some projection methods with the bb step sizes for variational inequalities since the appearance of the barzilai borwein bb step sizes strategy for unconstrained optimization problems it received more and more attention of the researchers it was applied in various fields of the nonlinear optimization problems and recently was also extended to optimization problems with bound constraints in this paper we further extend the bb step sizes to more general variational inequality vi problems i e we adopt them in projection methods under the condition that the underlying mapping of the vi problem is strongly monotone and lipschitz continuous and the modulus of strong monotonicity and the lipschitz constant satisfy some further conditions we establish the global convergence of the projection methods with bb step sizes a series of numerical examples are presented which demonstrate that the proposed methods are convergent under mild conditions and are more efficient than some classical projection like methods

adaptive mesh and algorithm refinement using direct simulation monte carlo adaptive mesh and algorithm refinement amar embeds a particle method within a continuum method at the finest level of an adaptive mesh refinement amr hierarchy the coupling between the particle region and the overlaying continuum grid is algorithmically equivalent to that between the fine and coarse levels of amr direct simulation monte carlo dsmc is used as the particle algorithm embedded within a godunov type compressible navier stokes solver several examples are presented and compared with purely continuum calculations

heterogeneous multiscale method for the modeling of complex fluids and micro fluidics the framework of the heterogeneous multiscale method hmm is used to develop numerical methods for the study of macroscale dynamics of fluids in situations where either the constitutive relation or the boundary conditions are not explicitly available and have to be inferred from microscopic models such as molecular dynamics continuum hydrodynamics is used as the macroscopic model while molecular dynamics serves as the microscopic model and is used to supply the necessary data e g the stress or the boundary condition for the macroscopic model scale separation is exploited so that the macroscopic variables can be evolved in macroscopic spatial temporal scales using data that are estimated from molecular dynamics simulation on microscale spatial temporal domains this naturally decouples the micro and macrospatial and temporal scales whenever possible applications are presented for models of complex fluids contact line dynamics and a simple model of non trivial fluid solid interactions

an assessment of time integration schemes for non linear dynamic equations abstract in the transient analysis of engineering structures by the finite element method there are several commonly used time integration schemes such as the method of central difference the newmark method the wilson method the houbolt method and the method of hilber hughes taylor in this paper these time integration schemes are applied to simple non linear dynamic systems and the performances of these schemes are compared it is found that there are significant differences in the accuracy of these schemes the result of some of these schemes are so inaccurate that it becomes clear that they are not suited for time integration over a long time duration the behaviour of these schemes is also compared with that of the fourth order runge kutta method which has been the preferred integration scheme for non linear systems of a single or a few degrees of freedom

algorithm refinement for the stochastic burgers equation in this paper we develop an algorithm refinement ar scheme for an excluded random walk model whose mean field behavior is given by the viscous burgers equation ar hybrids use the adaptive mesh refinement framework to model a system using a molecular algorithm where desired while allowing a computationally faster continuum representation to be used in the remainder of the domain the focus in this paper is the role of fluctuations on the dynamics in particular we demonstrate that it is necessary to include a stochastic forcing term in burgers equation to accurately capture the correct behavior of the system the conclusion we draw from this study is that the fidelity of multiscale methods that couple disparate algorithms depends on the consistent modeling of fluctuations in each algorithm and on a coupling such as algorithm refinement that preserves this consistency

alternative formulations of a combined trip generation trip distribution modal split and trip assignment model abstract the traditional four step model has been widely used in travel demand forecasting by considering trip generation trip distribution modal split and traffic assignment sequentially in a fixed order however this sequential approach suffers from the inconsistency among the level of service and flow values in each step of the procedure in the last two decades this problem has been addressed by many researchers who have sought to develop combined or integrated models that can consider travelers choice on different stages simultaneously and give consistent results in this paper alternative formulations including mathematical programming mp formulation and variational inequality vi formulations are provided for a combined travel demand model that integrates trip generation trip distribution modal split and traffic assignment using the random utility theory framework thus the proposed alternative formulations not only allow a systematic and consistent treatment of travel choice over different dimensions but also have behavioral richness qualitative properties of the formulations are also given to ensure the existence and uniqueness of the solution particularly the model is analyzed for a special but useful case where the probabilistic travel choices are assumed to be a hierarchical logit model furthermore a self adaptive goldstein levitin polyak glp projection algorithm is adopted for solving this special case

a self adaptive projection method with improved step size for solving variational inequalities in this paper we propose a new projection method for solving variational inequality problems which can be viewed as an improvement of the method of han and lo d r han hong k lo two new self adaptive projection methods for variational inequality problems computers mathematics with applications 43 2002 1529 1537 by adopting a new step size rule the method is as simple as han and lo s methods d r han hong k lo two new self adaptive projection methods for variational inequality problems computers mathematics with applications 43 2002 1529 1537 and other extra gradient type methods which uses only function evolutions and projections onto the feasible set we prove that under the condition that the underlying function is co coercive the sequence generated by the method converges to a solution of the variational inequality problem globally some preliminary computational results are reported which illustrated that the new method is more efficient than han and lo s method d r han hong k lo two new self adaptive projection methods for variational inequality problems computers mathematics with applications 43 2002 1529 1537

numerical integration of non linear elastic multi body systems this paper is concerned with the modelling of nonlinear elastic multi body systems discretized using the finite element method the formulation uses cartesian co ordinates to represent the position of each elastic body with respect to a single inertial frame the kinematic constraints among the various bodies of the system are enforced via the lagrange multiplier technique the resulting equations of motion are stiff non linear differential algebraic equations the integration of these equations presents a real challenge as most available techniques are either numerically unstable or present undesirable high frequency oscillations of a purely numerical origin an approach is proposed in which the equations of motion are discretized so that they imply conservation of the total energy for the elastic components of the system whereas the forces of constraint are discretized so that the work they perform vanishes exactly the combination of these two features of the discretization guarantees the stability of the numerical integration process for non linear elastic multi body systems examples of the procedure are presented

asymmetric variational inequality problems over product sets applications and iterative methods in this paper we i describe how several equilibrium problems can be uniformly modelled by a finite dimensional asymmetric variational inequality defined over a cartesian product of sets and ii investigate the local and global convergence of various iterative methods for solving such a variational inequality problem because of the special cartesian product structure these iterative methods decompose the original variational inequality problem into a sequence of simpler variational inequality subproblems in lower dimensions the resulting decomposition schemes often have a natural interpretation as some adjustment processes

design of energy conserving algorithms for frictionless dynamic contact problems this paper proposes a formulation of dynamic contact problems which enables exact algorithmic conservation of linear momentum angular momentum and energy in finite element simulations it is seen that a lagrange multiplier enforcement of an appropriate contact rate constraint produces these conservation properties a related method is presented in which a penalty regularization of the aforementioned rate constraint is utilized this penalty method sacrifices the energy conservation property but is dissipative under all conditions of changing contact so that the global algorithm remains stable notably it is also shown that augmented lagrangian iteration utilizing this penalty kernel reproduces the energy conserving i e lagrange multiplier solution to any desired degree of accuracy the result is a robust stable method even in the context of large deformations as is shown by some representative numerical examples in particular the ability of the formulation to produce accurate results where more traditional integration schemes fail is emphasized by the numerical simulations 1997 by john wiley sons ltd

numerical comparison between prediction correction methods for general variational inequalities in this paper we propose two prediction correction methods for solving general variational inequalities the first method makes use of a descent direction to produce the new iterate while the second method generates the corrector by the improved extragradient method under certain conditions the global convergence of both methods is proved it is proved theoretically that the lower bound of the progress obtained by the second method is greater than that by the first one numerical results are given to verify and compare the numerical efficiency of these two predictor corrector methods

a globally convergent newton method for solving strongly monotone variational inequalities variational inequality problems have been used to formulate and study equilibrium problems which arise in many fields including economics operations research and regional sciences for solving variational inequality problems various iterative methods such as projection methods and the nonlinear jacobi method have been developed these methods are convergent to a solution under certain conditions but their rates of convergence are typically linear in this paper we propose to modify the newton method for variational inequality problems by using a certain differentiable merit function to determine a suitable step length the purpose of introducing this merit function is to provide some measure of the discrepancy between the solution and the current iterate it is then shown that under the strong monotonicity assumption the method is globally convergent and under some additional assumptions the rate of convergence is quadratic limited computational experience indicates the high efficiency of the proposed method

explicit time integration algorithms for structural dynamics with optimal numerical dissipation abstract a new predictor corrector explicit time integration algorithm is presented for solving structural dynamics problems the basis of the algorithm is the implicit generalized method developed by the authors like its implicit parent the explicit generalized method is a one parameter family of algorithms in which the parameter defines the high frequency numerical dissipation for a given value of high frequency dissipation the explicit generalized method minimizes low frequency dissipation the algorithm can be utilized effectively for structural dynamics calculations in which numerical dissipation is needed to reduce spurious oscillations parameter values are given that enable physical damping to be treated explicitly while maintaining second order accuracy without the need for an expensive second corrector pass

comparison of some single step methods for the numerical solution of the structural dynamic equation this paper compares the performance of the ss22 and ss23 references 1 and 2 single step algorithms for the numerical solution of the second order structural dynamic equation and a related new algorithm ss32b applied to the equivalent first order system with sine and step forcing functions various aspects of stability relevant to these equations are discussed

generalized descent methods for asymmetric systems of equations we consider generalizations of the steepest descent algorithm for solving asymmetric systems of equations we first show that if the system is linear and is defined by the matrix m then the method converges if m2 is positive definite we also establish easy to verify conditions on the matrix m that ensure that m2 is positive definite and develop a scaling procedure that extends the class of matrices that satisfy the convergence conditions in addition we establish a local convergence result for nonlinear systems defined by uniformly monotone maps and discuss a class of general descent methods all of the methods that we consider reduce to standard nonlinear programming algorithms for equivalent optimization problems when the jacobian of the underlying problem map is symmetric we interpret the convergence conditions for the generalized steepest descent algorithms as restricting the degree of asymmetry of the problem map

an operator splitting method for variational inequalities with partially unknown mappings in this paper we propose a new operator splitting method for solving a class of variational inequality problems in which part of the underlying mappings are unknown this class of problems arises frequently from engineering economics and transportation equilibrium problems at each iteration by using the information observed from the system the method solves a system of nonlinear equations which is well defined under mild assumptions the global convergence of the method is proved and its efficiency is demonstrated with numerical examples

a mesoscopic bridging scale method for fluids and coupling dissipative particle dynamics with continuum finite element method a multiscale procedure to couple a mesoscale discrete particle model and a macroscale continuum model of incompressible fluid flow is proposed in this study we call this procedure the mesoscopic bridging scale mbs method since it is developed on the basis of the bridging scale method for coupling molecular dynamics and finite element models g j wagner w k liu coupling of atomistic and continuum simulations using a bridging scale decomposition j comput phys 190 2003 249 274 we derive the governing equations of the mbs method and show that the differential equations of motion of the mesoscale discrete particle model and finite element fe model are only coupled through the force terms based on this coupling we express the finite element equations which rely on the navier stokes and continuity equations in a way that the internal nodal fe forces are evaluated using viscous stresses from the mesoscale model the dissipative particle dynamics dpd method for the discrete particle mesoscale model is employed the entire fluid domain is divided into a local domain and a global domain fluid flow in the local domain is modeled with both dpd and fe method while fluid flow in the global domain is modeled by the fe method only the mbs method is suitable for modeling complex colloidal fluid flows where continuum methods are sufficiently accurate only in the large fluid domain while small local regions of particular interest require detailed modeling by mesoscopic discrete particles solved examples simple poiseuille and driven cavity flows illustrate the applicability of the proposed mbs method

long term stable time integration scheme for dynamic analysis of planar geometrically exact timoshenko beams abstract in this paper an energy momentum method for geometrically exact timoshenko type beam is proposed the classical time integration schemes in dynamics are known to exhibit instability in the non linear regime the so called timoshenko type beam with the use of rotational degree of freedom leads to simpler strain relations and simpler expressions of the inertial terms as compared to the well known bernoulli type model the treatment of the bernoulli model has been recently addressed by the authors in this present work we extend our approach of using the strain rates to define the strain fields to in plane geometrically exact timoshenko type beams the large rotational degrees of freedom are exactly computed the well known enhanced strain method is used to avoid locking phenomena conservation of energy momentum and angular momentum is proved formally and numerically the excellent performance of the formulation will be demonstrated through a range of examples

solving variational inequalities by a modified projection method with an effective step size in this paper we propose a new projection method for the solution of variational inequality problems the method is simple which uses only function evaluations and projections onto the feasible set we adopt a new step size rule and a new search direction in the new method under the mild conditions we prove the proposed method is globally convergent preliminary numerical results are reported

an iterative scheme for variational inequalities in this paper we introduce and study a general iterative scheme for the numerical solution of finite dimensional variational inequalities this iterative scheme not only contains as special cases the projection linear approximation and relaxation methods but also induces new algorithms then we show that under appropriate assumptions the proposed iterative scheme converges by establishing contraction estimates involving a sequence of norms in en induced by symmetric positive definite matrices gm thus in contrast to the above mentioned methods this technique allows the possibility of adjusting the norm at each step of the algorithm this flexibility will generally yield convergence under weaker assumptions

on the computation of ensemble averages for spatially non uniform particle systems when there is no clear separation between micro and macro scales ergodicity cannot be invoked to transform ensemble into volume averages in such cases it is necessary to use ensemble averaging directly a straightforward calculation of such averages converges slowly and therefore requires a large number of realizations of the system this paper describes a much more efficient method based on the use of a fourier expansion of the quantity to be averaged the advantages of the fourier approach are estimated in general terms and demonstrated explicitly with several examples for the specific problem of equal spheres in a viscous fluid the analytical estimates suggest that similar results can be expected for other situations as well it is shown both analytically and numerically that the variance of the fourier coefficients is in many cases significantly smaller than that of the direct method which leads to a much faster convergence of the former the paper also describes a method by which the probability distribution of a uniform ensemble can be biased so as to mimic that of a non uniform one with prescribed properties

a new modified goldstein levitin polyakprojection method for variational inequality problems in this paper we first show that the adjustment parameter in the step size choice strategy of the modified goldstein levitin polyak projection method proposed by he et al for asymmetric strongly monotone variational inequality problems can be bounded away from zero by a positive constant under this observation we propose a new step size rule which seems to be more practical and robust than the original one we show that the new modified method is globally convergent under the same conditions and report some computational results to illustrate the method

iterative methods for variational and complementarity problems in this paper we study both the local and global convergence of various iterative methods for solving the variational inequality and the nonlinear complementarity problems included among such methods are the newton and several successive overrelaxation algorithms for the most part the study is concerned with the family of linear approximation methods these are iterative methods in which a sequence of vectors is generated by solving certain linearized subproblems convergence to a solution of the given variational or complementarity problem is established by using three different yet related approaches the paper also studies a special class of variational inequality problems arising from such applications as computing traffic and economic spatial equilibria finally several convergence results are obtained for some nonlinear approximation methods

a stochastic mixed finite element heterogeneous multiscale method for flow in porous media a computational methodology is developed to efficiently perform uncertainty quantification for fluid transport in porous media in the presence of both stochastic permeability and multiple scales in order to capture the small scale heterogeneity a new mixed multiscale finite element method is developed within the framework of the heterogeneous multiscale method hmm in the spatial domain this new method ensures both local and global mass conservation starting from a specified covariance function the stochastic log permeability is discretized in the stochastic space using a truncated karhunen loeve expansion with several random variables due to the small correlation length of the covariance function this often results in a high stochastic dimensionality therefore a newly developed adaptive high dimensional stochastic model representation technique hdmr is used in the stochastic space this results in a set of low stochastic dimensional subproblems which are efficiently solved using the adaptive sparse grid collocation method asgc numerical examples are presented for both deterministic and stochastic permeability to show the accuracy and efficiency of the developed stochastic multiscale method

evaluation of kriging based surrogate models constructed from mesoscale computations of shock interaction with particles abstract macro scale computations of shocked particulate flows require closure laws that model the exchange of momentum energy between the fluid and particle phases closure laws are constructed in this work in the form of surrogate models derived from highly resolved mesoscale computations of shock particle interactions the mesoscale computations are performed to calculate the drag force on a cluster of particles for different values of mach number and particle volume fraction two kriging based methods viz the dynamic kriging method dkg and the modified bayesian kriging method mbkg are evaluated for their ability to construct surrogate models with sparse data i e using the least number of mesoscale simulations it is shown that if the input data is noise free the dkg method converges monotonically convergence is less robust in the presence of noise the mbkg method converges monotonically even with noisy input data and is therefore more suitable for surrogate model construction from numerical experiments this work is the first step towards a full multiscale modeling of interaction of shocked particle laden flows

a new family of explicit time integration methods for linear and non linear structural dynamics a new family of explicit single step time integration methods with controllable high frequency dissipation is presented for linear and non linear structural dynamic analyses the proposed methods are second order accurate and completely explicit with a diagonal mass matrix even when the damping matrix is not diagonal in the linear structural dynamics or the internal force vector is a function of velocities in the non linear structural dynamics stability and accuracy of the new explicit methods are analysed for the linear undamped damped cases furthermore the new methods are compared with other explicit methods

energy conserving and decaying algorithms in non linear structural dynamics a generalized formulation of the energy momentum methodwill be developed within the framework of the generalizedmethodwhich allows at the same time guaranteed conservation or decay of total energy and controllable numerical dissipation of unwanted high frequency response furthermore the latter algorithm will be extended by the consistently integrated constraints of energy and momentum conservation originally derived for the constraint energy momentum algorithm the goal of this general approach of implicit energyconserving and decaying time integration schemes is to compare these algorithms on the basis of an equivalent notation by the means of an overall algorithmic design and hence to investigate their numerical properties numerical stability and controllable numerical dissipation of high frequencies will be studied in application to non linear structural dynamics among the methods considered will be the newmark method the classical methods the energy momentum methodwith and without numerical dissipation the constraint energymomentum algorithm and the constraint energy method copyright 1999 john wiley sons ltd

efficient energy stable numerical schemes for a phase field moving contact line model in this paper we present two efficient energy stable schemes to solve a phase field model incorporating moving contact line the model is a coupled system that consists of incompressible navier stokes equations with a generalized navier boundary condition and cahn hilliard equation in conserved form in both schemes the projection method is used to deal with the navier stokes equations and stabilization approach is used for the non convex ginzburg landau bulk potential by some subtle explicit implicit treatments we obtain a linear coupled energy stable scheme for systems with dynamic contact line conditions and a linear decoupled energy stable scheme for systems with static contact line conditions an efficient spectral galerkin spatial discretization method is implemented to verify the accuracy and efficiency of proposed schemes numerical results show that the proposed schemes are very efficient and accurate we proposed two linear energy stable numerical schemes for a phase field moving contact line model discrete energy laws that are consistent to the continuous ones are proved the efficiency and fast convergence of the proposed schemes with spectral spatial approximations are numerically verified

on practical integration of semi discretized nonlinear equations of motion part 1 reasons for probable instability and improper convergence time integration is the most versatile method for analyzing the general case of nonlinear semi discretized equations of motion however the approximate responses of such analyses generally do not converge properly and might even display numerical instability this is a very significant shortcoming especially in practical time integration herein after illustrating that this phenomenon is viable even for very simple nonlinear dynamic models sources of the shortcoming are discussed in detail the conclusion is that in time integration of nonlinear dynamic mathematical models of physically stable structural systems responses may converge improperly for three major reasons these reasons are 1 inadequate number of iterations before terminating nonlinearity solutions 2 deficiencies in the formulation of some time integration methods and 3 the inherent behaviour of the models of some special dynamic systems in addition limitations on computational facilities and improper consideration of these limitations may impair the numerical stability and convergence of the computed responses the differences between static and dynamic analyses are also discussed from the viewpoint of the numerical errors induced by nonlinearity

conservation properties of a time fe method part ii time stepping schemes for non linear elastodynamics in the present paper one step implicit integration algorithms for non linear elastodynamics are developed the discretization process rests on galerkin methods in space and time in particular the continuous galerkin method applied to the hamiltonian formulation of semidiscrete non linear elastodynamics lies at the heart of the time stepping schemes algorithmic conservation of energy and angular momentum are shown to be closely related to quadrature formulas that are required for the calculation of time integrals we newly introduce the assumed strain method in time which enables the design of energy momentum conserving schemes and which can be interpreted as temporal counterpart of the well established assumed strain method for finite elements in space the numerical examples deal with quasi rigid motion as well as large strain motion copyright 2001 john wiley sons ltd

new step lengths in projection method for variational inequality problems in this paper we consider projection method for variational inequality problems first we give a new modification of recently proposed self adaptive step length rules with possibly minimal parameters then the resulting self adaptive projection method is proven to converge globally at a r linear rate provided that the underlying mapping is strongly monotone and lipschitz continuous preliminary numerical results confirm its flexibility and effectiveness

numerical approximations for a phase field moving contact line model with variable densities and viscosities we consider the numerical approximations of a two phase hydrodynamics coupled phase field model that incorporates the variable densities viscosities and moving contact line boundary conditions the model is a nonlinear coupled system that consists of incompressible navierstokes equations with the generalized navier boundary condition and the cahnhilliard equations with moving contact line boundary conditions by some subtle explicitimplicit treatments to nonlinear terms we develop two efficient unconditionally energy stable numerical schemes in particular a linear decoupled energy stable scheme for the system with static contact line condition and a nonlinear energy stable scheme for the system with dynamic contact line condition an efficient spectral galerkin spatial discretization is implemented to verify the accuracy and efficiency of proposed schemes various numerical results show that the proposed schemes are efficient and accurate

a front tracking method for viscous incompressible multi fluid flows abstract a method to simulate unsteady multi fluid flows in which a sharp interface or a front separates incompressible fluids of different density and viscosity is described the flow field is discretized by a conservative finite difference approximation on a stationary grid and the interface is explicitly represented by a separate unstructured grid that moves through the stationary grid since the interface deforms continuously it is necessary to restructure its grid as the calculations proceed in addition to keeping the density and viscosity stratification sharp the tracked interface provides a natural way to include surface tension effects both two and three dimensional full numerical simulations of bubble motion are presented

a relaxed projection method for variational inequalities this paper presents a modification of the projection methods for solving variational inequality problems each iteration of the proposed algorithm consists of projection onto a halfspace containing the given closed convex set rather than the latter set itself the algorithm can thus be implemented very easily and its global convergence to the solution can be established under suitable conditions

analytical and numerical study of coupled atomistic continuum methods for fluids the stability and convergence rate of coupled atomistic continuum methods are studied analytically and numerically these methods couple a continuum model with molecular dynamics through the exchange of boundary conditions in the continuum particle overlapping region different coupling schemes including velocity velocity flux velocity velocity flux and flux flux are studied it is found that the velocity velocity and flux velocity schemes are stable the flux flux scheme is weakly unstable the stability of the velocity flux scheme depends on the parameter t c which is the length of the time interval between successive exchange of boundary conditions it is stable when t c is small and unstable when t c is large for steady state problems the flux velocity scheme converges faster than the other coupling schemes

a modified projection method with a new direction for solving variational inequalities in this paper we propose a new projection method for solving variational inequality problems which can be viewed as an improvement of the method of li et al m li l z liao x m yuan a modified projection method for co coercive variational inequality european journal of operational research 189 2008 310 323 by adopting a new direction under the same assumptions as those in li et al 2008 we establish the global convergence of the proposed algorithm some preliminary computational results are reported which illustrated that the new method is more efficient than the method of li et al 2008

asymptotic upper bounds for the errors of richardson extrapolation with practical application in approximate computations the results produced by richardson extrapolation though in general very accurate are inexact numerical evaluation of this inexactness and implementation of the evaluation in practice are the objectives of this paper first considering linear changes of errors in the convergence plots asymptotic upper bounds are proposed for the errors then the achievement is extended to the results produced by richardson extrapolation and finally an error controlling procedure is proposed and successfully implemented in approximate computations originated in science and engineering copyright 2009 john wiley sons ltd

a sequential linear programming algorithm for solving monotone variational inequalities applied to strongly monotone variational inequalities newton s algorithm achieves local quadratic convergence in this paper it is shown how the basic newton method can be modified to yield an algorithm whose global convergence can be guaranteed by monitoring the monotone decrease of the gap function associated with the variational inequality each iteration consists in the solution of a linear program in the space of primal dual variables and of a linesearch convergence does not depend on strong monotonicity however under strong monotonicity and geometric stability assumptions the set of active constraints at the solution is implicitly identified and quadratic convergence is achieved

coarse variables of autonomous ode systems and their evolution abstract given an autonomous system of ordinary differential equations ode we consider developing practical models for the deterministic slow coarse behavior of the ode system two types of coarse variables are considered the first type consists of running finite time averages of phase functions approaches to construct the coarse evolution equation for this type are discussed and implemented on a forced lorenz system and a singularly perturbed system whose fast flow does not necessarily converge to an equilibrium we explore two strategies in one we compute locally invariant manifolds of the fast dynamics parameterized by the slow variables in the other the choice of our coarse variables automatically guarantees them to be slow in a precise sense this allows their evolution to be phrased in terms of averaging utilizing limit measures probability distributions of the fast flow coarse evolution equations are constructed based on these approaches and tested against coarse response of the microscopic models the second type of coarse variables are defined as non trivial scalar state functions that are required by design to evolve autonomously to the extent possible with the goal of being candidate state functions for unambiguously initializable coarse dynamics the question motivates a mathematical restatement in terms of a first order pde a computational approximation is developed and tested on the lorenz system and the hald hamiltonian system