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

a time domain decomposition iterative method for the solution of distributed linear quadratic optimal control problems we study a class of time domain decomposition based methods for the numerical solution of large scale linear quadratic optimal control problems our methods are based on a multiple shooting reformulation of the linear quadratic optimal control problem as a discrete time optimal control dtoc problem the optimality conditions for this dtoc problem lead to a linear block tridiagonal system the diagonal blocks are invertible and are related to the original linear quadratic optimal control problem restricted to smaller time subintervals this motivates the application of block gauss seidel gs type methods for the solution of the block tridiagonal systems numerical experiments show that the spectral radii of the block gs iteration matrices are larger than one for typical applications but that the eigenvalues of the iteration matrices decay to zero fast hence while the gs method is not expected to convergence for typical applications it can be effective as a preconditioner for krylov subspace methods this is confirmed by our numerical tests a byproduct of this research is the insight that certain instantaneous control techniques can be viewed as the application of one step of the forward block gs method applied to the dtoc optimality system

quenched annealed and functional large deviations for one dimensional random walk in random environment suppose that the integers are assigned random variables i taking values in the unit interval which serve as an environment this environment defines a random walk x n called a rwre which when at i moves one step to the right with probability i and one step to the left with probability 1 i when the i sequence is i i d greven and den hollander 1994 proved a large deviation principle for x n n conditional upon the environment with deterministic rate function we consider in this paper large deviations both conditioned on the environment quenched and averaged on the environment annealed for the rwre in the ergodic environment case the annealed rate function is the solution of a variational problem involving the quenched rate function and specific relative entropy we also give a detailed qualitative description of the resulting rate functions our techniques differ from those of greven and den hollander and allow us to present also a trajectorial quenched large deviation principle

stochastic dominance based comparison for system selection abstract we present two complementing selection procedures for comparing simulated systems based on the stochastic dominance relationship of a performance metric of interest the decision maker specifies an output quantile set representing a section of the distribution of the metric e g downside or upside risks or central tendencies as the basis for comparison the first procedure compares systems over the quantile set of interest by a first order stochastic dominance criterion the systems that are deemed nondominant in the first procedure could be compared by a weaker almost first order stochastic dominance criterion in the second procedure numerical examples illustrate the capabilities of the proposed procedures

large deviations for empirical process of mean field interacting particle system with unbounded jumps a large deviation system is established for the empirical processes of a mean field interacting particle system with unbounded jump rates under assumptions that are satisfied by many interesting models including the first and the second schlogl models the action functional obtained has a form that is very useful for applications

laplace approximations for sums of independent random vectors we consider expressions of the form zn e exp nf sn n where sn is the sum of n i i d random vectors with values in a banach space and f is a smooth real valued function by results of donsker varadhan and bahadur zabell one knows that lim 1 n log zn supxf x h x where h is the so called entropy function in an earlier paper a more precise evaluation of zn is given in the case where there was a unique point maximizing f h and the curvature at the maximum was nonvanishing the present paper treats the more delicate problem where these conditions fail to hold

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

quantile estimation in dependent sequences standard nonparametric estimators of quantiles based on order statistics can be used not only when the data are i i d but also when the data are drawn from a stationary i mixing process of continuous random variables however when the random variables are highly positively correlated the sample sizes needed for estimating extreme quantiles become computationally unmanageable this paper gives a practical scheme based on a maximum transformation in a two way layout of the data that reduces the sample size sufficiently to allow an experimenter to obtain a point estimate of an extreme quantile the paper gives three schemes that lead to confidence interval estimates for the quantile one uses a spectral analysis of the reduced sample the other two averaged group quantiles and nested group quantiles are extensions of the method of batched means to quantile estimation these two schemes give even greater data compaction than the first scheme none of the schemes requires that the process being simulated is regenerative

a quantile based approach to system selection abstract we propose a quantile based ranking and selection r s procedure for comparing a finite set of stochastic systems via simulation our r s procedure uses a quantile set of the simulated probability distribution of a performance characteristic of interest that best represents the most appropriate selection criterion as the basis for comparison since this quantile set may represent either the downside risk upside risk or central tendency of the performance characteristic the proposed approach is more flexible than the traditional mean based approach to r s we first present a procedure that selects the best system from among k systems and then we modified that procedure for the case where k 1 systems are compared against a standard system we present a set of experiments to highlight the flexibility of the proposed procedures

fast algorithm based on tt m fe system for space fractional allen cahn equations with smooth and non smooth solutions abstract in this article a fast algorithm based on time two mesh tt m finite element fe scheme which aims at solving nonlinear problems quickly is considered to numerically solve the nonlinear space fractional allen cahn equations with smooth and non smooth solutions the implicit second order scheme containing both implicit crank nicolson scheme and second order backward difference method is applied to time direction a fast tt m method is used to increase the speed of calculation and the fe method is developed to approximate the spacial direction the tt m fe algorithm includes the following main computing steps firstly a nonlinear implicit second order fe scheme on the time coarse mesh c is solved by a nonlinear iterative method secondly based on the chosen initial iterative value a linearized fe system on time fine mesh c is solved where some useful coarse numerical solutions are found by the lagrange s interpolation formula the analysis for both stability and a priori error estimates are made in detail finally three numerical examples with smooth and non smooth solutions are provided to illustrate the computational efficiency in solving nonlinear partial differential equations from which it is easy to find that the computing time can be saved

control variates for probability and quantile estimation in stochastic systems quantiles indicate the level of system performance that can be delivered with a specified probability while probabilities indicate the likelihood that a specified level of system performance can be achieved we present new estimators for use in simulation experiments designed to estimate such quantiles or probabilities of system performance all of the estimators exploit control variates to increase their precision which is especially important when extreme quantiles in the tails of the distribution of system performance or extreme probabilities near zero or one are of interest control variates are auxiliary random variables with known properties in this case known quantiles and a strong stochastic association with the performance measure of interest since transforming a control variate can increase its effectiveness we propose both continuous and discrete approximations to the optimal variance minimizing transformation for estimating probabilities and then invert the probability estimators to obtain corresponding quantile estimators we also propose a direct control variate quantile estimator that is not based on inverting a probability estimator an empirical study using queueing inventory and project planning examples shows that substantial reductions in mean squared error can be obtained when estimating the 0 9 0 95 and 0 99 quantiles

entropy limit theorems and variational principles for disordered lattice systems we study infinite volume limits and gibbs states of disordered lattice systems with bounded and continuous potentials our main tools are a generalization of relative entropy for random reference measures and a large deviation theory for nonstationary independent processes we find that many familiar results of invariant potentials such as large deviation theorems variational principles and equivalence of ensembles continue to hold for disordered models with suitably modified statements

stochastic homogenization of hamilton jacobi bellman equations we study the homogenization of some hamilton jacobi bellman equations with a vanishing second order term in a stationary ergodic random medium under the hyperbolic scaling of time and space imposing certain convexity growth and regularity assumptions on the hamiltonian we show the locally uniform convergence of solutions of such equations to the solution of a deterministic effective first order hamilton jacobi equation the effective hamiltonian is obtained from the original stochastic hamiltonian by a minimax formula our homogenization results have a large deviations interpretation for a diffusion in a random environment c 2006 wiley periodicals inc

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 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

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

technology scoring model for reflecting evaluator s perception within confidence limits abstract the main purpose of this study is to propose a new technology scoring model for reflecting the total perception scoring phenomenon which happens often in many evaluation settings a base model used is a logistic regression for non default prediction of a firm the point estimator used to predict the probability for non default based on this model does not consider the risk involved in the estimation error we propose to update the point estimator within its confidence interval using the evaluator s perception the proposed approach takes into account not only the risk involved in the estimation error of the point estimator but also the total perception scoring phenomenon empirical evidence of a better prediction ability of the proposed model is displayed in terms of the area under the roc curves additionally we showed that the proposed model can take advantage when it is applied to smaller data size it is expected that the proposed approach can be applied to various technology related decision makings such as r d investment alliance transfer and loan

large deviation properties of weakly interacting processes via weak convergence methods we study large deviation properties of systems of weakly interacting particles modeled by ito stochastic differential equations sdes it is known under certain conditions that the corresponding sequence of empirical measures converges as the number of particles tends to infinity to the weak solution of an associated mckean vlasov equation we derive a large deviation principle via the weak convergence approach the proof which avoids discretization arguments is based on a representation theorem weak convergence and ideas from stochastic optimal control the method works under rather mild assumptions and also for models described by sdes not of diffusion type to illustrate this we treat the case of sdes with delay

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

control variates for quantile estimation new point and interval estimators for quantiles that employ a control variate are introduced the properties of these estimators do not depend on the usual assumption of joint normality between the random variable of interest and the control illustrative examples for queueing and stochastic activity network models are given in those examples the new estimators are superior to the standard estimator in terms of the mean squared error of the point estimator and the length of the confidence interval

synchronization and random long time dynamics for mean field plane rotators we consider the natural langevin dynamics which is reversible with respect to the mean field plane rotator or classical spin xy measure it is well known that this model exhibits a phase transition at a critical value of the interaction strength parameter k in the limit of the number n of rotators going to infinity a fokker planck pde captures the evolution of the empirical measure of the system as n rightarrow infty at least for finite times and when the empirical measure of the system at time zero satisfies a law of large numbers the phase transition is reflected in the fact that the pde for k above the critical value has several stationary solutions notably a stable manifold in fact a circle of stationary solutions that are equivalent up to rotations these stationary solutions are actually unimodal densities parametrized by the position of their maximum the synchronization phase or center we characterize the dynamics on times of order n and we show substantial deviations from the behavior of the solutions of the pde in fact if the empirical measure at time zero converges as n rightarrow infty to a probability measure which is away from a thin set that we characterize and if time is speeded up by n the empirical measure reaches almost instantaneously a small neighborhood of the stable manifold to which it then sticks and on which a non trivial random dynamics takes place in fact the synchronization center performs a brownian motion with a diffusion coefficient that we compute our approach therefore provides for one of the basic statistical mechanics systems with continuum symmetry a detailed characterization of the macroscopic deviations from the large scale limit or law of large numbers due to finite size effects but the interest for this model goes beyond statistical mechanics since it plays a central role in a variety of scientific domains in which one aims at understanding synchronization phenomena

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

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

a multi domain spectral method for time fractional differential equations this paper proposes an approach for high order time integration within a multi domain setting for time fractional differential equations since the kernel is singular or nearly singular two main difficulties arise after the domain decomposition how to properly account for the history memory part and how to perform the integration accurately to address these issues we propose a novel hybrid approach for the numerical integration based on the combination of three term recurrence relations of jacobi polynomials and high order gauss quadrature the different approximations used in the hybrid approach are justified theoretically and through numerical examples based on this we propose a new multi domain spectral method for high order accurate time integrations and study its stability properties by identifying the method as a generalized linear method numerical experiments confirm hp convergence for both time fractional differential equations and time fractional partial differential equations

correlation induction techniques for estimating quantiles in simulation experiments a simulation based quantile estimator measures the level of system performance that can be delivered with a prespecified probability to estimate selected quantiles of the response of a finite horizon simulation we develop procedures based on correlation induction techniques for variance reduction with emphasis on antithetic variates and latin hypercube sampling these procedures achieve improved precision by controlling the simulation s random number inputs as an integral part of the experimental design the proposed multiple sample quantile estimator is the average of negatively correlated quantile estimators computed from disjoint samples of the simulation response where negative correlation is induced between corresponding responses in different samples while mutual independence of responses is maintained within each sample the proposed single sample quantile estimator is computed from negatively correlated simulation responses within one all inclusive sample the single sample estimator based on latin hypercube sampling is shown to be asymptotically normal and unbiased with smaller variance than the comparable directsimulation estimator based on independent replications similar asymptotic comparisons of the multiple sample and directsimulation estimators focus on bias and mean square error monte carlo results suggest that the proposed procedures can yield significant reductions in bias variance and mean square error when estimating quantiles of the completion time of a stochastic activity network

quenched free energy and large deviations for random walks in random potentials we study quenched distributions on random walks in a random potential on integer lattices of arbitrary dimension and with an arbitrary finite set of admissible steps the potential can be unbounded and can depend on a few steps of the walk directed undirected and stretched polymers as well as random walk in random environment are covered the restriction needed is on the moment of the potential in relation to the degree of mixing of the ergodic environment we derive two variational formulas for the limiting quenched free energy and prove a process level quenched large deviation principle ldp for the empirical measure as a corollary we obtain ldps for types of random walks in random environments not covered by earlier results 2012 wiley periodicals inc

a batching approach to quantile estimation in regenerative simulations this paper presents a new method called the batch quantile method for estimating quantiles in regenerative simulations the quantile estimator is consistent and asymptotically normal and the method can be easily implemented and does not require prior knowledge of the range of values the data will assume empirical studies show adequate coverage of confidence intervals when batches of 50 cycles or more are used

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

interacting diffusions on random graphs with diverging average degrees hydrodynamics and large deviations we consider systems of mean field interacting diffusions where the pairwise interaction structure is described by a sparse and potentially inhomogeneous random graph examples include the stochastic kuramoto model with pairwise interactions given by an erd s renyi graph our problem is to compare the bulk behavior of such systems with that of corresponding systems with dense nonrandom interactions for a broad class of interaction functions we find the optimal sparsity condition that implies that the two systems have the same hydrodynamic limit which is given by a mckean vlasov diffusion moreover we also prove matching behavior of the two systems at the level of large deviations our results extend classical results of dai pra and den hollander and provide the first examples of ldps for systems with sparse random interactions

quenched large deviations for interacting diffusions in random media the aim of the paper is to establish a large deviation principle ldp for the empirical measure of mean field interacting diffusions in a random environment the point is to derive such a result once the environment has been frozen quenched model the main theorem states that a ldp holds for every sequence of environment satisfying appropriate convergence condition with a rate function that does not depend on the disorder and is different from the rate function in the averaged model similar results concerning the empirical flow and local empirical measures are provided

analysis of a new finite difference local discontinuous galerkin method for the fractional diffusion wave equation in this paper a finite difference local discontinuous galerkin method for the fractional diffusion wave equation is presented and analyzed we first propose a new finite difference method to approximate the time fractional derivatives and give a semidiscrete scheme in time further we develop a fully discrete scheme for the fractional diffusion wave equation and prove that the method is unconditionally stable and convergent with order o hk 1 t 3 where k is the degree of piecewise polynomial extensive numerical examples are carried out to confirm the theoretical convergence rates

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

monotonic time discretized schemes in quantum control most of the numerical simulations in quantum bilinear control have used monotonically convergent algorithms of krotov introduced by tannor et al 15 of zhu rabitz 16 or their unified formulation in 17 however the properties of the discrete version of these procedures have not been yet tackled with we present in this paper a stable time and space discretization which preserves the monotonic properties of the monotonic algorithms numerical results show that the newly derived algorithms are stable and enable various experimentations

quantile and tolerance interval estimation in simulation this paper discusses two sequential procedures to construct proportional half width confidence intervals for a simulation estimator of the steady state quantile and tolerance intervals for a stationary stochastic process having the reasonable property that the autocorrelation of the underlying process approaches zero with increasing lag at each quantile to be estimated the marginal cumulative distribution function must be absolutely continuous in some neighborhood of that quantile with a positive continuous probability density function these algorithms sequentially increase the simulation run length so that the quantile and tolerance interval estimates satisfy pre specified precision requirements an experimental performance evaluation demonstrates the validity of these procedures

time domain decomposition of optimal control problems for the wave equation we consider the problem of boundary optimal control of a wave equation with boundary dissipation by the way of time domain decomposition of the corresponding optimality system we develop an iterative algorithm which shows that the decomposed optimality system corresponds to local in time optimal control problems which can be treated in parallel we show convergence of the algorithm finally we provide a time discretization which is reminiscent of an instantaneous control scheme we thereby also contribute to the problem of convergence of such schemes

large deviation estimates for a conditional probability distribution applications to random interaction gibbs measures let x i y i d be independent identically distributed random variables with arbitrary distribution we show that for almost every y i i the conditional law of the empirical field given y i i satisfies to large deviation inequalities this applies to the study of gibbs measures with random interaction in the case of some mean field models as well as of short range summable interaction we show that the pressure is nonrandom and is given by a variational formula these random gibbs measures have the same large deviation rate which does not depend on the particular realization of the interaction their local behaviour is described in terms of conditional probabilities given the interaction of solutions to the variational formula