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

space time spectral collocation method for one dimensional pde constrained optimisation this paper solves an optimal control problem governed by a parabolic pde using lagrangian multipliers necessary conditions are derived and then space time spectral collocation method is applied t

the determination of integral closures and geometric applications abstract we express explicitly the integral closures of some ring extensions this is done for all bring jerrard extensions of any degree as well as for all general extensions of degree 5 so far such an explicit expression is known only for degree 3 extensions as a geometric application we present explicitly the structure sheaf of every bring jerrard covering space in terms of coefficients of the equation defining the covering in particular we show that a degree 3 morphism y x is quasi etale if and only if c 1 o y is trivial details in theorem 5 3 we also try to get a geometric galoisness criterion for an arbitrary degree n finite morphism this is successfully done when n 3 and less satisfactorily done when n 5

reduced order modeling of parameterized pdes using time space parameter principal component analysis this paper presents a methodology for constructing low order surrogate models of finite element finite volume discrete solutions of parameterized steady state partial differential equations the construction of proper orthogonal decomposition modes in both physical space and parameter space allows us to represent high dimensional discrete solutions using only a few coefficients an incremental greedy approach is developed for efficiently tackling problems with high dimensional parameter spaces for numerical experiments and validation several non linear steady state convection diffusion reaction problems are considered first in one spatial dimension with two parameters and then in two spatial dimensions with two and five parameters in the two dimensional spatial case with two parameters it is shown that a 7 7 coefficient matrix is sufficient to accurately reproduce the expected solution while in the five parameters problem a 13 6 coefficient matrix is shown to reproduce the solution with sufficient accuracy the proposed methodology is expected to find applications to parameter variation studies uncertainty analysis inverse problems and optimal design

time evolution of two dimensional systems with infinitely many particles mutually interacting via very singular forces we study the existence and uniqueness of the time evolution via the newton law of a two dimensional system of infinitely many particles with very singular mutual interactions it is an improvement of the result by fritz and dobrushin given in comm math phys 57 67 81 1977 for inverse power like singular interactions

on non equilibrium dynamics of multidimensional infinite particle systems in the translation invariant case it is shown that for a set of full measure with respect to any translation invariant probability distribution on the space of initial configurarations of classical particle systems on d with interaction given by a smooth superstable potential of finite range there is a solution to the newtonian equations of motion provided that the specific energy and the particle density of the initial configuration exist a s

parallel algorithms for initial value problems for difference and differential equations abstract let yn be a trajectory defined sequentially by a nonlinear vector difference equation yn 1 fn 1 yn with y0 known a steffensen like iterative method is proposed which starts from a guessed sequence un 0 for the approximation of yn and allows certain computations to be performed in parallel the sequence u k 1 n is obtained from u k n by means of a formula of the form u k 1 n 1 v k 1 n 1 k 1 n 1 u k 1 n u k n here the vectors v k 1 n 1 and the matrices k 1 n 1 each require a function evaluation but can be computed in parallel with respect to n in a suitable interval nk mk the length of which depends on the number of processors available therefore for the algorithm to serve effectively the steffensen iteration must converge quickly and the machine used must possess a large number of processors finally note that the theory includes the case that the sequence yn is given by a one step ode solver

a review of algebraic multigrid abstract since the early 1990s there has been a strongly increasing demand for more efficient methods to solve large sparse unstructured linear systems of equations for practically relevant problem sizes classical one level methods had already reached their limits and new hierarchical algorithms had to be developed in order to allow an efficient solution of even larger problems this paper gives a review of the first hierarchical and purely matrix based approach algebraic multigrid amg amg can directly be applied for instance to efficiently solve various types of elliptic partial differential equations discretized on unstructured meshes both in 2d and 3d since amg does not make use of any geometric information it is a plug in solver which can even be applied to problems without any geometric background provided that the underlying matrix has certain properties

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

implicit space time domain decomposition methods for stochastic parabolic partial differential equations we introduce and study parallel space time domain decomposition methods for solving deterministic and stochastic parabolic equations traditional parallel algorithms solve parabolic problems time step by time step the parallelism is restricted to each time step and the algorithms are purely sequential in time in this paper we develop some overlapping schwarz methods whose subdomains cover both space and time variables and we show that the methods work well for stochastic parabolic equations discretized with an implicit stochastic galerkin method one and two level schwarz preconditioned recycling gmres methods are carefully investigated such that many components of the methods are reused when a large number of linear systems are solved the key elements of this approach include an ordering algorithm and two grouping algorithms we present some experimental results obtained on a parallel computer with more than one thousand processors

covers of algebraic varieties iii the discriminant of a cover of degree 4 and the trigonal construction in c e a general structure theorem for gorenstein covers of degrree d has been proved such a result has been used for characterizing covers of degree 3 4 and 5 see c e and cs respectively of algebraic varieties more precisely in section 4 resp 3 of c e it is proved that the total space x of each gorenstein cover e x y of degree 4 esp 3 is obta ned inside a suit bl p ojective bundle 7r 11 y of rank 2 resp 1 as the b se locus of a pencil relativ conics resp

stacks of trigonal curves in this paper we study the stack tg of smooth triple covers of a conic when g 5 this stack is embedded mg as the locus of trigonal curves we show that tg is a quotient ug g where g is a certain algebraic group and ug is an open subscheme of a g equivariant vector bundle over an open subscheme of a representation of g using this we compute the integral picard group of tg when g 1 the main tools are a result of miranda that describes a flat finite triple cover of a scheme s as given by a locally free sheaf e of rank two on s with a section of sym 3 e det e and a new description of the stack of globally generated locally free sheaves of fixed rank and degree on a projective line as a quotient stack in moduli theory stacks have often been constructed as quotients by group actions if an algebraic stack f is a quotient stack x g where g is an algebraic group acting on an algebraic variety x the geometry of f is the geometry of the action of g on x and one can apply to f the powerful techniques that have been developed for studying invariants of group actions in algebraic topology and algebraic geometry even just knowing that such a presentation exists even without an explicit description of the action can be useful but it is even better when the variety x and the group g are fairly simple so that this description may be used directly to study f this does not seem possible in many cases for example the stack mg of smooth curves of genus g is of the form x g but when g is large the space x is complicated and to our knowledge no general result about mg has been proved by exploiting this presentation of course in teichmuller theory one represents mg as a quotient of an action of the teichmuller group which is an infinite discrete group acting on a ball in c 3g 3 but this description is topological and it is hard to use it directly to prove algebraic geometric results about mg it has long been known that in characteristic different from 2 and 3 the stack m1 1 of elliptic curves is a quotient x gm where x is the complement of the hypersurface 4x 3 27y 2 0 in a 2 and gm acts with weights 4 and 6 this gives an easy proof of the fact due to mumford mum65 that the picard group of m1 1 is cyclic of order 12 in vis98 the second author gives a presentation of m2 as a quotient x gl2 where x is the scheme of smooth binary forms of degree 6 in two variables the action of gl2 on x is a twist of the customary one as an application he computes the chow ring of m2 this was generalized in av04 to the stack hg of hyperelliptic curves of genus g which has a presentation as a quotient xg gl2 if g is even or xg gmpgl2 if g is odd where x is the space of smooth binary forms of degree 2g 2 in two variables this allows

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

a theoretical framework for proving accuracy results for deferred corrections general techniques are described for proving accuracy results for deferred correction solutions to differential equations these techniques apply also to computational estimates of the local discretization error the proofs avoid the necessity of demonstrating the existence of asymptotic expansions of the global error in powers of some meshsize parameter

some remarks on nonequilibrium dynamics of infinite particle systems classical mechanics of infinitely many particles in dimensions one and two is considered particles interacting by a superstable pair potential of finite range the group of motion generated by newton s equations is constructed in the space of locally finite configurations with a logarithmic order of energy fluctuations at infinity a core of the liouville operator is also described results of dobrushin and the author and of marchioro pellegrinotti pulvirenti are improved

ample vector bundles of small δ genera a natural notion of delta genus for a generalized polarized manifold x e strictly related to its associated scroll is introduced and pairs x e with low are classified the stronger are the properties enjoyed by the vector bundle e the larger are the values of attained by the results

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 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

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

a parareal waveform relaxation algorithm for semi linear parabolic partial differential equations we report a new parallel iterative algorithm for semi linear parabolic partial differential equations pdes by combining a kind of waveform relaxation wr techniques into the classical parareal algorithm the parallelism can be simultaneously exploited by wr and parareal in different directions we provide sharp error estimations for the new algorithm on bounded time domain and on unbounded time domain respectively the iterations of the parareal and the wr are balanced to optimize the performance of the algorithm furthermore the speedup and the parallel efficiency of the new approach are analyzed numerical experiments are carried out to verify the effectiveness of the theoretic work

a second order parareal algorithm for fractional pdes we are concerned with using the parareal parallel in time algorithm for large scale odes system u t a u t d a u t f t arising frequently in semi discretizing time dependent pdes with spatial fractional operators where d 0 is a constant 0 1 and a is a spare and symmetric positive definite spd matrix the parareal algorithm is iterative and is characterized by two propagators f and g which are respectively associated with small temporal mesh size t and large temporal mesh size t the two mesh sizes satisfy t j t with j 2 being an integer which is called mesh ratio let t unit f and t unit g be respectively the computational cost of the two propagators for moving forward one time step then it is well understood that the speedup of the parareal algorithm namely e satisfies e o c log 1 where c t unit f t unit g and is the convergence factor a larger e corresponds a more efficient parareal solver for g backward euler and some choices of f previous studies show that can be a satisfactory quantity particularly for f 2nd order dirk diagonally implicit runge kutta it holds 1 3 for any choice of the mesh ratio j in this paper we continue to consider f 2nd order dirk but with a new choice for g the imex implicit explicit euler method where the implicit and explicit computation is respectively associated with a and d a compared to the widely used backward euler method this choice on the one hand increases c this point is apparent and interestingly on the other hand it can also make the convergence factor smaller 1 5 numerical results are provided to support our conclusions

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

multigrid method based on a space time approach with standard coarsening for parabolic problems in this work a space time multigrid method which uses standard coarsening in both temporal and spatial domains and combines the use of different smoothers is proposed for the solution of the heat equation in one and two space dimensions in particular an adaptive smoothing strategy based on the degree of anisotropy of the discrete operator on each grid level is the basis of the proposed multigrid algorithm local fourier analysis is used for the selection of the crucial parameter defining such an adaptive smoothing approach central differences are used to discretize the spatial derivatives and both implicit euler and crank nicolson schemes are considered for approximating the time derivative for the solution of the second order scheme we apply a double discretization approach within the space time multigrid method the good performance of the method is illustrated through several numerical experiments

using hierarchical matrices in the solution of the time fractional heat equation by multigrid waveform relaxation abstract this work deals with the efficient numerical solution of the time fractional heat equation discretized on non uniform temporal meshes non uniform grids are essential to capture the singularities of typical solutions of time fractional problems we propose an efficient space time multigrid method based on the waveform relaxation technique which accounts for the nonlocal character of the fractional differential operator to maintain an optimal complexity which can be obtained for the case of uniform grids we approximate the coefficient matrix corresponding to the temporal discretization by its hierarchical matrix h matrix representation in particular the proposed method has a computational cost of o k n m log m where m is the number of time steps n is the number of spatial grid points and k is a parameter which controls the accuracy of the h matrix approximation the efficiency and the good convergence of the algorithm which can be theoretically justified by a semi algebraic mode analysis are demonstrated through numerical experiments in both one and two dimensional spaces

on positivity of line bundles on enriques surfaces we study linear systems on enriques surfaces we prove rationality of seshadri constants of ample line bundles on enriques surfaces and provide lower bounds on these numbers we show the nonexistence of k very ample line bundles on enriques surfaces of degree 4k 4 for k 1 thus answering an old question of ballico and sommese introduction in this note we answer a question posed by ballico and sommese 1 see also terakawa 14 concerning the existence of k very ample line bundles of degree 4k 4 on enriques surfaces we show that such line bundles exist if and only if k 0 and that in fact the degree of a k very ample line bundle on an enriques surface is subject to a much more restrictive bound theorem 2 4 this has strong consequences for the local positivity of line bundles on enriques surfaces we address this questions in section 3 in particular we show that seshadri constants of ample line bundles on enriques surfaces are always rational 1 auxiliary material and notation the basic reference for this paper is the monograph 6 by cossec and dolgachev here we briefly recall properties of enriques surfaces needed for our considerations first of all an enriques surface is a surface y of kodaira dimension zero with no differential forms in particular ho ky 0 and the canonical bundle ky is a 2torsion element in pico y i e 2ky oy to alleviate notation it is convenient to use the following convention in the sequel we make frequent use of vanishing theorems and statements on adjoint line bundles since the canonical divisor ky is numerically trivial the adjoint line bundle ky l has the same numerical properties as l in particular if l is nef and big ample then so is ky l by the nakai moishezon criterion so in order to get vanishing statements for l we apply vanishing theorems to ky l and use the fact that 2ky is trivial in the sequel we do so without comment and hope to cause no confusion this way received by the editors june 1 2000 2000 mathematics subject classification primary 14j28 secondary 14c20 14e25

bisecant and trisecant curves on ruled surfaces abstract in this paper we study numerical classes of integral curves on a ruled surface p e associated to a stable bundle e the nef cone of a ruled surface is generated by the classes of a minimal section c 0 and a fiber f we compute the smallest integer b such that the class k c 0 b f contains a multisecant curve for k 2 and 3 also we find its consequence on the lange stability of s y m k e

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

gauss composition for p1 and the universal jacobian of the hurwitz space of double covers abstract in this paper we give an explicit description of the moduli space of line bundles on hyperelliptic curves including singular curves we study the universal jacobian j 2 g n of degree n line bundles over the hurwitz stack of double covers of p 1 by a curve of genus g our main results are the construction of a smooth irreducible universally closed but not separated moduli compactification j bd 2 g n of j 2 g n whose points we describe simply and explicitly as sections of certain vector bundles on p 1 a description of the global geometry and moduli properties of these stacks and a computation of the picard groups of j bd 2 g n and j 2 g n in the cases when n g is even an important ingredient of our work is the parametrization of line bundles on double covers by binary quadratic forms this parametrization generalizes the classical number theoretic correspondence between ideal classes of quadratic rings and integral binary quadratic forms which in particular gives the group law on integral binary quadratic forms first discovered by gauss

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

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

time evolution of an infinitely extended vlasov system with singular mutual interaction we study the time evolution of an infinitely extended system in the mean field approximation governed by the vlasov equation this system is confined in an unbounded cylinder by an external force singular on the border the mutual interaction is assumed singular at short distance as 1 r alpha with alpha 2 3 or alpha 1 in case of an external lorentz force and with a short range the initial density is assumed bounded differently from studies which assume initial data compact in space and or in velocities here we consider a system having infinite mass and an exponential bound on the velocities according to the maxwell boltzmann law

a parallel nearly implicit time stepping scheme across the space parallelism still remains the most mature convenient and natural way to parallelize large scale problems one of the major problems here is that implicit time stepping is often difficult to parallelize due to the structure of the system approximate implicit schemes have been suggested to circumvent the problem these schemes have attractive stability properties and they are also very well parallelizable the purpose of this article is to give an overall assessment of the parallelism of the method

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

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

eigenvalues and pseudo eigenvalues of toeplitz matrices the eigenvalues of a nonhermitian toeplitz matrix a are usually highly sensitive to perturbations having condition numbers that increase exponentially with the dimension n an equivalent statement is that the resolvent zz a of a toeplitz matrix may be much larger in norm than the eigenvalues alone would suggest exponentially large as a function of n even when z is far from the spectrum because of these facts the meaningfulness of the eigenvalues of nonhermitian toeplitz matrices for any but the most theoretical purposes should be considered suspect in many applications it is more meaningful to investigate the e pseudo eigenvalues the complex numbers z with ll zz a 11 l this paper analyzes the pseudospectra of toeplitz matrices and in particular relates them to the symbols of the matrices and thereby to the spectra of the associated toeplitz and laurent operators our results are reasonably complete in the triangular case and preliminary in the cases of nontriangular toeplitz matrices block toeplitz matrices and toeplitz like matrices with smoothly varying coefficients computed examples of pseudospectra are presented throughout and applications in numerical analysis are mentioned

towards essential improvement for the parareal tr and parareal gauss4 algorithms parareal is an iterative algorithm and is characterized by two propagators g and f which are respectively associated with large step size t and small step size t where t j t and j 2 is an integer for symmetric positive definite spd system u t a u t g t arising from semi discretizing time dependent pdes if we fix the g propagator to the backward euler method and choose for f some l stable time integrator it can be proven that the convergence factors of the corresponding parareal algorithms satisfy 1 3 j 2 and a 0 where a is the spectrum of the matrix a however this result does not hold when time integrators that lack l stability such as the trapezoidal rule and the 4th order gauss rk method are chosen as the f propagator the parareal algorithms using these two methods for the f propagator are denoted by parareal tr and parareal gauss4 in this paper we propose a strategy to let these two parareal algorithms possess such a uniform convergence property the idea is to choose an l stable propagator f and on each coarse time interval t n t n 1 we perform first two steps of f then followed by j 2 steps of f precisely for the trapezoidal rule we select the 2nd order sdirk method as the f propagator and for the 4th order gauss rk method we select the 4th order lobatto iii c method as the f propagator numerical results are given to support our theoretical conclusions

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

non equilibrium dynamics of three dimensional infinite particle systems we show existence and uniqueness for the solutions to the newton equations relative to a system of infinitely many particles in the space interacting by means of a positive and short range potential the initial conditions are chosen in a set sufficiently large to be the support of any reasonable non equilibrium state we extend previous results in one and two dimensions obtained by lanford and by fritz and dobrushin respectively many years ago

on the picard group of enriques surfaces an enriques surface over an algebraically closed field k of characteristic 4 2 is a non singular projective surface s with h s d s h2 s cs 0 and 2ks 0 the unramitied double cover defined by ks is a k3 surface r a non singular projective surface with hi r d r 0 kr 0 illusie has shown i that the group of divisors modulo numerical equivalence is isomorphic to the enriques lattice u e 8 1 where u and e s 1 denote respectively the unique even unimodular lattices of index of inertia 1 1 and 0 8 the purpose of this note is to use this isomorphism to study the picard group of s we prove the existence of certain configurations of irreducible curves of arithmetic genus 0 or 1 and deduce from them the existence of certain projective models for s and r for example we prove the following results

convergent iterative schemes for time parallelization parallel methods are usually not applied to the time domain because of the inherit sequentialness of time evolution but for many evolutionary problems computer simulation can benefit substantially from time parallelization methods in this paper we present several such algorithms that actually exploit the sequential nature of time evolution through a predictor corrector procedure this sequentialness ensures convergence of a parallel predictor corrector scheme within a fixed number of iterations the performance of these novel algorithms which are derived from the classical alternating schwarz method are illustrated through several numerical examples using the reservoir simulator athena

timestep acceleration of waveform relaxation dynamic iteration methods for treating certain classes of linear systems of differential equations are considered it is shown that the discretized picard lindelof waveform relaxation iteration can be accelerated by solving the defect equations with a larger timestep or by using a recursive procedure based on a succession of increasing timesteps a discussion of convergence is presented including analysis of a discrete smoothing property maintained by symmetric multistep methods applied to linear wave equations numerical experiments indicate that the method can speed convergence

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

coupling methods for heat transfer and heat flow operator splitting and the parareal algorithm abstract we propose an operator splitting method for coupling heat transfer and heat flow equations this work is motivated by the need to couple independent industrial heat transfer solvers e g the aura fluid software package and heat flow solvers e g openfoam such packages are often used to simulate the influence of solar heat in car bodies and are coupled by a b splitting techniques the main goal of this work is the acceleration of the coupled software system by iterative operator splitting methods and additional time parallelism using the parareal algorithm we present these new splitting techniques along with some novel convergence results and test the splitting parareal combination on various numerical problems

reye constructions for nodal enriques surfaces a classical reye congruence x is an enriques surface of rational equivalence class 3 7 in the grassmannian g 1 3 of lines of p 3 x is the locus of lines of p 3 which are included in two quadrics of w web of quadrics a generaiization to g 1 t is given 1 for each t 2 there exist enriques surfaces x of class t 3t 2 in g 1 t 2 the determinant of the dual of the universal bundle on x is o x 2e r k x with e isolated elliptic curve r 2 2 e r t 3 x parameterizes lines of p t which are included in a codimension 2 subsystem of w w linear system of quadrics of dimension 2 t the paper includes a description of the variety of trisecant lines to a smooth enriques surface of degree 10 p 5

a partially parallel in time fixed stress splitting method for biot s consolidation model in this work we study the parallel in time iterative solution of coupled flow and geomechanics in porous media modelled by a two field formulation of biot s equations in particular we propose a new version of the fixed stress splitting method which has been widely used as solution method of these problems this new approach forgets about the sequential nature of the temporal variable and considers the time direction as a further direction for parallelization the method is partially parallel in time we present a rigorous convergence analysis of the method and numerical experiments to demonstrate the robust behaviour of the algorithm

modular compactifications of the space of marked trigonal curves abstract we construct a sequence of modular compactifications of the space of marked trigonal curves by allowing the branch points to coincide to a given extent beginning with the standard admissible cover compactification the sequence first proceeds through contractions of the boundary divisors and then through flips of the so called maroni strata culminating in a fano model for even genera and a fano fibration for odd genera while the sequence of divisorial contractions arises from a more general construction the sequence of flips uses the particular geometry of triple covers we explicitly describe the mori chamber decomposition given by this sequence of flips

three divisible families of skew lines on a smooth projective quintic we give an example of a family of 15 skew lines on a quintic such that its class is divisible by 3 we study properties of the codes given by arrangements of disjoint lines on quintics

on quintic surfaces of general type the study of quintic surfaces is of special interest because 5 is the lowest degree of surfaces of general type the aim of this paper is to give a classification of the quintic surfaces of general type over the complex number field c we show that if s is an irreducible quintic surface of general type then it must be normal and it has only elliptic double or triple points as essential singularities then we classify all such surfaces in terms of the classification of the elliptic double and triple points we give many examples in order to verify the existence of various types of quintic surfaces of general type we also make a study of the double or triple covering of a quintic surface over p2 obtained by the projection from a triple or double point on the surface this reduces the classification of the surfaces to the classification of branch loci satisfying certain conditions finally we derive some properties of the hilbert schemes of some types of quintic surfaces

time evolution of infinitely many particles an existence theorem in this paper we deal with systems of infinitely many particles in 3 given by a two body short range potential and an external potential depending on the position of the particles we show the existence of dynamics for a set of initial configurations which has measure one with respect to the gibbs measure induced by a suitable family of hamiltonians

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