noise induced drift in stochastic differential equations with arbitrary friction and diffusion in the smoluchowski kramers limit we consider the dynamics of systems with arbitrary friction and diffusion these include as a special case systems for which friction and diffusion are connected by einstein fluctuation dissipation relation e g brownian motion we study the limit where friction effects dominate the inertia i e where the mass goes to zero smoluchowski kramers limit using the ito stochastic integral convention we show that the limiting effective langevin equations has different drift fields depending on the relation between friction and diffusion alternatively our results can be cast as different interpretations of stochastic integration in the limiting equation which can be parametrized by interestingly in addition to the classical ito 0 stratonovich 0 5 and anti ito 1 integrals we show that position dependent x and even stochastic integrals with 0 1 arise our findings are supported by numerical simulations

conditional probabilities in multiplicative noise processes we address the calculation of transition probabilities in multiplicative noise stochastic differential equations using a path integral approach we show the equivalence between the conditional probability and the propagator of a quantum particle with variable mass introducing a time reparametrization we are able to transform the problem of multiplicative noise fluctuations into an equivalent additive one we illustrate the method by showing the explicit analytic computation of the conditional probability of a harmonic oscillator in a nonlinear multiplicative environment

asymptotics of work distributions in nonequilibrium systems the asymptotic behavior of the work distribution in driven nonequilibrium systems is determined using the method of optimal fluctuations for systems described by langevin dynamics the corresponding euler lagrange equation together with the appropriate boundary conditions and an equation for the leading pre exponential factor are derived the method is applied to three representative examples and the results are used to improve the accuracy of free energy estimates based on the application of the jarzynski equation

the velocity tracking problem for navier stokes flows with bounded distributed controls we present some systematic approaches to the mathematical analysis and numerical approximation of the time dependent optimal control problem of tracking the velocity for navier stokes flows in bounded two dimensional domains with bounded distributed controls we study the existence of optimal solutions and derive an optimality system from which optimal solutions may be determined we also define and analyze semidiscrete in time and fully space time discrete approximations of the optimality system and a gradient method for the solution of the fully discrete system the results of some computational experiments are provided

a monte carlo method for solving unsteady adjoint equations traditionally solving the adjoint equation for unsteady problems involves solving a large structured linear system this paper presents a variation on this technique and uses a monte carlo linear solver the monte carlo solver yields a forward time algorithm for solving unsteady adjoint equations when applied to computing the adjoint associated with burgers equation the monte carlo approach is faster for a large class of problems while preserving sufficient accuracy

traces and determinants of strongly stochastic operators periodic orbit theory allows calculations of long time properties of chaotic systems from traces dynamical zeta functions and spectral determinants of deterministic evolution operators which are in turn evaluated in terms of periodic orbits for the case of stochastic dynamics a direct numerical evaluation of the trace of an evolution operator is possible as a multidimensional integral techniques for evaluating such path integrals are discussed using as an example the logistic map f x lambdax 1 x with moderate to strong additive gaussian noise rapid convergence is demonstrated for all values of lambda with strong noise as well as at fixed lambda 5 for all noise levels

energy representation for nonequilibrium brownian like systems steady states and fluctuation relations stochastic dynamics in the energy representation is used as a method to represent nonequilibrium brownian like systems it is shown that the equation of motion for the energy of such systems can be taken in the form of the langevin equation with multiplicative noise properties of the steady states are examined by solving the fokker planck equation for the energy distribution functions the generalized integral fluctuation theorem is deduced for the systems characterized by the shifted probability flux operator from this theorem a number of entropy and fluctuation relations such as the evans searles fluctuation theorem the hatano sasa identity and the jarzynski s equality are derived

boosting bayesian parameter inference of nonlinear stochastic differential equation models by hamiltonian scale separation parameter inference is a fundamental problem in data driven modeling given observed data that is believed to be a realization of some parameterized model the aim is to find parameter values that are able to explain the observed data in many situations the dominant sources of uncertainty must be included into the model for making reliable predictions this naturally leads to stochastic models stochastic models render parameter inference much harder as the aim then is to find a distribution of likely parameter values in bayesian statistics which is a consistent framework for data driven learning this so called posterior distribution can be used to make probabilistic predictions we propose a novel exact and very efficient approach for generating posterior parameter distributions for stochastic differential equation models calibrated to measured time series the algorithm is inspired by reinterpreting the posterior distribution as a statistical mechanics partition function of an object akin to a polymer where the measurements are mapped on heavier beads compared to those of the simulated data to arrive at distribution samples we employ a hamiltonian monte carlo approach combined with a multiple time scale integration a separation of time scales naturally arises if either the number of measurement points or the number of simulation points becomes large furthermore at least for one dimensional problems we can decouple the harmonic modes between measurement points and solve the fastest part of their dynamics analytically our approach is applicable to a wide range of inference problems and is highly parallelizable

ergodicity breaking in geometric brownian motion geometric brownian motion gbm is a model for systems as varied as financial instruments and populations the statistical properties of gbm are complicated by nonergodicity which can lead to ensemble averages exhibiting exponential growth while any individual trajectory collapses according to its time average a common tactic for bringing time averages closer to ensemble averages is diversification in this letter we study the effects of diversification using the concept of ergodicity breaking

an adjoint based jacobi type iterative method for elastic full waveform inversion problem full waveform inversion fwi is a promising technique that is capable of creating high resolution subsurface images of the earth from seismic data however it is computationally expensive especially when 3d elastic wave equation is considered in this paper an adjoint based computational algorithm has been proposed to address the computational challenges the important strategy in this work is to decouple the two lame parameters so that two half sized subproblems are resulted and solved separately mathematically the inverse problem is formulated as a pde constrained optimization problem in which the objective functional is defined as the misfit between observational and synthetic data the parameters to be recovered are the spatially varying lame parameters which are of great interests to geophysicists for the purpose of hydrocarbon exploration and subsurface imaging the gradient of the misfit functional with respect to the lame parameters is calculated by solving the adjoint elastic wave equation whilst the quasi newton method l bfgs is used to minimize the misfit functional numerical experiments demonstrated that the new method is accurate efficient and robust in recovering lame parameters from seismic data

finite element formulation of fluctuating hydrodynamics for fluids filled with rigid particles using boundary fitted meshes in this paper we present a finite element implementation of fluctuating hydrodynamics with a moving boundary fitted mesh for treating the suspended particles the thermal fluctuations are incorporated into the continuum equations using the landau and lifshitz approach 1 the proposed implementation fulfills the fluctuation dissipation theorem exactly at the discrete level since we restrict the equations to the creeping flow case this takes the form of a relation between the diffusion coefficient matrix and friction matrix both at the particle and nodal level of the finite elements brownian motion of arbitrarily shaped particles in complex confinements can be considered within the present formulation a multi step time integration scheme is developed to correctly capture the drift term required in the stochastic differential equation sde describing the evolution of the positions of the particles the proposed approach is validated by simulating the brownian motion of a sphere between two parallel plates and the motion of a spherical particle in a cylindrical cavity the time integration algorithm and the fluctuating hydrodynamics implementation are then applied to study the diffusion and the equilibrium probability distribution of a confined circle under an external harmonic potential

functional integral approach for multiplicative stochastic processes we present a functional formalism to derive a generating functional for correlation functions of a multiplicative stochastic process represented by a langevin equation we deduce a path integral over a set of fermionic and bosonic variables without performing any time discretization the usual prescriptions to define the wiener integral appear in our formalism in the definition of green s functions in the grassman sector of the theory we also study nonperturbative constraints imposed by becchi rouet and stora symmetry brs and supersymmetry on correlation functions we show that the specific prescription to define the stochastic process is wholly contained in tadpole diagrams therefore in a supersymmetric theory the stochastic process is uniquely defined since tadpole contributions cancels at all order of perturbation theory

risk assessment of scramjet unstart using adjoint based sampling methods we demonstrate an adjoint based approach for accelerating monte carlo estimation of risk and apply it to estimating the probability of unstart in a scramjet engine under uncertain conditions that are characterized by various gaussian and non gaussian distributions the adjoint equation is solved with respect to an objective function that is used to identify unstart and the adjoint solution is used to generate a linear approximation to the objective function this linear surrogate is used to divide the uncertain input parameters into three different strata corresponding to safe operation of the engine uncertain operation and unstart the probability of unstart within these strata is very different and as a result stratified sampling significantly increases the efficiency of the risk assessment procedure by reducing the variance of the estimator using this technique the monte carlo method was demonstrated to be accelerated by a factor of 5 4

currency target zone modeling an interplay between physics and economics we study the performance of the euro swiss franc exchange rate in the extraordinary period from september 6 2011 to january 15 2015 when the swiss national bank enforced a minimum exchange rate of 1 20 swiss francs per euro within the general framework built on geometric brownian motions and based on the analogy between brownian motion in finance and physics the first order effect of such a steric constraint would enter a priori in the form of a repulsive entropic force associated with the paths crossing the barrier that are forbidden nonparametric empirical estimates of drift and volatility show that the predicted first order analogy between economics and physics is incorrect the clue is to realize that the random walk nature of financial prices results from the continuous anticipation of traders about future opportunities whose aggregate actions translate into an approximate efficient market with almost no arbitrage opportunities with the swiss national bank s stated commitment to enforce the barrier traders anticipation of this action leads to a vanishing drift together with a volatility of the exchange rate that depends on the distance to the barrier this effect is described by krugman s model p r krugman target zones and exchange rate dynamics q j econ 106 669 1991 we present direct quantitative empirical evidence that krugman s theoretical model provides an accurate description of the euro swiss franc target zone motivated by the insights from the economic model we revise the initial economics physics analogy and show that within the context of hindered diffusion the two systems can be described with the same mathematics after all using a recently proposed extended analogy in terms of a colloidal brownian particle embedded in a fluid of molecules associated with the underlying order book we derive that close to the restricting boundary the dynamics of both systems is described by a stochastic differential equation with a very small constant drift and a linear diffusion coefficient as a side result we present a simplified derivation of the linear hydrodynamic diffusion coefficient of a brownian particle close to a wall

analysis and approximation for linear feedback control for tracking the velocity in navier stokes flows abstract some systematic approaches to the mathematical formulation and numerical resolution of the linear feedback control problem for tracking the velocity in navier stokes flows in a bounded two dimensional domain with bounded distributed control are presented semidiscrete in time and full space time discrete approximations are also studied some computational results are presented and compared with analogous results from optimal control theory

error estimates for transport problems with high peclet number using a continuous dependence assumption in this paper we discuss the behavior of stabilized finite element methods for the transient advection diffusion problem with dominant advection and rough data we show that provided a certain continuous dependence result holds for the quantity of interest independent of the peclet number this quantity may be computed using a stabilized finite element method in all flow regimes as an example of a stable quantity we consider the parameterized weak norm introduced in burman 2014 the same results may not be obtained using a standard galerkin method we consider the following stabilized methods continuous interior penalty cip and streamline upwind petrov galerkin supg the theoretical results are illustrated by computations on a scalar transport equation with no diffusion term rough data and strongly varying velocity field

trace formulas for stochastic evolution operators weak noise perturbation theory periodic orbit theory is all effective tool for the analysis of classical and quantum chaotic systems in this paper we extend this approach to stochastic systems in particular to mappings with additive noise the theory is cast in the standard field theoretic formalism and weak noise perturbation theory written in terms of feynman diagrams the result is a stochastic analog of the next to leading h corrections to the gutzwiller trace formula with long time averages calculated from periodic orbits of the deterministic system the perturbative corrections are computed analytically and tested numerically on a simple 1 dimensional system

chaotic field theory a sketch abstract spatio temporally chaotic dynamics of a classical field can be described by means of an infinite hierarchy of its unstable spatio temporally periodic solutions the periodic orbit theory yields the global averages characterizing the chaotic dynamics as well as the starting semiclassical approximation to the quantum theory new methods for computing corrections to the semiclassical approximation are developed in particular a nonlinear field transformation yields the perturbative corrections in a form more compact than the feynman diagram expansions

fluctuating multicomponent lattice boltzmann model current implementations of fluctuating lattice boltzmann equations flbes describe single component fluids in this paper a model based on the continuum kinetic boltzmann equation for describing multicomponent fluids is extended to incorporate the effects of thermal fluctuations the thus obtained fluctuating boltzmann equation is first linearized to apply the theory of linear fluctuations and expressions for the noise covariances are determined by invoking the fluctuation dissipation theorem directly at the kinetic level crucial for our analysis is the projection of the boltzmann equation onto the orthonormal hermite basis by integrating in space and time the fluctuating boltzmann equation with a discrete number of velocities the flbe is obtained for both ideal and nonideal multicomponent fluids numerical simulations are specialized to the case where mean field interactions are introduced on the lattice indicating a proper thermalization of the system

brownian motion with multiplicative noises revisited the langevin equation with multiplicative noise and a state dependent transport coefficient should always complemented with the proper interpretation rule of the noise such as the ito and stratonovich conventions although the mathematical relationship between the different rules and how to translate from one rule to another are well established the subject of which is a more physically natural rule still remains controversial in this communication we derive the overdamped langevin equation with multiplicative noise for brownian particles by systematically eliminating the fast degrees of freedom of the underdamped langevin equation the langevin equations obtained here vary depending on the choice of the noise conventions but they are different representations for an identical phenomenon the results apply to multi variable nonequilibrium non stationary systems and other general settings

stochastic dynamics of planar magnetic moments in a three dimensional environment abstract we study the stochastic dynamics of a two dimensional magnetic moment embedded in a three dimensional environment described by means of the stochastic landau lifshitz gilbert sllg equation we define a covariant generalization of this equation valid in the generalized stratonovich discretization prescription we present a path integral formulation that allows to compute any n point correlation function independently of the stochastic calculus used using this formalism we show the equivalence between the cartesian formulation with vectorial noise and the polar formulation with just one scalar fluctuation term in particular we show that for isotropic fluctuations the system is represented by an additive stochastic process despite of the multiplicative terms appearing in the original formulation of the sllg equation but for anisotropic fluctuations the noise turns out to be truly multiplicative

towards a parameter free method for high reynolds number turbulent flow simulation based on adaptive finite element approximation we present work towards a parameter free method for turbulent flow simulation based on adaptive finite element approximation of the navier stokes equations at high reynolds numbers in this model viscous dissipation is assumed to be dominated by turbulent dissipation proportional to the residual of the equations and skin friction at solid walls is assumed to be negligible compared to inertial effects the result is a computational model without empirical data where the only parameter is the local size of the finite element mesh under adaptive refinement of the mesh based on a posteriori error estimation output quantities of interest in the form of functionals of the finite element solution converge to become independent of the mesh resolution and thus the resulting method has no adjustable parameters no ad hoc design of the mesh is needed instead the mesh is optimised based on solution features in particular no bounder layer mesh is needed we connect the computational method to the mathematical concept of a dissipative weak solution of the euler equations as a model of high reynolds number turbulent flow and we highlight a number of benchmark problems for which the method is validated

nonlinear inhomogeneous fokker planck equations entropy and free energy time evolution we extend a recently introduced free energy formalism for homogeneous fokker planck equations to a wide and physically appealing class of inhomogeneous nonlinear fokker planck equations in our approach the free energy functional is expressed in terms of an entropic functional and an auxiliary potential both derived from the coefficients of the equation with reference to the introduced entropic functional we discuss the entropy production in a relaxation process towards equilibrium the properties of the stationary solutions of the considered fokker planck equations are also discussed

superconvergent functional output for time dependent problems using finite differences on summation by parts form finite difference operators satisfying the summation by parts sbp rules can be used to obtain high order accurate energy stable schemes for time dependent partial differential equations when the boundary conditions are imposed weakly by the simultaneous approximation term sat in general an sbp sat discretization is accurate of order p 1 with an internal accuracy of 2p and a boundary accuracy of p despite this it is shown in this paper that any linear functional computed from the time dependent solution will be accurate of order 2p when the boundary terms are imposed in a stable and dual consistent way the method does not involve the solution of the dual equations and superconvergent functionals are obtained at no extra computational cost four representative model problems are analyzed in terms of convergence and errors and it is shown in a systematic way how to derive schemes which gives superconvergent functional outputs

unification of classical nucleation theories via a unified ito stratonovich stochastic equation classical nucleation theory cnt is the most widely used framework to describe the early stage of first order phase transitions unfortunately the different points of view adopted to derive it yield different kinetic equations for the probability density function e g zeldovich frenkel or becker doring tunitskii equations starting from a phenomenological stochastic differential equation a unified equation is obtained in this work in other words cnt expressions are recovered by selecting one or another stochastic calculus moreover it is shown that the unified cnt thus obtained produces the same fokker planck equation as that from a recent update of cnt j f lutsko and m a duran olivencia j chem phys 138 244908 2013 10 1063 1 4811490 when mass transport is governed by diffusion finally we derive a general induction time expression along with specific approximations of it to be used under different scenarios in particular when the mass transport mechanism is governed by direct impingement volume diffusion surface diffusion or interface transfer

dynamical entropy for systems with stochastic perturbation dynamics of deterministic systems perturbed by random additive noise is characterized quantitatively since for such systems the kolmogorov sinai ks entropy diverges if the diameter of the partition tends to zero we analyze the difference between the total entropy of a noisy system and the entropy of the noise itself we show that this quantity is finite and non negative and we call it the dynamical entropy of the noisy system in the weak noise limit this quantity is conjectured to tend to the ks entropy of the deterministic system in particular we consider one dimensional systems with noise described by a finite dimensional kernel for which the frobenius perron operator can be represented by a finite matrix

weak noise approach to the logistic map using a nonperturbative weak noise approach we investigate the interference of noise and chaos in simple 1d maps we replace the noise driven 1d map by an area preserving 2d map modelling the poincare sections of a conserved dynamical system with unbounded energy manifolds we analyze the properties of the 2d map and draw conclusions concerning the interference of noise on the nonlinear time evolution we apply this technique to the standard period doubling sequence in the logistic map from the 2d area preserving analogue we in addition to the usual period doubling sequence obtain a series of period doubled cycles which are elliptic in nature these cycles are spinning off the real axis at parameters values corresponding to the standard period doubling events

random walk on distant mesh points monte carlo methods a new technique for obtaining monte carlo algorithms based on the markov chains with a finite number of states is suggested instead of the classical random walk on neighboring mesh points a general way of constructing monte carlo algorithms that could be called random walk on distant mesh points is considered it is applied to solve boundary value problems the numerical examples indicate that the new methods are less laborious and therefore more efficient

symmetric path integrals for stochastic equations with multiplicative noise a langevin equation with multiplicative noise is an equation schematically of the form dq dt f q e q xi where e q xi is gaussian white noise whose amplitude e q depends on q itself i show how to convert such equations into path integrals the definition of the path integral depends crucially on the convention used for discretizing time and i specifically derive the correct path integral when the convention used is the natural time symmetric one whose time derivatives are q sub t q sub t delta sub t delta t and coordinates are q sub t q sub t delta sub t 2 this is the convention that permits standard manipulations of calculus on the action like naive integration by parts it has sometimes been assumed in the literature that a stratonovich langevin equation can be quickly converted to a path integral by treating time as continuous but using the rule theta t 0 1 2 i show that this prescription fails when the amplitude e q is q dependent c 2000 the american physical society

subgrid modeling for convection diffusion reaction in one space dimension using a haar multiresolution analysis in this paper we propose and study a subgrid model for linear convection diffusion reaction equations with fractal rough coefficients the subgrid model is based on scale extrapolation of a modeling residual from coarser scales using a computed solution on a finest scale as reference we show in experiments that a solution with subgrid model on a scale h in most cases corresponds to a solution without subgrid model on a scale less than h 4 we also present error estimates for the modeling error in terms of modeling residuals

chemical langevin equation a path integral view of gillespie s derivation in 2000 gillespie rehabilitated the chemical langevin equation cle by describing two conditions that must be satisfied for it to yield a valid approximation of the chemical master equation cme in this work we construct an original path integral description of the cme and show how applying gillespie s two conditions to it directly leads to a path integral equivalent to the cle we compare this approach to the path integral equivalent of a large system size derivation and show that they are qualitatively different in particular both approaches involve converting many sums into many integrals and the difference between the two methods is essentially the difference between using the euler maclaurin formula and using riemann sums our results shed light on how path integrals can be used to conceptualize coarse graining biochemical systems and are readily generalizable

an adaptive finite element splitting method for the incompressible navier stokes equations we present an adaptive finite element method for the incompressible navier stokes equations based on a standard splitting scheme the incremental pressure correction scheme the presented method combines the efficiency and simplicity of a splitting method with the powerful framework offered by the finite element method for error analysis and adaptivity an a posteriori error estimate is derived which expresses the error in a goal functional of interest as a sum of contributions from spatial discretization time discretization and a term that measures the deviation of the splitting scheme from a pure galerkin scheme the computational error numerical examples are presented which demonstrate the performance of the adaptive algorithm and high quality efficiency indices it is further demonstrated that the computational error of the navier stokes momentum equation is linear in the size of the time step while the computational error of the continuity equation is quadratic in the size of the time step

asymptotic expansions for the escape rate of stochastically perturbed unimodal maps abstract the escape rate of a stochastic dynamical system can be found as an expansion in powers of the noise strength in the previous work the coefficients of such an expansion for a one dimensional map were fitted to a general form containing a few parameters these parameters were found to be related to the fractal structure of the repeller of the system the parameter the noise dimension remains to be interpreted this report presents new data for showing that the relation to the dimensions is more complicated than predicted in the earlier work and oscillates as a function of the map parameter in contrast to other dimension like quantities

adiabatic elimination for systems of brownian particles with nonconstant damping coefficients we discuss the problem of eliminating the momentum variable in the phase space langevin equations for a system of brownian particles in two related situations i position dependent damping and ii existence of hydrodynamic interactions we discuss the problems associated with the conventional elimination and we develop an alternative elimination procedure in the lagevin framework which leads to the correct smoluchowski equation we give a heuristic argument on the basis of stochastic differential equations for the smoluchowski limit and establish rigorously the limit for the general case of position dependent friction and diffusion coefficents

local kinetic interpretation of entropy production through reversed diffusion the time reversal of stochastic diffusion processes is revisited with emphasis on the physical meaning of the time reversed drift and the noise prescription in the case of multiplicative noise the local kinematics and mechanics of free diffusion are linked to the hydrodynamic description these properties also provide an interpretation of the pope ching formula for the steady state probability density function along with a geometric interpretation of the fluctuation dissipation relation finally the statistics of the local entropy production rate of diffusion are discussed in the light of local diffusion properties and a stochastic differential equation for entropy production is obtained using the girsanov theorem for reversed diffusion the results are illustrated for the ornstein uhlenbeck process

langevin dynamics for vector variables driven by multiplicative white noise a functional formalism we discuss general multidimensional stochastic processes driven by a system of langevin equations with multiplicative white noise in particular we address the problem of how time reversal diffusion processes are affected by the variety of conventions available to deal with stochastic integrals we present a functional formalism to build up the generating functional of correlation functions without any type of discretization of the langevin equations at any intermediate step the generating functional is characterized by a functional integration over two sets of commuting variables as well as grassmann variables in this representation time reversal transformation became a linear transformation in the extended variables simplifying in this way the complexity introduced by the mixture of prescriptions and the associated calculus rules the stochastic calculus is codified in our formalism in the structure of the grassmann algebra we study some examples such as higher order derivative langevin equations and the functional representation of the micromagnetic stochastic landau lifshitz gilbert equation

a new approach to computational turbulence modeling we present a new approach to computational fluid dynamics cfd using adaptive stabilized galerkin finite element methods with duality based a posteriori error control for chosen output quantities of interest we address the basic question of computability in cfd for a given flow what quantity is computable to what tole rance to what cost we focus on incompressible newtonian flow with medium to large rey nolds numbers involving both laminar and turbulent flow features we estimate a poste riori the output of the computed solution with the output based on the exact solution to the navier stokes equations thus circumventing introducing and modeling reynolds stresses in averaged navier stokes equations our basic tool is a representation formula for th e error in the quantity of interest in terms of a space time integral of the residual of a computed solution multiplied by weights related to derivatives of the solution of an associa ted dual problem with data connected to the output we use the error representation formula to derive an a posterori error estimate combining residuals with computed dual weights which is used for mesh adaptivity in space time with the objective of satisfying a give n error tolerance with minimal computational effort we show in concrete examples that outputs such as mean values in time of drag and lift of turbulent flow around a bluff body are c omputable on a pc with a tolerance of a few percent using a few hundred thousand mesh points in space we refer to our methodology as adaptive dns les where automatically by adaptivity certain features of the flow are resolved in a direct numerical simulation dns while certain other small scale turbulent features are left unresolved in a large eddy simulation les the stabilization of the galerkin method giving a weighted least square control of the residual acts as the subgrid model in the les the a posteriori error estimate takes into account both the error from discretization and the error from the subgrid model we pay particular attention to the stability of the dual solution from i perturbations repla cing the exact convection velocity by a computed velocity and ii computational solution of the dual problem which are the crucial aspects entering by avoiding using averaged navier stokes equations including reynolds stresses a crucial observation is that the contri bution from subgrid modeling in the a posteriori error estimation is small making it possib le to simulate aspects of turbulent flow without accurate modeling of reynolds stresses

on duality based a posteriori error estimation in various norms and linear functionals for large eddy simulation we derive a posteriori error estimates for the filtered velocity field in a large eddy simulation in various norms and linear functionals the a posteriori error estimates take the form of an integral in space time of a discretization residual and a modeling residual times a dual weight the discretization residual is directly computable and the modeling residual is estimated by a scale similarity model we approximate the dual weight by solving an associated linearized dual problem numerically computational examples from transition to turbulence in couette flow are presented

ito versus stratonovich a survey is given of the facts and fancies concerning the nonlinear langevin or ito equation actually it is merely a pre equation which becomes an equation when an interpretation rule is added the rules of ito and stratonovich differ but both are mathematically consistent and therefore equally admissible conventions the reason why they seem to lead to physical differences is that the langevin approach used to arrive at the equation involves a tacit assumption for systems with external noise this assumption can be justified and it is then clear that the stratonovich rule applies systems with internal noise however can only be properly described by a master equation and the ito stratonovich controversy never enters afterward one is free to model the resulting fluctuations either with an ito or a stratonovich scheme but that does not lead to any new information

noisy one dimensional maps near a crisis i weak gaussian white and colored noise we study one dimensional single humped maps near the boundary crisis at fully developed chaos in the presence of additive weak gaussian white noise by means of a new perturbation like method the quasi invariant density is calculated from the invariant density at the crisis in the absence of noise in the precritical regime where the deterministic map may show periodic windows a necessary and sufficient condition for the validity of this method is derived from the quasi invariant density we determine the escape rate which has the form of a scaling law and compares excellently with results from numerical simulations we find that deterministic transient chaos is stabilized by weak noise whenever the maximum of the map is of orderz 1 finally we extend our method to more general maps near a boundary crisis and to multiplicative as well as colored weak gaussian noise within this extended class of noises and for single humped maps with any fixed orderz 0 of the maximum in the scaling law for the escape rate both the critical exponents and the scaling function are universal

state dependent diffusion thermodynamic consistency and its path integral formulation the friction coefficient of a particle can depend on its position as it does when the particle is near a wall we formulate the dynamics of particles with such state dependent friction coefficients in terms of a general langevin equation with multiplicative noise whose evaluation requires the introduction of specific rules two common conventions the ito and the stratonovich provide alternative rules for evaluation of the noise but other conventions are possible we show that the requirement that a particle s distribution function approach the boltzmann distribution at long times dictates that a drift term must be added to the langevin equation this drift term is proportional to the derivative of the diffusion coefficient times a factor that depends on the convention used to define the multiplicative noise we explore the consequences of this result in a number of examples with spatially varying diffusion coefficients we also derive a path integral representation for arbitrary interpretation of the noise and use it in a perturbative study of correlations in a simple system

noisy one dimensional maps near a crisis ii general uncorrelated weak noise the escape rate for one dimensional noisy maps near a crisis is investigated a previously introduced perturbation theory is extended to very general kinds of weak uncorrelated noise including multiplicative white noise as a special case for single humped maps near the boundary crisis at fully developed chaos an asymptotically exact scaling law for the rate is derived it predicts that transient chaos is stabilized by basically any noise of appropriate strength provided the maximum of the map is of sufficiently large order a simple heuristic explanation of this effect is given the escape rate is discussed in detail for noise distributions of levy dichotomous and exponential type in the latter case the rate is dominated by an exponentially leading arrhenius factor in the deep precritical regime however the preexponential factor may still depend more strongly than any power law on the noise strength

towards efficient backward in time adjoint computations using data compression techniques abstract in the context of a posteriori error estimation for nonlinear time dependent partial differential equations the state of the practice is to use adjoint approaches which require the solution of a backward in time problem defined by a linearization of the forward problem one of the major obstacles in the practical application of these approaches is the need to store or recompute the forward solution to define the adjoint problem and to evaluate the error representation this study considers the use of data compression techniques to approximate forward solutions employed in the backward in time integration the development derives an error representation that accounts for the difference between the standard approach and the compressed approximation of the forward solution this representation is algorithmically similar to the standard representation and only requires the computation of the quantity of interest for the forward solution and the data compressed reconstructed solution i e scalar quantities that can be evaluated as the forward problem is integrated this approach is then compared with existing techniques such as checkpointing and time averaged adjoints finally we provide numerical results indicating the potential efficiency of our approach on a transient diffusion reaction equation and on the navier stokes equations these results demonstrate memory compression ratios up to 450 while maintaining reasonable accuracy in the error estimates

periodic orbit sum rules for billiards accelerating cycle expansions we show that the periodic orbit sums for two dimensional billiards satisfy an infinity of exact sum rules we demonstrate their utility by using the flow conservation sum rule to accelerate the convergence of cycle expansions for the overlapping three disc billiard the effectiveness of the approach is studied by applying the method on averages known explicitly by other sum rules the method is then applied to the lyapunov exponent

optimization and universality of brownian search in a basic model of quenched heterogeneous media the kinetics of a variety of transport controlled processes can be reduced to the problem of determining the mean time needed to arrive at a given location for the first time the so called mean first passage time mfpt problem the occurrence of occasional large jumps or intermittent patterns combining various types of motion are known to outperform the standard random walk with respect to the mfpt by reducing oversampling of space here we show that a regular but spatially heterogeneous random walk can significantly and universally enhance the search in any spatial dimension in a generic minimal model we consider a spherically symmetric system comprising two concentric regions with piecewise constant diffusivity the mfpt is analyzed under the constraint of conserved average dynamics that is the spatially averaged diffusivity is kept constant our analytical calculations and extensive numerical simulations demonstrate the existence of an optimal heterogeneity minimizing the mfpt to the target we prove that the mfpt for a random walk is completely dominated by what we term direct trajectories towards the target and reveal a remarkable universality of the spatially heterogeneous search with respect to target size and system dimensionality in contrast to intermittent strategies which are most profitable in low spatial dimensions the spatially inhomogeneous search performs best in higher dimensions discussing our results alongside recent experiments on single particle tracking in living cells we argue that the observed spatial heterogeneity may be beneficial for cellular signaling processes

stability of the dual navier stokes equations and efficient computation of mean output in turbulent flow using adaptive dns les we discuss aspects of adaptive dns les where adaptive finite element methods are used to accurately compute chosen output from a turbulent flow with the computational power of a pc the key to this break through is i application of the general approach to adaptive error control in galerkin methods based on duality coupled with ii crucial properties of turbulent flow allowing accurate computation of mean value quantities such as drag and lift without full resolution of all scales

mode coupling theory and the fluctuation dissipation theorem for nonlinear langevin equations with multiplicative noise in this letter we develop a mode coupling theory for a class of nonlinear langevin equations with multiplicative noise using a field theoretic formalism these equations are simplified models of realistic colloidal suspensions we prove that the derived equations are consistent with the fluctuation dissipation theorem we also discuss the generalization of the result given here to real fluids and the possible description of supercooled fluids in the ageing regime we demonstrate that the standard idealized mode coupling theory is not consistent with the fdt in a strict field theoretic sense

fluctuation response inequality out of equilibrium we present an approach to response around arbitrary out of equilibrium states in the form of a fluctuation response inequality fri we study the response of an observable to a perturbation of the underlying stochastic dynamics we find that the magnitude of the response is bounded from above by the fluctuations of the observable in the unperturbed system and the kullback leibler divergence between the probability densities describing the perturbed and the unperturbed system this establishes a connection between linear response and concepts of information theory we show that in many physical situations the relative entropy may be expressed in terms of physical observables as a direct consequence of this fri we show that for steady state particle transport the differential mobility is bounded by the diffusivity for a virtual perturbation proportional to the local mean velocity we recover the thermodynamic uncertainty relation tur for steady state transport processes finally we use the fri to derive a generalization of the uncertainty relation to arbitrary dynamics which involves higher order cumulants of the observable we provide an explicit example in which the tur is violated but its generalization is satisfied with equality

minimal repetition dynamic checkpointing algorithm for unsteady adjoint calculation adjoint equations of differential equations have seen widespread applications in optimization inverse problems and uncertainty quantification a major challenge in solving adjoint equations for time dependent systems has been the need to use the solution of the original system in the adjoint calculation and the associated memory requirement in applications where storing the entire solution history is impractical checkpointing methods have frequently been used however traditional checkpointing algorithms such as revolve require a priori knowledge of the number of time steps making these methods incompatible with adaptive time stepping we propose a dynamic checkpointing algorithm applicable when the number of time steps is a priori unknown our algorithm maintains a specified number of checkpoints on the fly as time integration proceeds for an arbitrary number of time steps the resulting checkpoints at any snapshot during the time integration have the optimal repetition number the efficiency of our algorithm is demonstrated both analytically and experimentally in solving adjoint equations this algorithm also has significant advantage in automatic differentiation when the length of execution is variable

a rule for quantizing chaos the authors find a real function delta e whose zeros approximate the quantum energy levels of a system with chaotic classical trajectories delta e is a finite sum over combinations of classical periodic orbits it is obtained from gutzwiller s infinite and divergent sum 1982 representing the spectral density in terms of periodic orbits by means of a resummation conjectured by analogy with a derivation of the riemann siegel formula for the riemann zeros they assess the practicality of the quantization condition

anomalous diffusion in stochastic systems with nonhomogeneously distributed traps the stochastic motion in a nonhomogeneous medium with traps is studied and diffusion properties of that system are discussed the particle is subjected to a stochastic stimulation obeying a general levy stable statistics and experiences long rests due to nonhomogeneously distributed traps the memory is taken into account by subordination of that process to a random time then the subordination equation is position dependent the problem is approximated by a decoupling of the medium structure and memory and exactly solved for a power law position dependence of the memory in the case of the gaussian statistics the density distribution and moments are derived depending on geometry and memory parameters the system may reveal both the subdiffusion and enhanced diffusion the similar analysis is performed for the levy flights where the finiteness of the variance follows from a variable noise intensity near a boundary two diffusion regimes are found in the bulk and near the surface the anomalous diffusion exponent as a function of the system parameters is derived

optimal growth trajectories with finite carrying capacity we consider the problem of finding optimal strategies that maximize the average growth rate of multiplicative stochastic processes for a geometric brownian motion the problem is solved through the so called kelly criterion according to which the optimal growth rate is achieved by investing a constant given fraction of resources at any step of the dynamics we generalize these finding to the case of dynamical equations with finite carrying capacity which can find applications in biology mathematical ecology and finance we formulate the problem in terms of a stochastic process with multiplicative noise and a nonlinear drift term that is determined by the specific functional form of carrying capacity we solve the stochastic equation for two classes of carrying capacity functions power laws and logarithmic and in both cases we compute the optimal trajectories of the control parameter we further test the validity of our analytical results using numerical simulations

a dynamical zeta function for pseudo riemannian foliations we investigate a generalization of geodesic random walks to pseudo riemannian foliations the main application we have in mind is to consider the logarithm of the associated zeta function as grand canonical partition function in a theory unifying aspects of general relativity quantum mechanics and dynamical systems

entire fredholm determinants for evaluation of semiclassical and thermodynamical spectra proofs that fredholm determinants of transfer operators for hyperbolic flows are entire can be extended to a large new class of multiplicative evolution operators we construct such operators both for the gutzwiller semiclassical quantum mechanics and for classical thermodynamic formalism and introduce a new functional determinant which is expected to be entire for axiom a flows and whose zeros coincide with the zeros of the gutzwiller voros zeta function

adaptive modeling of turbulent flow with residual based turbulent kinetic energy dissipation in this paper we first review our recent work on a new framework for adaptive turbulence simulation we model turbulence by weak solutions to the navier stokes equations that are wellposed with respect to mean value output in the form of functionals and we use an adaptive finite element method to compute approximations with a posteriori error control based on the error in the functional output we then derive a local energy estimate for a particular finite element method which we connect to related work on dissipative weak euler solutions with kinetic energy dissipation due to lack of local smoothness of the weak solutions the ideas are illustrated by numerical results where we observe a law of finite dissipation with respect to a decreasing mesh size

goal oriented error estimation and adaptivity for free boundary problems the domain map linearization approach in free boundary problems the accuracy of a goal quantity of interest depends on both the accuracy of the approximate solution and the accuracy of the domain approximation we develop duality based a posteriori error estimates for functional outputs of solutions of free boundary problems that include both sources of error the derivation of an appropriate dual problem linearized adjoint is however nonobvious for free boundary problems to derive an appropriate dual problem we present the domain map linearization approach in this approach the free boundary problem is first transformed into an equivalent problem on a fixed reference domain after which the dual problem is obtained by linearization with respect to the domain map we show for a bernoulli type free boundary problem that this dual problem corresponds to a poisson problem with a nonlocal robin type boundary condition furthermore we present numerical experiments that demonstrate the effectivity of the dual based error estimate and its usefulness in goal oriented adaptive mesh refinement

checkpointing schemes for adjoint codes application to the meteorological model meso nh the adjoint code of a nonlinear computer model calculates gradients along a trajectory that has to be known at integration time when the storage of the whole trajectory requires too large an amount of memory the calculation of the adjoint code is split and is done part by part from restart points called checkpoints griewank proposed a checkpointing method named revolve which provides an optimal logarithmic behavior with respect to time and memory requirement in this work some checkpointing schedules are proposed some of them correspond to special cases of revolve the user s preference is essential to choose between time and memory requirements this is a key point for adjoint codes of temporal models such as the meteorological model meso nh that may be used for weather forecasts when the computational time is the top priority a particular checkpointing scheme allows computation of the adjoint code with at most one extra integration of the model the memory requirement behaves then as the square root of the number of iterations of the model checkpointing schemes are tested on adjoint simulations of meso nh

trace formula for noisy flows a trace formula is derived for the fokker planck equation associated with ito stochastic differential equations describing noisy time continuous nonlinear dynamical systems in the weak noise limit the trace formula provides estimations of the eigenvalues of the fokker planck operator on the basis of the pollicott ruelle resonances of the noiseless deterministic system which is assumed to be non bifurcating at first order in the noise amplitude the effect of noise on a periodic orbit is given in terms of the period and the derivative of the period with respect to the pseudo energy of the onsager machlup freidlin wentzell scheme