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

garden of eden configurations for 2 d cellular automata with rule 2460n an important problem in cellular automata theory is the reversibility of a cellular automaton which is related to the existence of garden of eden configurations in cellular automata in this paper we study new local rules for two dimensional cellular automata over the ternary field z 3 the set of integers modulo three with some of their important characteristics we obtain necessary and sufficient conditions for the existence of garden of eden configurations for two dimensional ternary cellular automata also by making use of the matrix representation of two dimensional cellular automata we provide an algorithm to obtain the number of garden of eden configurations for two dimensional cellular automata defined by rule 2460n we present an application of the reversible two dimensional ternary cellular automata to cryptography

time evolution with the density matrix renormalization group algorithm a generic implementation for strongly correlated electronic systems a detailed description of the time step targeting time evolution method within the density matrix renormalization group algorithm is presented the focus of this publication is on the implementation of the algorithm and its generic application the case of one site excitations within a hubbard model is analyzed as a test for the algorithm using open chains and two leg ladder geometries the accuracy of the procedure in the case of the recently discussed holon doublon photo excitations of mott insulators is also analyzed performance and parallelization issues are discussed in addition the full open source code is provided as supplementary material

a polynomial based iterative method for linear parabolic equations a new polynomial based method pbm is developed to integrate multi dimensional linear parabolic initial boundary value problems it is based on l sb2 approximations to f z 1 exp z z f 0 1 over ellipses in the complex plane using expansions of f in chebychev polynomials the calculation of the fourier coefficients requires numerical integration over only a single line segment in the complex plane whose length and orientation depend on the step size and the parabolic operator itself the simplicity with which these coefficients are obtained rests on special properties of the chebychev polynomials most of the work in pbm consists of matrix vector multiplications involving a matrix l which arises from the spatial discretization of the differential operator to be specific pbm integrates the semi discrete problem u sb t l t u b t u b in r sp n and l in r sp n times n and requires only a modest amount of storage a few vectors of order n due to the analyticity of f it has good convergence properties and compares favorably to other standard methods from the classes of hopscotch alternating direction implicit adi and locally one dimensional lod schemes as measured by the cpu times required on a single cpu of a cray x mp 24 it is also competitive with crank nicolson which we couple with two proven iterative solvers i recommend pbm on problems which require fourth order spatial accuracy problems whose solutions contain significant high frequency components and problems whose operators cannot be split conveniently in an adi or lod fashion for example problems with mixed derivatives

reduced order modeling for time dependent optimization problems with initial value controls this paper presents a new reduced order model rom hessian approximation for linear quadratic optimal control problems where the optimal control is the initial value such problems arise in parameter identification and data assimilation where the parameters to be identified appear in the initial data and also as subproblems in multiple shooting formulations of more general optimal control problems the new rom hessians can provide a substantially better approximation than the underlying basic rom approximation and thus can substantially reduce the computing time needed to solve these optimal control problems the computation of a hessian vector product requires the solution of the linearized state equation with initial value given by the vector to which the hessian is applied followed by the solution of the second order adjoint equation projection based roms of these two linear differential equations are used to generate the hessian approximation while the objective function and gradient are compute

symplectic and multisymplectic numerical methods for maxwell s equations in this paper we compare the behaviour of one symplectic and three multisymplectic methods for maxwell s equations in a simple medium this is a system of pdes with symplectic and multisymplectic structures we give a theoretical discussion of how some numerical methods preserve the discrete versions of the local and global conservation laws and verify this behaviour in numerical experiments we also show that these numerical methods preserve the divergence furthermore we extend the discussion on dispersion for multi symplectic methods applied to pdes with one spatial dimension to include anisotropy when applying multi symplectic methods to maxwell s equations in two spatial dimensions lastly we demonstrate how varying the courant friedrichs lewy cfl number can cause the multi symplectic methods in our comparison to behave differently which can be explained by the study of backward error analysis for pdes

physics like models of computation abstract reversible cellular automata are computer models that embody discrete analogues of the classical physics notions of space time locality and microscopic reversibility they are offered as a step towards models of computation that are closer to fundamental physics

fluctuation driven computing on number conserving cellular automata a number conserving cellular automaton ncca is a cellular automaton in which the states of cells are denoted by integers and the sum of all of the numbers in a configuration is conserved throughout its evolution nccas have been widely used to model physical systems that are ruled by conservation laws of mass or energy imai et al 13 showed that the local transition function of ncca can be effectively translated into the sum of a binary flow function over pairs of neighboring cells in this paper we explore the computability of nccas in which the pairwise number flows are performed at fully asynchronous timings despite the randomness that is associated with asynchronous transitions useful computation still can be accomplished efficiently in the cellular automata through the active exploitation of fluctuations 18 specifically certain numbers may flow randomly fluctuating between forward and backward directions in the cellular space as if they were subject to brownian motion because random fluctuations promise a powerful resource for searching through a computational state space the brownian like flow of the numbers allows for efficient embedding of logic circuits into our novel asynchronous ncca

emergence of universal global behavior from reversible local transitions in asynchronous systems reversible computing usually focuses on how to establish a valid equivalence between the global reversibility and local reversibility in computational systems hitherto the equivalence has been precisely developed in cellular automata combinational circuits and quantum computers but it implicitly assumes that the underlying systems are synchronously timed alternative systems include the delay insensitive di circuits and asynchronous cellular automata acas where the local operations of each component cell may be executed independently at random times despite the randomness associated with asynchronous timings equivalence between the global and local reversibility can be simply achieved in both di circuits morita 2001 and acas lee et al 2003 provided that their local operations transitions are thoroughly serialized the complete exclusion of concurrency in local behavior however will profoundly depress the parallel processing efficiency of di circuits as well as the intrinsic massive parallelism of acas this paper aims at exploring what kind of complex global behavior may arise from the concurrent operations that are invertible at local level to this end we show that di circuits composed of reversible elements can actually exhibit universal input and output behavior with the universality emerging from the concurrency in reversible local operations likewise by further embedding all circuits into the cellular space the emergence of universal global transitions from reversible local transitions can be exactly identified in an aca which due to the bijectivity of its local function has significantly lower complexity as compared to other models

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

krylov subspace methods for solving large unsymmetric linear systems some algorithms based upon a projection process onto the krylov subspace k sub m span r sub 0 ar sub 0 a sup m 1 r sub 0 are developed generalizing the method of conjugate gradients to unsymmetric systems these methods are extensions of arnoldi s algorithm for solving eigenvalue problems the convergence is analyzed in terms of the distance of the solution to the subspace k sub m and some error bounds are established showing in particular a similarity with the conjugate gradient method for symmetric matrices when the eigenvalues are real several numerical experiments are described and discussed

spectral methods in time for parabolic problems a pseudospectral explicit scheme for solving linear periodic parabolic problems is described it has infinite accuracy both in time and in space the high accuracy is achieved while the time resolution parameter m m o 1 delta t for time marching algorithm and the space resolution parameter n n o 1 delta x must satisfy m o n 1 varepsilon varepsilon 0 compared to the common stability condition m o n 2 which must be satisfied in any explicit finite order time algorithm

a restarted krylov subspace method for the evaluation of matrix functions we show how the arnoldi algorithm for approximating a function of a matrix times a vector can be restarted in a manner analogous to restarted krylov subspace methods for solving linear systems of equations the resulting restarted algorithm reduces to other known algorithms for the reciprocal and the exponential functions we further show that the restarted algorithm inherits the superlinear convergence property of its unrestarted counterpart for entire functions and present the results of numerical experiments

communication complexity and intrinsic universality in cellular automata the notions of universality and completeness are central in the theories of computation and computational complexity however proving lower bounds and necessary conditions remains hard in most cases in this article we introduce necessary conditions for a cellular automaton to be universal according to a precise notion of simulation related both to the dynamics of cellular automata and to their computational power this notion of simulation relies on simple operations of space time rescaling and it is intrinsic to the model of cellular automata intrinsic universality the derived notion is stronger than turing universality but more uniform and easier to define and study our approach builds upon the notion of communication complexity which was primarily designed to study parallel programs and thus is as we show in this article particulary well suited to the study of cellular automata it allowed us to show by studying natural problems on the dynamics of cellular automata that several classes of cellular automata as well as many natural elementary examples were not intrinsically universal

classifying circular cellular automata we introduce a hierarchy of linear cellular automata based on their limiting behavior on spatially periodic configurations we show that it is undecidable to which class in the hierarchy a cellular automaton belongs in particular it is undecidable whether all spatially periodic configurations evolve to a fixed point furthermore there is no computable bound on the period lengths of these configurations our arguments are based on a non standard simulation of turing machines on circular cellular automata

rate of convergence of local linearization schemes for initial value problems there is a large variety of local linearization ll schemes for the numerical integration of initial value problems which differ with respect to the algorithm that is used in the numerical implementation of the local linear discretization however in contrast with the ll discretization the order of convergence of the ll schemes have not been studied so far in this paper a general result about that matter is given in addition a brief survey of the main implementations of the ll method is also presented

analyzing the positivity preservation of numerical methods for the liouville von neumann equation abstract the density matrix is a widely used tool in quantum mechanics in order to determine its evolution with respect to time the liouville von neumann equation must be solved however analytic solutions of this differential equation exist only for simple cases additionally if the equation is coupled to maxwell s equations to model light matter interaction the resulting equation set the maxwell bloch or maxwell liouville von neumann mln equations becomes nonlinear in these advanced cases numerical methods are required since the density matrix has certain mathematical properties the numerical methods applied should be designed to preserve those properties we establish the criterion that only methods that have a completely positive trace preserving cptp update map can be used in long term simulations subsequently we assess the three most widely used methods the matrix exponential me method the runge kutta rk method and the predictor corrector pc approach whether they provide this feature and demonstrate that only the update step of the matrix exponential method is a cptp map

adaptive residual subsampling methods for radial basis function interpolation and collocation problems we construct a new adaptive algorithm for radial basis functions rbfs method applied to interpolation boundary value and initial boundary value problems with localized features nodes can be added and removed based on residuals evaluated at a finer point set we also adapt the shape parameters of rbfs based on the node spacings to prevent the growth of the conditioning of the interpolation matrix the performance of the method is shown in numerical examples in one and two space dimensions with nontrivial domains

general design of reversible sequential machines based on reversible logic elements this paper proposes a scheme for decomposing reversible sequential machines rsms into simple reversible logic elements rles this scheme is generalized from the previous constructions of reversible turing machines rtms using various rles lee et al 2012 in which each rtm is composed by a uniform circuit of identical rsms associated with two states each rsm s operation in the circuit however is subject to strict conditions that prevent direct extension of their constructions to more general rsms our novel scheme to the contrary imposes no condition on the operations of an rsm which may take any arbitrary number of states moreover each rsm s construction can work asynchronously without a clock signal to synchronize all rles comprising the construction

design of 1 tape 2 symbol reversible turing machines based on reversible logic elements this paper proposes a novel scheme for constructing reversible turing machines rtms via various reversible logic elements a reversible logic element is similar to a conventional reversible logic gate except that the element also carries a 1 bit memory that can store binary states the usage of internal states allows much more efficient and straightforward constructions of rtms based on reversible elements than based on reversible logic gates in particular a remarkable feature of our constructions is that they can operate asynchronously whereby the clock signal which is indispensable for synchronizing all elements as in a reversible logic circuit can possibly be removed from the construction

a minimal lattice realization for multivariable systems abstract an algorithm for constructing a minimal lattice realization of a multivariable stable linear discrete time system is proposed the algorithm consists of four steps i realize the objective system in an input normal hessenberg form ii construct an lbr system associated with the objective system iii construct a lattice realization of the lbr system from its input normal hessenberg realization iv transform the lattice realization of the lbr system into one of the objective system

characterization of random fluctuation based computation in cellular automata a brownian cellular automaton bca is an asynchronous cellular automaton aca in which the timing of the local cell state transitions approaches a poisson point process and the global transitions between cell configurations can be formulated as an irreducible and reversible markov chain bcas exploit fluctuations to find solutions of a computation via a stochastic search process this characteristic has been shown to contribute to a decreased complexity of bcas lee and peper 2008 as compared to non brownian acas with equivalent functionality this paper proposes a computationally universal bca that is based on a conventional aca rather than the previously employed block cellular automata which have transition rules that update each cell simultaneously with its neighboring cells the proposed allows for a more natural update model in which no need arises to locally synchronize the cells in a neighborhood the resulting bca is significantly less complex than equivalent non block non brownian acas which demonstrates that random fluctuations can serve as a powerful resource for efficient computation on cellular automata

matrices moments and rational quadrature abstract many problems in science and engineering require the evaluation of functionals of the form f u a u t f a u where a is a large symmetric matrix u a vector and f a nonlinear function a popular and fairly inexpensive approach to determining upper and lower bounds for such functionals is based on first carrying out a few steps of the lanczos procedure applied to a with initial vector u and then evaluating pairs of gauss and gauss radau quadrature rules associated with the tridiagonal matrix determined by the lanczos procedure the present paper extends this approach to allow the use of rational gauss quadrature rules

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

invertible behavior in elementary cellular automata with memory elementary cellular automata ecas have been studied for their ability to generate complex global behavior despite their simplicity one variation of ecas is obtained by adding memory to each cell in a neighborhood this process generates a provisional configuration in which the application of an evolution rule establishes the dynamics of the system this version is known as an eca with memory ecam most previous work on ecams analyzed the complex behavior taking chaotic ecas however the present paper investigates reversible ecams as obtained from reversible and permutative ecas these ecas have at least one ancestor for every configuration thus the correct permutation of states may specify the memory function to obtain reversible ecams for permutative ecas which are often irreversible we demonstrate that the use of a quiescent state and the correct manipulation of de bruijn blocks produce reversible ecams

paraexp a parallel integrator for linear initial value problems a novel parallel algorithm for the integration of linear initial value problems is proposed this algorithm is based on the simple observation that homogeneous problems can typically be integrated much faster than inhomogeneous problems an overlapping time domain decomposition is utilized to obtain decoupled inhomogeneous and homogeneous subproblems and a near optimal krylov method is used for the fast exponential integration of the homogeneous subproblems we present an error analysis and discuss the parallel scaling of our algorithm the efficiency of this approach is demonstrated with numerical examples

a numerical method for the computation of the lyapunov exponents of nonlinear ordinary differential equations in this paper an alternative method to compute the lyapunov exponents of dynamical systems described by ordinary differential equations odes is introduced the lyapunov exponents are computed in terms of the solutions of two piecewise linear odes that approximate respectively the solutions of the original ode and its associated variational equation this approach is strongly connected with the local linearization ll method for odes and its major advantage is that these piecewise linear odes might be exactly integrated in a non simultaneous way the performance of the method is illustrated with a numerical example

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

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

stopping criteria for rational matrix functions of hermitian and symmetric matrices building upon earlier work by golub meurant strakos and tich we derive new a posteriori error bounds for krylov subspace approximations to f a b the action of a function f of a real symmetric or complex hermitian matrix a on a vector b to this purpose we assume that a rational function in partial fraction expansion form is used to approximate f and the krylov subspace approximations are obtained as linear combinations of galerkin approximations to the individual terms in the partial fraction expansion our error estimates come at very low computational cost in certain important special situations they can be shown to actually be lower bounds of the error our numerical results include experiments with the matrix exponential as used in exponential integrators and with the matrix sign function as used in lattice quantum chromodynamics simulations and demonstrate the accuracy of the estimates the use of our error estimates within acceleration procedures is also discussed

a convergence analysis for nonsymmetric lanczos algorithms a convergence analysis for the nonsymmetric lanczos algorithm is presented by using a tridiagonal structure of the algorithm some identities concerning ritz values and ritz vectors are established and used to derive approximation bounds in particular the analysis implies the classical results for the symmetric lanczos algorithm

an exponential integrator for advection dominated reactive transport in heterogeneous porous media we present an exponential time integrator in conjunction with a finite volume discretisation in space for simulating transport by advection and diffusion including chemical reactions in highly heterogeneous porous media representative of geological reservoirs these numerical integrators are based on the variation of constants solution and solving the linear system exactly this is at the expense of computing the exponential of the stiff matrix comprising the finite volume discretisation using real leja points or a krylov subspace technique compared to standard finite difference based time integrators we observe for a variety of example applications that numerical solutions with exponential methods are generally more accurate and require less computational cost they hence comprise an efficient and accurate method for simulating non linear advection dominated transport in geological formations

error analysis of krylov methods in a nutshell error and residual bounds for the matrix iteration methods bicg qmr fom and gmres are derived in a simple and unified way

on sigma lossless transfer functions and related questions abstract this paper is concerned with a systematic approach to the properties of lossless rational transfer functions in the discrete as well as in the continuous time case as a result a unifying framework is revealed where several known results fit naturally special attention is given to the embedding problem of the lyapunov equation in view of its direct application to generalized levinson algorithms

linear estimation of continuous discrete linear state space models with multiplicative noise this paper deals with the estimation of the state variable of continuous discrete linear state space models with multiplicative noise specifically the optimal minimum variance linear filter for that class of models is constructed moreover the solutions of the differential equations that describe the evolution of the two first conditional moments between observations are obtained and an algorithm for their numerical computation is also given the performance of the linear filter is illustrated by means of numerical experiments

on semiiterative methods generated by faber polynomials consider a nonsingular system of linear algebraic equationsa x b or in fixed point formx tx c where the eigenvalues oft are contained in some compact subset of the complex plane using faber polynomials iteration methods of the form y m mu m 0 left c ty m 1 right mu m 1 y m 1 ldots mu m k y m k m 0 m j 1 for allm 1 are constructed which include the chebyshev iterative method as well as the stationaryk step methods when is a rectangle we also develop four step methods which yield fast convergence in cases when the speed of convergence of the optimized chebyshev method is poor

exponential rosenbrock type methods we introduce a new class of exponential integrators for the numerical integration of large scale systems of stiff differential equations these so called rosenbrock type methods linearize the flow in each time step and make use of the matrix exponential and related functions of the jacobian in contrast to standard integrators the methods are fully explicit and do not require the numerical solution of linear systems we analyze the convergence properties of these integrators in a semigroup framework of semilinear evolution equations in banach spaces in particular we derive an abstract stability and convergence result for variable step sizes this analysis further provides the required order conditions and thus allows us to construct pairs of embedded methods we present a third order method with two stages and a fourth order method with three stages respectively the application of the required matrix functions to vectors are computed by krylov subspace approximations we briefly discuss these implementation issues and we give numerical examples that demonstrate the efficiency of the new integrators

grid free adaptive semi lagrangian advection using radial basis functions this paper proposes a new grid free adaptive advection scheme the resulting algorithm is a combination of the semi lagrangian method slm and the grid free radial basis function interpolation rbf the set of scattered interpolation nodes is subject to dynamic changes at run time based on a posteriori local error estimates a self adaptive local refinement and coarsening of the nodes serves to obtain enhanced accuracy at reasonable computational costs due to well known features of slm and rbf the method is guaranteed to be stable it has good approximation behaviour and it works for arbitrary space dimension numerical examples in two dimensions illustrate the performance of the method in comparison with existing grid based advection schemes

state space formulae for discrete time h optimization abstract state space formulae and proofs are given for ail the important steps in discrete time h 8 optimization the steps closely follow the algorithms for continuous time systems but there are some technically involved differences in the detail that make their derivation non trivial

meshfree exponential integrators for the numerical solution of time dependent partial differential equations a class of meshfree exponential integrators is proposed these methods are of particular interest in situations where the solution of the differential equation concentrates on a small part of the computational domain which may vary in time for the space discretization radial basis functions with compact support are suggested the reasons for this choice are the stability and robustness of the resulting interpolation procedure the time integration is performed with an exponential rosenbrock method the required matrix functions are computed by newton interpolation based on leja points the proposed integrators are fully adaptive in space and time numerical examples that illustrate the robustness and the good stability properties of the method are included

graph spectral image smoothing using the heat kernel a new method for smoothing both gray scale and color images is presented that relies on the heat diffusion equation on a graph we represent the image pixel lattice using a weighted undirected graph the edge weights of the graph are determined by the gaussian weighted distances between local neighboring windows we then compute the associated laplacian matrix the degree matrix minus the adjacency matrix anisotropic diffusion across this weighted graph structure with time is captured by the heat equation and the solution i e the heat kernel is found by exponentiating the laplacian eigensystem with time image smoothing is accomplished by convolving the heat kernel with the image and its numerical implementation is realized by using the krylov subspace technique the method has the effect of smoothing within regions but does not blur region boundaries we also demonstrate the relationship between our method standard diffusion based pdes fourier domain signal processing and spectral clustering experiments and comparisons on standard images illustrate the effectiveness of the method