time varying discrete riccati equation in terms of ben artzi gohberg dichotomy abstract necessary and sufficient conditions for the existence of the stabilizing solution to the time varying discrete riccati equation are derived in terms of the so called ben artzi gohberg dichotomy it is shown that the problem of such existence conditions is strongly related to that of unique solvability in vector valued l2 spaces of a system of singular difference equations which in this case is termed the extended hamiltonian system ehs connections between the existence of the stabilizing solution to the riccati equation and the bounded invertibility of an appropriate l2 operator are pointed out via the notion of disconjugacy

a martingale approach to homogenization of unbounded random flows we study the asymptotic behavior of brownian motion in steady unbounded incompressible random flows we prove an invariance principle for almost all realizations of random flows the key compactness result is obtained by moser s iterative scheme in pde theory

analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients we consider the problem of numerically approximating statistical moments of the solution of a time dependent linear parabolic partial differential equation pde whose coefficients and or forcing terms are spatially correlated random fields the stochastic coefficients of the pde are approximated by truncated karhunen loeve expansions driven by a finite number of uncorrelated random variables after approximating the stochastic coefficients the original stochastic pde turns into a new deterministic parametric pde of the same type the dimension of the parameter set being equal to the number of random variables introduced after proving that the solution of the parametric pde problem is analytic with respect to the parameters we consider global polynomial approximations based on tensor product total degree or sparse polynomial spaces and constructed by either a stochastic galerkin or a stochastic collocation approach we derive convergence rates for the different cases and present numerical results that show how these approaches are a valid alternative to the more traditional monte carlo method for this class of problems copyright 2009 john wiley sons ltd

error analysis of a stochastic collocation method for parabolic partial differential equations with random input data a stochastic collocation method for solving linear parabolic partial differential equations with random coefficients forcing terms and initial conditions is analyzed the input data are assumed to depend on a finite number of random variables unlike previous analyses a wider range of situations are considered including input data that depend nonlinearly on the random variables and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate the exponential decay of the interpolation error in the probability space for both finite element semidiscrete spatial discretizations and for finite element crank nicolson fully discrete space time discretizations ingredients in the convergence analysis include the proof of the analyticity with respect to the probabilistic parameters of the semidiscrete and fully discrete approximate solutions a numerical example is provided to illustrate the analyses

multigrid methods and sparse grid collocation techniques for parabolic optimal control problems with random coefficients an efficient computational framework to solve nonlinear parabolic optimal control problems with random coefficients is presented this framework allows us to investigate the influence of randomness or uncertainty of problem s parameters values on the control provided by the optimal control theory the proposed framework combines space time multigrid methods with sparse grid collocation techniques theoretical and numerical results of computation of stochastic optimal control solutions and formulation of mean control functions are presented

quenched invariance principles for random walks on percolation clusters we consider a supercritical bernoulli percolation model in d2 and study the simple symmetric random walk on the infinite percolation cluster the aim of this paper is to prove the almost sure q

harnack inequalities on weighted graphs and some applications to the random conductance model we establish elliptic and parabolic harnack inequalities on graphs with unbounded weights as an application we prove a local limit theorem for a continuous time random walk x in an environment of ergodic random conductances taking values in 0 infty satisfying some moment conditions

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

invariance principle for the random conductance model we study a continuous time random walk x in an environment of i i d random conductances mu e in 0 infty in mathbb z d we assume that mathbb p mu e 0 p c so that the bonds with strictly positive conductances percolate but make no other assumptions on the law of the e we prove a quenched invariance principle for x and obtain green s functions bounds and an elliptic harnack inequality

limit theory for random walks in degenerate time dependent random environments abstract we study continuous time variable speed random walks in random environments on z d d 2 where at time t the walk at x jumps across edge x y at time dependent rate a t x y the rates which we assume stationary and ergodic with respect to space time shifts are symmetric and bounded but possibly degenerate in the sense that the total jump rate from a vertex may vanish over finite intervals of time we formulate conditions on the environment under which the law of diffusively scaled random walk path tends to brownian motion for almost every sample of the rates the proofs invoke moser iteration to prove sublinearity of the corrector in pointwise sense a key additional input is a conversion of certain weighted energy norms to ordinary ones our conclusions apply to random walks on dynamical bond percolation and interacting particle systems as well as to random walks arising from the helffer sjostrand representation of gradient models with certain non strictly convex potentials

optimal control with stochastic pde constraints and uncertain controls abstract the optimal control of problems that are constrained by partial differential equations with uncertainties and with uncertain controls is addressed the lagrangian that defines the problem is postulated in terms of stochastic functions with the control function possibly decomposed into an unknown deterministic component and a known zero mean stochastic component the extra freedom provided by the stochastic dimension in defining cost functionals is explored demonstrating the scope for controlling statistical aspects of the system response one shot stochastic finite element methods are used to find approximate solutions to control problems it is shown that applying the stochastic collocation finite element method to the formulated problem leads to a coupling between stochastic collocation points when a deterministic optimal control is considered or when moments are included in the cost functional thereby forgoing the primary advantage of the collocation method over the stochastic galerkin method for the considered problem the application of the presented methods is demonstrated through a number of numerical examples the presented framework is sufficiently general to also consider a class of inverse problems and numerical examples of this type are also presented

comparison of quenched and annealed invariance principles for random conductance model we show that there exists an ergodic conductance environment such that the weak annealed invariance principle holds for the corresponding continuous time random walk but the quenched invariance principle does not hold

averaged control we analyze the problem of controlling parameter dependent systems we introduce the notion of averaged control according to which the quantity of interest is the average of the states with respect to the parameter first we consider the problem of controllability for linear finite dimensional systems and show that a necessary and sufficient condition for averaged controllability is an averaged rank condition in the spirit of the classical rank condition for linear control systems but involving averaged momenta of any order of the matrices generating the dynamics and representing the control action we also describe some open problems and directions of possible research in particular on the average controllability of evolution partial differential equations in this context we analyze also the averaged version of a classical optimal control problem for a parameter dependent elliptic equation and derive the corresponding optimality system

quenched invariance principles for walks on clusters of percolation or among random conductances in this work we principally study random walk on the supercritical infinite cluster for bond percolation on d we prove a quenched functional central limit theorem for the walk when d 4 we also prove a similar result for random walk among i i d random conductances along nearest neighbor edges of d when d 1

on the mixing time of a simple random walk on the super critical percolation cluster we study the robustness under perturbations of mixing times by studying mixing times of random walks in percolation clusters inside boxes in z d we show that for d 2 and p p c z d the mixing time of simple random walk on the largest cluster inside is n 2 thus the mixing time is robust up to a constant factor the mixing time bound utilizes the lovasz kannan average conductance method this is the first non trivial application of this method which yields a tight result

quenched invariance principle for random walks with time dependent ergodic degenerate weights we study a continuous time random walk xx on zdzd in an environment of dynamic random conductances taking values in 0 0 we assume that the law of the conductances is ergodic with respect to space time shifts we prove a quenched invariance principle for the markov process xx under some moment conditions on the environment the key result on the sublinearity of the corrector is obtained by moser s iteration scheme

extended nehari problem for time variant discrete systems abstract in this paper a suboptimal solution to the time variant discrete version of the extended nehari problem is given the main tool used to obtain such a solution with prescribed tolerance is the so called anticausal stabilizing solution to the reverse discrete time riccati equation

isoperimetry and heat kernel decay on percolation clusters we prove that the heat kernel on the infinite bernoulli percolation cluster in zd almost surely decays faster than t d 2 we also derive estimates on the mixing time for the random walk confined to a finite box our approach is based on local isoperimetric inequalities some of the results of this paper were previously announced in the note of mathieu and remy c r acad sci paris ser i math 332 2001 927 931

invariance principle for the random conductance model in a degenerate ergodic environment we study a continuous time random walk x on zd in an environment of random conductances taking values in 0 we assume that the law of the conductances is ergodic with respect to space shifts we prove a quenched invariance principle for x under some moment conditions of the environment the key result on the sublinearity of the corrector is obtained by moser s iteration scheme

anticausal stabilizing solution to discrete reverse time riccati equation abstract this paper is dealing with the problem of general conditions for the existence of the anticausal stabilizing solution to the discrete reverse time riccati equation the connection between the contracting property of the l2 input output operator of a linear internally stable system and the existence of the anticausal stabilizing solution to the above mentioned equation are pointed out

an anisotropic sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes an anisotropic sparse grid stochastic collocation method for solving partial differential equations with random coefficients and forcing terms input data of the model the method consists of a galerkin approximation in the space variables and a collocation in probability space on sparse tensor product grids utilizing either clenshaw curtis or gaussian knots even in the presence of nonlinearities the collocation approach leads to the solution of uncoupled deterministic problems just as in the monte carlo method this work includes a priori and a posteriori procedures to adapt the anisotropy of the sparse grids to each given problem these procedures seem to be very effective for the problems under study the proposed method combines the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation the first approach is effective for problems depending on random variables which weigh approximately equally in the solution while the benefits of the latter approach become apparent when solving highly anisotropic problems depending on a relatively small number of random variables as in the case where input random variables are karhunen loeve truncations of smooth random fields this work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates sub exponential convergence in the asymptotic regime and algebraic convergence in the preasymptotic regime with respect to the total number of collocation points it also shows that the anisotropic approximation breaks the curse of dimensionality for a wide set of problems numerical examples illustrate the theoretical results and are used to compare this approach with several others including the standard monte carlo in particular for moderately large dimensional problems the sparse grid approach with a properly chosen anisotropy seems to be very efficient and superior to all examined methods

a sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes a smolyak type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms input data of the model to compute solution statistics the sparse grid stochastic collocation method uses approximate solutions produced here by finite elements corresponding to a deterministic set of points in the random input space this naturally requires solving uncoupled deterministic problems as in the monte carlo method if the number of random variables needed to describe the input data is moderately large full tensor product spaces are computationally expensive to use due to the curse of dimensionality in this case the sparse grid approach is still expected to be competitive with the classical monte carlo method therefore it is of major practical relevance to understand in which situations the sparse grid stochastic collocation method is more efficient than monte carlo this work provides error estimates for the fully discrete solution using l q norms and analyzes the computational efficiency of the proposed method in particular it demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem number of input random variables in the final estimates the derived estimates are then used to compare the method with monte carlo indicating for which problems the former is more efficient than the latter computational evidence complements the present theory and shows the effectiveness of the sparse grid stochastic collocation method compared to full tensor and monte carlo approaches

a stochastic collocation method for elliptic partial differential equations with random input data this work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms these input data are assumed to depend on a finite number of random variables the method consists of a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it treats easily a wide range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction of the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method finally we include a section with developments posterior to the original publication of this work there we review sparse grid stochastic collocation methods which are effective collocation strategies for problems that depend on a moderately large number of random variables

stabilizing solution to the reverse discrete time riccati equation a matrix pencil based approach abstract we derive necessary and sufficient conditions for the existence of the stabilizing solution to the reverse discrete time riccati equation a particular type of algebraic riccati equation which is related to the solution of the discrete time version of the extended two block nehari problem the conditions are expressed in terms of the left stable deflating subspace of an associated symplectic matrix pencil in particular a maximum phase spectral factorization of the popov function is obtained under very relaxed conditions imposed on the initial data it is also proved that under a restrictive additional assumption the stabilizing solution to the reverse riccati equation reduces to the antistabilizing solution to the usual discrete time riccati equation a reliable numerical algorithm for computing the stabilizing solution to the reverse riccati equation is also presented together with formulae for the solution to the extended nehari problem

an invariance principle for diffusion in turbulence we prove an almost sure invariance principle for diffusion driven by velocities with unbounded stationary vector potentials the result generalizes to multiple particles motion driven by a common velocity field and independent molecular brownian motions

stochastic optimal robin boundary control problems of advection dominated elliptic equations in this work we deal with a stochastic optimal robin boundary control problem constrained by an advection diffusion reaction elliptic equation with advection dominated term we assume that the uncertainty comes from the advection field and consider a stochastic robin boundary condition as control function a stochastic saddle point system is formulated and proved to be equivalent to the first order optimality system for the optimal control problem based on which we provide the existence and uniqueness of the optimal solution as well as some results on stochastic regularity with respect to the random variables stabilized finite element approximations in physical space and collocation approximations in stochastic space are applied to discretize the optimality system a global error estimate in the product of physical space and stochastic space for the numerical approximation is derived illustrative numerical experiments are provided

spectral theory of the linear quadratic and h problems abstract an overview of the spectral theory of the linear quadratic and h problems is presented originally the spectral theory of the linear quadratic problem emerged out of the problem of characterizing the finite time escape of the riccati differential equation as a condition on the spectrum of a toeplitz plus hankel operator later and following an independent line of thought the spectral theory of the h problem emerged as a toeplitz plus hankel operator characterization of the smallest achievable tolerance in feedback design this common toeplitz plus hankel operator structure shared by the linear quadratic and h problems is elucidated by mapping the usual h frequency response specification into the time domain leading to an inequality bound on a quadratic functional precisely induced by the toeplitz plus hankel operator with this deep linear quadratic h connection at hand we derive a linear quadratic solution to the h problem and show that such issues as the adamjan arov krein problem pole zero cancellation in optimal h compensation etc can be fruitfully attacked using simple linear quadratic arguments

random walks in quenched i i d space time random environment are always a s diffusive we consider a general model of discrete time random walk x t on the lattice 1 in a random environment t x t x 1 with i i d components t x previous results on the a s validity of the central limit theorem for the quenched model required a small stochasticity condition in this paper we show that the result holds provided only that an obvious non degeneracy condition is met the proof is based on the analysis of a suitable generating function which allows to estimate l 2 norms by contour integrals

optimality of adaptive galerkin methods for random parabolic partial differential equations galerkin discretizations of a class of parametric and random parabolic partial differential equations pdes are considered the parabolic pdes are assumed to depend on a vector y y 1 y 2 of possibly countably many parameters y j which are assumed to take values in 1 1 well posedness of weak formulations of these parametric equations in suitable bochner spaces is established adaptive galerkin discretizations of the equation based on a tensor product of a generalized polynomial chaos in the parameter domain 1 1 n and of suitable wavelet bases in the time interval i 0 t and the spatial domain d r d are proposed and their optimality is established

robust optimal robin boundary control for the transient heat equation with random input data the problem of robust optimal robin boundary control for a parabolic partial dierential equation with uncertain input data is considered as a measure of robustness the variance of the random system response is included in two dierent cost functionals uncertainties in both the underlying state equation and the control variable are quantied through random elds the paper is mainly concerned with the numerical resolution of the problem to this end a gradient method is proposed a stochastic collocation nite element method is used for the numerical resolution of the associated state and adjoint state equations two numerical experiments illustrate the performance of the algorithm

anchored nash inequalities and heat kernel bounds for static and dynamic degenerate environments abstract we introduce anchored versions of the nash inequality they allow to control the l 2 norm of a function by dirichlet forms that are not uniformly elliptic we then use them to provide heat kernel upper bounds for diffusions in degenerate static and dynamic random environments as an example we apply our results to the case of a random walk with degenerate jump rates that depend on an underlying exclusion process at equilibrium

on estimating the derivatives of symmetric diffusions in stationary random environment with applications to ϕ interface model we consider diffusions on d or random walks on d in a random environment which is stationary in space and in time and with symmetric and uniformly elliptic coefficients we show existence and holder continuity of second space derivatives and time derivatives for the annealed kernels of such diffusions and give estimates for these derivatives in the case of random walks these estimates are applied to the ginzburg landau interface model

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

time varying discrete riccati equation some monotonicity results abstract using a fre chet derivative based approach some monotonicity and comparison results concerning the solutions of the time varying discrete time riccati equation are obtained connections with the existence of the semistabilizing solution are made explicit as well

generalized discrete time popov yakubovich theory abstract this paper deals with the problem of general conditions for the existence of a stabilizing solution to the so called kalman szego popov yakubovich system the well known popov yakubovich positivity condition is replaced with a more general one expressed in terms of the invertibility of an adequate operator this allows one to consider also game theoretical situations

mixing time for random walk on supercritical dynamical percolation we consider dynamical percolation on the d dimensional discrete torus znd of side length n where each edge refreshes its status at rate n 1 2 to be open with probability p we study random walk on the torus where the walker moves at rate 1 2d along each open edge in earlier work of two of the authors with a stauffer it was shown that in the subcritical case p pc zd the mixing time is n2 1 here the implied constants depend only on d and p we prove a quenched and hence annealed version of this conjecture up to a poly logarithmic factor under the assumption p 1 2 when p 0 we prove a version of this conjecture for an alternative notion of mixing time involving randomised stopping times the latter implies sharp up to poly logarithmic factors upper bounds on exit times of large balls throughout the supercritical regime our proofs are based on percolation results e g the grimmett marstrand theorem and an analysis of the volume biased evolving set process the key point is that typically the evolving set has a substantial intersection with the giant percolation cluster at many times this allows us to use precise isoperimetric properties of the cluster due to g pete to infer rapid growth of the evolving set which in turn yields the upper bound on the mixing time

detectability and stabilizability of time varying discrete time linear systems the concepts of detectability and stabilizability are explored for time varying systems we study duality invariance under feedback an extended version of the lemma of lyapunov existence of stabilizing feedback laws linear quadratic filtering and control and the existence of approximate canonical forms

random conductance models with stable like jumps heat kernel estimates and harnack inequalities abstract we establish two sided heat kernel estimates for random conductance models with non uniformly elliptic possibly degenerate stable like jumps on graphs these are long range counterparts of the well known two sided gaussian heat kernel estimates by m t barlow for nearest neighbor short range random walks on the supercritical percolation cluster unlike the cases of nearest neighbor conductance models we cannot use parabolic harnack inequalities since even elliptic harnack inequalities do not hold in the present setting as an application we establish the local limit theorem for the models