a self adaptive time integration algorithm for solving partial differential equations non uniform spatial grids are commonly used to resolve locally fast changing physical phenomena in space if a traditional explicit time integration scheme is used to advance solutions in the temporal dimension the step size is restricted by the stability criterion which is in turn dictated by the smallest grid spacing in the spatial dimensions it turns out that the excessively small time step enforced by the smallest spatial grid spacing in a domain is usually unnecessary for most of the grid points with larger grid spacing a new adaptive time integration method is introduced in this paper to improve the computational efficiency the basic idea is to use different time step sizes at different spatial grid points the stability criterion is still satisfied at all grid points by using different time step sizes in this way the time step size adjusts automatically based on the local spatial grid spacing complexity analysis and implementation details are also discussed in the paper the non linear burger s equation is used in the numerical experiment both the complexity analysis and numerical computations demostrate significant improvement of computational efficiency

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

controllability and observability of 2d thermal flow in bulk storage facilities using sensitivity fields abstractto control and observe spatially distributed thermal flow systems the controllable field and observable field around the actuator and sensor are of interest respectively for spatially distributed systems the classical systems theoretical concepts of controllability and observability are in general difficult to apply in this study sensitivity fields were used to analyse the behaviour from input to state and from initial state to output for the analysis of controllability and observability a large scale bulk storage facility with coupled thermal flow of air and agro products was used analysis of this system using the classical systems theory results in controllability and observability results that are dependent on the step size of the spatially discretised system due to matrix multiplications inaccurate results are calculated if the step size is too small our findings indicate that input state and initial state output sensitivity fields provide sufficient information about the contro

a linear domain decomposition method for partially saturated flow in porous media abstract the richards equation is a nonlinear parabolic equation that is commonly used for modelling saturated unsaturated flow in porous media we assume that the medium occupies a bounded lipschitz domain partitioned into two disjoint subdomains separated by a fixed interface this leads to two problems defined on the subdomains which are coupled through conditions expressing flux and pressure continuity at after an euler implicit discretisation of the resulting nonlinear subproblems a linear iterative l type domain decomposition scheme is proposed the convergence of the scheme is proved rigorously in the last part we present numerical results that are in line with the theoretical finding in particular the convergence of the scheme under mild restrictions on the time step size we further compare the scheme to other approaches not making use of a domain decomposition namely we compare to a newton and a picard scheme we show that the proposed scheme is more stable than the newton scheme while remaining comparable in computational time even if no parallelisation is being adopted after presenting a parametric study that can be used to optimise the proposed scheme we briefly discuss the effect of parallelisation and give an example of a four domain implementation

a new solution for the nonlinear diffusion convection equation the cauchy problem for a one dimensional nonlinear porous media diffusion convection differential equation is considered a self similar solution is derived that after the riccati transformation is described in terms of the airy functions the asymptotic behavior of the solution for a small amount of species introduced into the porous media is investigated

new exact solutions of heat conduction in metals we consider the way in which a solution to a class of nonlinear partial differential equationss u ut k u ux x approaches the similarity form the problem we solve is chosen for two main reasons first the equation above is of widespread use in modeling physical situations and second it provides a tractable but significant example of a free boundary problem

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

analysis of block parareal preconditioners for parabolic optimal control problems in this paper we describe block matrix algorithms for the iterative solution of a large scale linear quadratic optimal control problem involving a parabolic partial differential equation over a finite control horizon we consider an all at once discretization of the problem and formulate three iterative algorithms the first algorithm is based on preconditioning a symmetric positive definite reduced linear system involving only the unknown control variables however inner outer iterations are required the second algorithm modifies the first algorithm to avoid inner outer iterations by introducing an auxiliary variable it yields a symmetric indefinite system with a positive definite block preconditioner the third algorithm is the central focus of this paper it modifies the preconditioner in the second algorithm by a parallel in time preconditioner based on the parareal algorithm theoretical results show that the preconditioned algorithms have optimal convergence properties and parallel scalability numerical experiments confirm the theoretical results

stable parareal in time method for first and second order hyperbolic systems the parareal in time algorithm allows one to perform parallel simulations of time dependent problems this algorithm has been implemented on many types of time dependent problems with some success recent contributions have allowed one to extend the domain of application of the parareal in time algorithm so as to handle long time simulations of hamiltonian systems this improvement has managed to avoid the fatally large lack of accuracy of the plain parareal in time algorithm which does not conserve invariant quantities a somewhat similar difficulty occurs for problems where the solution lacks regularity either initially or during the evolution as is the case for hyperbolic systems of conservation laws in this paper we identify the reasons for instabilities of the parareal in time algorithm and propose a simple way to cure them we use the new method to solve a linear wave equation and a nonlinear burgers equation the results illustrate the stability of this variant of the parareal in time algorithm

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

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

dynamical analysis of distributed parameter tubular reactors this paper is dedicated to the dynamical analysis of tubular reactor models namely plug flow and axial dispersion reactors involving sequential reactions for which the kinetics only depends on the concentrations of the reactants involved in the reaction this analysis is performed by describing these models as infinite dimensional state space systems with bounded observation and control operators it is shown that these systems are exponentially stable and that a the plug flow reactor model is observable when the component concentrations are measured at the reactor output and observable with respect to its physical domain when only the product concentration is measured and it is reachable with respect to its physical domain when it is controlled at the reactor input and b any finite set of dominant modes of the axial dispersion reactor model is observable when either all the component concentrations or only the product concentration are measured and reachable by using appropriate sensors and actuators the dynamical properties of related finite dimensional models are also discussed

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

lq optimal boundary control of infinite dimensional systems with yosida type approximate boundary observation a class of boundary control systems with boundary observation is considered for which the unbounded operators often lead to technical difficulties an extended model for this class of systems is described and analyzed which involves no unbounded operator except for the dynamics generator a method for the resolution of the lq optimal control problem for this model is described and the solution provides a stabilizing feedback for the nominal system with unbounded operators in the sense that in closed loop the state trajectories converge to zero exponentially fast the model consists of an extended abstract differential equation whose state components are the boundary input the state up to an affine transformation and a yosida type approximation of the output of the nominal system it is shown that under suitable conditions the model is well posed and in particular that the dynamics operator is the generator of a c 0 semigroup moreover the model is shown to be observable and to carry controllability stabilizability and detectability properties from the nominal system a general method of resolution based on the problem of spectral factorization of a multi dimensional operator valued spectral density is described in order to solve a lq optimal control problem for this model it is expected that this approach will lead hopefully to a good trade off between the cost of modeling and the efficiency of methods of resolution of control problems for such systems

multigrid methods for pde optimization research on multigrid methods for optimization problems is reviewed optimization problems considered include shape design parameter optimization and optimal control problems governed by partial differential equations of elliptic parabolic and hyperbolic type

eigenvalues of the barenblatt pattle similarity solution in nonlinear diffusion we consider the large time behaviour of the nonlinear diffusion equation u t r 1 r r 1 u u r u 0 0 for certain types of compact initial data we show that the solution approaches the barenblatt pattle similarity solution through an infinite sequence of negative real powers of t which can be found in explicit form these together with their interactive product terms determine the power series development of u r t as t

parallelization in time for thermo viscoplastic problems in extrusion of aluminium summary the parareal algorithm 14 is a parallel approach for solving numerically systems of ordinary dierential equations odes by exploiting parallelism across the steps of the numerical integrator the method performs well for dissipative problems and problems of uid structure interaction 8 we consider here a convergence analysis for the method and we report the performance achieved from the parallelization of a stokes navier stokes code via the parareal algorithm copyright c 2000 john wiley sons ltd

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

legendre spectral collocation in space and time for pdes spectral methods solve partial differential equations numerically with errors bounded by an exponentially decaying function of the number of modes when the solution is analytic for time dependent problems almost all focus has been on low order finite difference schemes for the time derivative and spectral schemes for spatial derivatives spectral methods which converge spectrally in both space and time have appeared recently this paper shows that a legendre spectral collocation method of tang and xu for the heat equation converges exponentially quickly when the solution is analytic we also derive a condition number estimate of the method another space time spectral scheme which is easier to implement is proposed numerical experiments verify the theoretical results

waveform relaxation for the computational homogenization of multiscale magnetoquasistatic problems this paper proposes the application of the waveform relaxation method to the homogenization of multiscale magnetoquasistatic problems in the monolithic heterogeneous multiscale method the nonlinear macroscale problem is solved using the newtonraphson scheme the resolution of many mesoscale problems per gau point allows to compute the homogenized constitutive law and its derivative by finite differences in the proposed approach the macroscale problem and the mesoscale problems are weakly coupled and solved separately using the finite element method on time intervals for several waveform relaxation iterations the exchange of information between both problems is still carried out using the heterogeneous multiscale method however the partial derivatives can now be evaluated exactly by solving only one mesoscale problem per gau point

spectral factorization and lq optimal regulation for multivariable distributed systems a necessary and sufficient condition is proved for the existence of a bistable spectral factor with entries in the distributed proper stable transfer function algebra x1d49c in the context of distributed multivariable convolution systems with no delays a by product is the existence of a normalized coprime fraction of the transfer function of such a possibly unstable system with entries in the algebra of fractions over x1d49c we next study semigroup state space systems sgb with bounded sensing and control having a transfer function with entries in and consider its standard lq optimal regulation problem having an optimal state feedback operator k0 for a system sgb a formula is given relating any spectral factor of a transfer function coprime fraction power spectral density to k0 a by product is the description of any normalized coprime fraction of the transfer function in terms of k0 finally we describe an alternative way of finding the solution operator k0 of the lq problem using spectral factor

fully reliable error control for first order evolutionary problems abstract this work is focused on the application of functional type a posteriori error estimates and corresponding indicators to a class of time dependent problems we consider the algorithmic part of their derivation and implementation and also discuss the numerical properties of these bounds that comply with obtained numerical results this paper examines two different methods of solution approximation for evolutionary models i e a time marching technique and a space time approach the first part of the study presents an algorithm for global minimisation of the majorant on each of discretisation time cylinders time slabs the effectiveness of this approach to error estimation is confirmed by extensive numerical tests in the second part of the publication the application of functional error estimates is discussed with respect to a space time approach it is followed by a set of extensive numerical tests that demonstrates the efficiency of proposed error control method the numerical results obtained in this paper rely on the implementation carried out using open source software which allows formulating the problem in a weak setting to work in a variational framework appears to be very natural for functional error estimates due to their derivation method the search for the optimal parameters for the majorant is done by a global functional minimisation which to the authors knowledge is the first work using this technique in an evolutionary framework

on monotone iteration and schwarz methods for nonlinear parabolic pdes the schwarz alternating method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains the solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each of the subdomains in this paper proofs of convergence of some schwarz alternating methods for nonlinear parabolic problems which are known to have solutions by the monotone method also known as the method of subsolutions and supersolutions are given in particular an additive schwarz method for scalar as well some coupled nonlinear pdes are shown to converge to the solution on finitely many subdomains in the coupled system case each subdomain pde is linear decoupled and can be solved concurrently with other subdomain pdes these results are applicable to several models in population biology the convergence behavior is illustrated by two numerical examples

a spatially second order alternating direction implicit adi method for solving three dimensional parabolic interface problems abstract a new matched alternating direction implicit adi method is proposed in this paper for solving three dimensional 3d parabolic interface problems with discontinuous jumps and complex interfaces this scheme inherits the merits of its ancestor for two dimensional problems while possesses several novel features such as a non orthogonal local coordinate system for decoupling the jump conditions two side estimation of tangential derivatives at an interface point and a new douglas rachford adi formulation that minimizes the number of perturbation terms to attack more challenging 3d problems in time discretization this new adi method is found to be of first order and stable in various experiments in space discretization the matched adi method achieves the second order accuracy based on simple cartesian grids for various irregularly shaped surfaces and spatial temporal dependent jumps computationally the matched adi method is as efficient as the fastest implicit scheme based on the geometrical multigrid for solving 3d parabolic equations in the sense that its complexity in each time step scales linearly with respect to the spatial degree of freedom n i e o n furthermore unlike iterative methods the adi method is an exact or non iterative algebraic solver which guarantees to stop after a certain number of computations for a fixed n therefore the proposed matched adi method provides a very promising tool for solving 3d parabolic interface problems

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

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

communication aware adaptive parareal with application to a nonlinear hyperbolic system of partial differential equations in the strong scaling limit the performance of conventional spatial domain decomposition techniques for the parallel solution of pdes saturates when sub domains become small halo communication and other overheard come to dominate a potential path beyond this scaling limit is to introduce domain decomposition in time with one such popular approach being the parareal algorithm which has received a lot of attention due to its generality and potential scalability low efficiency particularly on convection dominated problems has however limited the adoption of the method in this paper we introduce a new strategy communication aware adaptive parareal caap to overcome some of the challenges with caap we choose time subdomains short enough that convergence of the parareal algorithm is quick yet long enough that the overheard of communicating time subdomain interfaces does not induce a new limit to parallel speed up furthermore we propose an adaptive work scheduling algorithm that overlaps consecutive parareal cycles and decouples the number of time subdomains and number of active node groups in an efficient manner to allow for comparatively high parallel eciency we demonstrate the viability of caap trough the parallel in time integration of a hyperbolic system of pdes in the form of the two dimensional nonlinear shallow water wave equation solved using a 3rd order accurate weno rk discretization for the computational cheap approximate operator needed as a preconditioner in the parareal corrections we use a lower order roe type discretization time parallel integration of purely hyperbolic type evolution problems is traditionally considered impractical trough large scale numerical experiments we demonstrate that with caap it is possible not only to obtain time parallel speedup on this class of evolution problems but also that we may obtain parallel acceleration beyond what is possible using conventional spatial domain decomposition techniques alone the approach is widely applicable for parallel in time integration over long time domains regardless of the class of evolution problem

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

lq optimal control of infinite dimensional systems by spectral factorization abstract we consider the standard lq optimal control of an exponentially stabilizable and detectable infinite dimensional semigroup state space system with bounded sensing and control it is reported that the reachable restriction of the lq optimal state feedback operator can be identified 1 by solving a spectral factorization problem delivering a bistable spectral factor with entries in the distributed proper stable transfer function algebra and 2 by obtaining any constant solution of an operator diophantine equation over the properties of the restricted solution feedback operator are discussed it is shown in particular that the lq optimal state feedback operator and its reachable restriction coincide whenever the unreachable state components are unobservable these theoretical results are applied to a simple model of heat diffusion with collocated sensor and actuator leading to an approximation procedure converging exponentially fast to the lq optimal state feedback operator this procedure is based on approximate spectral factorization by symmetric extraction and on simple residue calculus

a parareal method for time fractional differential equations in this paper a parareal method is proposed for the parallel in time integration of time fractional differential equations tfdes it is a generalization of the original parareal method proposed for classic differential equations to match the global feature of fractional derivatives the new method has in the correction step embraced the history part of the solution we provide a convergence analysis under the assumption of lipschitz stability conditions we use a multi domain spectral integrator to build the serial solvers and numerical results demonstrate the feasibility of the new approach and confirm the convergence analysis studies also show that both the coarse resolution and the nature of the differential operators can affect the performance

a block krylov subspace implementation of the time parallel paraexp method and its extension for nonlinear partial differential equations a parallel time integration method for nonlinear partial differential equations is proposed it is based on a new implementation of the paraexp method for linear partial differential equations pdes employing a block krylov subspace method for nonlinear pdes the algorithm is based on our paraexp implementation within a waveform relaxation the initial value problem is solved iteratively on a complete time interval nonlinear terms are treated as source terms provided by the solution from the previous iteration at each iteration the problem is decoupled into independent subproblems by the principle of superposition the decoupled subproblems are solved fast by exponential integration based on a block krylov method the new time integration is demonstrated for the one dimensional advection diffusion equation and the viscous burgers equation numerical experiments confirm excellent parallel scaling for the linear advection diffusion problem and good scaling in case the nonlinear burgers equation is simulated

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

variational space time elements for large scale systems abstract in this paper we introduce a new galerkin based formulation for transient continuum problems governed by partial differential equations in space and time therefore we aim at a direct finite element discretization of the space time suitable for massive parallel analysis of the arising large scale problem the proposed formulation is applied to thermal mechanical and fluid systems as well as to a kuramoto sivashinsky problem representing the general class of higher order formulations in material science using nurbs based shape functions we verify whenever possible the conservation properties of the formulation finally a series of examples demonstrate the applicability to all systems presented in this paper

a matched peaceman rachford adi method for solving parabolic interface problems a new peaceman rachford alternating direction implicit pr adi method is proposed in this work for solving two dimensional 2d parabolic interface problems with discontinuous solutions the classical adi schemes are known to be inaccurate for handling interfaces this motivates the development of a matched douglas adi d adi method in the literature in which the finite difference is locally corrected according to the jump conditions however the unconditional stability of the matched adi method cannot be maintained if the d adi is simply replaced by the pr adi to stabilize the computation the tangential derivative approximations in the jump conditions decomposition are substantially improved in this paper moreover a new temporal discretization is adopted for formulating the pr adi method which involves less perturbation terms stability analysis is conducted through eigenvalue spectrum analysis which demonstrates the unconditional stability of the proposed method the matched pr adi method achieves second order of accuracy in space in all tested problems with complex geometries and jumps while maintaining the efficiency of the adi the proposed pr adi method is found to be more accurate than the d adi method in time integration even though its formal temporal order is limited in the matched adi framework

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

optimal linear quadratic control of coupled parabolic hyperbolic pdes abstractthis paper focuses on the optimal control design for a system of coupled parabolic hypebolic partial differential equations by using the infinite dimensional state space description and the corresponding operator riccati equation some dynamical properties of the coupled system of interest are analysed to guarantee the existence and uniqueness of the solution of the linear quadratic lq optimal control problem a state lq feedback operator is computed by solving the operator riccati equation which is converted into a set of algebraic and differential riccati equations thanks to the eigenvalues and the eigenvectors of the parabolic operator the results are applied to a non isothermal packed bed catalytic reactor the lq optimal controller designed in the early portion of the paper is implemented for the original nonlinear model numerical simulations are performed to show the controller performances

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

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

a new multigrid method for unconstrained parabolic optimal control problems abstract a second order leapfrog finite difference scheme in time is proposed and developed for solving the first order necessary optimality system of the distributed parabolic optimal control problems different from available approaches the proposed leapfrog scheme for the two point boundary optimality system is shown to be unconditionally stable and provides a second order accuracy though the classical leapfrog scheme usually is unstable moreover the proposed leapfrog scheme provides a feasible structure that leads to an effective implementation of a fast solver under the multigrid framework a detailed mathematical proof for the stability of the proposed scheme is provided in terms of a new norm that is more suitable and stronger to characterize the convergence than the l 2 norm often used in literature numerical experiments show that the proposed scheme significantly outperforms the widely used second order backward time differentiation approach and the resultant fast solver demonstrates a mesh independent convergence as well as a linear time complexity

convergence analysis of two level space time additive schwarz method for parabolic equations the additive schwarz as method was originally designed for preconditioning elliptic problems and recently the method was extended successfully as a coupled space time preconditioner for parabolic problems however the existing theory for the additive schwarz method doesn t apply directly to the space time discretized problems in this paper we develop an optimal convergence theory for the two level space time additive schwarz preconditioner we obtain lower and upper bounds for the spectrum of the as preconditioned operator and their dependency on the fine coarse mesh sizes in space and time the number of subdomains and the window size i e the number of coupled time steps some numerical experiments are reported to confirm the theory

multigrid methods for parabolic distributed optimal control problems multigrid schemes that solve parabolic distributed optimality systems discretized by finite differences are investigated accuracy properties of finite difference approximation are discussed and validated two multigrid methods are considered which are based on a robust relaxation technique and use two different coarsening strategies semicoarsening and standard coarsening the resulting multigrid algorithms show robustness with respect to changes of the value of v the weight of the cost of the control is sufficiently small fourier mode analysis is used to investigate the dependence of the linear twogrid convergence factor on v and on the discretization parameters results of numerical experiments are reported that demonstrate sharpness of fourier analysis estimates a multigrid algorithm that solves optimal control problems with box constraints on the control is considered

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

optimal control design for time varying catalytic reactors a riccati equation based approach the linear quadratic lq optimal control problem is studied for a partial differential equation model of a time varying catalytic reactor first the dynamical properties of the linearised model are studied next an lq control feedback is computed by using the corresponding operator riccati differential equation whose solution can be obtained via a related matrix riccati partial differential equation finally the designed controller is applied to the non linear reactor system and tested numerically

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 multilevel method for molecular dynamics simulations we present a novel multiscale approach for molecular dynamics simulations it is aimed at bridging discrete scales with either coarse grained discrete or continuum scales the method consists of the waveform relaxation scheme aimed at capturing the high frequency response of atomistic vibrations and a coarse scale solution in space and time aimed at resolving smooth features in both space and time domains of the discrete medium the method is implicit in space and time possesses superior stability properties and consequently enables larger time steps governed by accuracy considerations of coarse scale quantities of interest performance studies on polymer melts have shown significant speed up compared to the classical explicit methods in particular on parallel machines

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

parallel matrix function evaluation via initial value ode modeling the purpose of this article is to propose ode based approaches for the numerical evaluation of matrix functions f a a question of major interest in the numerical linear algebra for that we model f a as the solution at a finite time t of a time dependent equation we use parallel algorithms such as the parareal method on the time interval 0 t in order to solve the obtained evolution equation when f a is reached as a stable steady state it can be computed by combining parareal algorithms and optimal control techniques numerical illustrations are given

large time solution of a nonlinear diffusion equation in this paper we consider the way in which the solution to a class of initial boundary value problems for the plane nonlinear diffusion equation frac partial partial t frac partial partial x u beta frac partial u partial x t geqslant 0 x geqslant 0 beta 0 approaches the similarity form as t rightarrow infty the problem we solve is chosen for two main reasons firstly the equation above is of widespread use in modelling physical situations and secondly it provides a tractable but significant example of a free boundary problem

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

lq boundary control of a diffusion convection reaction system in this work the boundary control of a distributed parameter system modelled by linear parabolic partial differential equations pdes with spatially varying coefficients is studied an infinite dimensional state space setting is considered and an exact transformation of the boundary actuation is realised to obtain an evolutionary model the evolutionary model which incorporates the spatially varying coefficients of the underlying set of the pdes is used for subsequent linear quadratic regulator synthesis the formulated linear quadratic state feedback controller is applied to a nonlinear model of the reactor and its performance is studied

time parallel multiscale multiphysics framework we introduce the time parallel compound wavelet matrix method tpcwm for modeling the temporal evolution of multiscale and multiphysics systems the method couples time parallel tp and cwm methods operating at different spatial and temporal scales we demonstrate the efficiency of our approach on two examples a chemical reaction kinetic system and a non linear predator prey system our results indicate that the tpcwm technique is capable of accelerating time to solution by 2 3 orders of magnitude and is amenable to efficient parallel implementation

decreasing the temporal complexity for nonlinear implicit reduced order models by forecasting abstract implicit numerical integration of nonlinear odes requires solving a system of nonlinear algebraic equations at each time step each of these systems is often solved by a newton like method which incurs a sequence of linear system solves most model reduction techniques for nonlinear odes exploit knowledge of a system s spatial behavior to reduce the computational complexity of each linear system solve however the number of linear system solves for the reduced order simulation often remains roughly the same as that for the full order simulation we propose exploiting knowledge of the model s temporal behavior to 1 forecast the unknown variable of the reduced order system of nonlinear equations at future time steps and 2 use this forecast as an initial guess for the newton like solver during the reduced order model simulation to compute the forecast we propose using the gappy pod technique the goal is to generate an accurate initial guess so that the newton solver requires many fewer iterations to converge thereby decreasing the number of linear system solves in the reduced order model simulation

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

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

an efficient parareal algorithm for a class of time dependent problems with fractional laplacian time dependent diffusion equations with fractional laplacian have received considerable attention in recent years for which numerical methods play an important role because a simple and analytic solution is often unavailable we analyze in this paper a parareal algorithm for this kind of problem which realizes parallel in time computation the algorithm is iterative and uses the 3rd order sdirk singly diagonally implicit runge kutta method with a small step size t as the f propagator and the implicit explicit euler method with a large step size t as the g propagator the two step sizes satisfy t t j with j 2 being an integer using the implicit explicit euler method as the g propagator potentially improves the parallel efficiency but complicates the convergence analysis by employing some technical analysis we provide a sharp estimate of the convergence rate which is independent of the mesh ratio j and the distribution of the eigenvalues of the coefficient matrix an extension of the results to problems with time periodic conditions is also given several numerical experiments are carried out to verify the theoretical results

an improved iterative hdg approach for partial differential equations abstract we propose and analyze an iterative high order hybridized discontinuous galerkin ihdg discretization for linear partial differential equations we improve our previous work 45 in several directions 1 the improved ihdg approach converges in a finite number of iterations for the scalar transport equation 2 it is unconditionally convergent for both the linearized shallow water system and the convection diffusion equation 3 it has improved stability and convergence rates 4 we uncover a relationship between the number of iterations and time stepsize solution order meshsize and the equation parameters this allows us to choose the time stepsize such that the number of iterations is approximately independent of the solution order and the meshsize and 5 we provide both strong and weak scalings of the improved ihdg approach up to 16 384 cores a connection between ihdg and time integration methods such as parareal and implicit explicit methods are discussed extensive numerical results for linear and nonlinear pdes are presented to verify the theoretical findings

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

indirect multiple shooting for nonlinear parabolic optimal control problems with control constraints we discuss the indirect multiple shooting approach for the solution of pde based parabolic optimal control problems with control constraints the method is formulated within an abstract function space setting and uses a space time galerkin finite element discretization the emphasis is on the embedding of indirect multiple shooting into the optimal control framework as well as the detailed description of the discretization within the pde context numerical results for linear and nonlinear model problems with and without control constraints illustrate the efficient use of indirect multiple shooting particularly in cases where other standard methods fail

similarity solutions to three dimensional nonlinear diffusion equations the similarity transformation method has been used to solve three dimensional nonlinear diffusion equations the general lie group is calculated exact solutions to the cylindrical and spherical symmetry cases are found and their relations to some real physical processes are discussed

sufficient conditions for a partially hyperbolic attractor to be a homoclinic class we prove that a partially hyperbolic attractor with two dimensional central direction is a homoclinic class if it exhibits a hyperbolic periodic orbit o and a lorenz like singularity with wu ws o such that ws is dense in

multigrid algorithms for inverse problems with linear parabolic pde constraints we present a multigrid algorithm for the solution of source identification inverse problems constrained by variable coefficient linear parabolic partial differential equations we consider problems in which the inversion variable is a function of space only we consider the case of l 2 tikhonov regularization the convergence rate of our algorithm is mesh independent even in the case of no regularization this feature makes the method algorithmically robust to the value of the regularization parameter and thus useful for the cases in which we seek high fidelity reconstructions the inverse problem is formulated as a pde constrained optimization we use a reduced space approach in which we eliminate the state and adjoint variables and we iterate in the inversion parameter space using conjugate gradients we precondition the hessian with a v cycle multigrid scheme the multigrid smoother is a two step stationary iterative solver that inexactly inverts an approximate hessian by iterating exclusively in the high frequency subspace using a high pass filter we analyze the performance of the scheme for the constant coefficient case with full observations we analytically calculate the spectrum of the reduced hessian and the smoothing factor for the multigrid scheme the forward and adjoint problems are discretized using a backward euler finite difference scheme the overall complexity of our inversion algorithm is mathcal o n tn n log 2n where n is the number of grid points in space and n t is the number of time steps we provide numerical experiments that demonstrate the effectiveness of the method for different diffusion coefficients and values of the regularization parameter we also provide heuristics and we conduct numerical experiments for the case with variable coefficients and partial observations we observe the same complexity as in the constant coefficient case finally we examine the effectiveness of using the reduced space solver as a preconditioner for a full space solver

solution of lambda omega systems theta schemes and multigrid methods the numerical solution of a time dependent reaction diffusion lambda omega system discretized by theta schemes and finite differences is considered stability and accuracy of finite difference theta schemes for this system are established to solve the time implicit evolution equations a nonlinear multigrid method is applied the convergence properties of this solver are investigated considering a linearized lambda omega model

a leapfrog multigrid algorithm for the optimal control of parabolic pdes with robin boundary conditions we develop a second order finite difference scheme for solving the first order necessary optimality systems arising from the optimal control of parabolic pdes with robin boundary conditions under the framework of matrix analysis the proposed leapfrog scheme is shown to be unconditionally stable and second order convergent for both time and spatial variables without the requirement of the classical courant friedrichs lewy cfl condition on the spatial and temporal mesh step sizes moreover the developed leapfrog scheme provides a well structured discrete algebraic system that allows us to establish an effective multigrid iterative fast solver the resultant multigrid solver demonstrates a mesh independent convergence rate and a linear time complexity numerical experiments are provided to illustrate the accuracy and efficiency of the proposed leapfrog scheme

the sil nikov problem exponential expansion strong λ lemma c1 linearization and homoclinic bifurcation on introduit une nouvelle theorie appelee le developpement exponentiel et on explore ses interactions avec la theorie de la variete invariante locale les lemmes et la theorie de linerarisation reguliere