stability of numerical methods for delay differential equations abstract consider the following delay differential equation dde y t y t y t t t t 0 with the initial condition y t for t t 0 where and are such that 0 1 0 2 has a unique solution y t the author gives sufficient conditions for the asymptotic stability of the equation 0 1 for which he introduces new definitions of numerical stability the approach is reminiscent of that from the nonlinear stiff ordinary differential equation ode field he investigates stability properties of the class of one point collocation rules in particular the backward euler method turns out to be stable with respect to all the given definitions

on contact anosov flows exponential decay of correlations for co 4 contact anosov flows is established this implies in particular exponential decay of correlations for all smooth geodesic flows in strictly negative curvature

optimal mixing and optimal stirring for fixed energy fixed power or fixed palenstrophy flows we consider passive scalar mixing by a prescribed divergence free velocity vector field in a periodic box and address the following question starting from a given initial inhomogeneous distribution of passive tracers and given a certain energy budget power budget or finite palenstrophy budget what incompressible flow field best mixes the scalar quantity we focus on the optimal stirring strategy recently proposed by lin et al optimal stirring strategies for passive scalar mixing j fluid mech 675 465 2011 10 1017 s0022112011000292 that determines the flow field that instantaneously maximizes the depletion of the h 1 mix norm in this work we bridge some of the gap between the best available a priori analysis and simulation results after recalling some previous analysis we present an explicit example demonstrating finite time perfect mixing with a finite energy constraint on the stirring flow on the other hand using a recent result by wirosoetisno et al long time stability of a classi

stability analysis of numerical schemes for stochastic differential equations stochastic differential equations sdes represent physical phenomena dominated by stochastic processes as for deterministic ordinary differential equations odes various numerical schemes are proposed for sdes in this paper we study the stability of numerical schemes for scalar sdes with respect to the mean square norm which we call ms stability we will show some figures of the ms stability domain or regions for some numerical schemes and present numerical results which confirm it this notion is an extension of absolute stability in numerical methods for odes

on circular flows linear stability and damping abstract in this article we establish linear inviscid damping with optimal decay rates around 2d taylor couette flow and similar monotone flows in an annular domain b r 2 0 b r 1 0 r 2 following recent results by wei zhang and zhao 10 we establish stability in weighted norms which allow for a singularity formation at the boundary and additionally provide a description of the blow up behavior

critical non sobolev regularity for continuity equations with rough velocity fields abstract we present a divergence free vector field in the sobolev space h 1 such that the flow associated to the field does not belong to any sobolev space the vector field is deterministic but constructed as the realization of a random field combining simple elements this construction illustrates the optimality of recent quantitative regularity estimates as it gives a straightforward example of a well posed flow which has nevertheless only very weak regularity

on landau damping going beyond the linearized study has been a longstanding problem in the theory of landau damping in this paper we establish exponential landau damping in analytic regularity the damping phenomenon is reinterpreted in terms of transfer of regularity between kinetic and spatial variables rather than exchanges of energy phase mixing is the driving mechanism the analysis involves new families of analytic norms measuring regularity by comparison with solutions of the free transport equation new functional inequalities a control of non linear echoes sharp deflection estimates and a newton approximation scheme our results hold for any potential no more singular than coulomb or newton interaction the limit cases are included with specific technical effort as a side result the stability of homogeneous equilibria of the non linear vlasov equation is established under sharp assumptions we point out the strong analogy with the kam theory and discuss physical implications finally we extend these results to some gevrey non analytic distribution functions

linear inviscid damping and enhanced viscous dissipation of shear flows by using the conjugate operator method abstract we study the large time behavior of solutions to two dimensional euler and navier stokes equations linearized about shear flows of the mixing layer type in the unbounded channel t r under a simple spectral stability assumption on a self adjoint operator we prove a local form of the linear inviscid damping that is uniform with respect to small viscosity we also prove a local form of the enhanced viscous dissipation that takes place at times of order 1 3 being the small viscosity to prove these results we use a hamiltonian approach following the conjugate operator method developed in the study of schrodinger operators combined with a hypocoercivity argument to handle the viscous case

almost sure enhanced dissipation and uniform in diffusivity exponential mixing for advection diffusion by stochastic navier stokes we study the mixing and dissipation properties of the advection diffusion equation with diffusivity 0 kappa ll 1 and advection by a class of random velocity fields on mathbb t d d 2 3 including solutions of the 2d navier stokes equations forced by sufficiently regular in space non degenerate white in time noise we prove that the solution almost surely mixes exponentially fast uniformly in the diffusivity kappa namely that there is a deterministic exponential rate independent of kappa such that all mean zero h 1 initial data decays exponentially fast in h 1 at this rate with probability one this implies almost sure enhanced dissipation in l 2 specifically that there is a deterministic uniform in kappa exponential decay in l 2 after time t rsim left log kappa right both the mathcal o left log kappa right time scale and the uniform in kappa exponential mixing are optimal for lipschitz velocity fields this work is also a major step in our program on scalar mixing and lagrangian chaos necessary for a rigorous proof of the batchelor power spectrum of passive scalar turbulence

exponential mean square stability of numerical methods for systems of stochastic differential equations this paper is concerned with exponential mean square stability of the classical stochastic theta method and the so called split step theta method for stochastic systems first we consider linear autonomous systems under a sufficient and necessary condition for exponential mean square stability of the exact solution it is proved that the two classes of theta methods with q 0 5 are exponentially mean square stable for all positive step sizes and the methods with q 0 5 still unconditionally preserves the exponential mean square stability of the underlying systems but the stochastic theta method does not have this property finally we consider stochastic differential equations with jumps some similar results are derived

diffusion and mixing in fluid flow we study enhancement of diffusive mixing on a compact riemannian manifold by a fast incompressible flow our main result is a sharp description of the class of flows that make the deviation of the solution from its average arbitrarily small in an arbitrarily short time provided that the flow amplitude is large enough the necessary and suficient condition on such flows is expressed naturally in terms of the spectral properties of the dynamical system associated with the flow in particular we find that weakly mixing flows always enhance dissipation in this sense the proofs are based on a general criterion for the decay of the semigroup generated by an operator of the form g ial with a negative unbounded self adjoint operator g a self adjoint operator l and parameter a 1 in particular they employ the rage theorem describing evolution of a quantum state belonging to the continuous spectral subspace of the hamiltonian related to a classical theorem of wiener on fourier transforms of measures applications to quenching in reaction diffusion equations are also considered

a maximum principle applied to quasi geostrophic equations we study the initial value problem for dissipative 2d quasi geostrophic equations proving local existence global results for small initial data in the super critical case decay of lp norms and asymptotic behavior of viscosity solution in the critical case our proofs are based on a maximum principle valid for more general flows

random forests and networks analysis wilson proceedings of the twenty eight annual acm symposium on the theory of computing pp 296 303 1996 in the 1990s described a simple and efficient algorithm based on loop erased random walks to sample uniform spanning trees and more generally weighted trees or forests spanning a given graph this algorithm provides a powerful tool in analyzing structures on networks and along this line of thinking in recent works avena and gaudilliere in a proof of the transfer current theorem in absence of reversibility in stat probab lett 142 17 22 2018 avena and gaudilliere in j theor probab 2017 https doi org 10 1007 s10959 017 0771 3 avena et al in approximate and exact solutions of intertwining equations though random spanning forests 2017 arxiv 1702 05992v1 avena et al in intertwining wavelets or multiresolution analysis on graphs through random forests 2017 arxiv 1707 04616 to appear in acha 2018 we focused on applications of spanning rooted forests on finite graphs the resulting main conclusions are reviewed in this paper by collecting related theorems algorithms heuristics and numerical experiments a first foundational part on determinantal structures and efficient sampling procedures is followed by four main applications 1 a random walk based notion of well distributed points in a graph 2 a framework to describe metastable like dynamics in finite settings by means of markov intertwining dualities 3 coarse graining schemes for networks and associated processes 4 wavelets like pyramidal algorithms for graph signals

stable mixing estimates in the infinite peclet number limit abstract we consider a passive scalar f advected by a strictly monotone shear flow and with a diffusivity parameter 1 we prove an estimate on the homogeneous h 1 norm of f that combines both the l 2 enhanced diffusion effect at a sharp rate proportional to 1 3 and the sharp mixing decay proportional to t 1 of the h 1 norm of f when 0 in particular the estimate is stable in the infinite peclet number limit as 0 to the best of our knowledge this is the first result of this kind since the work of kelvin in 1887 on the couette flow the two key ingredients in the proof are an adaptation of the hypocoercivity method and the use of a vector field j that commutes with the transport part of the equation the l 2 norm of jf together with the l 2 norm of f produces a suitable upper bound for the h 1 norm of the solution that gives the extra decay factor of t 1

numerical methods for nonlinear stochastic differential equations with jumps we present and analyse two implicit methods for ito stochastic differential equations sdes with poisson driven jumps the first method ssbe is a split step extension of the backward euler method the second method cssbe arises from the introduction of a compensated martingale form of the poisson process we show that both methods are amenable to rigorous analysis when a one sided lipschitz condition rather than a more restrictive global lipschitz condition holds for the drift our analysis covers strong convergence and nonlinear stability we prove that both methods give strong convergence when the drift coefficient is one sided lipschitz and the diffusion and jump coefficients are globally lipschitz on the way to proving these results we show that a compensated form of the euler maruyama method converges strongly when the sde coefficients satisfy a local lipschitz condition and the pth moment of the exact and numerical solution are bounded for some p 2 under our assumptions both ssbe and cssbe give well defined unique solutions for sufficiently small stepsizes and ssbe has the advantage that the restriction is independent of the jump intensity we also study the ability of the methods to reproduce exponential mean square stability in the case where the drift has a negative one sided lipschitz constant this work extends the deterministic nonlinear stability theory in numerical analysis we find that ssbe preserves stability under a stepsize constraint that is independent of the initial data cssbe satisfies an even stronger condition and gives a generalization of b stability finally we specialize to a linear test problem and show that cssbe has a natural extension of deterministic a stability the difference in stability properties of the ssbe and cssbe methods emphasizes that the addition of a jump term has a significant effect that cannot be deduced directly from the non jump literature

stability of analytical and numerical solutions of nonlinear stochastic delay differential equations this paper concerns the stability of analytical and numerical solutions of nonlinear stochastic delay differential equations sddes we derive sufficient conditions for the stability contractivity and asymptotic contractivity in mean square of the solutions for nonlinear sddes the results provide a unified theoretical treatment for sddes with constant delay and variable delay including bounded and unbounded variable delays then the stability contractivity and asymptotic contractivity in mean square are investigated for the backward euler method it is shown that the backward euler method preserves the properties of the underlying sddes the main results obtained in this work are different from those of razumikhin type theorems indeed our results hold without the necessity of constructing or finding an appropriate lyapunov functional

stability of the drift implicit and double implicit milstein schemes for nonlinear sdes abstract this paper examines the stability of numerical solutions of nonlinear stochastic differential equations sdes with non global lipschitz continuous coefficients two implicit milstein schemes called drift implicit milstein scheme and double implicit milstein scheme are considered to simulate the underlying sdes it is proved that the schemes can preserve the stability and contractivity in mean square of the underlying systems

balanced implicit methods for stiff stochastic systems this paper introduces some implicitness in stochastic terms of numerical methods for solving stiff stochastic differential equations and especially a class of fully implicit methods the balanced methods their order of strong convergence is proved numerical experiments compare the stability properties of these schemes with explicit ones

exponential self similar mixing by incompressible flows we study the problem of the optimal mixing of a passive scalar under the action of an incompressible flow in two space dimensions the scalar solves the continuity equation with a divergence free velocity field which satisfies a bound in the sobolev space ws p where s 0 and 1 p the mixing properties are given in terms of a characteristic length scale called the mixing scale we consider two notions of mixing scale one functional expressed in terms of the homogeneous sobolev norm h 1 the other geometric related to rearrangements of sets we study rates of decay in time of both scales under self similar mixing for the case s 1 and 1 p including the case of lipschitz continuous velocities and the case of physical interest of enstrophy constrained flows we present examples of velocity fields and initial configurations for the scalar that saturate the exponential lower bound established in previous works on the time decay of both scales we also present several consequences for the geometry of regular lagrangian flows associated to sobolev velocity fields

universal mixers in all dimensions abstract we construct universal mixers incompressible flows that mix arbitrarily well general solutions to the corresponding transport equation in all dimensions this mixing is exponential in time i e essentially optimal for any initial condition with at least some regularity and we also show that a uniform mixing rate for all initial conditions cannot be achieved the flows are time periodic and uniformly in time bounded in spaces w s p for a range of s p that includes points with s 1 and p 2

linear inviscid damping for a class of monotone shear flow in sobolev spaces in this paper we prove the decay estimates of the velocity and h1 scattering for the two dimensional linearized euler equations around a class of monotone shear flow in a finite channel our result is consistent with the decay rate predicted by case in 1960 2016 wiley periodicals inc

first order strong approximations of scalar sdes defined in a domain we are interested in strong approximations of one dimensional sdes which have non lipschitz coefficients and which take values in a domain under a set of general assumptions we derive an implicit scheme that preserves the domain of the sdes and is strongly convergent with rate one moreover we show that this general result can be applied to many sdes we encounter in mathematical finance and bio mathematics we will demonstrate flexibility of our approach by analyzing classical examples of sdes with sublinear coefficients cir cev models and wright fisher diffusion and also with superlinear coefficients 3 2 volatility ait sahalia model our goal is to justify an efficient multilevel monte carlo method for a rich family of sdes which relies on good strong convergence properties

exponential decay of correlations for finite horizon sinai billiard flows we prove exponential decay of correlations for the billiard flow associated with a two dimensional finite horizon lorentz gas i e the sinai billiard flow with finite horizon along the way we describe the spectrum of the generator of the corresponding semi group mathcal l t of transfer operators i e the resonances of the sinai billiard flow on a suitable banach space of anisotropic distributions

linear inviscid damping for monotone shear flows recently there has been much interest in damping phenomena for kinetic equations following the seminal works of mouhot villani on landau damping and of bedrossian masmoudi on inviscid damping around couette flow in this talk i present a proof of linear inviscid damping for the 2d euler equations around general monotone shear flows in the framework of sobolev regularity here i consider both the settings of an infinite periodic and a finite periodic channel with impermeable walls in the latter case i explain the non negligible effect of boundary conditions on the attainable regularity and stability results speaker sciencesconf org equadiff2015 64596

preserving stability implicit euler method for nonlinear volterra and neutral functional differential equations in banach space this paper is concerned with the contractivity and asymptotic stability properties of the implicit euler method iem for nonlinear functional differential equations fdes these properties are first analyzed for volterra fdes and then the analysis is extended to the case of neutral fdes nfdes such an extension is particularly important since nfdes are more general and have received little attention in the literature the main result we establish is that the iem with linear interpolation can completely preserve these stability properties of the analytical solution to such fdes

exponential stability in p th mean of solutions and of convergent euler type solutions of stochastic delay differential equations one concept of the stability of a solution of an evolutionary equation relates to the sensitivity of the solution to perturbations in the initial data there are other stability concepts notably those concerned with persistent perturbations results are presented on the stability in p th mean of solutions of stochastic delay differential equations with multiplicative noise and of stochastic delay difference equations the difference equations are of a type found in numerical analysis and we employ our results to obtain mean square stability criteria for the solution of the euler maruyama discretization of stochastic delay differential equations the analysis proceeds as follows we show that an inequality of halanay type derivable via comparison theory can be employed to derive conditions for p th mean stability of a solution we then produce a discrete analogue of the halanay type theory that permits us to develop a p th mean stability analysis of analogous stochastic difference equations the application of the theoretical results is illustrated by deriving mean square stability conditions for solutions and numerical solutions of a constant coefficient linear test equation