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Despite great progress in seeking accurate numerical approximator to nonlinear structural seismic response prediction using deep learning approaches, tedious training process and large volume of structural response data under earthquakes for training and validation are often prohibitively accessible. In our methodology, the main innovation can be seen in the incorporation of deep neural networks (DNNs) into a classical numerical integration method by using a hybridized integration time‐stepper. In this way, the linear physics information of the structure and the obscure nonlinear dynamics are smoothly combined. We propose to use residual network (ResNet) to learn time‐stepping schemes specifically for the nonlinear state variables of the system. Our Physics‐DNN hybridized integration (PDHI) time‐stepping scheme provides important advantages over current pure data‐driven approaches, including (i) a flexible framework incorporating known time‐invariant physics information, (ii) requirement of structural seismic response data being circumvented by simple short bursts of trajectories collected from underlying nonlinear components, and (iii) efficiency in training and validation process. Besides, our results indicate that a simple feedforward or convolutional architecture outperforms recurrent networks to fulfill the requirement of prediction accuracy as well as long‐range memory in structural dynamic analysis. Several numerical and experimental examples are presented to demonstrate the performance of the method. [ABSTRACT FROM AUTHOR]

We describe a proof‐of‐concept development and application of a phase‐averaging technique to the nonlinear rotating shallow‐water equations on the sphere, discretised using compatible finite‐element methods. Phase averaging consists of averaging the nonlinearity over phase shifts in the exponential of the linear wave operator. Phase averaging aims to capture the slow dynamics in a solution that is smoother in time (in transformed variables), so that larger timesteps may be taken. We overcome the two key technical challenges that stand in the way of studying the phase averaging and advancing its implementation: (1) we have developed a stable matrix exponential specific to finite elements and (2) we have developed a parallel finite averaging procedure. Following recent studies, we consider finite‐width phase‐averaging windows, since the equations have a finite timescale separation. In our numerical implementation, the averaging integral is replaced by a Riemann sum, where each term can be evaluated in parallel. This creates an opportunity for parallelism in the timestepping method, which we use here to compute our solutions. Here, we focus on the stability and accuracy of the numerical solution. We confirm that there is an optimal averaging window, in agreement with theory. Critically, we observe that the combined time discretisation and averaging error is much smaller than the time discretisation error in a semi‐implicit method applied to the same spatial discretisation. An evaluation of the parallel aspects will follow in later work. [ABSTRACT FROM AUTHOR]

This work compares two Nitsche‐type approaches to treat non‐conforming triangulations for a high‐order discontinuous Galerkin (DG) solver for the acoustic conservation equations. The first approach (point‐to‐point interpolation) uses inconsistent integration with quadrature points prescribed by a primary element. The second approach uses consistent integration by choosing quadratures depending on the intersection between non‐conforming elements. In literature, some excellent properties regarding performance and ease of implementation are reported for point‐to‐point interpolation. However, we show that this approach can not safely be used for DG discretizations of the acoustic conservation equations since, in our setting, it yields spurious oscillations that lead to instabilities. This work presents a test case in that we can observe the instabilities and shows that consistent integration is required to maintain a stable method. Additionally, we provide a detailed analysis of the method with consistent integration. We show optimal spatial convergence rates globally and in each mesh region separately. The method is constructed such that it can natively treat overlaps between elements. Finally, we highlight the benefits of non‐conforming discretizations in acoustic computations by a numerical test case with different fluids. [ABSTRACT FROM AUTHOR]

In this paper a thorough analysis is carried out of the type of order reduction that Lawson methods exhibit when used to integrate nonlinear initial boundary value problems. In particular, we focus on nonlinear reaction‐diffusion problems, and therefore, this study is important in a large number of practical applications modeled by this type of nonlinear equations. A theoretical study of the local and global error of the total discretization of the problem is carried out, taking into account both, the error coming from the space discretization and that due to the integration in time. These results are also corroborated by the numerical experiments performed in this paper. [ABSTRACT FROM AUTHOR]

We introduce a numerical method for the solution of the time‐dependent Schrödinger equation with a smooth potential, based on its reformulation as a Volterra integral equation. We present versions of the method both for periodic boundary conditions, and for free space problems with compactly supported initial data and potential. A spatially uniform electric field may be included, making the solver applicable to simulations of light‐matter interaction. The primary computational challenge in using the Volterra formulation is the application of a spacetime history dependent integral operator. This may be accomplished by projecting the solution onto a set of Fourier modes, and updating their coefficients from one time step to the next by a simple recurrence. In the periodic case, the modes are those of the usual Fourier series, and the fast Fourier transform (FFT) is used to alternate between physical and frequency domain grids. In the free space case, the oscillatory behavior of the spectral Green's function leads us to use a set of complex‐frequency Fourier modes obtained by discretizing a contour deformation of the inverse Fourier transform, and we develop a corresponding fast transform based on the FFT. Our approach is related to pseudospectral methods, but applied to an integral rather than the usual differential formulation. This has several advantages: it avoids the need for artificial boundary conditions, admits simple, inexpensive, high‐order implicit time marching schemes, and naturally includes time‐dependent potentials. We present examples in one and two dimensions showing spectral accuracy in space and eighth‐order accuracy in time for both periodic and free space problems. © 2020 Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

We propose and analyze computationally a new fictitious domain method, based on higher order space‐time finite element discretizations, for the simulation of the nonstationary, incompressible Navier–Stokes equations on evolving domains. The physical domain is embedded into a fixed computational mesh such that arbitrary intersections of the moving domain's boundaries with the background mesh occur. The potential of such cut finite element techniques for higher order space‐time finite element methods has rarely been studied in the literature so far and deserves further elucidation. The key ingredients of the approach are the weak formulation of Dirichlet boundary conditions by Nitsche's method, the flexible and efficient integration over all types of intersections of cells by moving boundaries and the spatial extension of the discrete physical quantities to the entire computational background mesh including fictitious (ghost) subdomains of fluid flow. Thereby, an expensive remeshing and adaptation of the sparse matrix data structure are avoided and the computations are accelerated. To prevent spurious oscillations caused by irregular intersections of mesh cells, a penalization, defining also implicitly the extension to ghost domains, is added. These techniques are embedded in an arbitrary order, discontinuous Galerkin discretization of the time variable and an inf‐sup stable discretization of the spatial variables. The parallel implementation of the matrix assembly is described. The optimal order convergence properties of the algorithm are illustrated in a numerical experiment for an evolving domain. The well‐known 2d benchmark of flow around a cylinder as well as flow around moving obstacles with arising cut cells and fictitious domains are considered further. [ABSTRACT FROM AUTHOR]

Exponential time integrators are well‐established discretization methods for time semilinear systems of ordinary differential equations. These methods use φ−functions, which are matrix functions related to the exponential. This work introduces an algorithm to speed up the computation of the φ−function action over vectors for two‐dimensional (2D) matrices expressed as a Kronecker sum. For that, we present an auxiliary exponential‐related matrix function that we express using Kronecker products of one‐dimensional matrices. We exploit state‐of‐the‐art implementations of φ−functions to compute this auxiliary function's action and then recover the original φ−action by solving a Sylvester equation system. Our approach allows us to save memory and solve exponential integrators of 2D+time problems in a fraction of the time traditional methods need. We analyze the method's performance considering different linear operators and with the nonlinear 2D+time Allen–Cahn equation. [ABSTRACT FROM AUTHOR]

This article develops high‐order implicit time‐stepping methods combined with the fourth‐order central essentially‐non‐oscillatory (CENO) scheme for stiff three‐dimensional computational fluid dynamics problems having disparate characteristic time scales. Both aerodynamic and magnetohydrodynamic problems are considered on three‐dimensional multiblock body‐fitted grids with hexahedral cells. Several implicit time integration methods of third‐ and fourth‐order accuracy are considered, including the multistep backward differentiation formulas (BDF4), multistage explicitly singly diagonally implicit Runge‐Kutta (ESDIRK4), and Rosenbrock‐type methods (ROS34POW2). The resulting nonlinear algebraic system of equations is solved via a preconditioned Jacobian‐free inexact Newton–Krylov method with additive Schwarz preconditioning using block‐based incomplete LU decomposition. The performance of the high‐order implicit time‐stepping methods on smooth and stiff problems is compared with a standard fourth‐order explicit Runge‐Kutta (RK4) method. It is shown that the Rosenbrock methods, despite their advantage of only requiring the solution of linear systems, have significant drawbacks in terms of robustness issues for highly nonlinear compressible flows. The implicit BDF4 and ESDIRK4 methods are found to be much more efficient than the explicit fourth‐order RK4 method for a stiff resistive magnetohydrodynamic (MHD) problem discretized with the fourth‐order CENO method. When applied to the problem of vortex shedding governed by the Navier–Stokes equations, an A‐stable ESDIRK4 scheme proved to be the more robust and accurate implicit time‐marching scheme and was able to offer significant speedup compared with the RK4 method. Initial results are also shown for high‐order implicit time integration applied to two problems with discontinuities. The current study represents the first to achieve high‐order implicit time integration for MHD, enabling large time steps and substantial speedups for stiff MHD problems with high‐order accuracy, and it also represents the first to establish high‐order implicit time integration for high‐order CENO in space. [ABSTRACT FROM AUTHOR]

A model for studying Transient Inertia–Gravity And Rossby wave dynamics (TIGAR) is introduced. The presented horizontal component of the model solves the nonlinear rotating shallow‐water equations on the sphere using Hough harmonics. Spectral modelling using Hough harmonics as basis functions describes atmospheric dynamics in terms of physically identifiable structures: Rossby and inertia–gravity eigensolutions of linearized primitive equations. This offers an attractive framework for detangling gravity wave dynamics in high‐resolution simulations. Accurate computations are achieved through the use of higher order integrating factor and exponential time‐differencing methods, leading to a major increase in computational efficiency and stability. A comparison with classical time‐stepping schemes shows accuracy improvements of several orders of magnitude at no additional computational cost. In particular, stability gains are achieved through enhanced accuracy and efficiency in the computation of gravity waves, rather than through their damping. In the new framework, reduced models using Rossby and gravity waves aimed at studying dynamical aspects of data assimilation or wave interactions are easily implemented. [ABSTRACT FROM AUTHOR]

In high‐resolution numerical weather prediction models, fast‐moving acoustic waves must be treated in a stable manner. Among the implicit–explicit (IMEX) schemes used for the solution of these models, horizontally explicit, vertically implicit (HEVI) methods show good stability and scalability on massively parallel machines. In this work, we present two classes of exponential time‐differencing (ETD) methods for atmospheric models that use a HEVI splitting strategy, one being a three‐stage method with the addition of artificial diffusion, the second based on a Strang splitting approach. The stability properties of the methods are analyzed and numerical examples are provided, which compare time‐step restrictions and cost‐to‐accuracy ratios of the new methods with those for existing approaches. [ABSTRACT FROM AUTHOR]

We consider the mildly stiff and stiff inhomogeneous linear initial value Problems sharing constant coefficients. Exponential Runge–Kutta methods are considered to tackle this problem. For this type of problem, we were able to save a function evaluation (stage) per step compared to the best available methods. This is important, as seen in various computational experiments where our current approach outperforms older ones. [ABSTRACT FROM AUTHOR]

We generalize a previously published numerical approach for the one‐dimensional (1D) nonlinear Schrödinger (NLS) equation based on a multidomain spectral method on the whole real line in two ways: first, a fully explicit fourth‐order method for the time integration, based on a splitting scheme and an implicit Runge‐Kutta method for the linear part, is presented. Second, the 1D code is combined with a Fourier spectral method in the transverse variable both for elliptic and hyperbolic NLS equations. As an example we study the transverse stability of the Peregrine solution, an exact solution to the 1D NLS equation and thus a y‐independent solution to the 2D NLS. It is shown that the Peregine solution is unstable agains all standard perturbations, and that some perturbations can even lead to a blow‐up for the elliptic NLS equation. [ABSTRACT FROM AUTHOR]

Boundary value problems for the Euclidean Hodge‐Laplacian in three dimensions −ΔHL:=curlcurl−graddiv lead to variational formulations set in subspaces of H(curl,Ω)∩H(div,Ω), Ω⊂R3 a bounded Lipschitz domain. Via a representation formula and Calderón identities, we derive corresponding first‐kind boundary integral equations set in trace spaces of H1(Ω), H(curl,Ω), and H(div,Ω). They give rise to saddle‐point variational formulations and feature kernels whose dimensions are linked to fundamental topological invariants of Ω. Kernels of the same dimensions also arise for the linear systems generated by low‐order conforming Galerkin (BE) discretization. On their complements, we can prove stability of the discretized problems, nevertheless. We prove that discretization does not affect the dimensions of the kernels and also illustrate this fact by numerical tests. [ABSTRACT FROM AUTHOR]

Summary: A definition for functions of multidimensional arrays is presented. The definition is valid for third‐order tensors in the tensor t‐product formalism, which regards third‐order tensors as block circulant matrices. The tensor function definition is shown to have similar properties as standard matrix function definitions in fundamental scenarios. To demonstrate the definition's potential in applications, the notion of network communicability is generalized to third‐order tensors and computed for a small‐scale example via block Krylov subspace methods for matrix functions. A complexity analysis for these methods in the context of tensors is also provided. [ABSTRACT FROM AUTHOR]

In this article, we propose and analyze some efficient numerical methods for the solution of a class of stiff stochastic partial differential equations (SPDEs) with additive noise. First, we apply the polynomial chaos (PC) expansion to treat the randomness, which results to a set of deterministic stiff PDEs. Then this system of PDEs is discretized in space by the Fourier pseudo‐spectral method, enabling us to use the FFT algorithm to reduce computational cost efficiently. To overcome the instability in discretizing time component of resulting system of ordinary differential equations (ODEs), we apply the exponential time differencing (ETD), integrating factor Runge‐Kutta (IFRK) and modified exponential time differencing Runge‐Kutta (ETDRKB) methods. We also analyze the strong error bound of numerical approximation in a framework of Lipschitz nonlinearities for the method based on the ETDRKB scheme. Some numerical experiments are included to demonstrate the performance and convergence of methods. Numerical results show that the method based on the ETDRKB scheme is more efficient for stiff SPDEs. [ABSTRACT FROM AUTHOR]

Summary: In the present paper, we propose Krylov‐based methods for solving large‐scale differential Sylvester matrix equations having a low‐rank constant term. We present two new approaches for solving such differential matrix equations. The first approach is based on the integral expression of the exact solution and a Krylov method for the computation of the exponential of a matrix times a block of vectors. In the second approach, we first project the initial problem onto a block (or extended block) Krylov subspace and get a low‐dimensional differential Sylvester matrix equation. The latter problem is then solved by some integration numerical methods such as the backward differentiation formula or Rosenbrock method, and the obtained solution is used to build the low‐rank approximate solution of the original problem. We give some new theoretical results such as a simple expression of the residual norm and upper bounds for the norm of the error. Some numerical experiments are given in order to compare the two approaches. [ABSTRACT FROM AUTHOR]

Krylov subspace approximations to the matrix exponential are popularly used with full orthogonalization instead of incomplete orthogonalization, even though the latter strategy is known to reduce the cost by truncating the recurrences of the modified Gram-Schmidt process. This study combines such a strategy with an adaptive step-by-step integration scheme that allows both the stepsize and the dimension of the Krylov subspace to vary. A convergence analysis is done. Numerical results on test problems drawn from systems biology and computer systems show a significant speedup over the standard implementation with full orthogonalization and fixed dimension. [ABSTRACT FROM AUTHOR]

In a singular limit, the Klein-Gordon (KG) equation can be derived from the Klein-Gordon-Zakharov (KGZ) system. We point out that for the original system posed on a d-dimensional torus, the solutions of the KG equation do not approximate the solutions of the KGZ system. The KG system has to be modified to make correct predictions about the dynamics of the KGZ system. We explain that this modification is not necessary for the approximation result for the whole space [ABSTRACT FROM AUTHOR]

We consider the parallel factorization of sparse finite element matrices on distributed memory machines. Our method is based on a nested dissection approach combined with a cyclic re-distribution of the interface Schur complements. We present a detailed definition of the parallel method, and the well-posedness and the complexity of the algorithm are analyzed. A lean and transparent functional interface to existing finite element software is defined, and the performance is demonstrated for several representative examples. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, a high-order accurate numerical method for two-dimensional semilinear parabolic equations is presented. We apply a Galerkin-Legendre spectral method for discretizing spatial derivatives and a spectral collocation method for the time integration of the resulting nonlinear system of ordinary differential equations. Our formulation can be made arbitrarily high-order accurate in both space and time. Optimal a priori error bound is derived in the L2-norm for the semidiscrete formulation. Extensive numerical results are presented to demonstrate the convergence property of the method, show our formulation have spectrally accurate in both space and time. John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Reduced-order models for linear time-invariant dynamical systems are considered, and the error between the full-order model and the reduced-order model solutions is characterized. Based on the analytical representation of the error, an a posteriori error indicator is proposed that combines a Krylov-based exponential integrator and an a posteriori residual-based estimate. Numerical experiments illustrate the quality of the error estimator. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This work is concerned with the numerical solution of large-scale linear matrix equations [ABSTRACT FROM AUTHOR]

SUMMARY This paper is concerned with the numerical solution of symmetric large-scale Lyapunov equations with low-rank right-hand sides and coefficient matrices depending on a parameter. Specifically, we consider the situation when the parameter dependence is sufficiently smooth, and the aim is to compute solutions for many different parameter samples. On the basis of existing results for Lyapunov equations and parameter-dependent linear systems, we prove that the tensor containing all solution samples typically allows for an excellent low multilinear rank approximation. Stacking all sampled equations into one huge linear system, this fact can be exploited by combining the preconditioned CG method with low-rank truncation. Our approach is flexible enough to allow for a variety of preconditioners based, for example, on the sign function iteration or the alternating direction implicit method. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY The aim of this paper is to analyze efficient numerical methods for time integration of European option pricing models. When spatial discretization is adopted, the resulting problem consists of an ordinary differential equation that can be approximated by means of exponential Runge-Kutta integrators, where the matrix-valued functions are computed by the so-called shift-and-invert Krylov method. To our knowledge, the use of this numerical approach is innovative in the framework of option pricing, and it reveals to be very attractive and efficient to solve the problem at hand. In this respect, we propose some a posteriori estimates for the error in the shift-and-invert approximation of the core-functions arising in exponential integrators. The effectiveness of these error bounds is tested on several examples of interest. They can be adopted as a convenient stopping criterion for implementing the exponential Runge-Kutta algorithm in order to perform time integration. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY The focus of this paper is two-dimensional computational modelling of water flow in unsaturated soils consisting of weakly conductive disconnected inclusions embedded in a highly conductive connected matrix. When the inclusions are small, a two-scale Richards' equation-based model has been proposed in the literature taking the form of an equation with effective parameters governing the macroscopic flow coupled with a microscopic equation, defined at each point in the macroscopic domain, governing the flow in the inclusions. This paper is devoted to a number of advances in the numerical implementation of this model. Namely, by treating the micro-scale as a two-dimensional problem, our solution approach based on a control volume finite element method can be applied to irregular inclusion geometries, and, if necessary, modified to account for additional phenomena (e.g. imposing the macroscopic gradient on the micro-scale via a linear approximation of the macroscopic variable along the microscopic boundary). This is achieved with the help of an exponential integrator for advancing the solution in time. This time integration method completely avoids generation of the Jacobian matrix of the system and hence eases the computation when solving the two-scale model in a completely coupled manner. Numerical simulations are presented for a two-dimensional infiltration problem. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY In this paper, the numerical evaluation of matrix functions expressed in partial fraction form is addressed. The shift-and-invert Krylov method is analyzed, with special attention to error estimates. Such estimates give insights into the selection of the shift parameter and lead to a simple and effective restart procedure. Applications to the class of Mittag-Leffler functions are presented. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY We propose a time-exact Krylov-subspace-based method for solving linear ordinary differential equation systems of the form y ′ = − Ay + g( t) and y ′ ′ = − Ay + g( t), where y( t) is the unknown function. The method consists of two stages. The first stage is an accurate piecewise polynomial approximation of the source term g( t), constructed with the help of the truncated singular value decomposition. The second stage is a special residual-based block Krylov subspace method. The accuracy of the method is only restricted by the accuracy of the piecewise polynomial approximation and by the error of the block Krylov process. Because both errors can, in principle, be made arbitrarily small, this yields, at some costs, a time-exact method. Numerical experiments are presented to demonstrate efficiency of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY This paper describes the development of explicit time stepping methods for linear PDEs that are specifically designed to cope with the stiffness of the system of ODEs that results from spatial discretization. As stiffness is caused by the contrasting behavior of coupled components of the solution, it is proposed to adopt a componentwise approach in which each coefficient of the solution in an appropriate basis is computed using an individualized approximation of the solution operator. This has been accomplished by Krylov subspace spectral (KSS) methods, which use techniques from 'matrices, moments and quadrature' to approximate bilinear forms involving functions of matrices via block Gaussian quadrature rules. These forms correspond to coefficients with respect to the chosen basis of the application of the solution operator of the PDE to the solution at an earlier time. In this paper, it is proposed to substantially enhance the efficiency of KSS methods through the prescription of quadrature nodes on the basis of asymptotic analysis of the recursion coefficients produced by block Lanczos iteration for each Fourier coefficient as a function of frequency. The potential of this idea is illustrated through numerical results obtained from the application of the modified KSS methods to diffusion equations and wave equations. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY A fast exponential time integration scheme is considered for pricing European and double barrier options in jump-diffusion models. After spatial discretization, the option pricing problem is transformed into the product of a matrix exponential and a vector, while the matrix bears a Toeplitz structure. The shift-and-invert Arnoldi method is then employed for fast approximation to such operation. Owing to the Toeplitz structure, the computational cost can be reduced by the fast Fourier transform. Furthermore, the discretized form of option pricing problem satisfies a given condition such that the error bound of the shift-and-invert Arnoldi approximation is unrelated to the norm of the matrix. Numerical tests are carried out to compare the proposed method with other option pricing methods. Copyright © 2010 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We give a geometric framework for analysing iterative methods on singular linear systems A= and apply them to Krylov subspace methods. The idea is to decompose the method into the ℛ( A) component and its orthogonal complement ℛ( A), where ℛ( A) is the range of A. We apply the framework to GMRES, GMRES( k) and GCR( k), and derive conditions for convergence without breakdown for inconsistent and consistent singular systems. The approach also gives a geometric interpretation and different proofs of the conditions obtained by Brown and Walker for GMRES. We also give examples arising in the finite difference discretization of two-point boundary value problems of an ordinary differential equation. Copyright © 2010 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The Lanczos method with shift-invert technique is exploited to approximate the symmetric positive semidefinite Toeplitz matrix exponential. The complexity is lowered by the Gohberg-Semencul formula and the fast Fourier transform. Application to the numerical solution of an integral equation is studied. Numerical experiments are carried out to demonstrate the effectiveness of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

For large square matrices A and functions f, the numerical approximation of the action of f(A) to a vector v has received considerable attention in the last two decades. In this paper we investigate theextended Krylov subspace method, a technique that was recently proposed to approximate f(A)v for A symmetric. We provide a new theoretical analysis of the method, which improves the original result for A symmetric, and gives a new estimate for A nonsymmetric. Numerical experiments confirm that the new error estimates correctly capture the linear asymptotic convergence rate of the approximation. By using recent algorithmic improvements, we also show that the method is computationally competitive with respect to other enhancement techniques. Copyright © 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

General stationary iterative methods with a singular matrix M for solving range-Hermitian singular linear systems are presented, some convergence conditions and the representation of the solution are also given. It can be verified that the general Ortega–Plemmons theorem and Keller theorem for the singular matrix M still hold. Furthermore, the singular matrix M can act as a good preconditioner for solving range-Hermitian linear systems. Numerical results have demonstrated the effectiveness of the general stationary iterations and the singular preconditioner M. Copyright © 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We consider restricted rational Lanczos approximations to matrix functions representable by some integral forms. A convergence analysis that stresses the effectiveness of the proposed method is developed. Error estimates are derived. Numerical experiments are presented. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We consider linear matrix equations where the linear mapping is the sum of a standard Lyapunov operator and a positive operator. These equations play a role in the context of stochastic or bilinear control systems. To solve them efficiently one can fall back on known efficient methods developed for standard Lyapunov equations. In this paper, we describe a direct and an iterative method based on this idea. The direct method is applicable if the generalized Lyapunov operator is a low-rank perturbation of a standard Lyapunov operator; it is related to the Sherman–Morrison–Woodbury formula. The iterative method requires a stability assumption; it uses convergent regular splittings, an alternate direction implicit iteration as preconditioner, and Krylov subspace methods. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We study large-scale, continuous-time linear time-invariant control systems with a sparse or structured state matrix and a relatively small number of inputs and outputs. The main contributions of this paper are numerical algorithms for the solution of large algebraic Lyapunov and Riccati equations and linear-quadratic optimal control problems, which arise from such systems. First, we review an alternating direction implicit iteration-based method to compute approximate low-rank Cholesky factors of the solution matrix of large-scale Lyapunov equations, and we propose a refined version of this algorithm. Second, a combination of this method with a variant of Newton's method (in this context also called Kleinman iteration) results in an algorithm for the solution of large-scale Riccati equations. Third, we describe an implicit version of this algorithm for the solution of linear-quadratic optimal control problems, which computes the feedback directly without solving the underlying algebraic Riccati equation explicitly. Our algorithms are efficient with respect to both memory and computation. In particular, they can be applied to problems of very large scale, where square, dense matrices of the system order cannot be stored in the computer memory. We study the performance of our algorithms in numerical experiments. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The paper deals with the application of the restricted-denominator rational Krylov method, recently discussed in (BIT 2004; 44(3):595–615; SIAM J. Sci. Comput. 2005; 27:1438–1457), to the computation of the action of the so-called ϕ-functions, which play a fundamental role in several modern exponential integrators. The analysis here presented is devoted in particular to the construction of error estimates of easy practical use. Copyright © 2007 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The paper deals with Krylov methods for approximating functions of matrices via interpolation. In this frame residual smoothing techniques based on quasi-kernel polynomials are considered. Theoretical results as well as numerical experiments illustrate the effectiveness of our approach. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The Yee-method is a simple and elegant way of solving the time-dependent Maxwell's equations. On the other hand, this method has some inherent drawbacks too. The main one is that its stability requires a very strict upper bound for the possible time-steps. This is why, during the last decade, the main goal was to construct such methods that are unconditionally stable. This means that the time-step can be chosen based only on accuracy instead of stability considerations. In this paper we give a uniform treatment of methods that use the same spatial staggered grid approximation as the classical Yee-method. Three other numerical methods are discussed: the Namiki–Zheng–Chen–Zhang alternating direction implicit method (NZCZ), the Kole–Figge-de Raedt method (KFR) and a Krylov-space method. All methods are discussed with non-homogeneous material parameters. We show how the existing finite difference numerical methods are based on the approximation of a matrix exponential. With this formulation we prove the unconditional stability of the NZCZ method without any computer algebraic tool. Moreover, we accelerate the Krylov-space method with a skew-symmetric formulation of the semi-discretized equations. Our main goal is to compare the methods from the point of view of the computational speed. This question is investigated in ID numerical tests. Copyright © 2005 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Fast algorithms, based on the unsymmetric look-ahead Lanczos and the Arnoldi process, are developed for the estimation of the functional Φ(ƒ)=uTƒ(A) v for fixed u, v and A, where A∈ℜn×n is a large-scale unsymmetric matrix. Numerical results are presented which validate the comparable accuracy of both approaches. Although the Arnoldi process reaches convergence more quickly in some cases, it has greater memory requirements, and may not be suitable for especially large applications. Copyright © 2003 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Given the generalized symmetric eigenvalue problem Ax=λMx, with A semidefinite and M definite, we analyse some algebraic formulations for the approximation of the smallest non-zero eigenpairs, assuming that a sparse basis for the null space is available. In particular, we consider the inexact version of the Shift-and-Invert Lanczos method, and we show that apparently different algebraic formulations provide the same approximation iterates, under some natural hypotheses. Our results suggest that alternative strategies need to be explored to really take advantage of the special problem setting, other than reformulating the algebraic problem. Experiments on a real application problem corroborate our theoretical findings. Copyright © 2002 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper we use the theory of Faber polynomials for solving N-dimensional linear initial value problems. In particular, we use Faber polynomials to approximate the evolution operator creating the so-called exponential integrators. We also provide a consistence and convergence analysis. Some tests where we compare our methods with some Krylov exponential integrators are finally shown. Copyright © 2002 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper we compare Krylov subspace methods with Faber series expansion for approximating the matrix exponential operator on large, sparse, non-symmetric matrices. We consider in particular the case of Chebyshev series, corresponding to an initial estimate of the spectrum of the matrix by a suitable ellipse. Experimental results upon matrices with large size, arising from space discretization of 2D advection–diffusion problems, demonstrate that the Chebyshev method can be an effective alternative to Krylov techniques. Copyright © 2002 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper deals with two-step W-methods for stiff initial value problems which allow the parallel solution of the stage equations. Special methods of order 3 and 4 are tested for both sequential and parallel implementations and compared with efficient sequential codes. The results show that the methods are already competitive in a sequential implementation and have a high level of parallelization especially in combination with Krylov solvers for linear equations of large dimensions. [ABSTRACT FROM AUTHOR]

In this paper we compare Krylov subspace methods with Chebyshev series expansion for approximating the matrix exponential operator on large, sparse, symmetric matrices. Experimental results upon negative-definite matrices with very large size, arising from (2D and 3D) FE and FD spatial discretization of linear parabolic PDEs, demonstrate that the Chebyshev method can be an effective alternative to Krylov techniques, especially when memory bounds do not allow the storage of all Ritz vectors. We also discuss the sensitivity of Chebyshev convergence to extreme eigenvalue approximation, as well as the reliability of various a priori and a posteriori error estimates for both methods. Copyright © 2000 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]