fast methods for the eikonal and related hamilton jacobi equations on unstructured meshes the fast marching method is a numerical algorithm for solving the eikonal equation on a rectangular orthogonal mesh in o m log m steps where m is the total number of grid points the scheme relies on an upwind finite difference approximation to the gradient and a resulting causality relationship that lends itself to a dijkstra like programming approach in this paper we discuss several extensions to this technique including higher order versions on unstructured meshes in rn and on manifolds and connections to more general static hamilton jacobi equations

fast sweeping algorithms for a class of hamilton jacobi equations we derive a godunov type numerical flux for the class of strictly convex homogeneous hamiltonians that includes h p q sqrt ap 2 bq 2 2cpq c 2 ab we combine our godunov numerical fluxes with simple gauss seidel type iterations for solving the corresponding hamilton jacobi hj equations the resulting algorithm is fast since it does not require a sorting strategy as found e g in the fast marching method in addition it providesa way to compute solutions to a class of hj equations for which the conventional fast marching method is not applicable our experiments indicate convergence after a few iterations even in rather difficult cases

a parallel two scale method for eikonal equations numerous applications of eikonal equations prompted the development of many efficient numerical algorithms the heap cell method hcm is a recent serial two scale technique that has been shown to have advantages over other serial state of the art solvers for a wide range of problems a chacon and a vladimirsky siam j sci comput 34 2012 pp a547 a578 this paper presents a parallelization of hcm for a shared memory architecture the numerical experiments in r 3 show that the parallel hcm exhibits good algorithmic behavior and scales well resulting in a very fast and practical solver

on transformations of graded matrices with applications to stiff ode s estimates concerning the spectrum of a graded matrix and other information useful for a reliable and efficient handling of certain complications in the numerical treatment of some stiff ode s can be inexpensively obtained from the factorized jacobian the validity of the estimates is studied by considering them as the first step in a block lr algorithm which may be of interest in its own right its convergence properties are examined

point island dynamics under fixed rate deposition abstract we consider the dynamics of point islands during submonolayer deposition in which the fragmentation of subcritical size islands is allowed to understand asymptotics of solutions we use methods of centre manifold theory and for globalisation we employ results from the theories of compartmental systems and of asymptotically autonomous dynamical systems we also compare our results with those obtained by making the quasi steady state assumption

a viscosity solutions approach to shape from shading the problem of recovering a lambertian surface from a single two dimensional image may be written as a first order nonlinear equation which presents the disadvantage of having several continuous and even smooth solutions a new approach based on hamilton jacobi bellman equations and viscosity solutions theories enables one to study non uniqueness phenomenon and thus to characterize the surface among the various solutions a consistent and monotone scheme approximating the surface is constructed thanks to the dynamic programming principle and numerical results are presented

quasi steady state and nearly invariant sets we propose the notion of nearly invariant set for an ordinary differential equation to give a clear mathematical meaning to the notion of quasi steady state qss in equations describing chemically reacting systems we show that this approach is sensible in view of the intended applications moreover the approach leads to a heuristic for finding small parameters to satisfy a qss assumption the results are illustrated by several examples

a fast marching algorithm for the factored eikonal equation the eikonal equation is instrumental in many applications in several fields ranging from computer vision to geoscience this equation can be efficiently solved using the iterative fast sweeping fs methods and the direct fast marching fm methods however when used for a point source the original eikonal equation is known to yield inaccurate numerical solutions because of a singularity at the source in this case the factored eikonal equation is often preferred and is known to yield a more accurate numerical solution one application that requires the solution of the eikonal equation for point sources is travel time tomography this inverse problem may be formulated using the eikonal equation as a forward problem while this problem has been solved using fs in the past the more recent choice for applying it involves fm methods because of the efficiency in which sensitivities can be obtained using them however while several fs methods are available for solving the factored equation the fm method is available only for the original eikonal equation in this paper we develop a fast marching algorithm for the factored eikonal equation using both first and second order finite difference schemes our algorithm follows the same lines as the original fm algorithm and requires the same computational effort in addition we show how to obtain sensitivities using this fm method and apply travel time tomography formulated as an inverse factored eikonal equation numerical results in two and three dimensions show that our algorithm solves the factored eikonal equation efficiently and demonstrate the achieved accuracy for computing the travel time we also demonstrate a recovery of a 2d and 3d heterogeneous medium by travel time tomography using the eikonal equation for forward modeling and inversion by gauss newton

fast marching methods for stationary hamilton jacobi equations with axis aligned anisotropy the fast marching method fmm has proved to be a very efficient algorithm for solving the isotropic eikonal equation because it is a minor modification of dijkstra s algorithm for finding the shortest path through a discrete graph fmm is also easy to implement in this paper we describe a new class of hamilton jacobi hj pdes with axis aligned anisotropy which satisfy a causality condition for standard finite difference schemes on orthogonal grids and can hence be solved using the fmm the only modification required to the algorithm is in the local update equation for a node this class of hj pdes has applications in anelliptic wave propagation and robotic path planning and brief examples are included since our class of hj pdes and grids permit asymmetries we also examine some methods of improving the efficiency of the local update that do not require symmetric grids and pdes finally we include explicit update formulas for variations of the eikonal equation that use the manhattan euclidean and infinity norms on orthogonal grids of arbitrary dimension and with variable node spacing

invariant grids for reaction kinetics abstract in this paper we construct low dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold mim mim is based on a formulation of the condition of invariance as an equation and its solution by newton iterations a grid based version of mim is developed the method of invariant grids we describe the newton method and the relaxation method for the invariant grids construction the problem of the grid correction is fully decomposed into the problems of the grid s nodes correction the edges between the nodes appear only in the calculation of the tangent spaces this fact determines high computational efficiency of the method of invariant grids the method is illustrated by two examples the simplest catalytic reaction michaelis menten mechanism and the hydrogen oxidation the algorithm of analytical continuation of the approximate invariant manifold from the discrete grid is proposed generalizations to open systems are suggested the set of methods covered makes it possible to effectively reduce description in chemical kinetics

0 1 2 chva tal gomory cuts given the integer polyhedronp t conv x n ax b wherea m n andb m achvatal gomory cg cut is a valid inequality forp 1 of the type ax b for some such that a n in this paper we study 0 1 2 cg cuts arising for 0 1 2 m we show that the associated separation problem 0 1 2 sep is equivalent to finding a minimum weight member of a binary clutter this implies that 0 1 2 sep is np complete in the general case but polynomially solvable whena is related to the edge path incidence matrix of a tree we show that 0 1 2 sep can be solved in polynomial time for a convenient relaxation of the systemax b this leads to an efficient separation algorithm for a subclass of 0 1 2 cg cuts which often contains wide families of strong inequalities forp 1 applications to the clique partitioning asymmetric traveling salesman plant location acyclic subgraph and linear ordering polytopes are briefly discussed

quasi steady state reduction for compartmental systems abstract we present a method to determine an asymptotic reduction in the sense of tikhonov and fenichel for singularly perturbed compartmental systems in the presence of slow transport it turns out that the reduction can be derived from the individual interaction terms alone we apply the result to spatially discretized reaction diffusion systems and obtain based on the reduced discretized systems a heuristic to reduce reaction diffusion systems in presence of slow diffusion

an efficient discontinuous galerkin method on triangular meshes for a pedestrian flow model in this paper we develop a discontinuous galerkin method on triangular meshes to solve the reactive dynamic user equilibrium model for pedestrian flows the pedestrian density in this model is governed by the conservation law in which the flow flux is implicitly dependent on the density through the eikonal equation to solve the eikonal equation efficiently at each time level we use the fast sweeping method two numerical examples are then used to demonstrate the effectiveness of the algorithm copyright 2008 john wiley sons ltd

determining small parameters for quasi steady state for a parameter dependent system of ordinary dierential equations we present a systematic approach to the determination of parameter values near which singular perturbation scenarios in the sense of tikhonov and fenichel arise we call these special values tikhonov parameter values the principal application we intend is to equations that describe chemical reactions in the context of quasi steady state or partial equilibrium settings such equations have rational or even polynomial right hand side we determine the structure of the set of tikhonov parameter values as a semi algebraic set and present an algorithmic approach to their explicit determination using groebner bases examples and applications which include the irreversible and reversible michaelis menten systems illustrate that the approach is rather easy to implement msc 2010 92c45 34e15 80a30 13p10

reversible equivalent monomolecular tau leaping methods provide for efficient and approximate time stepping of chemical reaction systems modeled by continuous time discrete state stochastic dynamics we investigate the application of leaping methods for small number and stiff systems i e systems whose dynamics involve different time scales and have some molecular species present in very small numbers specifically in the range 0 to 10 we propose a new explicit leaping scheme reversible equivalent monomolecular tau remm which shows considerable promise in the simulation of such systems the remm scheme is based on the fact that the exact solution of the two prototypical monomolecular reversible reactions s1 s2 and s 0 as a function of time takes a simple form involving binomial and or poisson random variables the remm method involves approximating bimolecular reversible reactions by suitable monomolecular reversible reactions as well as considering each reversible pair of reactions in the system to be operating in isolation during the time step we illustrate the use of the remm method through a number of biologically motivated examples and compare its performance to those of the implicit and trapezoidal implicit algorithms in most cases considered remm appears to perform better than these two methods while having the important advantage of being computationally faster due to the explicit nature of the method furthermore when stepsize is increased the remm exhibits a more robust performance than the implicit or the trapezoidal implicit for small number stiff problems

high order essentially nonsocillatory schemes for hamilton jacobi equations hamilton jacobi h j equations are frequently encountered in applications e g in control theory and differential games h j equations are closely related to hyperbolic conservation laws in one space dimension the former is simply the integrated version of the latter similarity also exists for the multidimensional case and this is helpful in the design of difference approximations in this paper high order essentially nonoscillatory eno schemes for h j equations are investigated which yield uniform high order accuracy in smooth regions and sharply resolve discontinuities in the derivatives the eno scheme construction procedure is adapted from that for hyperbolic conservation laws the schemes are numerically tested on a variety of one dimensional and two dimensional problems including a problem related to control optimization and high order accuracy in smooth regions good resolution of discontinuities in the derivatives and convergence to viscosity solutions are observed

rescue of the quasi steady state approximation in a model for oscillations in an enzymatic cascade a three variable model describing the oscillatory activity of a cascade of enzyme reactions is analyzed a quasi steady state approximation reduces the three equations to a system of two equations which admits only a stable steady state this apparent failure of the quasi steady state approximation to describe the limit cycle oscillations observed in the full three variable system is analyzed in detail we first show that the oscillations occur in the full system provided the michaelis constants are sufficiently small we then develop a method for determining the correct limit for application of the quasi steady state approximation the leading problem consists of two equations for a conservative oscillator and a higher order analysis is required in order to determine the amplitude of the limit cycle oscillations finally we observe a good agreement when comparing exact numerical and approximate bifurcation diagrams

new approach to the stability of chemical reaction networks piecewise linear in rates lyapunov functions piecewise linear in rates pwlr lyapunov functions are introduced for a class of chemical reaction networks crns in addition to their simple structure these functions are robust with respect to arbitrary monotone reaction rates of which mass action is a special case the existence of such functions ensures the convergence of trajectories towards equilibria and can be used to establish their asymptotic stability with respect to the corresponding stoichiometric compatibility class we give the definition of these lyapunov functions prove their basic properties and provide algorithms for constructing them examples are provided relationship with consensus dynamics are discussed and future directions are elaborated

an outer approximation algorithm for a class of mixed integer nonlinear programs an outer approximation algorithm is presented for solving mixed integer nonlinear programming problems of a particular class linearity of the integer or discrete variables and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure based on principles of decomposition outer approximation and relaxation the proposed algorithm effectively exploits the structure of the problems and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a mixed integer linear master program convergence and optimality properties of the algorithm are presented as well as a general discussion on its implementation numerical results are reported for several example problems to illustrate the potential of the proposed algorithm for programs in the class addressed in this paper finally a theoretical comparison with generalized benders decomposition is presented on the lower bounds predicted by the relaxed master programs

deterministic and stochastic models of enzymatic networks applications to pharmaceutical research the intracellular transducing device consists of complex networks of enzymatic reactions unfortunately the mathematical models commonly used to describe them are still unsatisfactory and unreliable even at the level of reproducing simple reaction schemes the improvement of mathematical models is necessary and can follow different approaches still poorly employed such as the modeling of spatial structures and phenomena time delays stochastic perturbations only to cite the most relevant ones in this paper we show some recent results related to the total quasi steady state approximation tqssa in a deterministic scenario moreover we show some possible applications of the tqssa in a stochastic scheme

lax friedrichs sweeping scheme for static hamilton jacobi equations we propose a simple fast sweeping method based on the lax friedrichs monotone numerical hamiltonian to approximate viscosity solutions of arbitrary static hamilton jacobi equations in any number of spatial dimensions by using the lax friedrichs numerical hamiltonian we can easily obtain the solution at a specific grid point in terms of its neighbors so that a gauss seidel type nonlinear iterative method can be utilized furthermore by incorporating a group wise causality principle into the gauss seidel iteration by following a finite group of characteristics we have an easy to implement sweeping type and fast convergent numerical method however unlike other methods based on the godunov numerical hamiltonian some computational boundary conditions are needed in the implementation we give a simple recipe which enforces a version of discrete min max principle some convergence analysis is done for the one dimensional eikonal equation extensive 2 d and 3 d numerical examples illustrate the efficiency and accuracy of the new approach to our knowledge this is the first fast numerical method based on discretizing the hamilton jacobi equation directly without assuming convexity and or homogeneity of the hamiltonian

polyhedral lyapunov functions structurally ensure global asymptotic stability of dynamical networks iff the jacobian is non singular for a vast class of dynamical networks including chemical reaction networks crns with monotonic reaction rates the existence of a polyhedral lyapunov function plf implies structural i e parameter free local stability global structural stability is ensured under the additional assumption that each of the variables chemical species concentrations in crns is subject to a spontaneous infinitesimal dissipation this paper solves the open problem of global structural stability in the absence of the infinitesimal dissipation showing that the existence of a plf structurally ensures global convergence if and only if the system jacobian passes a structural non singularity test it is also shown that if the jacobian is structurally non singular under positivity assumptions for the system partial derivatives the existence of an equilibrium is guaranteed for systems subject to positivity constraints it is shown that if the system admits a plf under structural non singularity assumptions global convergence within the positive orthant is structurally ensured while the existence of an equilibrium can be proven by means of a linear programming test and the computation of a piecewise linear in rate lyapunov function

the quasi steady state assumption a case study in perturbation the quasi steady state assumption qssa of biochemistry is studied as an approximation that is important in itself and also that exemplifies an approach to ode systems with an initial fast transient simple estimates of the two relevant time scales of the underlying problem are made these estimates lead to a dimensionless formulation containing a different small parameter from that which is customary earlier results on the qssa are surveyed in the context of the new nondimensionalization it is shown that more comprehensive and simpler error estimates can now be made some general methodological hints are drawn from this example

two approximations of solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games the associated initial value problems almost never have global time classical solutions and one must deal with suitable generalized solutions the correct class of generalized solutions has only recently been established by the authors this article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates analogous results are proved by similar means for the method of vanishing viscosity author

volume determination for bulk materials in bunkers a simple model for the determination of the shape of large granular piles in complicated geometries is discussed an eikonal formulation of the problem is proposed two distinct cases arise in cylindrical geometries i e if both container and possible obstacles have vertical walls the problem is equivalent to a two dimensional travel time problem with obstacles while in general geometries this analogy breaks down in the first case classical one sided discretizations are generalized to handle obstacles without loss in accuracy in the second case a fast and efficient numerical method is proposed implemented and tested the discrete problems are solved through fast marching

the deficiency zero theorem and global asymptotic stability for a class of chemical reaction networks with arbitrary time delays abstract the deficiency zero theorem dzt is extended to be applied to chemical reaction networks crns with time delays the time evolution of concentrations of species in which networks is described by delay differential equations ddes the extended dzt provides a sufficient condition for a positive equilibrium point of ddes to exist and be locally asymptotically stable furthermore it is proven that the absence of boundary equilibria is a sufficient condition for the positive equilibrium point to be globally asymptotically stable these results are natural analogues of theorems proven for crns without time delays

numerical discretization of the first order hamilton jacobi equation on triangular meshes we present and analyze several ways of discretizing first order hamilton jacobi equations on unstructured meshes we first discuss two godunov type hamiltonians the first one is an extension of a result by bardi and osher where a particular decomposition of the initial condition is assumed and we point out its practical limits the other one arises from a particular decomposition of the hamiltonian despite its complexity this decomposition enables us to construct a lax friedrichs hamiltonian these schemes all share common properties they are consistent monotonic and independent of the geometrical interpretation of the piecewise linear initial condition under these assumptions and classical ones on the mesh we show these schemes are convergent we describe their high order extensions using the eno technique and provide numerical illustrations 1996 john wiley sons inc

a reaction diffusion system with fast reversible reaction abstract we consider a reaction diffusion system which models a fast reversible reaction between two mobile reactants and prove convergence of the solutions as the reaction rate tends to infinity where the limiting problem is given by a diffusion equation with nonlinear diffusion since the rate function has no sign the usual methods to obtain a priori estimates in the case of irreversible reactions do not apply we deduce instead a priori estimates from computations based on lyapunov function techniques

a fast sweeping method for eikonal equations in this paper a fast sweeping method for computing the numerical solution of eikonal equations on a rectangular grid is presented the method is an iterative method which uses upwind difference for discretization and uses gauss seidel iterations with alternating sweeping ordering to solve the discretized system the crucial idea is that each sweeping ordering follows a family of characteristics of the corresponding eikonal equation in a certain direction simultaneously the method has an optimal complexity of o n for n grid points and is extremely simple to implement in any number of dimensions monotonicity and stability properties of the fast sweeping algorithm are proven convergence and error estimates of the algorithm for computing the distance function is studied in detail it is shown that 2 n gauss seidel iterations is enough for the distance function in n dimensions an estimation of the number of iterations for general eikonal equations is also studied numerical examples are used to verify the analysis

a reaction network approach to the theory of acoustic wave turbulence abstract we propose a new approach to study the long time dynamics of the wave kinetic equation in the statistical description of acoustic turbulence the approach is based on rewriting the discrete version of the wave kinetic equation in the form of a chemical reaction network then employing techniques used to study the global attractor conjecture to investigate the long time dynamics of the newly obtained chemical system we show that the solution of the chemical system converges to an equilibrium exponentially in time in addition a resonance broadening modification of the acoustic wave kinetic equation is also studied with the same technique for the near resonance equation if the resonance broadening frequency is larger than a threshold the solution of the system goes to infinity as time evolves

computational study of fast methods for the eikonal equation a computational study of the fast marching and the fast sweeping methods for the eikonal equation is given it is stressed that both algorithms should be considered as direct as opposed to iterative methods on realistic grids fast sweeping is faster than fast marching for problems with simple geometry for strongly nonuniform problems and or complex geometry the situation may be reversed finally fully second order generalizations of methods of this type for problems with obstacles are proposed and implemented

fast sweeping methods for eikonal equations on triangular meshes the original fast sweeping method which is an efficient iterative method for stationary hamilton jacobi equations relies on natural ordering provided by a rectangular mesh we propose novel ordering strategies so that the fast sweeping method can be extended efficiently and easily to any unstructured mesh to that end we introduce multiple reference points and order all the nodes according to their l p metrics to those reference points we show that these orderings satisfy the two most important properties underlying the fast sweeping method 1 these orderings can cover all directions of information propagating efficiently 2 any characteristic can be decomposed into a finite number of pieces and each piece can be covered by one of the orderings we prove the convergence of the new algorithm the computational complexity of the algorithm is nearly optimal in the sense that the total computational cost consists of o m flops for iteration steps and o m rm log m flops for sorting at the predetermined initialization step which can be efficiently optimized by adopting a linear time sorting method where m is the total number of mesh points extensive numerical examples demonstrate that the new algorithm converges in a finite number of iterations independent of mesh size

data driven non markovian closure models abstract this paper has two interrelated foci i obtaining stable and efficient data driven closure models by using a multivariate time series of partial observations from a large dimensional system and ii comparing these closure models with the optimal closures predicted by the mori zwanzig mz formalism of statistical physics multilayer stochastic models msms are introduced as both a generalization and a time continuous limit of existing multilevel regression based approaches to closure in a data driven setting these approaches include empirical model reduction emr as well as more recent multi layer modeling it is shown that the multilayer structure of msms can provide a natural markov approximation to the generalized langevin equation gle of the mz formalism a simple correlation based stopping criterion for an emr msm model is derived to assess how well it approximates the gle solution sufficient conditions are derived on the structure of the nonlinear cross interactions between the constitutive layers of a given msm to guarantee the existence of a global random attractor this existence ensures that no blow up can occur for a broad class of msm applications a class that includes non polynomial predictors and nonlinearities that do not necessarily preserve quadratic energy invariants the emr msm methodology is first applied to a conceptual nonlinear stochastic climate model of coupled slow and fast variables in which only slow variables are observed it is shown that the resulting closure model with energy conserving nonlinearities efficiently captures the main statistical features of the slow variables even when there is no formal scale separation and the fast variables are quite energetic second an msm is shown to successfully reproduce the statistics of a partially observed generalized lotka volterra model of population dynamics in its chaotic regime the challenges here include the rarity of strange attractors in the model s parameter space and the existence of multiple attractor basins with fractal boundaries the positivity constraint on the solutions components replaces here the quadratic energy preserving constraint of fluid flow problems and it successfully prevents blow up

valid linear inequalities for fixed charge problems many problems in the operations research management science literature can be formulated with both zero one and continuous variables however the exact optimization of such mixed zero one models remains a computational challenge in this paper we propose to study mixed problems from a mathematical point of view that is similar in spirit to recent research on purely combinatorial problems that has investigated systems of defining linear inequalities or facets of the underlying polytope at least two numerical studies have validated this line of research computationally and the advances in the problem solving capabilities are considerable we expect that similar gains are possible for the mixed zero one problem more precisely we consider the mixed integer programs whose feasible region x is composed of i a simple additive constraint in the continuous variables xj for j 1 2 n and ii constraints 0 xj mjyj defined by binary variables yj for j 1 2 n this type of feasible region arises in a variety of mixed integer problems and particularly in network problems with fixed charges on the arcs we derive two classes of facet defining linear inequalities of the convex hull of x and show that the second of these classes gives a complete description of the convex hull when mj m for all j we also develop methods to detect violated inequalities from these classes so that these facets can be used as cutting planes to strengthen the formulations of certain mixed integer problems

fast marching methods fast marching methods are numerical schemes for computing solutions to the nonlinear eikonal equation and related static hamilton jacobi equations based on entropy satisfying upwind schemes and fast sorting techniques they yield consistent accurate and highly efficient algorithms they are optimal in the sense that the computational complexity of the algorithms is o n log n where n is the total number of points in the domain the schemes are of use in a variety of applications including problems in shape offsetting computing distances from complex curves and surfaces shape from shading photolithographic development computing first arrivals in seismic travel times construction of shortest geodesics on surfaces optimal path planning around obstacles and visibility and reflection calculations in this paper we review the development of these techniques including the theoretical and numerical underpinnings provide details of the computational schemes including higher order versions and demonstrate the techniques in a collection of different areas

sensitivity and network topology in chemical reaction systems in living cells biochemical reactions are catalyzed by specific enzymes and connect to one another by sharing substrates and products forming complex networks in our previous studies we established a framework determining the responses to enzyme perturbations only from network topology and then proved a theorem called the law of localization explaining response patterns in terms of network topology in this paper we generalize these results to reaction networks with conserved concentrations which allows us to study any reaction system we also propose network characteristics quantifying robustness we compare e coli metabolic network with randomly rewired networks and find that the robustness of the e coli network is significantly higher than that of the random networks

input to state stability of rate controlled biochemical networks in this paper the study of the class of biochemical systems known as zero deficiency networks is extended to the case of time varying kinetic parameters we show that the resulting class of nonlinear systems with inputs satisfies a notion of input to state stability uniformly over a set of parameters in particular the input to state stability estimates allow us to characterize the robustness of zero deficiency networks with respect to perturbations in the parameters as well as study their stability when the reaction rates are controlled by an independent process

piecewise linear lyapunov functions for structural stability of biochemical networks we consider the problem of assessing structural stability of biochemical reaction networks with monotone reaction rates namely of establishing if all the networks with a certain structure are stable regardless of specific parameter values we investigate stability by absorbing the network equations in a linear differential inclusion and seeking for a polyhedral lyapunov function proper to the considered network structure a numerical recursive procedure is devised to test stability for a wide class of mono and bimolecular reaction networks which we name unitary the procedure is shown to be very efficient since due to the particular structure of the problem it requires iterations in the space of integer valued matrices we also consider a similar less conservative procedure that allows us to test even when the lyapunov function cannot be found whether the system evolution is structurally bounded in this case we absorb the equations in a positive linear differential inclusion to show the effectiveness of the proposed procedure we report the outcomes of both a stability and a boundedness test for many non trivial biochemical reaction networks and we analyze well established models in the literature

nonlinear optimization by successive linear programming successive linear programming slp which is also known as the method of approximation programming solves nonlinear optimization problems via a sequence of linear programs this paper reports on promising computational results with slp that contrast with the poor performance indicated by previously published comparative tests the paper provides a detailed description of an efficient reliable slp algorithm along with a convergence theorem for linearly constrained problems and extensive computational results it also discusses several alternative strategies for implementing slp the computational results show that slp compares favorably with the generalized reduced gradient code grg2 and with minos grg it appears that slp will be most successful when applied to large problems with low degrees of freedom

preclusion of switch behavior in networks with mass action kinetics we study networks taken with mass action kinetics and provide a jacobian criterion that applies to an arbitrary network to preclude the existence of multiple positive steady states within any stoichiometric class for any choice of rate constants we are concerned with the characterization of injective networks that is networks for which the species formation rate function is injective in the interior of the positive orthant within each stoichiometric class we show that a network is injective if and only if the determinant of the jacobian of a certain function does not vanish the function consists of components of the species formation rate function and a maximal set of independent conservation laws the determinant of the function is a polynomial in the species concentrations and the rate constants linear in the latter and its coefficients are fully determined the criterion also precludes the existence of degenerate steady states further we relate injectivity of a network to that of the network obtained by adding outflow or degradation reactions for all species

the g scheme a framework for multi scale adaptive model reduction the numerical solution of mathematical models for reaction systems in general and reacting flows in particular is a challenging task because of the simultaneous contribution of a wide range of time scales to the system dynamics however the dynamics can develop very slow and very fast time scales separated by a range of active time scales an opportunity to reduce the complexity of the problem arises when the fast active and slow active time scales gaps becomes large we propose a numerical technique consisting of an algorithmic framework named the g scheme to achieve multi scale adaptive model reduction along with the integration of the differential equations des the method is directly applicable to initial value odes and by using the method of lines pdes we assume that the dynamics is decomposed into active slow fast and when applicable invariant subspaces the g scheme introduces locally a curvilinear frame of reference defined by a set of orthonormal basis vectors with corresponding coordinates attached to this decomposition the evolution of the curvilinear coordinates associated with the active subspace is described by non stiff des whereas that associated with the slow and fast subspaces is accounted for by applying algebraic corrections derived from asymptotics of the original problem adjusting the active des dynamically during the time integration is the most significant feature of the g scheme since the numerical integration is accomplished by solving a number of des typically much smaller than the dimension of the original problem with corresponding saving in computational work to demonstrate the effectiveness of the g scheme we present results from illustrative as well as from relevant problems

ordered upwind methods for static hamilton jacobi equations theory and algorithms we develop a family of fast methods for approximating the solutions to a wide class of static hamilton jacobi pdes these fast methods include both semi lagrangian and fully eulerian versions numerical solutions to these problems are typically obtained by solving large systems of coupled nonlinear discretized equations our techniques which we refer to as ordered upwind methods oums use partial information about the characteristic directions to decouple these nonlinear systems greatly reducing the computational labor our techniques are considered in the context of control theoretic and front propagation problems we begin by discussing existing oums focusing on those designed for isotropic problems we then introduce a new class of oums which decouple systems for general anisotropic problems we prove convergence of one such scheme to the viscosity solution of the corresponding hamilton jacobi pde next we introduce a set of finite differences methods based on an analysis of the role played by anisotropy in the context of front propagation and optimal trajectory problems the performance of the methods is analyzed and computational experiments are performed using test problems from computational geometry and seismology

properties of the michaelis menten mechanism in phase space abstract we study the two dimensional reduction of the michaelis menten reaction of enzyme kinetics first we prove the existence and uniqueness of a slow manifold between the horizontal and vertical isoclines second we determine the concavity of all solutions in the first quadrant third we establish the asymptotic behaviour of all solutions near the origin which generally is not given by a taylor series finally we determine the asymptotic behaviour of the slow manifold at infinity to this end we show that the slow manifold can be constructed as a centre manifold for a fixed point at infinity

a network dynamics approach to chemical reaction networks abstracta treatment of a chemical reaction network theory is given from the perspective of nonlinear network dynamics in particular of consensus dynamics by starting from the complex balanced assumption the reaction dynamics governed by mass action kinetics can be rewritten into a form which allows for a very simple derivation of a number of key results in the chemical reaction network theory and which directly relates to the thermodynamics and port hamiltonian formulation of the system central in this formulation is the definition of a balanced laplacian matrix on the graph of chemical complexes together with a resulting fundamental inequality this immediately leads to the characterisation of the set of equilibria and their stability furthermore the assumption of complex balancedness is revisited from the point of view of kirchhoff s matrix tree theorem both the form of the dynamics and the deduced behaviour are very similar to consensus dynamics and provide additional perspectives to the latte

computational singular perturbation analysis of stochastic chemical systems with stiffness abstract computational singular perturbation csp is a useful method for analysis reduction and time integration of stiff ordinary differential equation systems it has found dominant utility in particular in chemical reaction systems with a large range of time scales at continuum and deterministic level on the other hand csp is not directly applicable to chemical reaction systems at micro or meso scale where stochasticity plays an non negligible role and thus has to be taken into account in this work we develop a novel stochastic computational singular perturbation scsp analysis and time integration framework and associated algorithm that can be used to not only construct accurately and efficiently the numerical solutions to stiff stochastic chemical reaction systems but also analyze the dynamics of the reduced stochastic reaction systems the algorithm is illustrated by an application to a benchmark stochastic differential equation model and numerical experiments are carried out to demonstrate the effectiveness of the construction

efficient algorithms for globally optimal trajectories we present serial and parallel algorithms for solving a system of equations that arises from the discretization of the hamilton jacobi equation associated to a trajectory optimization problem of the following type a vehicle starts at a prespecified point x sub o and follows a unit speed trajectory x t inside a region in spl rscr sup m until an unspecified time t that the region is exited a trajectory minimizing a cost function of the form spl int sub 0 sup t r x t dt q x t is sought the discretized hamilton jacobi equation corresponding to this problem is usually solved using iterative methods nevertheless assuming that the function r is positive we are able to exploit the problem structure and develop one pass algorithms for the discretized problem the first algorithm resembles dijkstra s shortest path algorithm and runs in time o n log n where n is the number of grid points the second algorithm uses a somewhat different discretization and borrows some ideas from a variation of dial s shortest path algorithm 1969 that we develop here it runs in time o n which is the best possible under some fairly mild assumptions finally we show that the latter algorithm can be efficiently parallelized for two dimensional problems and with p processors its running time becomes o n p provided that p o spl radic n log n

multiple equilibria in complex chemical reaction networks ii the species reaction graph for mass action kinetics the capacity for multiple equilibria in an isothermal homogeneous continuous flow stirred tank reactor is determined by the structure of the underlying network of chemical reactions we suggest a new graph theoretical method for discriminating between complex reaction networks that can admit multiple equilibria and those that cannot in particular we associate with each network a species reaction graph which is similar to reaction network representations drawn by biochemists and we show that if the graph satisfies certain weak conditions the differential equations corresponding to the network cannot admit multiple equilibria em no matter what values the rate constants take because these conditions are very mild they amount to powerful and quite delicate necessary conditions that a network must satisfy if it is to have the capacity to engender multiple equilibria broad qualitative results of this kind are especially apt for individual reaction rate constants are rare

nonlinear dynamics of networks the groupoid formalism a formal theory of symmetries of networks of coupled dynamical systems stated in terms of the group of permutations of the nodes that preserve the network topology has existed for some time global network symmetries impose strong constraints on the corresponding dynamical systems which affect equilibria periodic states heteroclinic cycles and even chaotic states in particular the symmetries of the network can lead to synchrony phase relations resonances and synchronous or cycling chaos symmetry is a rather restrictive assumption and a general theory of networks should be more flexible a recent generalization of the group theoretic notion of symmetry replaces global symmetries by bijections between certain subsets of the directed edges of the network the input sets now the symmetry group becomes a groupoid which is an algebraic structure that resembles a group except that the product of two elements may not be defined the groupoid formalism makes it possible to extend group theoretic methods to more general networks and in particular it leads to a complete classification of robust patterns of synchrony in terms of the combinatorial structure of the network many phenomena that would be nongeneric in an arbitrary dynamical system can become generic when constrained by a particular network topology a network of dynamical systems is not just a dynamical system with a high dimensional phase space it is also equipped with a canonical set of observables the states of the individual nodes of the network moreover the form of the underlying ode is constrained by the network topology which variables occur in which component equations and how those equations relate to each other the result is a rich and new range of phenomena only a few of which are yet properly understood

weak dynamical nonemptiability and persistence of chemical kinetics systems a frequently desirable characteristic of chemical kinetics systems is that of persistence the property that if all the species are initially present then none of them may tend toward extinction it is known that solutions of deterministically modeled mass action systems may approach only portions of the boundary of the positive orthant which correspond to semilocking sets alternatively called siphons consequently most recent work on persistence of these systems has been focused on these sets in this paper we focus on a result which states that for a conservative mass action system persistence holds if every critical semilocking set is dynamically nonemptiable and the system contains no nested locking sets we will generalize this result by introducing the notion of a weakly dynamically nonemptiable semilocking set and making novel use of the well known farkas lemma we will also connect this result to known results regarding complex balanced systems and systems with facets

finite volume schemes for hamilton jacobi equations we introduce two classes of monotone finite volume schemes for hamilton jacobi equations the corresponding approximating functions are piecewise linear defined on a mesh consisting of triangles the schemes are shown to converge to the viscosity solution of the hamilton jacobi equation

hilbert s 6th problem exact and approximate hydrodynamic manifolds for kinetic equations the problem of the derivation of hydrodynamics from the boltz mann equation and related dissipative systems is formulated as the problem of slow invariant manifold in the space of distributions we review a few in stances where such hydrodynamic manifolds were found analytically both as the result of summation of the chapman enskog asymptotic expansion and by the direct solution of the invariance equation these model cases comprising grad s moment systems both linear and nonlinear are studied in depth in order to gain understanding of what can be expected for the boltzmann equa tion particularly the dispersive dominance and saturation of dissipation rate of the exact hydrodynamics in the short wave limit and the viscosity modifica tion at high divergence of the flow velocity are indicated as severe obstacles to the resolution of hilbert s 6th problem furthermore we review the derivation of the approximate hydrodynamic manifold for the boltzmann equation using newton s iteration and avoiding smallness parameters and compare this to the exact solutions additionally we discuss the problem of projection of the boltzmann equation onto the approximate hydrodynamic invariant manifold using entropy concepts finally a set of hypotheses is put forward where we describe open questions and set a horizon for what can be derived exactly or proven about the hydrodynamic manifolds for the boltzmann equation in the future

the confined primal integral a measure to benchmark heuristic minlp solvers against global minlp solvers it is a challenging task to fairly compare local solvers and heuristics against each other and against global solvers how does one weigh a faster termination time against a better quality of the found solution in this paper we introduce the confined primal integral a new performance measure that rewards a balance of speed and solution quality it emphasizes the early part of the solution process by using an exponential decay thereby it avoids that the order of solvers can be inverted by choosing an arbitrarily large time limit we provide a closed analytic formula to compute the confined primal integral a posteriori and an incremental update formula to compute it during the run of an algorithm for the latter we show that we can drop one of the main assumptions of the primal integral namely the knowledge of a fixed reference solution to compare against furthermore we prove that the confined primal integral is a transitive measure when comparing local solves with different final solution values finally we present a computational experiment where we compare a local minlp solver that uses certain classes of cutting planes against a solver that does not both versions show very different tendencies w r t average running time and solution quality and we use the confined primal integral to argue which of the two is the preferred setting

the role of slow system dynamics in predicting the degeneracy of slow invariant manifolds the case of vdp relaxation oscillations abstract the development of dissipative time scales which are much faster than the rest force the solution of a system of 1st order odes to land and then evolve on a slow invariant manifold the slow evolution on this manifold can be approximated by a reduced system which is constructed with asymptotic analysis and is free of the fast scales in this work the dynamics of the reduced system are compared to the fast and slow dynamics of the original system the comparison is based on the van der pol system which is typical of those exhibiting relaxation oscillation behavior and is characterized by successive dissipative and explosive phases special attention is given to the dissipative phase during which the solution evolves on a slow invariant manifold and the fast slow time scale gap tends to contract it is shown that one cannot predict from the dynamics of the van der pol system the approaching explosive phase and the associated degeneracy of the manifold such a prediction is possible only through the dynamics of the reduced system which involves an explosive component that is absent in the original system this explosive component accelerates as higher order effects are incorporated in the reduced system resulting in a smaller gap between the fast time scale of the original system and the fast time scale of the reduced one this feature sets a limit on the accuracy singular perturbation techniques can deliver even in the presence of a reasonably large fast slow time scale gap in the dynamics of the original system given a fast slow system predicting forthcoming changes in the dimensions of the manifold is very important when it is desired to identify the underlying major physical mechanisms or to control the process

quasi steady state approximation for a reaction diffusion system with fast intermediate abstract we consider a prototype reaction diffusion system which models a network of two consecutive reactions in which chemical components a and b form an intermediate c which decays into two products p and q such a situation often occurs in applications and in the typical case when the intermediate is highly reactive the species c is eliminated from the system by means of a quasi steady state approximation in this paper we prove the convergence of the solutions in l 2 as the decay rate of the intermediate tends to infinity for all bounded initial data even in the case of initial boundary layers the limiting system is indeed the one which results from formal application of the qssa the proof combines the recent l 2 approach to reaction diffusion systems having at most quadratic reaction terms with local l bounds which are independent of the decay rate of the intermediate we also prove existence of global classical solutions to the initial system

correction to structure and stability of certain chemical networks and applications to the kinetic proofreading model of t cell receptor signal transduction deals with the theory of structure stability robustness and stabilization for an appealing class of nonlinear systems which arises in the analysis of chemical networks the results given here extend but are also heavily based upon certain previous work by feinberg 1977 1979 1987 1995 and horn and jackson 1972 of which a self contained and streamlined exposition is included the theoretical conclusions are illustrated through an application to the kinetic proofreading model proposed by mckeithan 1995 for t cell receptor signal transduction

needed an empirical science of algorithms deductive algorithmic science has reached a high level of sophistication but its worst case and average case results seldom tell us how well an algorithm is actually going to work in practice i argue that an empirical science of algorithms is a viable alternative i respond to misgivings about an empirical approach including the prevalent notion that only a deductive treatment can be theoretical or sophisticated np completeness theory for instance is interesting partly because it has significant if unacknowledged empirical content an empirical approach requires not only rigorous experimental design and analysis but also the invention of empirically based explanatory theories i give some examples of recent work that partially achieves this aim

exponential time decay of solutions to reaction cross diffusion systems of maxwell stefan type the large time asymptotics of weak solutions to maxwell stefan diffusion systems for chemically reacting fluids with different molar masses and reversible reactions are investigated the diffusion matrix of the system is generally neither symmetric nor positive definite but the equations admit a formal gradient flow structure which provides entropy free energy estimates the main result is the exponential decay to the unique equilibrium with a rate that is constructive up to a finite dimensional inequality the key elements of the proof are the existence of a unique detailed balance equilibrium and the derivation of an inequality relating the entropy and the entropy production the main difficulty comes from the fact that the reactions are represented by molar fractions while the conservation laws hold for the concentrations the idea is to enlarge the space of n partial concentrations by adding the total concentration viewed as an independent variable thus working with n 1 variables further results concern the existence of global bounded weak solutions to the parabolic system and an extension of the results to complex balance systems

markov chain approximations for deterministic control problems with affine dynamics and quadratic cost in the control we consider the construction of markov chain approximations for an important class of deterministic control problems the emphasis is on the construction of schemes that can be easily implemented and which possess a number of highly desirable qualitative properties the class of problems covered is that for which the control is affine in the dynamics and with quadratic running cost this class covers a number of interesting areas of application including problems that arise in large deviations risk sensitive and robust control robust filtering and certain problems in computer vision examples are given as well as a proof of convergence

asymptotic analysis of two reduction methods for systems of chemical reactions abstract this paper concerns two methods for reducing large systems of chemical kinetics equations namely the method of intrinsic low dimensional manifolds ildms due to maas and pope combust flame 88 1992 239 and an iterative method due to fraser j chem phys 88 1988 4732 and further developed by roussel and fraser j chem phys 93 1990 1072 both methods exploit the separation of fast and slow reaction time scales to find low dimensional manifolds in the space of species concentrations where the long term dynamics are played out the asymptotic expansions of these manifolds e 0 where e measures the ratio of the reaction time scales are compared with the asymptotic expansion of m e the slow manifold given by geometric singular perturbation theory it is shown that the expansions of the ildm and m e agree up to and including terms of o e the former has an error at o e 2 that is proportional to the local curvature of m 0 the error vanishes if and only if the curvature is zero everywhere the iterative method generates term by term the asymptotic expansion of m e starting from m 0 the ith application of the algorithm yields the correct expansion coefficient at o e i while leaving the lower order coefficients invariant thus after l applications the expansion is accurate up to and including the terms of o e l the analytical results are illustrated on a planar system from enzyme kinetics michaelis menten henri and a model planar system due to davis and skodje

from enzyme kinetics to epidemiological models with michaelis menten contact rate design of nonstandard finite difference schemes we consider the basic sir epidemiological model with the michaelis menten formulation of the contact rate from the study of the michaelis menten basic enzymatic reaction we design two types of nonstandard finite difference nsfd schemes for the sir model exact related schemes based on the lambert w function and schemes obtained by using mickens s rules of more complex denominator functions for discrete derivatives and nonlocal approximations of nonlinear terms we compare and investigate the performance of the two types of schemes by showing that they are dynamically consistent with the continuous model numerical simulations that support the theory and demonstrate computationally the power of nsfd schemes are presented

multistencils fast marching methods a highly accurate solution to the eikonal equation on cartesian domains a wide range of computer vision applications require an accurate solution of a particular hamilton jacobi hj equation known as the eikonal equation in this paper we propose an improved version of the fast marching method fmm that is highly accurate for both 2d and 3d cartesian domains the new method is called multistencils fast marching msfm which computes the solution at each grid point by solving the eikonal equation along several stencils and then picks the solution that satisfies the upwind condition the stencils are centered at each grid point and cover its entire nearest neighbors in 2d space two stencils cover 8 neighbors of the point whereas in 3d space six stencils cover its 26 neighbors for those stencils that are not aligned with the natural coordinate system the eikonal equation is derived using directional derivatives and then solved using higher order finite difference schemes the accuracy of the proposed method over the state of the art fmm based techniques has been demonstrated through comprehensive numerical experiments

classical quasi steady state reduction a mathematical characterization abstract we discuss parameter dependent polynomial ordinary differential equations that model chemical reaction networks by classical quasi steady state qss reduction we understand the following familiar heuristically motivated mathematical procedure set the rate of change for certain a priori chosen variables equal to zero and use the resulting algebraic equations to obtain a system of smaller dimension for the remaining variables this procedure will generally be valid only for certain parameter ranges we start by showing that the reduction is accurate if and only if the corresponding parameter is what we call a qss parameter value and that the reduction is approximately accurate if and only if the corresponding parameter is close to a qss parameter value the qss parameter values can be characterized by polynomial equations and inequations hence parameter ranges for which qss reduction is valid are accessible in an algorithmic manner a defining characteristic of a qss parameter value is that the algebraic variety defined by the qss relations is invariant for the differential equation a closer investigation of the associated systems shows the existence of further invariant sets here singular perturbations enter the picture in a natural manner we compare qss reduction and singular perturbation reduction and show that while they do not agree in general they do up to lowest order in a small parameter for a quite large and relevant class of examples this observation in turn allows the computation of qss reductions even in cases where an explicit resolution of the polynomial equations is not possible

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

a uniformly second order fast sweeping method for eikonal equations a uniformly second order method with a local solver based on the piecewise linear discontinuous galerkin formulation is introduced to solve the eikonal equation with dirichlet boundary conditions the method utilizes an interesting phenomenon referred as the superconvergence phenomenon that the numerical solution of monotone upwind schemes for the eikonal equation is first order accurate on both its value and gradient when the solution is smooth this phenomenon greatly simplifies the local solver based on the discontinuous galerkin formulation by reducing its local degrees of freedom from two 1 d or three 2 d or four 3 d to one with the information of the gradient frozen when considering the eikonal equation with point source conditions we further utilize a factorization approach to resolve the source singularities of the eikonal by decomposing it into two parts either multiplicatively or additively one part is known and captures the source singularities the other part serves as a correction term that is differentiable at the sources and satisfies the factored eikonal equations we extend the second order method to solve the factored eikonal equations to compute the correction term with second order accuracy then recover the eikonal with second order accuracy numerical examples are presented to demonstrate the performance of the method

instantaneous limits of reversible chemical reactions in presence of macroscopic convection abstract chemically reacting systems frequently involve fast reversible reactions additional slow reactions as well as mass transport due to macroscopic convection in this situation the passage to infinite reaction speed is a means to reduce the complexity of the reaction kinetics and to avoid the need for explicit values of the rate constants thereby the large stiffness of the original system of differential equations is also removed in the present paper this instantaneous reaction limit is studied for systems with independent fast reversible reactions where the rate functions are given by mass action kinetics under realistic assumptions the limiting dynamical system is derived and convergence of the solutions is obtained as the rate constants tend to infinity the proof is based on lyapunov functions techniques and exploits the structure of rate functions that results from mass action kinetics this approach is complementary to the quasi steady state approximation which is often applied in chemical engineering the differences are illustrated by means of a classical enzyme substrate reaction scheme

analysis of autocatalytic networks in biology autocatalytic networks in particular the glycolytic pathway constitute an important part of the cell metabolism changes in the concentration of metabolites and catalyzing enzymes during the lifetime of the cell can lead to perturbations from its nominal operating condition we investigate the effects of such perturbations on stability properties e g the extent of regions of attraction of a particular family of autocatalytic network models numerical experiments demonstrate that systems that are robust with respect to perturbations in the parameter space have an easily verifiable in terms of proof complexity region of attraction properties motivated by the computational complexity of optimization based formulations we take a compositional approach and exploit a natural decomposition of the system induced by the underlying biological structure into a feedback interconnection of two input output subsystems a small subsystem with complicating nonlinearities and a large subsystem with simple dynamics this decomposition simplifies the analysis of large pathways by assembling region of attraction certificates based on the input output properties of the subsystems it enables numerical as well as analytical construction of block diagonal lyapunov functions for a large family of autocatalytic pathways

fast numerical methods for bernoulli free boundary problems the numerical solution of the free boundary bernoulli problem is addressed an iterative method based on a level set formulation and boundary element method is proposed issues related to the implementation the accuracy and the generality of the method are discussed the efficiency of the approach is illustrated by numerical results