is analyzed.FindingsIn the problem of the pot-type reactor, when SOR preconditioner supported by Eisenstat’s technique is applied to BiCGStab and BiCGStab2, the elapsed time can be reduced dramatically. However, in the problem of the three-phase transformer, the iterative process of the linear solvers with SOR preconditioner is not terminated, whereas the iterative process of linear solvers with ILU preconditioner is terminated. The preconditioner that can be supported by Eisenstat’s technique is not necessarily appropriate for the problem to derive from TP-FEM.Originality/valueIn this paper, the affinity of preconditioned linear solver supported by Eisenstat’s technique for the nonsymmetric linear system derived from TP-FEM is demonstrated.

On Euler preconditioned SHSS iterative method for a class of complex symmetric linear systems

In this paper, we propose an Euler preconditioned single-step HSS (EP-SHSS) iterative method for solving a broad class of complex symmetric linear systems. The proposed method can be applied not only to the non-singular complex symmetric linear systems but also to the singular ones. The convergence (semi-convergence) properties of the proposed method are carefully discussed under suitable restrictions. Furthermore, we consider the acceleration of the EP-SHSS method by preconditioned Krylov subspace method and discuss the spectral properties of the corresponding preconditioned matrix. Numerical experiments verify the effectiveness of the EP-SHSS method either as a solver or as a preconditioner for solving both non-singular and singular complex symmetric linear systems.

The numerical solution of forward and inverse Robin problems for Laplace\u2019s equation

The Robin boundary value problem for Laplace’s equation in the elliptic region (which is a forward problem) and its related inverse problem can be used to reconstruct Robin coefficients from measurements on a partial boundary (inverse problem). We present a numerical solution of the forward problem that uses a boundary integral equation method, and we propose a fast solver based on one that reduces the computational complexity to mathcal{O}(Nlog (N)), where N is the size of the data. We compute the solution of the inverse problem using a preconditioned Krylov subspace method where the preconditioner is based on a block matrix decomposition. The structure of the matrix is then exploited to solve the direct problem. Numerical examples are presented to illustrate the effectiveness of the proposed approach.

Implementation of interior-point methods for LP based on Krylov subspace iterative solvers with inner-iteration preconditioning

We apply novel inner-iteration preconditioned Krylov subspace methods to the interior-point algorithm for linear programming (LP). Inner-iteration preconditioners recently proposed by Morikuni and Hayami enable us to overcome the severe ill-conditioning of linear equations solved in the final phase of interior-point iterations. The Krylov subspace methods do not suffer from rank-deficiency and therefore no preprocessing is necessary even if rows of the constraint matrix are not linearly independent. By means of these methods, a new interior-point recurrence is proposed in order to omit one matrix-vector product at each step. Extensive numerical experiments are conducted over diverse instances of 140 LP problems including the Netlib, QAPLIB, Mittelmann and Atomizer Basis Pursuit collections. The largest problem has 434,580 unknowns. It turns out that our implementation is more robust than the standard public domain solvers SeDuMi (Self-Dual Minimization), SDPT3 (Semidefinite Programming Toh-Todd-Tütüncü) and the LSMR iterative solver in PDCO (Primal-Dual Barrier Method for Convex Objectives) without increasing CPU time. The proposed interior-point method based on iterative solvers succeeds in solving a fairly large number of LP instances from benchmark libraries under the standard stopping criteria. The work also presents a fairly extensive benchmark test for several renowned solvers including direct and iterative solvers.

Parameter-robust preconditioning for the optimal control of the wave equation

In this paper, we propose and analyze a new matching-type Schur complement preconditioner for solving the discretized first-order necessary optimality conditions that characterize the optimal control of wave equations. Coupled with this is a recently developed second-order implicit finite difference scheme used for the full space-time discretization of the optimality system of PDEs. Eigenvalue bounds for the preconditioned system are derived, which provide insights into the convergence rates of the preconditioned Krylov subspace method applied. Numerical examples are presented to validate our theoretical analysis and demonstrate the effectiveness of the proposed preconditioner, in particular its robustness with respect to very small regularization parameters, and all mesh sizes in the spatial variables.

Critical Review of Preconditioners for Helmholtz Equation and their Spectral Analysis

Objectives: The study reviews the iterative solvers for solving Helmholtz equation. It also surveys the preconditioners for Krylov subspace methods for discrete Helmholtz equation. Numerical experiments based conclusions are drawn, after a thorough comparison of various preconditioners. Methods/Statistical Analysis: Five point finite difference scheme for two-dimensional Helmholtz equation is employed to obtain discrete analogue of Helmholtz equation. Krylov subspace method for is considered for iterations. Matrix based preconditioner ILU with variety of fill-in and operator based preconditioner complex shifted Laplace preconditioner (CSLP) are tested. Different shifts are proposed and experimented in CSLP. Optimization of shifts is recommended on basis of results. Spectral analysis is performed to validate results. Findings: For small wave number problem, ILU type preconditioned performs better and is fairly comparable with CSLP. However, increasing wave number makes unaffordable in terms of computational cost. The pure imaginary shift (0,1) and complex shift (1, / 4 π ) costsless time and number of iterations compared to other choices of shift. These choices are recommended when preconditioner is inverted directly. Approximate inversion may affect iterations, but will save solve time for large problems. Extensive spectral analysis, validating results, highlights this work. Application/Improvements: Dependency upon wave number is decreased with this recommended choice of shifts in CSLP. The number of iterations increases at very low rate when wave number is increased at rate of 2. Keywords: Helmholtz Equation, Iterative Methods, Krylov Subspace Methods, Preconditioners, Spectral Analysis

FAST SECOND-ORDER ACCURATE DIFFERENCE SCHEMES FOR TIME DISTRIBUTED-ORDER AND RIESZ SPACE FRACTIONAL DIFFUSION EQUATIONS

The aim of this paper is to develop fast second-order accurate difference schemes for solving one- and two-dimensional time distributed-order and Riesz space fractional diffusion equations. We adopt the same measures for one- and two-dimensional problems as follows: we first transform the time distributed-order fractional diffusion problem into the multi-term time-space fractional diffusion problem with the composite trapezoid formula. Then, we propose a second-order accurate difference scheme based on the interpolation approximation on a special point to solve the resultant problem. Meanwhile, the unconditional stability and convergence of the new difference scheme in $L_2$-norm are proved. Furthermore, we find that the discretizations lead to a series of Toeplitz systems which can be efficiently solved by Krylov subspace methods with suitable circulant preconditioners. Finally, numerical results are presented to show the effectiveness of the proposed difference methods and demonstrate the fast convergence of our preconditioned Krylov subspace methods.

Efficient basis updating for parametric nonlinear model order reduction

AbstractNonlinear model reduction is used to speed up the numerical solution of finite element models for vibration analysis of structures undergoing large deflections. This speed up is highly desired in design and optimization applications where parametric models are considered and the outcoming high‐dimensional differential equation must be solved multiple times. A first step is to approximate the solution vector i.e. the displacements of the nodes by a linear combination of some basis vectors. One common choice for these basis vectors is a combination of vibration modes and static modal derivatives. However, these vectors depend on parameter values of the parameterized system. This contribution shows how these basis vectors can be updated in an efficient manner. The vibration modes are updated by an inverse free preconditioned Krylov subspace method while the static derivatives are updated by a preconditioned conjugate gradient solver. A case study with a parametric beam gives a first insight into the performance of the proposed method.

The MTA: An Advanced and Versatile Thermal Simulator for Integrated Systems

Fast and accurate thermal analysis is crucial for determining the propagation of heat and tracking the formation of hotspots in integrated circuits (ICs). Existing academic thermal analysis tools primarily use compact models to accelerate thermal simulations but are limited to linear problems on relatively simple circuit geometries. The Manchester Thermal Analyzer (MTA) is a comprehensive tool that allows for fast and highly accurate linear and nonlinear thermal simulations of complex physical structures including the IC, the package, and the heatsink. The MTA is targeted for 2.5/3-D IC designs but also handles standard planar ICs. The MTA discretizes the heat equation in space using the finite element method and performs the time integration with unconditionally stable implicit time stepping methods. To improve the computational efficiency without sacrificing accuracy, the MTA features adaptive spatiotemporal refinement. The large-scale linear systems that arise during the simulation are solved with fast preconditioned Krylov subspace methods. The MTA supports thermal analysis of realistic integrated systems and surpasses the computational capabilities and performance of existing academic thermal simulators. For example, the simulation of a processor in a package attached to a heat sink, modeled by a computational grid consisting of over 3 million nodes, takes less than 3 min. The MTA is fully parallel and publicly available. 1

An efficient GPU version of the preconditioned GMRES method

In a large number of scientific applications, the solution of sparse linear systems is the stage that concentrates most of the computational effort. This situation has motivated the study and development of several iterative solvers, among which preconditioned Krylov subspace methods occupy a place of privilege. In a previous effort, we developed a GPU-aware version of the GMRES method included in ILUPACK, a package of solvers distinguished by its inverse-based multilevel ILU preconditioner. In this work, we study the performance of our previous proposal and integrate several enhancements in order to mitigate its principal bottlenecks. The numerical evaluation shows that our novel proposal can reach important run-time reductions.

Preconditioned Krylov subspace methods for solving radiative transfer problems with scattering and reflection

Two Krylov subspace methods, the GMRES and the BiCGSTAB, are analyzed for solving the linear systems arising from the mixed finite element discretization of the discrete ordinates radiative transfer equation. To increase their convergence rate and stability, the Jacobi and block Jacobi methods are used as preconditioners for both Krylov subspace methods. Numerical experiments, designed to test the effectiveness of the (preconditioned) GMRES and the BiCGSTAB, are performed on various radiative transfer problems: (i) transparent, (ii) absorption dominant, (iii) scattering dominant, and (iv) with specular reflection. It is observed that the BiCGSTAB is superior to the GMRES, with lower iteration counts, solving times, and memory consumption. In particular, the BiCGSTAB preconditioned by the block Jacobi method performed best amongst the set of other solvers. To better understand the discrete systems for radiative problems (i) to (iv), an eigenvalue spectrum analysis has also been performed. It revealed that the linear system conditioning deteriorates for scattering media problems in comparison to absorbing or transparent media problems. This conditioning further deteriorates when reflection is involved.

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

SummaryPreconditioned 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.

An integral equation approach for the solution of the Stokes flow with Hermite surfaces

An integral equation approach based on bi-cubic Hermite surfaces has been proposed to solve the three-dimensional complex Stokes flow problems. The numerical solutions are obtained by utilizing the boundary collocation method as well as the continuous distribution of Stokeslets, which are the fundamental solutions of the steady Stokes equations. The quadrilateral surface elements are represented with the bi-cubic Hermite functions that allow the continuous variation of the surface normal vectors between neighbouring elements. The unique normal vectors at the vertices are obtained using the MeanWeighted by Sine and Edge Length Reciprocal (MWSELR) approach. The singular integrations are evaluated using the Tanh−Sinh quadrature rule, meanwhile non-singular integrals are evaluated using the Gauss-Legendre quadrature rule. For the case of freely moving/falling rigid bodies, the integral equations are fully coupled with the equation of motion and the resulting algebraic equations are solved iteratively using the preconditioned Krylov subspace methods provided within the PETSc library. The accuracy of the numerical algorithm is initially verified for the three-dimensional unbounded Stokes flow around a sphere. Then the algorithm is applied to the more complex problems such as the sedimentation of spherical particles and helical swimmers. The numerical simulations show parallel scalability for a large number of processors.