stallings foldings and subgroups of free groups we re cast in a more combinatorial and computational form the topological approach of j stallings to the study of subgroups of free groups

analysis of the parareal time parallel time integration method the parareal algorithm is a method to solve time dependent problems parallel in time it approximates parts of the solution later in time simultaneously to parts of the solution earlier in time in this paper the relation of the parareal algorithm to space time multigrid and multiple shooting methods is first briefly discussed the focus of the paper is on new convergence results that show superlinear convergence of the algorithm when used on bounded time intervals and linear convergence for unbounded intervals

mixed boundary integral methods for helmholtz transmission problems in this paper we propose a hybrid between direct and indirect boundary integral methods to solve a transmission problem for the helmholtz equation in lipschitz and smooth domains we present an exhaustive abstract study of the numerical approximation of the resulting system of boundary integral equations by means of galerkin methods some particular examples of convergent schemes in the smooth case in two dimensions are given finally we extend the results to a thermal scattering problem in a half plane with several obstacles and provide numerical results that illustrate the accuracy of our methods depending on the regularity of the interface

the probability distribution of word maps on finite groups abstract word maps provide a wealth of information about finite groups we examine the connection between the probability distribution induced by a word map and the underlying structure of a finite group we show that a finite group is nilpotent if and only if every surjective word map has fibers of uniform size moreover we show that probability distributions themselves are sufficient to identify nilpotent groups and these same distributions can be used to determine abelian groups up to isomorphism in addition we answer a question of amit and vishne

data sparse approximation to the operator valued functions of elliptic operator llin previous papers the arithmetic of hierarchical matrices has been described which allows us to compute the inverse for instance of finite element stiffness matrices discretising an elliptic operator l the required computing time is up to logarithmic factors linear in the dimension of the matrix in particular this technique can be used for the computation of the discrete analogue of a resolvent zi l 1 z c in the present paper we consider various operator functions the operator exponential e tl negative fractional powers l the cosine operator function cos t l l k and finally the solution operator of the lyapunov equation using the dunford cauchy representation we get integrals which can be discretised by a quadrature formula which involves the resolvents z k i l 1 mentioned above we give error estimates which are partly exponentially partly polynomially decreasing

expansion of random graphs new proofs new results we present a new approach to showing that random graphs are nearly optimal expanders this approach is based on recent deep results in combinatorial group theory it applies to both regular and irregular random graphs let gamma be a random d regular graph on n vertices and let lambda be the largest absolute value of a non trivial eigenvalue of its adjacency matrix it was conjectured by alon combinatorica 6 2 83 96 1986 that a random d regular graph is almost ramanujan in the following sense for every varepsilon 0 lambda 0 a a s lambda rho varepsilon where rho is the spectral radius of the universal cover of omega when omega is regular we get a bound of rho 0 84 and for an arbitrary omega we prove a nearly optimal upper bound of sqrt 3 rho this is a substantial improvement upon all known results by friedman linial puder lubetzky sudakov vu and addario berry griffiths

exponentially convergent algorithms for the operator exponential with applications to inhomogeneous problems in banach spaces new exponentially convergent algorithms for the operator exponential generated by a strongly positive operator a in a banach space x are proposed these algorithms are based on representations by a dunford cauchy integral along paths enveloping the spectrum of a combined with a proper quadrature involving a short sum of resolvents where the choice of the integration path dramatically affects desired features of the algorithms a parabola and a hyperbola are analyzed as the integration paths and scales of estimates of dependence on the smoothness of initial data i e of the initial vector and of the inhomogeneous right hand side are obtained one of the algorithms possesses an exponential convergence rate for the operator exponential e at for all t ge 0 including the initial point this allows one to construct an exponentially convergent algorithm for inhomogeneous initial value problems the algorithm is parallelizable it turns out that the resolvent must be modified in order to get numerically stable algorithms near the initial point the efficiency of the proposed method is demonstrated by numerical examples

the exponentially convergent trapezoidal rule it is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line the mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity it is linked with computational methods all across scientific computing including algorithms related to inverse laplace transforms special functions complex analysis rational approximation integral equations and the computation of functions and eigenvalues of matrices and operators

matrix group integrals surfaces and mapping class groups i u n since the 1970 s physicists and mathematicians who study random matrices in the gue or goe models are aware of intriguing connections between integrals of such random matrices and enumeration of graphs on surfaces we establish a new aspect of this theory for random matrices sampled from the group u n of unitary matrices more concretely we study measures induced by free words on u n let fr be the free group on r generators to sample a random element from u n according to the measure induced by w fr one substitutes the r letters in w by r independent haar random elements from u n the main theme of this paper is that every moment of this measure is determined by families of pairs f where is an orientable surface with boundary and f is a map from to the bouquet ofr circles which sends the boundary components of to powers of w a crucial role is then played by euler characteristics of subgroups of the mapping class group of as corollaries we obtain asymptotic bounds on the moments we show that the measure on u n bears information about the number of solutions to the equation u1 v1 ug vg w in the free group and deduce that one can hear the stable commutator length of a word through its unitary word measures

alternating splitting waveform relaxation method and its successive overrelaxation acceleration for the large sparse implicit linear initial value problem we present a block successive overrelaxation scheme for the alternating direction implicit waveform relaxation method to further accelerate its convergence speed and discuss the convergence property of the resulting iteration method in detail numerical implementations about several non hermitian implicit linear initial value problems show that the alternating direction implicit waveform relaxation method is very effective and the block successive overrelaxation technique really accelerates its convergence speed

time discretization via laplace transformation of an integro differential equation of parabolic type we consider the discretization in time of an inhomogeneous parabolic integro differential equation with a memory term of convolution type in a banach space setting the method is based on representing the solution as an integral along a smooth curve in the complex plane which is evaluated to high accuracy by quadrature using the approach in recent work of lopez fernandez and palencia this reduces the problem to a finite set of elliptic equations with complex coefficients which may be solved in parallel the method is combined with finite element discretization in the spatial variables to yield a fully discrete method the paper is a further development of earlier work by the authors which on the one hand treated purely parabolic equations and on the other an evolution equation with a positive type memory term

an algorithm with polylog parallel complexity for solving parabolic partial differential equations the standard numerical algorithms for solving parabolic partial differential equations are inherently sequential in the time direction this paper describes an algorithm for the time accurate solution of certain classes of parabolic partial differential equations that can be parallelized in both time and space it has a serial complexity that is proportional to the serial complexities of the best known algorithms the algorithm is a variant of the multigrid waveform relaxation method where the scalar ordinary differential equations that make up the kernel of computation are solved using a cyclic reduction type algorithm experimental results obtained on a massively parallel multiprocessor are presented

characters of symmetric groups sharp bounds and applications we provide new estimates on character values of symmetric groups which hold for all characters and which are in some sense best possible it follows from our general bound that if a permutation s n has at most n o 1 cycles of length m then 1 1 m o 1 for all irreducible characters of s n this is a far reaching generalization of a result of fomin and lulov

spectral theory for linearized p laplace equations abstract we continue and completely set up the spectral theory initiated in castorina et al 2009 5 for the linearized operator arising from p u f u 0 we establish existence and variational characterization of all the eigenvalues and by a weak harnack inequality we deduce holder continuity for the corresponding eigenfunctions this regularity being sharp the morse index of a positive solution can be now defined in the classical way and we will illustrate some qualitative consequences one should expect to deduce from such information in particular we show that zero morse index or more generally non degenerate solutions on the annulus are radial

multigrid methods for implicit runge kutta and boundary value method discretizations of parabolic pdes advanced time discretization schemes for stiff systems of ordinary differential equations odes such as implicit runge kutta and boundary value methods have many appealing properties however the resulting systems of equations can be quite large and expensive to solve many techniques exploiting the structure of these systems have been developed for general odes for spatial discretizations of time dependent partial differential equations pdes these techniques are in general not sufficient and also the structure arising from spatial discretization has to be taken into consideration we show here that for time dependent parabolic problems this can be done by multigrid methods as in the stationary elliptic case the key to this approach is the use of a smoother that updates several unknowns at a spatial grid point simultaneously the overall cost is essentially proportional to the cost of integrating a scalar ode for each grid point combination of the multigrid principle with both time stepping and waveform relaxation techniques is described together with a convergence analysis numerical results are presented for the isotropic heat equation and a general diffusion equation with variable coefficients

a spectral order method for inverting sectorial laplace transforms laplace transforms which admit a holomorphic extension to some sector strictly containing the right half plane and exhibiting a potential behavior are considered a spectral order parallelizable method for their numerical inversion is proposed the method takes into account the available information about the errors arising in the evaluations several numerical illustrations are provided

interweaving pfasst and parallel multigrid the parallel full approximation scheme in space and time pfasst introduced by emmett and minion in 2012 is an iterative strategy for the temporal parallelization of odes and discretized pdes as the name suggests pfasst is similar in spirit to a space time full approximation scheme multigrid method performed over multiple time steps in parallel however since the original focus of pfasst was on the performance of the method in terms of time parallelism the solution of any spatial system arising from the use of implicit or semi implicit temporal methods within pfasst have simply been assumed to be solved to some desired accuracy completely at each substep and each iteration by some unspecified procedure it hence is natural to investigate how iterative solvers in the spatial dimensions can be interwoven with the pfasst iterations and whether this strategy leads to a more efficient overall approach this paper presents an initial investigation on the relative performance of different strategies for cou

a fast algorithm for parabolic pde based inverse problems based on laplace transforms and flexible krylov solvers we consider the problem of estimating parameters in large scale weakly nonlinear inverse problems for which the underlying governing equations is a linear time dependent parabolic partial differential equation a major challenge in solving these inverse problems using newton type methods is the computational cost associated with solving the forward problem and with repeated construction of the jacobian which represents the sensitivity of the measurements to the unknown parameters forming the jacobian can be prohibitively expensive because it requires repeated solutions of the forward and adjoint time dependent parabolic partial differential equations corresponding to multiple sources and receivers we propose an efficient method based on a laplace transform based exponential time integrator combined with a flexible krylov subspace approach to solve the resulting shifted systems of equations efficiently our proposed solver speeds up the computation of the forward and adjoint problems thus yielding significant speedup in total inversion time we consider an application from transient hydraulic tomography tht which is an imaging technique to estimate hydraulic parameters related to the subsurface from pressure measurements obtained by a series of pumping tests the algorithms discussed are applied to a synthetic example taken from tht to demonstrate the resulting computational gains of this proposed method

convergence of the parallel chaotic waveform relaxation method for stiff systems in this paper we establish several algorithms for parallel chaotic waveform relaxation methods for solving linear ordinary differential systems based on some given models under some different assumptions on the coefficient matrix a and its multisplittings we obtain corresponding sufficient conditions of convergence of the algorithms also a discussion on convergence speed comparison of synchronous and asynchronous algorithms is given

a diagonalization based parareal algorithm for dissipative and wave propagation problems we present a new parareal algorithm based on a diagonalization technique proposed recently the algorithm uses a single implicit runge kutta method with the same small step size for both the mat

a schwarz waveform moving mesh method an r refinement moving mesh method is considered for solving time dependent partial differential equations pdes the resulting coupled system consisting of the physical pde and a moving mesh pde is solved by a schwarz waveform relaxation method in particular the computational space time domain is decomposed into overlapping subdomains and the solution obtained by iteratively solving the system of pdes on each subdomain dirichlet boundary conditions are used to pass solution information between neighboring regions the efficacy of this approach is demonstrated for some model problems for problems where the solutions evolve on disparate time scales in different regions of the spatial domain this approach demonstrates the significant savings in computational time and effort which are possible

convergence analysis for three parareal solvers we analyze in this paper the convergence properties of the parareal algorithm for the symmetric positive definite problem mathbf u a mathbf u g the coarse propagator mathcal g is fixed to the backward euler method and three time integrators are chosen for the mathcal f propagator the trapezoidal rule the third order diagonal implicit runge kutta rk dirk method and the fourth order gauss rk method it is well known that the parareal euler algorithm using the backward euler method for mathcal f and mathcal g converges rapidly but less is known when one uses for mathcal f the trapezoidal rule or the fourth order gauss rk method especially when the mesh ratio j delta t delta t is small we show that for a specified lambda max the maximal eigenvalue of a or its upper bound there exists some critical j rm cri such that the parareal solvers derived from these three choices of mathcal f converge as fast as parareal euler provided j geq j rm cri

optimized schwarz waveform relaxation for advection reaction diffusion equations in two dimensions optimized schwarz waveform relaxation methods have been developed over the last decade for the parallel solution of evolution problems they are based on a decomposition in space and an iteration where only subproblems in space time need to be solved each subproblem can be simulated using an adapted numerical method for example with local time stepping or one can even use a different model in different subdomains which makes these methods very suitable also from a modeling point of view for rapid convergence however it is important to use effective transmission conditions between the space time subdomains and for best performance these transmission conditions need to take the physics of the underlying evolution problem into account the optimization of these transmission conditions leads to mathematically hard best approximation problems of homographic functions we study in this paper in detail the best approximation problem for the case of linear advection reaction diffusion equations in two spatial dimensions we prove comprehensively best approximation results for transmission conditions of robin and ventcel higher order type which can also be used in the various limits for example for the heat equation since we include in our analysis a positive low frequency limiter both in space and time we give for each case closed form asymptotic values for the parameters which can directly be used in implementations of these algorithms and which guarantee asymptotically best performance of the iterative methods we finally show extensive numerical experiments including cases not covered by our analysis for example decompositions with cross points we use q1 finite element discretizations in space and forward and backward euler discretizations in time other discretization could also have been considered since all our analysis is at the continuous level and in all cases we measure performance corresponding to our analysis

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

a frequency domain parallel method for the numerical approximation of parabolic problems a naturally parallelizable numerical method is introduced and analyzed for parabolic partial differential equations instead of solving the problem in the space time formulation we propose to solve it in the space frequency formulation existence and uniqueness are given error estimates are given for each single frequency also given is a full estimate for errors coming from truncation numerical quadrature rule in fourier inversion and finite element discretization in the space frequency domain a numerical experiment on a parallel mimd machine is also given

h matrix approximation for the operator exponential with applications we develop a data sparse and accurate approximation to parabolic solution operators in the case of a rather general elliptic part given by a strongly p positive operator 4

a parallel method for time discretization of parabolic problems based on contour integral representation and quadrature we treat the time discretization of an initial value problem for a homogeneous abstract parabolic equation by first using a representation of the solution as an integral along the boundary of a sector in the right half of the complex plane then transforming this into a real integral on the finite interval and finally applying a standard quadrature formula to this integral the method requires the solution of a finite set of elliptic problems with complex coefficients which are independent and may therefore be done in parallel the method is combined with spatial discretization by finite elements

multigrid waveform relaxation on spatial finite element meshes the discrete time case the efficiency of numerically solving time dependent partial differential equations on parallel computers can be greatly improved by computing the solution on many time levels simultaneously the theoretical properties of one such method namely the discrete time multigrid waveform relaxation method are investigated for systems of ordinary differential equations obtained by spatial finite element discretisation of linear parabolic initial boundary value problems the results are compared to the corresponding continuous time results the theory is illustrated for a one dimensional and a two dimensional model problem and checked against results obtained by numerical experiments

convergence analysis of a periodic like waveform relaxation method for initial value problems via the diagonalization technique we present in this paper a time parallel algorithm for dot u f t u with initial value u 0 u 0 by using the waveform relaxation wr technique and the diagonalization technique with a suitable parameter alpha the wr technique generates a functional sequence u k t via the dynamic iterations dot u k f t u k u k 0 alpha u k t alpha u k 1 t u 0 and at convergence we get u infty t u t each wr iterate represents a periodic like differential equation which is very suitable for applying the diagonalization technique yielding direct parallel in time computation the parameter alpha controls both the roundoff error arising from the diagonalization procedure and the convergence factor of the wr iterations and we perform a detailed analysis for the influence of the parameter alpha on the method we show that the roundoff error is proportional to epsilon 2n 1 max alpha 2 alpha 2 n t vardelta t and epsilon is the machine precision and the convergence factor can be bounded by alpha e tl 1 alpha e tl where l ge 0 is the one sided lipschitz constant of f we also perform a convergence analysis at the discrete level and the effect of temporal discretizations is explored our analysis includes the heat and wave equations as special cases numerical results are given to support our findings

on sor waveform relaxation methods waveform relaxation is a numerical method for solving large scale systems of ordinary differential equations on parallel computers it differs from standard iterative methods in that it computes the solution on many time levels or along a continuous time interval simultaneously this paper deals with the acceleration of the standard waveform relaxation method by successive overrelaxation sor techniques in particular different sor acceleration schemes based on multiplication with a scalar parameter or convolution with a time dependent function are described and theoretically analyzed the theory is applied to a one dimensional and two dimensional model problem and checked against results obtained by numerical experiments

analysis of two parareal algorithms for time periodic problems the parareal algorithm which permits us to solve evolution problems in a time parallel fashion has created a lot of attention over the past decade the algorithm has its roots in the multiple shooting method for boundary value problems which in the parareal algorithm is applied to initial value problems with a particular coarse approximation of the jacobian matrix it is therefore of interest to formulate parareal type algorithms for time periodic problems which also couple the end of the time interval with the beginning and to analyze their performance in this context we present and analyze two parareal algorithms for time periodic problems one with a periodic coarse problem and one with a nonperiodic coarse problem an interesting advantage of the algorithm with the nonperiodic coarse problem is that no time periodic problems need to be solved during the iteration since on the time subdomains the problems are not time periodic either we prove for both linear and nonlinear problems convergence

word maps conjugacy classes and a noncommutative waring type theorem let w w x xd 1 be a nontrivial group word we show that if g is a sufficiently large finite simple group then every element g e g can be expressed as a product of three values of w in g this improves many known results for powers commutators as well as a theorem on general words obtained in 19 the proof relies on probabilistic ideas algebraic geometry and character theory our methods which apply the zeta function g s x eirrg xo s giye rise to various additional results of independent interest including applications to conjectures of ore and thompson

a parallel method for backward parabolic problems based on the laplace transformation a parallel method for time discretization of backward parabolic problems is proposed the problem is reformulated to a set of helmholtz type problems with a parameter on a suitably chosen contour in the complex plane after solving the resulting elliptic equations which can be solved in parallel we obtain a regularized solution with high frequency terms cut off by the inverse laplace transforms without requiring the knowledge of the eigenfunctions of the differential operator since the regularized solution is obtained without artificial perturbation and high frequency components of the noise are suppressed the quality of the solution is improved significantly compared to those obtained by other methods two different numerical inversions of laplace transforms with an arbitrary high order of accuracy and spectral accuracy respectively are used error estimates and numerical examples are presented

criteria for equidistribution of solutions of word equations on sl 2 abstract we study equidistribution of solutions of word equations of the form w x y g in the family of finite groups sl 2 q we provide criteria for equidistribution in terms of the trace polynomial of w this allows us to get an explicit description of certain classes of words possessing the equidistribution property and show that this property is generic within these classes

residual restarting and richardson iteration for the matrix exponential a well known problem in computing some matrix functions iteratively is the lack of a clear commonly accepted residual notion an important matrix function for which this is the case is the matrix exponential suppose the matrix exponential of a given matrix times a given vector has to be computed we develop the approach of druskin greenbaum and knizhnerman siam j sci comput 19 1998 pp 38 54 and interpret the sought after vector as the value of a vector function satisfying the linear system of ordinary differential equations odes whose coefficients form the given matrix the residual is then defined with respect to the initial value problem for this ode system the residual introduced in this way can be seen as a backward error we show how the residual can be computed efficiently within several iterative methods for the matrix exponential this resolves the question of reliable stopping criteria for these methods further we show that the residual concept can be used to construct new residual based iterative methods in particular a variant of the richardson method for the new residual appears to provide an efficient way to restart krylov subspace methods for evaluating the matrix exponential

measure preserving words are primitive a word w 2 fk the free group on k generators is called primitive if it belongs to some basis of fk associated with w and a finite group g is the word map w g g g defined on the direct product of k copies of g we call w measure preserving if given uniform measure on g g the image of this word map induces uniform measure on g for every finite group g it is easy to see that every primitive word is measure preserving and several authors have conjectured that the two properties are in fact equivalent here we prove this conjecture the main ingredients of the proof include random coverings of stallings graphs algebraic extensions of free groups and mobius inversions our methods yield the stronger result that a subgroup of fk is measure preserving iff it is a free factor as an interesting corollary of this result we resolve a question on the profinite topology of free groups and show that the primitive elements of fk form a closed set in this topology

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

an exponentially convergent algorithm for nonlinear differential equations in banach spaces an exponentially convergent approximation to the solution of a nonlinear first order differential equation with an operator coefficient in banach space is proposed the algorithm is based on an equivalent volterra integral equation including the operator exponential generated by the operator coefficient the operator exponential is represented by a dunford cauchy integral along a hyperbola enveloping the spectrum of the operator coefficient and then the integrals involved are approximated using the chebyshev interpolation and an appropriate sinc quadrature numerical examples are given which confirm theoretical results

c1 α local regularity of weak solutions of degenerate elliptic equations abstract it is demonstrated the local c 1 alpha nature of weak solutions of elliptic equations of the type 1 1 in the introduction under the degeneracy or singularity assumptions a sub 1 a sub 3

multigrid waveform relaxation on spatial finite element meshes the continuous time case the waveform relaxation method and its multigrid acceleration are studied as solution procedures for the system of ordinary differential equations obtained by finite element discretisation of a linear parabolic initial boundary value problem the convergence properties of the continuous time algorithm are theoretically investigated on finite length and infinite length time intervals in addition quantitative convergence estimates and numerical results are presented for one dimensional and two dimensional model problems

a space time multigrid method for parabolic partial differential equations we consider the solution of parabolic partial differential equations pdes in standard time stepping techniques multigrid can be used as an iterative solver for the elliptic equations arising at each discrete time step by contrast the method presented in this paper treats the whole of the space time problem simultaneously thus the multigrid operations of smoothing and coarse grid correction are defined on all of the space time variables of a given grid level the method is characterized by a coarsening strategy with prolongation and restriction operators which depend at each grid level on the degree of anisotropy of the discretization stencil numerical results for the one and two dimensional heat equations are presented and are shown to agree closely with predictions from fourier mode analysis

computing periodic solutions of linear differential algebraic equations by waveform relaxation we propose an algorithm which is based on the waveform relaxation wr approach to compute the periodic solutions of a linear system described by differential algebraic equations for this kind of two point boundary problems we derive an analytic expression of the spectral set for the periodic wr operator we show that the periodic wr algorithm is convergent if the supremum value of the spectral radii for a series of matrices derived from the system is less than 1 numerical examples where discrete waveforms are computed with a backward difference formula further illustrate the correctness of the theoretical work in this paper

waveform relaxation with fast direct methods as preconditioner for a restricted class of parabolic pdes one can devise a practical numerical solver with a parallel complexity that is theoretically optimal the method uses a multidimensional fft to decouple the unknowns in the spatial domain into independent scalar odes these are discretized to give recurrence relations in the time dimension solved by parallel cyclic reduction this is the fft cr algorithm we discuss the use of fft cr as a preconditioner to iteratively solve more general parabolic pdes this approach naturally leads to a waveform relaxation scheme waveform relaxation was developed as an iterative method for solving large systems of odes it is the continuous in time analogue of stationary iterative methods for linear algebraic equations using the fft cr solver as a preconditioner preserves most of the potential for concurrency that accounts for the attractiveness of waveform relaxation with simple preconditioners like jacobi or red black gauss seidel while showing an important advantage the convergence rate of the resulting iteration is independent of the mesh size used in the spatial discretization the method can be accelerated by applying an appropriate scaling of the system before preconditioning

convergence analysis of waveform relaxation methods for neutral differential functional systems in this paper the problems of convergence and superlinear convergence of continuous time waveform relaxation method applied to volterra type systems of neutral functional differential equations are discussed under a lipschitz condition with time and delay dependent right hand side imposed on the so called splitting function more suitable conditions about convergence and superlinear convergence of continuous time wr method are obtained we also investigate the initial interval acceleration strategy for the practical implementation of the continuous time waveform relaxation method i e discrete time waveform relaxation method it is shown by numerical results that this strategy is efficacious and has the essential acceleration effect for the whole computation process

laplace transform method for parabolic problems with time dependent coefficients the laplace transform method has proven to be very efficient for dealing with parabolic problems whose coefficients are time independent and it is easily parallelizable however the method has not been proven to be applicable to linear problems whose coefficients are time dependent the reason is that the laplace transform of two time dependent functions leads to a convolution of the laplace transformed functions in the dual variable in this paper we propose a laplace transform method to linear parabolic problems with time dependent coefficients which is as efficient as the method for parabolic problems with time independent coefficients several numerical results are provided which support the efficiency of the proposed scheme

some orbits of free words that are determined by measures on finite groups abstract every word in a free group f induces a probability measure on every finite group in a natural manner it is an open problem whether two words that induce the same measure on every finite group necessarily belong to the same orbit of aut f a special case of this problem when one of the words is the primitive word x was settled positively by the third author and parzanchevski 15 here we extend this result to the case where one of the words is x d or x y d for an arbitrary d z

topology of finite graphs this paper derives from a course in group theory which i gave at berkeley in 1982 i wanted to prove the standard theorems on free groups and discovered that after a few preliminaries the notion of locally injective map or immersion of graphs was very useful this enables one to see in an effective easy algorithmic way just what happens with finitely generated free groups one can understand in this way 1 howson s theorem that if a and b are finitely generated subgroups of a free group then a b is finitely generated and 2 m hall s theorem that finitely generated subgroups of free groups are closed in the profinite topology during this course s m gersten came up with a simple proof of h neumann s inequality on the ranks in howson s theorem one of the ideas in gersten s proof was to use core graphs graphs with no trees hanging on subsequently i found that some consequences of a paper of greenberg s could be proved using core graphs and covering translations of immersions the most striking such result is that if a and b are finitely generated subgroups of a free group and if a c b is of finite index in both a and b then a b is of finite index in a v b the subgroup generated by a ub

analysis of a new space time parallel multigrid algorithm for parabolic problems we present and analyze a new space time parallel multigrid method for parabolic equations the method is based on arbitrarily high order discontinuous galerkin discretizations in time and a finite element discretization in space the key ingredient of the new algorithm is a block jacobi smoother we present a detailed convergence analysis when the algorithm is applied to the heat equation and determine asymptotically optimal smoothing parameters a precise criterion for semi coarsening in time or full coarsening and give an asymptotic two grid contraction factor estimate we then explain how to implement the new multigrid algorithm in parallel and show with numerical experiments its excellent strong and weak scalability properties

computing a alpha log a and related matrix functions by contour integrals new methods are proposed for the numerical evaluation of f mathbf a or f mathbf a b where f mathbf a is a function such as mathbf a 1 2 or log mathbf a with singularities in infty 0 and mathbf a is a matrix with eigenvalues on or near 0 infty the methods are based on combining contour integrals evaluated by the periodic trapezoid rule with conformal maps involving jacobi elliptic functions the convergence is geometric so that the computation of f mathbf a b is typically reduced to one or two dozen linear system solves which can be carried out in parallel

congruence topologies on the mapping class group abstract let s be the pure mapping class group of a connected orientable surface s of negative euler characteristic for c a class of finite groups let 1 s c be the pro c completion of the fundamental group of s the c congruence completion s c of s is the profinite completion induced by the embedding s out 1 s c in this paper we begin a systematic study of such completions for different c we show that the combinatorial structure of the profinite groups s c closely resemble that of s a fundamental question is how c congruence completions compare with pro c completions even though in general e g for c the class of finite solvable groups s c is not even virtually a pro c group we show that for z 2 c g s 2 and s open there is a natural epimorphism from the c congruence completion s 2 c of the abelian level of order 2 to its pro c completion s 2 c in particular this is an isomorphism for the class of finite groups and for the class of 2 groups moreover in these two cases the result also holds for a closed surface

commutator maps measure preservation and t systems let g be a finite simple group we show that the commutator map g g g is almost equidistributed as g 1 this somewhat surprising result has many applications it shows that a for a subset x g we have 1 x g 2 x g o 1 namely is almost measure preserving from this we deduce that almost all elements g 2 g can be expressed as commutators g x y where x y generate g this enables us to solve some open problems regarding t systems and the product replacement algorithm pra graph we show that the number of t systems in g with two generators tends to infinity as g 1 this settles a conjecture of guralnick and pak a similar result follows for the number of connected components of the pra graph of g with two generators some of our results apply for more general finite groups and more general word maps our methods are based on representation theory combining classical character theory with recent results on character degrees and values in finite simple groups in particular the so called witten zeta function g s p 2irr g 1 s plays a key role in the proofs

word maps and waring type problems waring s problem asks whether every natural number is a sum of g k fcth powers where g is a suitable function this was solved affirmatively by hubert in 1909 optimizing g k has been a central problem in additive number theory ever since see na for more detail and background recently there has been considerable interest in group theoretic analogues of this phenomenon where the aim is to present group elements as short products of certain special elements these special elements can be powers or commutators or values of a general word w or elements of a special conjugacy class in the group the motivation for this is sometimes topological let g be a topologically finitely generated pro p group using expressions of elements of the commutator subgroup g1 as bounded products of commutators serre showed that g is closed and deduced that every finite index subgroup of g is open a recent deep result of nikolov and segal nsi ns2 shows that this holds for every finitely generated profinite group again the core of the proof is presenting group elements as short products of values of certain words

wavelet in time multigrid in space preconditioning of parabolic evolution equations two space time variational formulations of linear parabolic evolution equations are considered one is symmetric and elliptic on the trial space while the other is not in each case a space time petrov galerkin discretization using suitable tensor product trial and test functions leads to a large linear system of equations the well posedness of this system with respect to parabolic norms induces a canonical preconditioner for the algebraic equations that arise after a choice of basis for the iterative resolution of this algebraic system with parallelization in the temporal direction we propose a sparse algebraic wavelet in time transformation on possibly nonuniform temporal meshes this transformation approximately block diagonalizes the preconditioner and the individual spatial blocks can then be inverted for instance by standard spatial multigrid methods in parallel the performance of the preconditioner is documented in a series of numerical experiments

monotone waveform relaxation for systems of nonlinear differential algebraic equations we present a monotone waveform relaxation algorithm which produces very tight upper and lower bounds of the transient response of a class of systems described by nonlinear differential algebraic equations daes that satisfy certain lipschitz conditions the choice of initial iteration is critical and we give two methods of finding it we show that the class of systems in which monotone convergence of waveform relaxation is possible is actually larger than previously reported numerical experiments are given to confirm the monotonicity of convergence of the algorithm

on time domain simulation of lossless transmission lines with nonlinear terminations a time domain approach is presented to solve nonlinear circuits with lossless transmission lines mathematically the circuits are described by a special kind of nonlinear differential algebraic equations daes with multiple constant delays in order to directly compute these delay systems in time domain decoupling by waveform relaxation wr is applied to the systems for the relaxation based method we provide a new convergence proof numerical experiments are given to illustrate the novel approach

toward parallel coarse grid correction for the parareal algorithm in this paper we present an idea toward parallel coarse grid correction cgc for the parareal algorithm it is well known that such a cgc procedure is often the bottleneck of speedup of the parareal algorithm for an ode system with initial value condition u 0 u 0 the idea can be explained as follows first we apply the mathcal g propagator to the same ode system but with a special condition u 0 alpha u t where alpha in mathbb r is a crux parameter second in each iteration of the parareal algorithm the cgc procedure will be carried out by the so called diagonalization technique established recently the parameter alpha controls both the roundoff error arising from such a diagonalization technique and the convergence rate of the resulting parareal algorithm we show that there exists some threshold alpha such that the parareal algorithm with diagonalization based cgc possesses the same convergence rate as that of the parareal algorithm with classical cgc if alpha leq alph

compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity we study the branch of semistable and unstable solutions i e those whose morse index is at most 1 of the dirichlet boundary value problem u f x 1 u 2 on a bounded domain n which models among other things a simple electrostatic microelectromechanical system mems device we extend the results of 11 relating to the minimal branch by obtaining compactness along unstable branches for 1 n 7 on any domain and for a large class of permittivity profiles f we also show the remarkable fact that powerlike profiles f x x can push back the critical dimension n 7 of this problem by establishing compactness for the semistable branch on the unit ball also for n 8 and as long as as a byproduct we are able to follow the second branch of the bifurcation diagram and prove the existence of a second solution for in a natural range in all these results the conditions on the space dimension and on the power of the profile are essentially sharp 2007 wiley periodicals inc