on variational inequalities with maximal monotone operators and multivalued perturbing terms in sobolev spaces with variable exponents abstract we are concerned in this paper with variational inequalities of the form a u v u f u v u l v u v k u k where a is a maximal monotone operator f is an integral multivalued lower order term and k is a closed convex set in a sobolev space of variable exponent we study both coercive and noncoercive inequalities in the noncoercive case a sub supersolution approach is followed to obtain the existence and some other qualitative properties of solutions between sub and supersolutions

an efficient implicit fem scheme for fractional in space reaction diffusion equations fractional differential equations are becoming increasingly used as a modelling tool for processes associated with anomalous diffusion or spatial heterogeneity however the presence of a fractional differential operator causes memory time fractional or nonlocality space fractional issues that impose a number of computational constraints in this paper we develop efficient scalable techniques for solving fractional in space reaction diffusion equations using the finite element method on both structured and unstructured grids via robust techniques for computing the fractional power of a matrix times a vector our approach is show cased by solving the fractional fisher and fractional allen cahn reaction diffusion equations in two and three spatial dimensions and analyzing the speed of the traveling wave and size of the interface in terms of the fractional power of the underlying laplacian operator

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

a micro macro parareal algorithm application to singularly perturbed ordinary differential equations we introduce a micro macro parareal algorithm for the time parallel integration of multiscale in time systems the algorithm first computes a cheap but inaccurate solution using a coarse propagator simulating an approximate slow macroscopic model which is iteratively corrected using a fine scale propagator accurately simulating the full microscopic dynamics this correction is done in parallel over many subintervals thereby reducing the wall clock time needed to obtain the solution compared to the integration of the full microscopic model over the complete time interval we provide a numerical analysis of the algorithm for a prototypical example of a micro macro model namely singularly perturbed ordinary differential equations we show that the computed solution converges to the full microscopic solution when the parareal iterations proceed only if special care is taken during the coupling of the microscopic and macroscopic levels of description the error bound depends on the modeling error

parareal multiscale methods for highly oscillatory dynamical systems we introduce a new strategy for coupling the parallel in time parareal iterative methodology with multiscale integrators following the parareal framework the algorithm computes a low cost approximation of all slow variables in the system using an appropriate multiscale integrator which is refined using parallel fine scale integrations convergence is obtained using an alignment algorithm for fast phase like variables the method may be used either to enhance the accuracy and range of applicability of the multiscale method in approximating only the slow variables or to resolve all the state variables the numerical scheme does not require that the system is split into slow and fast coordinates moreover the dynamics may involve hidden slow variables for example due to resonances we propose an alignment algorithm for almost periodic solutions in which case convergence of the parareal iterations is proved the applicability of the method is demonstrated in numerical examples

on a sub supersolution method for variational inequalities with leray lions operators in variable exponent spaces abstract in this paper we consider a sub supersolution method for variational inequalities with leray lions operators in sobolev spaces with variable exponents existence and qualitative properties of solutions between sub and supersolutions are established

elliptic inequalities with multi valued operators existence comparison and related variational hemivariational type inequalities abstract we study multi valued elliptic variational inclusions in a bounded domain r n of the form u k 0 a u i k u f u f u where a is a second order quasilinear elliptic operator of leray lions type k is a closed convex subset of some sobolev space i k is the indicator function related to k and i k denoting its subdifferential the lower order multi valued operators f and f are generated by multi valued upper semicontinuous functions f r 2 r 0 and f r 2 r 0 respectively with our main goals are as follows first we provide an existence theory for the above multi valued variational inequalities second we establish an enclosure and comparison principle based on appropriately defined sub supersolutions and prove the existence of extremal solutions third by means of the sub supersolution method provided here we are going to show that rather general classes of variational hemivariational type inequalities turn out to be only subclasses of the above general multi valued elliptic variational inequalities which in a way fills a gap in the current literature where these kind of problems are studied independently finally the existence of extremal solutions will allow us to deal with classes of multi valued function f and f that are neither lower nor upper semicontinuous which in turn will provide a tool to obtain existence results for variational hemivariational type inequalities whose clarke s generalized directional derivative may in addition discontinuously depend on the function we are looking for this paper though of surveying nature provides an analytical framework that allows to present in a unifying way and to extend a number of recent results due to the authors

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

on the convergence of solutions of inclusions containing maximal monotone and generalized pseudomonotone mappings abstract we are concerned in this paper with the existence boundedness and the convergence of solutions to a sequence of inclusions a k u b k u l k where a k is a maximal monotone mapping b k is a generalized pseudomonotone mapping defined on a reflexive banach space x and l k x we study appropriate kinds of convergence for a k and b k such that a limit of a sequence of solutions of these inclusions is also a solution of the limit inclusion

monotonic parareal control for quantum systems following encouraging experimental results in quantum control numerical simulations have known significant improvements through the introduction of efficient optimization algorithms yet the computational cost still prevents using these procedures for high dimensional systems often present in quantum chemistry using parareal framework we present here a time parallelization of these schemes which allows us to reduce significantly their computational cost while still finding convenient controls

asymptotic behavior of positive solutions of a nonlinear integral system abstract in this paper we study the properties of positive solutions of an integral system in r n u x r n v p y u q y x y n y d y v x r n v q y u p y x y n y d y we prove the radial symmetry of positive solutions by means of the method of moving planes in integral forms which is established by chen li ou in addition by using a regularity lifting lemma we obtain an integrability result under the assumption p q 1 0 we prove that those solutions are the ground states finally we estimate the decay rates of such solutions when x

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

a 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

signorini type variational inequality with state dependent discontinuous multi valued boundary operators abstract we consider an elliptic variational inequality of signorini type in a bounded domain r n of the form find u k and 2 u u such that a u v u s v u d 0 v k where a is a quasilinear elliptic operator in divergence form k is the nonempty closed convex set representing the thin obstacle signorini problem on s and w 1 p l p denotes the trace operator the multi valued boundary operator is generated by the multifunction s 2 s s where r r r is a function such that s r s is supposed to be locally lipschitz for all r r while r r s is allowed to be discontinuous and 2 r s stands for clarke s generalized gradient of with respect to its second argument the novelty of this paper is that the multifunction r 2 r s may discontinuously depend on r in a certain specified way which gives rise to a new class of discontinuous nonmonotone multi valued boundary operators that is rich in structure to cover a wide range of multi valued constitutive laws on the contact surface s such as e g multi valued adhesion and friction laws that are not necessarily of subdifferential type our main goal is to provide an analytical framework to prove existence enclosure and comparison results for this new class of multi valued boundary variational inequalities that includes the theory of variational hemivariational inequalities as special case

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

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

positive solutions to integral systems with weight and bessel potentials abstract in this paper we consider the integral system with weight and the bessel potentials u x r n g x y u y p v y q y d y v x r n g x y v y p u y q y d y where u v 0 0 0 n p q 2 and g x is the bessel potential of order first we get the integrability by regularity lifting lemma then we also establish the regularity of the positive solutions afterwards by the method of moving planes in integral forms we show that the positive solutions are radially symmetric and monotone decreasing about the origin finally by an extension of the idea of lei 14 and analytical techniques we get the decay rates of solutions when x

decay symmetry and existence of solutions of semilinear elliptic systems our utmost aim was originally to establish the existence of solutions of system however in this effort we got sidetracked to consider other questions in this way we come to interesting results on asymptotic behavior of solutions on symmetry properties of such solutions and the existence of ground states

parallel in time algorithm with spectral subdomain enhancement for volterra integral equations nsf of china 11201077 11071203 91130002 nsf of fujian province 2012j01007 fuzhou university 0460022456 hong kong research grant council gif grants hong kong baptist university frg grants

efficient stochastic galerkin methods for random diffusion equations we discuss in this paper efficient solvers for stochastic diffusion equations in random media we employ generalized polynomial chaos gpc expansion to express the solution in a convergent series and obtain a set of deterministic equations for the expansion coefficients by galerkin projection although the resulting system of diffusion equations are coupled we show that one can construct fast numerical methods to solve them in a decoupled fashion the methods are based on separation of the diagonal terms and off diagonal terms in the matrix of the galerkin system we examine properties of this matrix and show that the proposed method is unconditionally stable for unsteady problems and convergent for steady problems with a convergent rate independent of discretization parameters numerical examples are provided for both steady and unsteady random diffusions to support the analysis

an asymptotic parallel in time method for highly oscillatory pdes we present a new time stepping algorithm for nonlinear pdes that exhibit scale separation in time of a highly oscillatory nature the algorithm combines the parareal method a parallel in time scheme introduced in j l lions y maday and g turinici c r acad sci paris ser i math 332 2001 pp 661 668 with techniques from the heterogeneous multiscale method cf w e and b engquist notices amer math soc 50 2003 pp 1062 1070 which make use of the slow asymptotic structure of the equations a j majda and p embid theoret comput fluid dyn 11 1998 pp 155 169 we present error bounds based on the analysis in m j gander and e hairer in domain decomposition methods in science and engineering xvii springer berlin 2008 pp 45 56 and g bal in domain decomposition methods in science and engineering springer berlin 2005 pp 425 432 that demonstrate convergence of the method a complexity analysis also demonstrates that the parallel speedup increases

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

fast parareal iterations for fractional diffusion equations abstract numerical methods for fractional pdes is a hot topic recently this work is concerned with the parareal algorithm for system of odes u t a u t f that arising from semi discretizations of time dependent fractional diffusion equations with nonsymmetric riemann liouville fractional derivatives the spatial semi discretization of this kind of fractional derivatives often results in a coefficient matrix a with spectrum a satisfying a s c r where 0 is a measure of dissipativity of the differential equations to accelerate the parareal algorithm we propose a scaled model u t 1 a u t f with 0 to serve the coarse grid correction which is an important component of our parareal algorithm given and we derive a sharp bound of the convergence factor of the parareal iterations moreover by minimizing such a bound we get optimized scaling factor opt it is shown that compared to 1 i e the classical implementation pattern of the coarse grid correction the optimized scaling factor significantly improves the convergence rate numerical examples are presented to support the theoretical finding