the leja method revisited backward error analysis for the matrix exponential the leja method is a polynomial interpolation procedure that can be used to compute matrix functions in particular computing the action of the matrix exponential on a given vector is a typical application this quantity is required e g in exponential integrators the leja method essentially depends on three parameters the scaling parameter the location of the interpolation points and the degree of interpolation we present here a backward error analysis that allows us to determine these three parameters as a function of the prescribed accuracy additional aspects that are required for an efficient and reliable implementation are discussed numerical examples illustrating the performance of our matlab code are included

numerical low rank approximation of matrix differential equations abstract the efficient numerical integration of large scale matrix differential equations is a topical problem in numerical analysis and of great importance in many applications standard numerical methods applied to such problems require an unduly amount of computing time and memory in general based on a dynamical low rank approximation of the solution a new splitting integrator is proposed for a quite general class of stiff matrix differential equations this class comprises differential lyapunov and differential riccati equations that arise from spatial discretizations of partial differential equations the proposed integrator handles stiffness in an efficient way and it preserves the symmetry and positive semidefiniteness of solutions of differential lyapunov equations numerical examples that illustrate the benefits of this new method are given in particular numerical results for the efficient simulation of the weather phenomenon el nino are presented

embedded minimal surfaces of finite topology in this paper we prove that any complete embedded minimal surface m in r 3 with finite topology and compact boundary possibly empty is conformally a compact riemann surface m with boundary punctured in a finite number of interior points and that m can be represented in terms of meromorphic data on its conformal completion m in particular we demonstrate that m is a minimal surface of finite type and describe how this property permits a classification of the asymptotic behavior ofm

embedded minimal surfaces with an infinite number of ends we prove the existence of an infinite family of periodic properly embedded minimal surfaces with an infinite number of annular ends 25 refs 29 figs

on the performance of exponential integrators for problems in magnetohydrodynamics exponential integrators have been introduced as an efficient alternative to explicit and implicit methods for integrating large stiff systems of differential equations over the past decades these methods have been studied theoretically and their performance was evaluated using a range of test problems while the results of these investigations showed that exponential integrators can provide significant computational savings the research on validating this hypothesis for large scale systems and understanding what classes of problems can particularly benefit from the use of the new techniques is in its initial stages resistive magnetohydrodynamic mhd modeling is widely used in studying large scale behavior of laboratory and astrophysical plasmas in many problems numerical solution of mhd equations is a challenging task due to the temporal stiffness of this system in the parameter regimes of interest in this paper we evaluate the performance of exponential integrators on large mhd problems and compare them to a state of the art implicit time integrator both the variable and constant time step exponential methods of epirk type are used to simulate magnetic reconnection and the kevin helmholtz instability in plasma performance of these methods which are part of the epic software package is compared to the variable time step variable order bdf scheme included in the cvode part of sundials library we study performance of the methods on parallel architectures and with respect to magnitudes of important parameters such as reynolds lundquist and prandtl numbers we find that the exponential integrators provide superior or equal performance in most circumstances and conclude that further development of exponential methods for mhd problems is warranted and can lead to significant computational advantages for large scale stiff systems of differential equations such as mhd

a massively parallel exponential integrator for advection diffusion models this work considers the real leja points method relpm m caliari m vianello l bergamaschi interpolating discrete advection diffusion propagators at spectral leja sequences j comput appl math 172 2004 79 99 for the exponential integration of large scale sparse systems of odes generated by finite element or finite difference discretizations of 3 d advection diffusion models we present an efficient parallel implementation of relpm for polynomial interpolation of the matrix exponential propagators exp dta v and f dta v f z exp z 1 z a scalability analysis of the most important computational kernel inside the code the parallel sparse matrix vector product has been performed as well as an experimental study of the communication overhead as a result of this study an optimized parallel sparse matrix vector product routine has been implemented the resulting code shows good scaling behavior even when using more than one thousand processors the numerical results presented on a number of very large test cases gives experimental evidence that relpm is a reliable and efficient tool for the simulation of complex hydrodynamic processes on parallel architectures

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

the structure of singly periodic minimal surfaces the only properly embedded minimal surface with more than one topological end and a cont inuous g roup of symmetries is the catenoid considering periodic minimal surfaces with more than one end a surface in n 3 is periodic if it is invariant under an infinite discrete g roup of isometrics that act freely on 1r3 the only classical examples are those discovered by riemann the riemann examples possess a quite special set of properties 1 they have an infinite number of flat annular ends 1 2 they are invariant under a nontrivial t ranslat ion t 3 the surfaces t h a v e genus 1 and 2 ends 4 the surfaces t have total curvature equal to 8 zr mot iva ted by the riemann examples we established the existence of an infinite family of proper ly embedded periodic minimal surfaces d k each with an infinite number of ends 1 these surfaces which are discussed below see sect 1 2 on rigidity also have an infinite number of flat annular ends and are invariant under a t ranslat ion t the quot ient of these surfaces by t is a surface of genus 2 k 1 with two ends and total curvature 8 r e k 1 k l we have been able to twis t each of these new surfaces that is we construct surfaces with the same properties that are invariant under screw motions t r where r is a rota t ion a round an axis in the direction of t see sect 1 3 below together with the catenoid and the examples of riemann these new surfaces comprise all the known properly embedded minimal surfaces with more than one topological end whose symmetry g roup contains an infinite discrete subgroup these examples share m a n y of the properties of the riemann examples a major goal of this paper is to show that this is not accidental

direct calculation of accurate siegert eigenvalues the roots of the hankel determinant built with the taylor coefficients of the regularized logarithmic derivative of the eigenfunction give both the bound state and siegert eigenvalues resonances of separable quantum mechanical systems we obtain accurate positions and widths for the resonances of some commonly chosen test examples

comparative performance of exponential implicit and explicit integrators for stiff systems of odes exponential integrators have enjoyed a resurgence of interest in recent years but there is still limited understanding of how their performance compares with that of state of the art integrators most notably the commonly used newton krylov implicit methods in this paper we present comparative performance analysis of krylov based exponential implicit and explicit integrators on a suite of stiff test problems and demonstrate that exponential integrators have computational advantages compared to the other methods particularly as problems become larger and more stiff we argue that the faster convergence of the krylov iteration within exponential integrators accounts for the main proportion of the computational savings that they provide and illustrate how the structure of these methods ensures such efficiency in addition we demonstrate the computational advantages of the newly introduced tokman and loffeld 2010 17 exponential propagation runge kutta epirk fifth order methods the detailed analysis of the performance of the methods that is presented provides guidelines for the construction and implementation of efficient exponential methods and the quantitative comparisons inform the selection of appropriate schemes for other problems

interpolating discrete advection diffusion propagators at leja sequences we propose and analyze the relpm real leja points method for evaluating the propagator tb v via matrix interpolation polynomials at spectral leja sequences here b is the large sparse nonsymmetric matrix arising from stable 2d or 3d finite difference discretization of linear advection diffusion equations and z is the entire function z ez 1 z the corresponding stiff differential system y t by t g y 0 y 0 is solved by the exact time marching scheme y i 1 y i t i t i b by i g i 0 1 where the time step is controlled simply via the variation percentage of the solution and can be large numerical tests show substantial speed ups up to one order of magnitude with respect to a classical variable step size crank nicolson solver

on the algorithm by al mohy and higham for computing the action of the matrix exponential a posteriori roundoff error estimation abstract the algorithm by al mohy and higham 2011 2 computes an approximation to e a b for given a and b where a is an n by n matrix and b is for example a vector of dimension n it uses a scaling together with a truncated taylor series approximation to the exponential of the scaled matrix in this paper a method is developed for estimating the roundoff error of the computed solution an asymptotic expansion of this error for small values of the unit roundoff is the basis of the method the roundoff error is further expressed in terms of sums of rounding errors which occur during the computation a second approximation to e a b which is computed with a lower precision than the first one is used to evaluate these rounding errors in practice the result is an upper bound on the normwise relative roundoff error further an algorithm is proposed for computing the error bound the cost for performing this algorithm depends on the type of problem and the accuracy which is required for the error estimate in case that all computations are performed with standard precisions this cost can be expressed in terms of the number of computed matrix vector products and is bounded from above by two times the cost for computing e a b

a new class of exponential propagation iterative methods of runge kutta type epirk we propose a new class of the exponential propagation iterative methods of runge kutta type epirk the epirk schemes are exponential integrators that can be competitive with explicit and implicit methods for integration of large stiff systems of odes introducing the new more general than previously proposed ansatz for epirk schemes allows for more flexibility in deriving computationally efficient high order integrators recent extension of the theory of b series to exponential integrators 1 is used to derive classical order conditions for schemes up to order five an algorithm to systematically solve these conditions is presented and several new fifth order schemes are constructed several numerical examples are used to verify the order of the methods and to illustrate the performance of the new schemes

summation of the eigenvalue perturbation series by multi valued pade approximants application to resonance problems and double wells quadratic pade approximants are used to obtain energy levels both for the anharmonic oscillator x2 2 lambda x4 and for the double well x2 2 lambda x4 in the first case the complex valued energy of the resonances is reproduced by summation of the real terms of the perturbation series the second case is treated formally as an anharmonic oscillator with a purely imaginary frequency we use the expansion around the central maximum of the potential to obtain a complex perturbation series on the unphysical sheet of the energy function then we perform an analytical continuation of this solution to the neighbouring physical sheet taking into account the supplementary branch of quadratic approximants in this way we can reconstruct the real energy by summation of the complex series such an unusual approach eliminates the double degeneracy of states that makes ordinary perturbation theory around the minima of the double well potential incorrect

the anharmonic oscillator perturbation series for cubic and quartic energy distortion twenty five terms of the perturbation series are calculated for the first five energy levels of an oscillator with lambda x4 energy distortion and a similar calculation is carried out for the first six levels for lambda x3 distortion for the positive quartic the alternating series is summed using the aitken delta 2 transformation for lambda from 0 1 to 100 an exponential integral model and a truncated binomial model are used to transform the series for negative distortion energies and the complex energy eigenvalues are calculated for lambda from 0 01 to 1

a splitting approach for the kadomtsev petviashvili equation we consider a splitting approach for the kadomtsev petviashvili equation with periodic boundary conditions and show that the necessary interpolation procedure can be efficiently implemented the error made by this numerical scheme is compared to exponential integrators which have been shown in klein and roidot 2011 2 to perform best for stiff solutions of the kadomtsev petviashvili equation since many classic high order splitting methods do not perform well we propose a stable extrapolation method in order to construct an efficient numerical scheme of order four in addition the conservation properties and the possibility of order reduction for certain initial values for the numerical schemes under consideration are investigated

on krylov subspace approximations to the matrix exponential operator krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper we derive error bounds via a functional calculus of arnoldi and lanczos methods that reduces the study of krylov subspace approximations of functions of matrices to that of linear systems of equations as a side result we obtain error bounds for galerkin type krylov methods for linear equations namely the biconjugate gradient method and the full orthogonalization method for krylov approximations to matrix exponentials we show superlinear error decay from relatively small iteration numbers onwards depending on the geometry of the numerical range the spectrum or the pseudospectrum the convergence to exp tau a v is faster than that of corresponding krylov methods for the solution of linear equations i tau a x v which usually arise in the numerical solution of stiff ordinary differential equations odes we therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use krylov approximations to the exponential function of the jacobian instead of solving linear or nonlinear systems of equations in every time step

explicit exponential runge kutta methods for semilinear parabolic problems the aim of this paper is to analyze explicit exponential runge kutta methods for the time integration of semilinear parabolic problems the analysis is performed in an abstract banach space framework of sectorial operators and locally lipschitz continuous nonlinearities we commence by giving a new and short derivation of the classical nonstiff order conditions for exponential runge kutta methods but the main interest of our paper lies in the stiff case by expanding the errors of the numerical method in terms of the solution we derive new order conditions that form the basis of our error bounds for parabolic problems we show convergence for methods up to order four and we analyze methods that were recently presented in the literature these methods have classical order four but they do not satisfy some of the new conditions therefore an order reduction is expected we present numerical experiments which show that this order reduction in fact arises in practical examples based on our new conditions we finally construct methods that do not suffer from order reduction

a rigidity theorem for properly embedded minimal surfaces in bf r sp 3 we consider the question of when an intrinsic isometry of a properly embedded minimal surface is induced by an ambient isometry we prove it always extends when the surface has at least two ends there are several interesting theorems and conjectures on the rigidity of complete surfaces in r 3 which satisfy some natural geometric constraint perhaps the best known theorem of this type is the cohn vossen theorem which shows that a compact riemannian surface of positive curvature has a unique immersion into r a similar result is conjectured for complete surfaces of nonpositive or nonnegative curvature with the additional hypothesis that the surface has finite area it is also conjectured that tight surfaces are rigid in this paper we prove a rigidity theorem for properly embedded minimal surfaces in r 3 which have more than one end this theorem states that the inclusion map of the surface into r3 represents the unique isometric minimal immersion of such a surface up to a rigid motion of r 3 in particular it follows from our theorem that every intrinsic isometry of this type of surface extends to an isometry of r 3 in 1 we derive a geometric condition on an immersed minimal surface which guarantees that the surface is minimally rigid we prove that if the surface intersects a plane transversally along an immersed closed curve then any other isometric minimal immersion of the surface into r differs from the original immersion by an ambient isometry in 2 we obtain some asymptotic properties of solutions of the minimal surface equation over annular planar domains in 3 we prove that if a properly embedded minimal surface m in r3 has more than one end then m is transverse

fourth order time stepping for kadomtsev petviashvili and davey stewartson equations purely dispersive partial differential equations such as the korteweg de vries equation the nonlinear schrodinger equation and higher dimensional generalizations thereof can have solutions which develop a zone of rapid modulated oscillations in the region where the corresponding dispersionless equations have shocks or blow up to numerically study such phenomena fourth order time stepping in combination with spectral methods is beneficial in resolving the steep gradients in the oscillatory region we compare the performance of several fourth order methods for the kadomtsev petviashvili and the davey stewartson equations two integrable equations in 2 1 dimensions these methods are exponential time differencing integrating factors time splitting implicit runge kutta and driscoll s composite runge kutta method the accuracy in the numerical conservation of integrals of motion is discussed

new adaptive exponential propagation iterative methods of runge kutta type exponential integrators have emerged as an efficient alternative to commonly used time integrators recently a new class of exponential propagation iterative methods of runge kutta type epirk has been introduced m tokman j comput phys 230 2011 pp 8762 8778 these schemes possess a structure that makes them computationally advantageous compared to other exponential methods in addition the general epirk formulation offers flexibility that allows derivation of new efficient techniques in this paper we use this feature to derive new epirk methods which are particularly designed to take advantage of the adaptive krylov algorithm j niesen and w wright http arxiv org abs 0907 4631 2010 the adaptive krylov method significantly reduces the computational complexity of evaluating products of matrix varphi functions and vectors necessary for implementing an exponential integrator we present the derivation of the new adaptive epirk methods construct new schemes and illustrate the co

analysis of some krylov subspace approximations to the matrix exponential operator in this note a theoretical analysis of some krylov subspace approximations to the matrix exponential operation exp a v is presented and a priori and a posteriors error estimates are established several such approximations are considered the main idea of these techniquesis to approximately project the exponential operator onto a small krylov subspace and to carry out the resulting small exponential matrix computation accurately this general approach which has been used with success in several applications provides a systematic way of defining high order explicit type schemes for solving systems of ordinary differential equations or time dependent partial differential equations

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

an iterative solution method for solving f a x b using krylov subspace information obtained for the symmetric positive definite matrix a abstract the conjugate gradients method generates successive approximations x i for the solution of the linear system ax b where a is symmetric positive definite and usually sparse it will be shown how intermediate information obtained by the conjugate gradients cg algorithm or by the closely related lanczos algorithm can be used to solve f a x b iteratively in an efficient way for suitable functions f the special case f a a 2 is discussed in particular we also consider the problem of solving ax b for different right hand sides b a variant on a well known algorithm for that problem is proposed which does not seem to suffer from the usual loss of orthogonality in the standard cg and lanczos algorithms

on the fly backward error estimate for matrix exponential approximation by taylor algorithm abstract in this paper we show that it is possible to estimate the backward error for the approximation of the matrix exponential on the fly without the need to precompute in high precision quantities related to specific accuracies in this way the scaling parameter s and the degree m of the truncated taylor series the underlying method are adapted to the input matrix and not to a class of matrices sharing some spectral properties such as with the classical backward error analysis the result is a very flexible method which can approximate the matrix exponential at any desired accuracy moreover the risk of overscaling that is the possibility to select a value s larger than necessary is mitigated by a new choice as the sum of two powers of two finally several numerical experiments in matlab with data in double and variable precision and with different tolerances confirm that the method is accurate and often faster than available good alternatives

the anharmonic oscillator complex eigenvalues for the ground state with negative quartic or cubic energy distortion the perturbed ground state eigenvalues for the harmonic oscillator with negative quartic and with cubic energy distortion are calculated by numerical integration these agree with the author s results using an asymptotic series

a new approach to constructing efficient stiffly accurate epirk methods the structural flexibility of the exponential propagation iterative methods of runge kutta type epirk enables construction of particularly efficient exponential time integrators while the epirk methods have been shown to perform well on stiff problems all of the schemes proposed up to now have been derived using classical order conditions in this paper we extend the stiff order conditions and the convergence theory developed for the exponential rosenbrock methods to the epirk integrators we derive stiff order conditions for the epirk methods and develop algorithms to solve them to obtain specific schemes moreover we propose a new approach to constructing particularly efficient epirk integrators that are optimized to work with an adaptive krylov algorithm we use a set of numerical examples to illustrate the computational advantages that the newly constructed epirk methods offer compared to previously proposed exponential integrators

complex variable hvpt complex variable hypervirial perturbation theory is applied to the case of oscillator and coulomb potentials perturbed by a single term potential of the form vxn or vrn respectively the trial calculations reported show that this approach can produce accurate complex energies for resonant states via a simple and speedy calculation and can also be useful in studies of pt symmetry and tunnelling resonance effects

a new class of split exponential propagation iterative methods of runge kutta type sepirk for semilinear systems of odes abstract in this paper the framework of the exponential propagation iterative methods of runge kutta type epirk is extended to construct split epirk sepirk integrators for semilinear systems of odes the structure of the sepirk methods possesses the flexibility and generality that allows construction of very efficient schemes we demonstrate how bicolored trees based b series can be used to derive the order conditions for the new integrators an algorithm is developed to solve the order conditions up to order three and several new schemes with desirable properties are proposed the numerical results illustrate the advantages offered by the new class of integrators the experiments also address the comparative performance of split vs non split epirk methods and the question of improving efficiency by optimizing coefficients of the sepirk schemes it is shown that specific schemes can be custom built to improve computational efficiency for a particular problem

strong coupling expansion analysis of the false vacuum decay rate of the lattice phi4 model in 1 1 dimensions strong coupling expansion is performed for the lattice 4 model in 1 1 dimensions because the strong coupling limit itself is not solvable we employed numerical calculations so as to set up unperturbed eigensystems restricting the number of hilbert space bases we performed linked cluster expansion up to 11th order we carried out an alternative simulation by means of the density matrix renormalization group thereby we confirmed that our series expansion data with a convergence acceleration trick are in good agreement with the simulation result through the analytic continuation to the domain of negative biquadratic interaction we obtain the false vacuum decay rate contrary to common belief that the tunnelling phenomenon lies out of perturbative treatments our series expansion reproduces the instanton theory behaviour for a high potential barrier for a shallow barrier in contrast our result tells us that the relaxation is no longer described by an instanton but the decay rate acquires notable enhancement

operator based preconditioning of stiff hyperbolic systems we introduce an operator based scheme for preconditioning stiff components encountered in implicit methods for hyperbolic systems of pdes posed on regular grids the method is based on a directional splitting of the implicit operator followed by a characteristic decomposition of the resulting directional parts this approach allows for the solution of any number of characteristic components from the entire system to only the fastest stiffness inducing waves we apply the preconditioning method to stiff hyperbolic systems arising in magnetohydrodynamics and gas dynamics we then present numerical results showing that this preconditioning scheme works well on problems where the underlying stiffness results from the interaction of fast transient waves with slowly evolving dynamics scales well to large problem sizes and numbers of processors and allows for additional customization based on the specific problems under study

a fully implicit numerical method for single fluid resistive magnetohydrodynamics we present a nonlinearly implicit conservative numerical method for integration of the single fluid resistive mhd equations the method uses a high order spatial discretization that preserves the solenoidal property of the magnetic field the fully coupled pde system is solved implicitly in time providing for increased interaction between physical processes as well as additional stability over explicit time methods a high order adaptive time integration is employed which in many cases enables time steps ranging from one to two orders of magnitude larger than those constrained by the explicit cfl condition we apply the solution method to illustrative examples relevant to stiff magnetic fusion processes which challenge the efficiency of explicit methods we provide computational evidence showing that for such problems the method is comparably accurate with explicit time simulations while providing a significant runtime improvement due to its increased temporal stability

computing the action of the matrix exponential with an application to exponential integrators a new algorithm is developed for computing e ta b where a is an n times n matrix and b is n times n 0 with n 0 ll n the algorithm works for any a its computational cost is dominated by the formation of products of a with n times n 0 matrices and the only input parameter is a backward error tolerance the algorithm can return a single matrix e ta b or a sequence e t ka b on an equally spaced grid of points t k it uses the scaling part of the scaling and squaring method together with a truncated taylor series approximation to the exponential it determines the amount of scaling and the taylor degree using the recent analysis of al mohy and higham siam j matrix anal appl 31 2009 pp 970 989 which provides sharp truncation error bounds expressed in terms of the quantities a k 1 k for a few values of k where the norms are estimated using a matrix norm estimator shifting and balancing are used as preprocessing steps to reduce the cost of the algorithm numerical experiments show that the algorithm performs in a numerically stable fashion across a wide range of problems and analysis of rounding errors and of the conditioning of the problem provides theoretical support experimental comparisons with matlab codes based on krylov subspace chebyshev polynomial and laguerre polynomial methods show the new algorithm to be sometimes much superior in terms of computational cost and accuracy an important application of the algorithm is to exponential integrators for ordinary differential equations it is shown that the sums of the form sum k 0 p varphi k a u k that arise in exponential integrators where the varphi k are related to the exponential function can be expressed in terms of a single exponential of a matrix of dimension n p built by augmenting a with additional rows and columns and the algorithm of this paper can therefore be employed

nonlinear magnetohydrodynamics simulation using high order finite elements a conforming representation composed of 2d finite elements and finite fourier series is applied to 3d nonlinear non ideal magnetohydrodynamics using a semi implicit time advance the self adjoint semi implicit operator and variational approach to spatial discretization are synergistic and enable simulation in the extremely stiff conditions found in high temperature plasmas without sacrificing the geometric flexibility needed for modeling laboratory experiments growth rates for resistive tearing modes with experimentally relevant lundquist number are computed accurately with time steps that are large with respect to the global alfven time and moderate spatial resolution when the finite elements have basis functions of polynomial degree p two or larger an error diffusion method controls the generation of magnetic divergence error convergence studies show that this approach is effective for continuous basis functions with p 2 where the number of test functions for the divergence control terms is less than the number of degrees of freedom in the expansion for vector fields anisotropic thermal conduction at realistic ratios of parallel to perpendicular conductivity is computed accurately with p 3 without mesh alignment a simulation of tearingmode evolution for a shaped toroidal tokamak equilibrium demonstrates the effectiveness of the algorithm in nonlinear conditions and its results are used to verify the accuracy of the numerical anisotropic thermal conduction in 3d magnetic topologies