set inversion via interval analysis for nonlinear bounded error estimation abstract in the context of bounded error estimation one is interested in characterizing the set of all the values of the parameters to be estimated that are consistent with the data in the sense that the errors between the data and model outputs fall within prior bounds while the problem can be considered as solved when the model output is linear in the parameters the situation is far less advanced in the general nonlinear case in this paper the problem of nonlinear bounded error estimation is viewed as one of set inversion an original algorithm is proposed based upon interval analysis that makes it possible to characterize the feasible set for the parameters by enclosing it between internal and external unions of boxes the convergence of the algorithm is proved and the algorithm is applied to two test cases the results obtained are compared with those provided by signomial analysis

strong solvability of interval max plus systems and applications to optimal control abstract this paper considers the strong solvability of interval max plus systems the solvable entry and solvable interval are introduced to establish the necessary and sufficient condition for strong solvability of interval max plus systems it is proved that such a criterion is equivalent to the criterion proposed by cechlarova and cuninghame green the interval strong solution of interval max plus systems is described and the necessary and sufficient condition for uniquely strong solvability is given the method of testing strong solvability of interval max plus systems is constructive and leads to an effective algorithm to illustrate our results a workshop optimal control problem is addressed

robust state estimation for singularly perturbed systems abstractthis paper deals with the design of interval observers for singularly perturbed linear systems the full order system is first decoupled into slow and fast subsystems then using the cooperativity theory an interval observer is designed for the slow and fast subsystems assuming that the measurement noise and the disturbances are bounded and the singular perturbed parameter is uncertain this decoupling leads to two observers that estimate the lower and upper bounds for the feasible state domain a numerical example shows the efficiency of the proposed technique

mixed spectral finite elements for the linear elasticity system in unbounded domains in this paper we present a mixed formulation of a spectral element approximation of the linear elasticity system after studying the main features of this approach we construct perfectly matched layers pmls for modeling unbounded domains then algorithmic issues are discussed and numerical results are given

an interval newton method we introduce an interval newton method for bounding solutions of systems of nonlinear equations it entails three subalgorithms the first is a gauss seidel type step the second is a real noninterval newton iteration the third solves the linearized equations by elimination we explain why each subalgorithm is desirable and how they fit together to provide solutions in as little as one third or one quarter the time required by krawczyk s method 7 in our implementations

on global identifiability for arbitrary model parametrizations it is a fundamental problem of identification to be able even before the data have been analyzed to decide if all the free parameters of a model structure can be uniquely recovered from data this is the issue of global identifiability in this contribution we show how global identifiability for an arbitrary model structure basically with analytic non linearities can be analyzed using concepts and algorithms from differential algebra it is shown how the question of global structural identifiability is reduced to the question of whether the given model structure can be rearranged as a linear regression an explicit algorithm to test this is also given furthermore the question of persistent excitation for the input can also be tested explicitly is a similar fashion the algorithms involved are very well suited for implementation in computer algebra one such implementation is also described

growth of sobolev norms in the cubic nonlinear schrodinger equation with a convolution potential fix s 1 colliander et al proved in invent math 181 39 113 2010 the existence of solutions of the cubic defocusing nonlinear schrodinger equation in the two torus whose s sobolev norm undergoes arbitrarily large growth as time evolves in this paper we generalize their result to the cubic defocusing nonlinear schrodinger equation with a convolution potential moreover we show that the speed of growth is the same as the one obtained for the cubic defocusing nonlinear schrodinger equation in guardia and kaloshin growth of sobolev norms in the cubic defocusing nonlinear schrodinger equation to appear in the journal of the european mathematical society 2012

brief paper interval observer design for consistency checks of nonlinear continuous time systems this paper deals with fault detection for nonlinear continuous time systems a procedure based on interval analysis is proposed to build a guaranteed qlpv quasi linear parameter varying approximation of the nonlinear model the interval qlpv approximation makes it possible to derive two point observers which estimate respectively the lower and the upper bound of the state vector using cooperativity theory a set guaranteed to contain the actual value of the residual is then designed the modelling uncertainties and measurement errors are taken into account at the design stage the proposed methodology is illustrated through numerical simulations

dynamic earthquake rupture simulations on nonplanar faults embedded in 3d geometrically complex heterogeneous elastic solids dynamic propagation of shear ruptures on a frictional interface in an elastic solid is a useful idealization of natural earthquakes the conditions relating discontinuities in particle velocities across fault zones and tractions acting on the fault are often expressed as nonlinear friction laws the corresponding initial boundary value problems are both numerically and computationally challenging in addition seismic waves generated by earthquake ruptures must be propagated for many wavelengths away from the fault therefore reliable and efficient numerical simulations require both provably stable and high order accurate numerical methods we present a high order accurate finite difference method for a enforcing nonlinear friction laws in a consistent and provably stable manner suitable for efficient explicit time integration b dynamic propagation of earthquake ruptures along nonplanar faults and c accurate propagation of seismic waves in heterogeneous media with free surface topography we solve the first order form of the 3d elastic wave equation on a boundary conforming curvilinear mesh in terms of particle velocities and stresses that are collocated in space and time using summation by parts sbp finite difference operators in space boundary and interface conditions are imposed weakly using penalties by deriving semi discrete energy estimates analogous to the continuous energy estimates we prove numerical stability the finite difference stencils used in this paper are sixth order accurate in the interior and third order accurate close to the boundaries however the method is applicable to any spatial operator with a diagonal norm satisfying the sbp property time stepping is performed with a 4th order accurate explicit low storage runge kutta scheme thus yielding a globally fourth order accurate method in both space and time we show numerical simulations on band limited self similar fractal faults revealing the complexity of rupture dynamics on rough faults

p sv wave propagation in heterogeneous media velocity stress finite difference method i present a finite difference method for modeling p sv wave propagation in heterogeneous media this is an extension of the method i previously proposed for modeling sh wave propagation by using velocity and stress in a discrete grid the two components of the velocity cannot be defined at the same node for a complete staggered grid the stability condition and the p wave phase velocity dispersion curve do not depend on the poisson s ratio while the s wave phase velocity dispersion curve behavior is rather insensitive to the poisson s ratio therefore the same code used for elastic media can be used for liquid media where s wave velocity goes to zero and no special treatment is needed for a liquid solid interface typical physical phenomena arising with p sv modeling such as surface waves are in agreement with analytical results the weathered layer and corner edge models show in seismograms the same converted phases obtained by previous authors this method gives stable results for step discontinuities as shown for a liquid layer above an elastic half space the head wave preserves the correct amplitude finally the corner edge model illustrates a more complex geometry for the liquid solid interface as the poisson s ratio v increases from 0 25 to 0 5 the shear converted phases are removed from seismograms and from the time section of the wave field

growth of sobolev norms in time dependent semiclassical anharmonic oscillators abstract we consider the semiclassical schrodinger equation on r d given by i t 2 2 w l x v t x where w l is an anharmonic trapping of the form w l x 1 2 l j 1 d x j 2 l l 2 is an integer and is a semiclassical small parameter we construct a smooth potential v t x bounded in time with its derivatives and an initial datum such that the sobolev norms of the solution grow at a logarithmic speed for all times of order log 1 2 1 the proof relies on two ingredients first we construct an unbounded solution to a forced mechanical anharmonic oscillator then we exploit semiclassical approximation with coherent states to obtain growth of sobolev norms for the quantum system which are valid for semiclassical time scales

paper identifiability and distinguishability concepts in electrochemistry although the notions of structural identifiability and distinguishability are now well accepted in some domains of mathematical modeling such as econometrics biology and pharmacokinetics much remains to be done in others the relevance of these notions to the building of phenomenological models in electrochemistry is illustrated by considering the volmer heyrovsky reaction and three mechanisms for dissolution passivation it is shown that some classical methods for collecting data cannot ensure suitable identifiability and distinguishability properties in some cases simple modifications of the experimental procedure can remove these ambiguities in others one may resort to more recent techniques of investigation

transfer of energy to high frequencies in the cubic defocusing nonlinear schrodinger equation we consider the cubic defocusing nonlinear schrodinger equa tion on the two dimensional torus we exhibit smooth solutions for which the support of the conserved energy moves to higher fourier modes this behavior is quantified by the growth of higher sobolev norms given any 1 k 1 s 1 we construct smooth initial data u0 withu0 h s k at some time t this growth occurs despite the hamiltonian s bound onu t h 1 and despite the conservation of the quantityu t l2

boundedness of solutions for non linear quasi periodic differential equations with liouvillean frequency abstract we study the boundedness of solutions for non linear quasi periodic differential equations with liouvillean frequencies we proved that if the forcing is quasi periodic in time with two frequencies which is not super liouvillean then all solutions of the equation are bounded the proof is based on action angle variables and modified kam theory

interval systems of max separable linear equations for a given matrix interval a a a and a given vector interval b b b the notation a x b if max represents an interval system of linear max separable equations we define and characterize several types of solutions and solvability of interval systems of max separable linear equations

distributed set membership observers for interconnected multi rate systems abstract this paper presents a new distributed observer for interconnected multi rate systems the developed observer belongs to the family of set membership estimators and the use of zonotopes is proposed to mathematically describe the sets a choice motivated by the available mathematical background in operations such as intersections combinations and linear manipulations the main features of the proposed distributed observer are a the actual state of any subsystem always belongs to the computed sets b the volume of these sets is minimized in real time c under equivalent assumptions the performance of the observer approaches to that of an analogous distributed kalman filter

a perfectly matched layer for the time dependent wave equation in heterogeneous and layered media a mathematical analysis of the perfectly matched layer pml for the time dependent wave equation in heterogeneous and layered media is presented we prove the stability of the pml for discontinuous media with piecewise constant coefficients and derive energy estimates for discontinuous media with piecewise smooth coefficients we consider a computational setup consisting of smaller structured subdomains that are discretized using high order accurate finite difference operators for approximating spatial derivatives the subdomains are then patched together into a global domain by a weak enforcement of interface conditions using penalties in order to ensure the stability of the discrete pml it is necessary to transform the interface conditions to include the auxiliary variables in the discrete setting the transformed interface conditions are crucial in deriving discrete energy estimates analogous to the continuous energy estimates thus proving stability and convergence of the numerical method finally we present numerical experiments demonstrating the stability of the pml in a layered medium and high order accuracy of the proposed interface conditions

subdivision direction selection in interval methods for global optimization the role of the interval subdivision selection rule is investigated in branch and bound algorithms for global optimization the class of rules that allows convergence for the model algorithm is characterized and it is shown that the four rules investigated satisfy the conditions of convergence a numerical study with a wide spectrum of test problems indicates that there are substantial differences between the rules in terms of the required cpu time the number of function and derivative evaluations and space complexity and two rules can provide substantial improvements in efficiency

bayesian state estimation on finite horizons the case of linear state space model abstract the finite impulse response fir filter and infinite impulse response filter including the kalman filter kf are generally considered as two different types of state estimation methods in this paper the sequential bayesian philosophy is extended to a filter design using a fixed amount of most recent measurements with the aim of exploiting the fir structure and unifying some basic fir filters with the kf specifically the conditional mean and covariance of the posterior probability density functions are first derived to show the fir counterpart of the kf to remove the dependence on initial states the corresponding likelihood is further maximized and realized iteratively it shows that the maximum likelihood modification unifies the existing unbiased fir filters by tuning a weighting matrix moreover it converges to the kalman estimate with the increase of horizon length and can thus be considered as a link between the fir filtering and the kf several important properties including stability and robustness against errors of noise statistics are illustrated finally a moving target tracking example and an experiment with a three degrees of freedom helicopter system are introduced to demonstrate effectiveness

brief paper robust set membership state estimation application to underwater robotics this paper proposes a new observer for estimating the state vector of a nonlinear system this observer which is robust with respect to outliers assumes that the measurement errors as well as the number of outliers that could occur within a given time window are bounded the principle of the approach is to use interval analysis to deal properly with the nonlinearities involved in the system without any linearization or approximation and to propagate through the time in a forward and backward manner the assumptions made about outliers a test case related to the localization and control of an underwater robot is also proposed to illustrate the efficiency of the approach

using interval arithmetic to prove that a set is path connected in this paper we give a numerical algorithm able to prove whether a set s described by nonlinear inequalities is path connected or not to our knowledge no other algorithm numerical or symbolic is able to deal with this type of problem the proposed approach uses interval arithmetic to build a graph which has exactly the same number of connected components as s examples illustrate the principle of the approach

a linear system theoretic view of discrete event processes and its use for performance evaluation in manufacturing a discrete event system is a system whose behavior can be described by means of a set of time consuming activities performed according to a prescribed ordering events correspond to starting or ending some activity an analogy between linear systems and a class of discrete event systems is developed following this analogy such discrete event systems can be viewed as linear in the sense of an appropriate algebra the periodical behavior of closed discrete event systems i e involving a set of repeatedly performed activities can be totally characterized by solving an eigenvalue and eigenvector equation in this algebra this problem is numerically solved by an efficient algorithm which basically consists of finding the shortest paths from one node to all other nodes in a graph the potentiality of this approach for the performance evaluation of flexible manufacturing systems is emphasized the case of a flowshop like production process is analyzed in detail

set membership state and parameter estimation for systems described by nonlinear differential equations this paper investigates the use of guaranteed methods to perform state and parameter estimation for nonlinear continuous time systems in a bounded error context a state estimator based on a prediction correction approach is given where the prediction step consists in a validated integration of an initial value problem for an ordinary differential equation ivp for ode using interval analysis and high order taylor models while the correction step uses a set inversion technique the state estimator is extended to solve the parameter estimation problem an illustrative example is presented for each part

invariant tori of duffing type equations abstract in this paper we prove the existence of invariant tori and thus the boundedness of all solutions and the existence of quasiperiodic solutions for a class of duffing type equations d 2 x dt 2 x 2n 1 j 0 2n p j t x j 0 n 1 wherepj t j 0 1 n are 1 periodicc 1 functions and wherepj t j n 1 2n are 1 periodicc 2 functions

on time dependent schrodinger equations global well posedness and growth of sobolev norms in this paper we consider time dependent schrodinger linear pdes of the form i t l t where l t is a continuous family of self adjoint operators we give conditions for well posedness and polynomial growth for the evolution in abstract sobolev spaces if l t h v t where v t is a perturbation smooth in time and h is a self adjoint positive operator whose spectrum can be enclosed in spectral clusters whose distance is increasing we prove that the sobolev norms of the solution grow at most as t when t for any 0 if v t is analytic in time we improve the bound to log t for some 0 the proof follows the strategy due to howland joye and nenciu of the adiabatic approximation of the flow we recover most of known results and obtain new estimates for several models including 1 degree of freedom schrodinger operators on r and schrodinger operators on zoll manifolds

solutions to range dependent benchmark problems by the finite difference method an explicit second order finite difference scheme has been used to solve the elastic wave equation in the time domain solutions are presented for the perfect wedge the lossless penetrable wedge and the plane parallel waveguide that have been proposed as benchmarks by the acoustical society of america good agreement with reference solutions is obtained if the media is discretized at 20 gridpoints per wavelength there is a major discrepancy up to 20 db in reference source level because the reference solutions are normalized to the source strength at 1 m in the model but the finite difference solutions are normalized to the source strength at 1 m in a homogeneous medium the finite difference method requires computational times between 10 and 20 h on a super minicomputer without an array processor the method has the advantage of providing phase information and when run for a pulse source of providing insight into the evolution of the wave field and energy partitioning more complex models including velocity gradients and strong lateral heterogeneities can be solved with no additional computational effort the method has also been formulated to include shear wave effects

stable and high order accurate boundary treatments for the elastic wave equation on second order form stable and high order accurate boundary treatments for the elastic wave equation on second order form

interval analysis theory and applications we give an overview on applications of interval arithmetic among others we discuss verification methods for linear systems of equations nonlinear systems the algebraic eigenvalue problem initial value problems for odes and boundary value problems for elliptic pdes of second order we also consider the item software in this field and give some historical remarks

brief paper box particle filtering for nonlinear state estimation using interval analysis in recent years particle filters have been applied to a variety of state estimation problems a particle filter is a sequential monte carlo bayesian estimator of the posterior density of the state using weighted particles the efficiency and accuracy of the filter depend mostly on the number of particles used in the estimation and on the propagation function used to re allocate weights to these particles at each iteration if the imprecision i e bias and noise in the available information is high the number of particles needs to be very large in order to obtain good performances this may give rise to complexity problems for a real time implementation this kind of imprecision can easily be represented by interval data if the maximum error is known handling interval data is a new approach successfully applied to different real applications in this paper we propose an extension of the particle filter algorithm able to handle interval data and using interval analysis and constraint satisfaction techniques in standard particle filtering particles are punctual states associated with weights whose likelihoods are defined by a statistical model of the observation error in the box particle filter particles are boxes associated with weights whose likelihood is defined by a bounded model of the observation error experiments using actual data for global localization of a vehicle show the usefulness and the efficiency of the proposed approach

validated solutions of initial value problems for ordinary differential equations compared to standard numerical methods for initial value problems ivps for ordinary differential equations odes validated methods for ivps for odes have two important advantages if they return a solution to a problem then 1 the problem is guaranteed to have a unique solution and 2 an enclosure of the true solution is produced the authors survey taylor series methods for validated solutions of ivps for odes describe several such methods in a common framework and identify areas for future research

bernoulli particle box particle filters for detection and tracking in the presence of triple measurement uncertainty this work presents sequential bayesian detection and estimation methods for nonlinear dynamic stochastic systems using measurements affected by three sources of uncertainty stochastic set theoretic and data association uncertainty following mahler s framework for information fusion the paper develops the optimal bayes filter for this problem in the form of the bernoulli filter for interval measurements two numerical implementations of the optimal filter are developed the first is the bernoulli particle filter pf which turns out to require a large number of particles in order to achieve a satisfactory performance for the sake of reduction in the number of particles the paper also develops an implementation based on box particles referred to as the bernoulli box pf a box particle is a random sample that occupies a small and controllable rectangular region of nonzero volume in the target state space manipulation of boxes utilizes the methods of interval analysis the two implementations are compared numerically and found to perform remarkably well the target is reliably detected and the posterior probability density function of the target state is estimated accurately the bernoulli box pf however when designed carefully is computationally more efficient

an algorithm for solving two sided interval system of max plus linear equations we study six types of solutions weak strong tolerance control left localized and right localized solutions to a two sided interval system of max plus linear equations with the same vector of variables on both sides of the equations and obtain their corresponding solvability conditions these conditions are in the form of two sided systems of max plus linear inequalities which can be derived as a union of a number of interval inclusion linear systems therefore we could work on the interval inclusion linear systems instead of directly solving these six types of solutions themselves an optimization problem with the two sided interval system of max plus linear constraints may be solved using the convexity of each solution set of the interval inclusion linear systems

roots of unity and unbounded motions of an asymmetric oscillator r 2 z the general questions that we have in mind are under what conditions ona b and f t can we say that 1 has unbounded solutions are there equationsin the class 1 where unbounded and periodic solutions can coexist equation 1 was first considered by dancer in 2 3 and 4 and byfucik in 7 they studied the periodic and dirichlet boundary valueproblems for 1 and looked at this equation as a model of the so called equations with jumping nonlinearities we sum up some of the results in 2 3 consider the family of curves in the a b parameter space 1 a 1 b 2p p 1 2 2 when a b is not in any of these curves then 1 has at least one 2 periodic solution however if a b lies in one of these curves there existfunctions f t for which 1 has no periodic solution more results on theperiodic problem for 1 have been obtained by lazer and mckenna 8 fabry 5 and fabry and fonda 6 the known results on the existenceof periodic solutions already shed some light on the boundedness problem actually the second theorem of massera implies that if 1 has no periodic

model predictive control for perturbed max plus linear systems abstract model predictive control mpc is a popular controller design technique in the process industry conventional mpc uses linear or nonlinear discrete time models recently we have extended mpc to a class of discrete event systems that can be described by a model that is linear in the max algebra in our previous work we have only considered mpc for the deterministic noise free case without modeling errors in this paper we extend our previous results on mpc for max plus linear systems to cases with noise and or modeling errors we show that under quite general conditions the resulting optimization problems can be solved very efficiently

interface waves in almost incompressible elastic materials we study the problem of two elastic half planes in contact and the stoneley interface wave that may exist at the interface between two different elastic materials emphasis being put on the case when the half planes are almost incompressible we show that numerical simulations involving interface waves require an unexpectedly high number of grid points per wavelength as the materials become more incompressible let and be the lame parameters and densities of the first and second half plane respectively a theoretical study shows that if k is a real constant k k k and 0 then for an accurate solution the required number of grid points per wavelength scales as 1 p where p is the order of accuracy of the numerical method this requirement becomes very restrictive close to the incompressible limit especially for lower order methods i e a small p the theoretical findings are supported by numerical experiments that illustrate the demanding resolution requirement as well as the superiority of higher order methods the scaling is also seen to hold for a more general choice of lame parameters numerical experiments when one of the half planes is a vacuum are also presented where the higher resolution requirement is illustrated in a numerical solution of lamb s problem

interval strong solutions of interval systems of max plus linear equations abstract this paper considers the existence and uniqueness of interval strong solutions of interval systems of max plus linear equations a necessary and sufficient condition for the existence of interval strong solutions is presented the proof of the existence of interval strong solutions is constructive and results in a formula for computing such solutions a necessary and sufficient condition for the uniqueness of interval strong solutions is established by testing the unique solvability of a finite number of subsystems rather than all subsystems on this basis a polynomial algorithm is developed to verify the uniqueness of interval strong solutions

fourth order finite difference p sv seismograms i describe the properties of a fourth order accurate space second order accurate time two dimensional p sk finite difference scheme based on the madariagavirieux staggered grid formulation the numerical scheme is developed from the first order system of hyperbolic elastic equations of motion and constitutive laws expressed in particle velocities and stresses the madariaga virieux staggered grid scheme has the desirable quality that it can correctly model any variation in material properties including both large and small poisson s ratio materials with minimal numerical dispersion and numerical anisotropy dispersion analysis indicates that the shortest wavelengths in the model need to be sampled at 5 gridpoints wavelength the scheme can be used to accurately simulate wave propagation in mixed acoustic elastic media making it ideal for modeling marine problems explicitly calculating both velocities and stresses makes it relatively simple to initiate a source at the free surface or within a layer and to satisfy free surface boundary conditions benchmark comparisons of finite difference and analytical solutions to lamb s problem are almost identical as are comparisons of finite difference and reflectivity solutions for elastic elastic and acoustic elastic layered models

brief paper rigorous parameter reconstruction for differential equations with noisy data we present a method that given a data set a finitely parametrized system of ordinary differential equations odes and a search space of parameters discards portions of the search space that are inconsistent with the model ode and data the method is completely rigorous as it is based on validated integration of the vector field as a consequence no consistent parameters can be lost during the pruning phase for data sets with moderate levels of noise this yields a good reconstruction of the underlying parameters several examples are included to illustrate the merits of the method

technical communique injectivity analysis using interval analysis application to structural identifiability this paper presents a new numerical algorithm based on interval analysis able to prove that a differentiable function f email protected r n r n is injective this algorithm also performs a partition of the domain a in subsets a i where for all email protected a i the cardinality of f 1 f x is constant in the context of parameter estimation we show how this algorithm provides an efficient and numerical method to study the structural identifiability of parametric models

sequential imputations and bayesian missing data problems abstract for missing data problems tanner and wong have described a data augmentation procedure that approximates the actual posterior distribution of the parameter vector by a mixture of complete data posteriors their method of constructing the complete data sets is closely related to the gibbs sampler both required iterations and similar to the em algorithm convergence can be slow we introduce in this article an alternative procedure that involves imputing the missing data sequentially and computing appropriate importance sampling weights in many applications this new procedure works very well without the need for iterations sensitivity analysis influence analysis and updating with new data can be performed cheaply bayesian prediction and model selection can also be incorporated examples taken from a wide range of applications are used for illustration

growth of sobolev norms for the analytic nls on t2 we consider the completely resonant non linear schrodinger equation on the two dimensional torus with any analytic gauge invariant nonlinearity fix s 1 we show the existence of solutions of this equation which achieve arbitrarily large growth of hs sobolev norms we also give estimates for the time required to attain this growth

interval systems of max separable linear equations an interval system of equations is weakly solvable if at least one of its subsystems is solvable and it is strongly solvable if all its subsystems are solvable we give necessary and sufficient conditions enabling efficient testing of weak and strong solvability over max plus and max min algebras

long time instability and unbounded sobolev orbits for some periodic nonlinear schrodinger equations we study the energy cascade problematic for some nonlinear schrodinger equations on mathbb t 2 in terms of the growth of sobolev norms we define the notion of long time strong instability and establish its connection to the existence of unbounded sobolev orbits this connection is then explored for a family of cubic schrodinger nonlinearities that are equal or closely related to the standard polynomial one u 2u most notably we prove the existence of unbounded sobolev orbits for a family of hamiltonian cubic nonlinearities that includes the resonant cubic nls equation a k a the first birkhoff normal form

technical communique nonlinear bounded error state estimation of continuous time systems this paper presents a first study on the application of interval analysis and consistency techniques to state estimation of continuous time systems described by nonlinear ordinary differential equations the approach is presented in a bounded error context and the resulting methodology is illustrated by an example

interval analysis and dioid application to robust controller design for timed event graphs this paper deals with feedback controller synthesis for timed event graphs where the number of initial tokens and time delays are only known to belong to intervals we discuss here the existence and the computation of a robust controller set for uncertain systems that can be described by parametric models the unknown parameters of which are assumed to vary between known bounds each controller is computed in order to guarantee that the closed loop system behavior is greater than the lower bound of a reference model set and is lower than the upper bound of this set the synthesis presented here is mainly based on dioid interval analysis and residuation theory

local smoothing property and strichartz inequality for schrodinger equations with potentials superquadratic at infinity abstract we study smoothing properties for time dependent schrodinger equations i u t 1 2 u v x u u 0 l 2 r n with potentials which satisfy v x o x m at infinity m 2 we show that the solution u t x is 1 m times differentiable with respect to x at almost all t r and explain that this is the result of the fact that the sojourn time of classical particles with energy in arbitrary compact set is less than ct 1 m during 0 t when is very large we also show strichartz s inequality with derivative loss for such potentials and give its application to nonlinear schrodinger equations

boundary estimates for the elastic wave equation in almost incompressible materials we study the half plane problem for the elastic wave equation subject to a free surface boundary condition with particular emphasis on almost incompressible materials a normal mode analysis is developed to estimate the solution in terms of the boundary data showing that the problem is boundary stable the dependence on the material properties which is difficult to analyze by the energy method is made transparent by our estimates the normal mode technique is used to analyze the influence of truncation errors in a finite difference approximation our analysis explains why the number of grid points per wave length must be increased when the shear modulus mu becomes small compared to the first lame parameter lambda that is for almost incompressible materials when the surface waves are scaled to have unit wave length our analysis predicts that the grid size must be proportional to mu lambda 1 2 for a second order method for a fourth order method the grid size can be proportional t

birkhoff coordinates for the toda lattice in the limit of infinitely many particles with an application to fpu abstract in this paper we study the birkhoff coordinates cartesian action angle coordinates of the toda lattice with periodic boundary condition in the limit where the number n of the particles tends to infinity we prove that the transformation introducing such coordinates maps analytically a complex ball of radius r n in discrete sobolev analytic norms into a ball of radius r n with r r 0 independent of n if and only if 2 then we consider the problem of equipartition of energy in the spirit of fermi pasta ulam we deduce that corresponding to initial data of size r n 2 0 r 1 and with only the first fourier mode excited the energy remains forever in a packet of fourier modes exponentially decreasing with the wave number finally we consider the original fpu model and prove that energy remains localized in a similar packet of fourier modes for times one order of magnitude longer than those covered by previous results which is the time of formation of the packet the proof of the theorem on birkhoff coordinates is based on a new quantitative version of a vey type theorem by kuksin and perelman which could be interesting in itself

interval bounds on the solutions of semi explicit index one daes part 1 analysis this article presents two methods for computing interval bounds on the solutions of nonlinear semi explicit index one differential algebraic equations daes part 1 presents theoretical developments while part 2 discusses implementation and numerical examples the primary theoretical contributions are 1 an interval inclusion test for existence and uniqueness of a solution and 2 sufficient conditions in terms of differential inequalities for two functions to describe componentwise upper and lower bounds on this solution point wise in the independent variable the first proposed method applies these results sequentially in a two phase algorithm analogous to validated integration methods for ordinary differential equations the second method unifies these steps to characterize bounds as the solutions of an auxiliary system of daes efficient implementations of both are described using interval computations and demonstrated on numerical examples

stable difference approximations for the elastic wave equation in second order formulation we consider the three dimensional elastic wave equation for an isotropic heterogeneous material subject to a stress free boundary condition building on our recently developed theory for difference methods for second order hyperbolic systems h o kreiss n a petersson j ystrom siam j numer anal 40 2002 pp 1940 1967 we develop an explicit second order accurate technique which is stable for all ratios of longitudinal over transverse phase velocities the spatial discretization is self adjoint and the stability is obtained through an energy estimate seismic events are often modeled using singular source terms and we devise a technique to place sources independently of the grid while retaining second order accuracy away from the source several numerical examples are given

a tutorial on particle filters for online nonlinear non gaussian bayesian tracking increasingly for many application areas it is becoming important to include elements of nonlinearity and non gaussianity in order to model accurately the underlying dynamics of a physical system moreover it is typically crucial to process data on line as it arrives both from the point of view of storage costs as well as for rapid adaptation to changing signal characteristics in this paper we review both optimal and suboptimal bayesian algorithms for nonlinear non gaussian tracking problems with a focus on particle filters particle filters are sequential monte carlo methods based on point mass or particle representations of probability densities which can be applied to any state space model and which generalize the traditional kalman filtering methods several variants of the particle filter such as sir asir and rpf are introduced within a generic framework of the sequential importance sampling sis algorithm these are discussed and compared with the standard ekf through an illustrative example

computing minimal volume credible sets using interval analysis application to bayesian estimation given a random vector p with a probability density function pdf pi p a credible set pi p with level alpha is a set that contains p with a probability alpha the problem of characterizing minimal volume credible sets which correspond to level sets for pi is considered it is only assumed that the expression of pi p results from a combination of elementary operators and elementary functions sin cos abs etc this paper provides an algorithm able to compute accurate inner and outer approximations of minimal volume credible sets in a guaranteed way the approach is based on interval analysis and an application to nonlinear parameter estimation in a bayesian context is treated a windows solver associated with the presented algorithm is made available at http www ensieta fr e3i2 jaulin bayes zip

zonotopes and kalman observers state bounding observation based on zonotopes is the subject of this paper dealing with zonotopes is motivated by set operations resulting in simple matrix calculations with regard to the often huge number of facets and vertices of the equivalent polytopes discrete time ltv lpv systems with state and measurement uncertainties are considered based on a new zonotope size criterion called f w radius and by merging optimal and robust observer gain designs a zonotopic kalman filter zkf is proposed with a proof of robust convergence the notion of covariation is introduced and results in an explicit bridge between the zonotopic set membership and the stochastic paradigms for kalman filtering no intersection is used and the influence of the reduction operator limiting to a tunable maximum the size of the matrices involved in the zonotopic set computations is fully taken into account in the lmi based robust stability analysis a numerical example illustrates the effectiveness of the proposed zkf

interval max plus systems of linear equations abstract in this paper we shall deal with solvability of interval systems of linear equations in max plus algebra max plus algebra is an algebraic structure in which classical addition and multiplication are replaced by and where a b max a b a b a b the notation a x b represents an interval system of linear equations where a a a b b b are given interval matrix and interval vector respectively and a solution is from a given interval vector x x x we can define several types of solvability of interval systems in this paper we define six types of solvability of interval max plus systems of linear equations and give necessary and sufficient conditions for them

interval state estimation for a class of nonlinear systems the goal of this technical note is to design interval observers for a class of nonlinear continuous time systems the first part of this work shows that it is usually possible to design an interval observer for linear systems by means of linear time invariant changes of coordinates even if the system is not cooperative this result is extended to a class of nonlinear systems using partial exact linearisations the proposed observers guarantee to enclose the set of system states that is consistent with the model the disturbances and the measurement noise moreover it is only assumed that the measurement noise and the disturbances are bounded without any additional information such as stationarity uncorrelation or type of distribution the proposed observer is illustrated through numerical simulations

stable and high order accurate difference methods for the elastic wave equation in discontinuous media in this paper we develop a stable and systematic procedure for numerical treatment of elastic waves in discontinuous and layered media we consider both planar and curved interfaces where media parameters are allowed to be discontinuous the key feature is the highly accurate and provably stable treatment of interfaces where media discontinuities arise we discretize in space using high order accurate finite difference schemes that satisfy the summation by parts rule conditions at layer interfaces are imposed weakly using penalties by deriving lower bounds of the penalty strength and constructing discrete energy estimates we prove time stability we present numerical experiments in two space dimensions to illustrate the usefulness of the proposed method for simulations involving typical interface phenomena in elastic materials the numerical experiments verify high order accuracy and time stability

bounds on the reachable sets of nonlinear control systems the computation of rigorous enclosures of the reachable sets of nonlinear control systems is considered with a focus on applications for which speed is crucial low computational costs make interval methods based on differential inequalities an attractive option unfortunately such methods are prone to large overestimation and often produce useless results in practice from physical considerations however it is common that some crude set is known to contain the reachable set we establish a general bounding result based on differential inequalities which enables the effective use of such sets during the bounding procedure in the case where this set is a convex polyhedron an efficient implementation using interval computations is developed using readily available physical information from practical examples this method is shown to provide significant advantages over alternative methods in terms of both efficiency and accuracy

loop detection of mobile robots using interval analysis this paper proposes an original set membership approach for loop detection of mobile robots in the situation where proprioceptive sensors only are available to detect loops the new concepts of thet plane which is a two dimensional space with time coordinates are introduced intervals of functions or tubes are then used to represent uncertain trajectories and tests are provided in order to eliminate parts of the t plane that do not correspond to any loop an experiment with an actual underwater robot is proposed in order to illustrate the principle and the efficiency of the approach

stable and accurate wave propagation in discontinuous media a time stable discretization is derived for the second order wave equation with discontinuous coefficients the discontinuity corresponds to inhomogeneity in the underlying medium and is treated by splitting the domain each homogeneous sub domain is discretized using narrow diagonal summation by parts operators and then patched to its neighbors by using a penalty method leading to fully explicit time integration this discretization yields a time stable and efficient scheme the analysis is verified by numerical simulations in one dimension using high order finite difference discretizations and in three dimensions using an unstructured finite volume discretization

spatial planning a configuration space approach this paper presents algorithms for computing constraints on the position of an object due to the presence of ther objects this problem arises in applications that require choosing how to arrange or how to move objects without collisions the approach presented here is based on characterizing the position and orientation of an object as a single point in a configuration space in which each coordinate represents a degree of freedom in the position or orientation of the object the configurations forbidden to this object due to the presence of other objects can then be characterized as regions in the configuration space called configuration space obstacles the paper presents algorithms for computing these configuration space obstacles when the objects are polygons or polyhedra

stable and accurate schemes for the compressible navier stokes equations minimal stencil width discretizations of combined mixed and non mixed second order derivatives are analyzed with respect to accuracy and stability we show that these discretizations lead to stability for cauchy problems with a careful boundary treatment we also show that the stability holds for initial boundary value problems the analysis is verified by numerical simulations of burgers and navier stokes equations in two and three space dimensions

quasiperiodic motions in superquadratic time periodic potentials it is shown that for a large class of potentials on the line with superquadratic growth at infinity and with the additional time periodic dependence all possible motions under the influence of such potentials are bounded for all time and that most in a precise sense motions are in fact quasiperiodic the class of potentials includes as very particular examples the exponential polynomial and much more this extends earlier results and gives an answer to a problem posed by littlewood in the mid 1960 s along the way machinery is developed for estimating the action angle transformation directly in terms of the potential and also some apparently new identities involving singular integrals are derived

guaranteed nonlinear estimation using constraint propagation on sets bounded error estimation is the estimation of the parameter or state vector of a model from experimental data under the assumption that some suitably de ned errors should belong to some prior feasible sets when the model outputs are linear in the vector to be estimated a number of methods are available to enclose all estimates that are consistent with the data into simple sets such as ellipsoids orthotopes or parallelotopes thereby providing guaranteed set estimates in the nonlinear case the situation is much less developed and there are very few methods that produce such guaranteed estimates in this paper the discrete time problem is cast into the more general framework of constraint satisfaction problems algorithms rather classical in the area of interval constraint propagation are extended by replacing intervals by more general subsets of real vector spaces this makes it possible to propose a new algorithm that contracts the feasible domains for each uncertain variable optimally i e no smaller domain could be obtained and e ciently the resulting methodology is illustrated on discrete time nonlinear state estimation the state at time k is estimated either from past measurement only or from all measurements assumed to be available from the start even in the causal case prior information on the future value of the state and output vectors due for instance to physical constraints is readily taken into account

minimal max realization of convex sequences we show that the minimal dimension of a linear realization over the max semiring of a convex sequence is equal to the minimal size of a decomposition of the sequence as a supremum of discrete affine maps the minimal dimensional realization of any convex realizable sequence can thus be found in linear time the result is based on a bound in terms of minors of the hankel matrix

long time dynamics of schrodinger and wave equations on flat tori abstract we consider a class of linear time dependent schrodinger equations and quasi periodically forced nonlinear hamiltonian wave klein gordon and schrodinger equations on arbitrary flat tori for the linear schrodinger equation we prove a t 0 upper bound for the growth of the sobolev norms as the time goes to infinity for the nonlinear hamiltonian pdes we construct families of time quasi periodic solutions both results are based on clusterization properties of the eigenvalues of the laplacian on a flat torus and on suitable separation properties of the singular sites of schrodinger and wave operators which are integers in space time fourier lattice close to a cone or a paraboloid thanks to these properties we are able to apply delort abstract theorem 20 to control the speed of growth of the sobolev norms and berti corsi procesi abstract nash moser theorem 8 to construct quasi periodic solutions

minimal realization in the max algebra is an extended linear complementarity problem we demonstrate that the minimal state space realization problem in the max algebra can be transformed into an extended linear complementarity problem elcp we use an algorithm that finds all solutions of an elcp to find all equivalent minimal state space realizations of a single input single output siso discrete event system we also give a geometrical description of the set of all minimal realizations of a siso max linear discrete event system

wave propagation in anisotropic elastic materials and curvilinear coordinates using a summation by parts finite difference method we develop a fourth order accurate finite difference method for solving the three dimensional elastic wave equation in general heterogeneous anisotropic materials on curvilinear grids the proposed method is an extension of the method for isotropic materials previously described in the paper by sjogreen and petersson 2012 11 the proposed method discretizes the anisotropic elastic wave equation in second order formulation using a node centered finite difference method that satisfies the principle of summation by parts the summation by parts technique results in a provably stable numerical method that is energy conserving we also generalize and evaluate the super grid far field technique for truncating unbounded domains unlike the commonly used perfectly matched layers pml the super grid technique is stable for general anisotropic material because it is based on a coordinate stretching combined with an artificial dissipation as a result the discretization satisfies an energy estimate proving that the numerical approximation is stable we demonstrate by numerical experiments that sufficiently wide super grid layers result in very small artificial reflections applications of the proposed method are demonstrated by three dimensional simulations of anisotropic wave propagation in crystals