global optimization of linear hybrid systems with explicit transitions the global optimization of hybrid systems described by linear time varying ordinary differential equations is examined a method to construct convex relaxations of general nonlinear bolza type objective functions or constraints subject to an embedded hybrid system with explicit transitions is presented the optimization problem can be solved using gradient based algorithms in a branch and bound framework that is shown to be infinitely convergent when the implied state bounds are employed

stepsize selection in the rigorous defect control of taylor series methods abstract modern numerical methods for initial value problems ivps in ordinary differential equations ode produce a piecewise differentiable numerical solution the defect or residual evaluated at it induces a perturbed ode that is satisfied exactly by this solution defect control methods try to ensure that a norm of the defect is bounded by a user specified tolerance this paper justifies and implements a simple and effective stepsize selection strategy and an overall method for both validated and non validated defect control of an explicit taylor series method for ivp odes in validated mode this method guarantees that the defect is bounded by a user specified tolerance

on taylor model based integration of odes interval methods for verified integration of initial value problems ivps for odes have been used for more than 40 years for many classes of ivps these methods are able to compute guaranteed error bounds for the flow of an ode where traditional methods provide only approximations to a solution overestimation however is a potential drawback of verified methods for some problems the computed error bounds become overly pessimistic or the integration even breaks down the dependency problem and the wrapping effect are particular sources of overestimations in interval computations berz and his coworkers have developed taylor model methods which extend interval arithmetic with symbolic computations the latter is an effective tool for reducing both the dependency problem and the wrapping effect by construction taylor model methods appear particularly suitable for integrating nonlinear odes we analyze taylor model based integration of odes and compare taylor model methods with traditional enclosure methods for ivps for odes

constrained zonotopes this article introduces a new class of sets called constrained zonotopes that can be used to enclose sets of interest for estimation and control the numerical representation of these sets is sufficient to describe arbitrary convex polytopes when the complexity of the representation is not limited at the same time this representation permits the computation of exact projections intersections and minkowski sums using very simple identities efficient and accurate methods for computing an enclosure of one constrained zonotope by another of lower complexity are provided the advantages and disadvantages of these sets are discussed in comparison to ellipsoids parallelotopes zonotopes and convex polytopes in halfspace and vertex representations moreover extensive numerical comparisons demonstrate significant advantages over other classes of sets in the context of set based state estimation and fault detection

survey of interval algorithms for ordinary differential equations interval algorithms compute an interval valued function in which the solution to a system of ordinary differential equations is guaranteed to lie interval algorithms use differential inequalities finite difference approximations with remainders taylor series defect corrections or contractive iterations

bounding the solutions of parameter dependent nonlinear ordinary differential equations this paper presents two techniques for generating rigorous bounds on the solution of parameter dependent nonlinear ordinary differential equations the first technique is an extension of differential inequalities that enables the construction of tight time varying state bounds by utilizing prior knowledge of the solution of the differential equations the second technique provides a method for constructing pointwise in time convex and concave relaxations of the image of the solution of a system of parameter dependent differential equations on subsets of a euclidean space two examples are presented to demonstrate the construction of the bounds the examples include a brief discussion of a computer program written to automatically generate the bounds

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

status of the differential transformation method abstract further to a recent controversy on whether the differential transformation method dtm for solving a differential equation is purely and solely the traditional taylor series method it is emphasized that the dtm is currently used often only as a technique for analytically calculating the power series of the solution in terms of the initial value parameters sometimes a piecewise analytic continuation process is implemented either in a numerical routine e g within a shooting method or in a semi analytical procedure e g to solve a boundary value problem emphasized also is the fact that at the time of its invention the currently used basic ingredients of the dtm that transform a differential equation into a difference equation of same order that is iteratively solvable were already known for a long time by the traditional taylor method users notably in the elaboration of software packages numerical routines for automatically solving ordinary differential equations at now the defenders of the dtm still ignore the though much better developed studies of the traditional taylor method users who in turn seem to ignore similarly the existence of the dtm the dtm has been given an apparent strong formalization set on the same footing as the fourier laplace or mellin transformations though often used trivially it is easily attainable and easily adaptable to different kinds of differentiation procedures that has made it very attractive hence applications to various problems of the taylor method and more generally of the power series method including noninteger powers has been sketched it seems that its potential has not been exploited as it could be after a discussion on the reasons of the misunderstandings which have caused the controversy the preceding topics are concretely illustrated it is concluded that for the sake of clarity when the dtm is applied to odes it should be mentioned that the dtm exactly coincides with the traditional taylor method contrary to what is currently written

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

a cohomological classification of vector bundles on smooth affine threefolds we give a cohomological classification of vector bundles of rank 2 on a smooth affine threefold over an algebraically closed field having characteristic unequal to 2 as a consequence we deduce that cancellation holds for rank 2 vector bundles on such varieties the proofs of these results involve three main ingredients first we give a description of the first nonstable a1 homotopy sheaf of the symplectic group second these computations can be used in concert with f morel s a1 homotopy classification of vector bundles on smooth affine schemes and obstruction theoretic techniques stemming from a version of the postnikov tower in a1 homotopy theory to reduce the classification results to cohomology vanishing statements third we prove the required vanishing statements

a time dependent hamilton jacobi formulation of reachable sets for continuous dynamic games we describe and implement an algorithm for computing the set of reachable states of a continuous dynamic game the algorithm is based on a proof that the reachable set is the zero sublevel set of the viscosity solution of a particular time dependent hamilton jacobi isaacs partial differential equation while alternative techniques for computing the reachable set have been proposed the differential game formulation allows treatment of nonlinear systems with inputs and uncertain parameters because the time dependent equation s solution is continuous and defined throughout the state space methods from the level set literature can be used to generate more accurate approximations than are possible for formulations with potentially discontinuous solutions a numerical implementation of our formulation is described and has been released on the web its correctness is verified through a two vehicle three dimensional collision avoidance example for which an analytic solution is available

the mayer vietoris principle for grothendieck witt groups of schemes we prove localization and zariski mayer vietoris for higher gro thendieck witt groups alias hermitian k groups of schemes admitting an ample family of line bundles no assumption on the characteristic is needed and our schemes can be singular along the way we prove additivity fibration and approximation theorems for the hermitian k theory of exact categories with weak equivalences and duality

numerical simulations of water gas flow in heterogeneous porous media with discontinuous capillary pressures by the concept of global pressure we present an approach and numerical results for a new formulation modeling immiscible compressible two phase flow in heterogeneous porous media with discontinuous capillary pressures the main feature of this model is the introduction of a new global pressure and it is fully equivalent to the original equations the resulting equations are written in a fractional flow formulation and lead to a coupled degenerate system which consists of a nonlinear parabolic the global pressure equation and a nonlinear diffusion convection one the saturation equation with nonlinear transmission conditions at the interfaces that separate different media the resulting system is discretized using a vertex centred finite volume method combined with pressure and flux interface conditions for the treatment of heterogeneities an implicit euler approach is used for time discretization a godunov type method is used to treat the convection terms and the diffusion terms are discretized by piecewise linear conforming finite elements we present numerical simulations for three one dimensional benchmark tests to demonstrate the ability of the method to approximate solutions of water gas equations efficiently and accurately in nuclear underground waste disposal situations

basis functions for general hsieh clough tocher triangles complete or reduced in this note we give the basis functions shape functions for hsieh clough tocher triangles complete or reduced in order to take the best advantage of symmetry and for simplicity we derive these functions for a general triangle by using simultaneously the area co ordinates and the so called eccentricity parameters we conclude with some remarks concerning the implementation

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

sensitivity analysis of uncertain dynamic systems using set valued integration we present an extension of set valued integration to enable efficient sensitivity analysis of parameter dependent ordinary differential equation ode systems using both the forward and adjoint me

three phase compressible flow in porous media we describe the construction of total differential td three phase data for the implementation of the exact global pressure formulation for the modeling of three phase compressible flow in porous media this global formulation is preferred since it reduces the coupling between the pressure and saturation equations compared to phase or weighted formulations it simplifies the numerical analysis of the problem and boosts its computational efficiency however this global pressure approach exists only for three phase data relative permeabilities capillary pressures which satisfy a td condition such td three phase data are determined by the choice of a global capillary pressure function and a global mobility function which take both saturations and global pressure level as argument boundary conditions for global capillary pressure and global mobility are given such that the corresponding three phase data are consistent with a given set of three two phase data the numerical construction of global capillary pressure and global mobility functions by c 1 and c 0 finite element is then performed using bi laplacian and laplacian interpolation examples of the corresponding td three phase data are given for a compressible and an incompressible case

a hybrid bounding method for computing an over approximation for the reachable set of uncertain nonlinear systems in this paper we show how to compute an over approximation for the reachable set of uncertain nonlinear continuous dynamical systems by using guaranteed set integration we introduce two ways to do so the first one is a full interval method which handles whole domains for set computation and relies on state of the art validated numerical integration methods the second one relies on comparison theorems for differential inequalities in order to bracket the uncertain dynamics between two dynamical systems where there is no uncertainty since the derived bracketing systems are piecewise ck differentiable functions validated numerical integration methods cannot be used directly hence our contribution resides in the use of hybrid automata to model the bounding systems we give a rule for building these automata and we show how to run them and address mode switching in a guaranteed way in order to compute the over approximation for the reachable set the computational cost of our method is also analyzed and shown to be smaller that the one of classical interval techniques sufficient conditions are given which ensure the epsiv practical stability of the enclosures given by our hybrid bounding method two examples are also given which show that the performance of our method is very promising

interval bounds on the solutions of semi explicit index one daes part 2 computation 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 odes 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 set valued integration of nonlinear dynamic systems using affine set parameterizations many set valued integration algorithms for parametric ordinary differential equations odes implement a combination of taylor series expansion with either interval arithmetic or taylor model arithmetic due to the wrapping effect the diameter of the solution set enclosures computed with these algorithms typically diverges to infinity on finite integration horizons even though the ode trajectories themselves may be asymptotically stable this paper starts by describing a new discretized set valued integration algorithm that uses a predictor validation approach to propagate generic affine set parameterizations whose images are guaranteed to enclose the ode solution set sufficient conditions are then derived for this algorithm to be locally asymptotically stable in the sense that the computed enclosures are guaranteed to remain stable on infinite time horizons when applied to a dynamic system in the neighborhood of a locally asymptotically stable periodic orbit or equilibrium point the key requireme

approximate reduction of heterogenous nonlinear models with differential hulls we present a model reduction technique for a class of nonlinear ordinary differential equation ode models of heterogeneous systems where heterogeneity is expressed in terms of classes of state variables having the same dynamics structurally but which are characterized by distinct parameters to this end we first build a system of differential inequalities that provides lower and upper bounds for each original state variable but such that it is homogeneous in its parameters then we use two methods for exact aggregation of odes to exploit this homogeneity yielding a smaller model of size independent of the number of heterogeneous classes we apply this technique to two case studies a multiclass queuing network and a model of epidemics spread

motivic spheres and the image of the suslin hurewicz map we show that an old conjecture of a a suslin characterizing the image of a hurewicz map from quillen k theory in degree n to milnor k theory in degree n admits an interpretation in terms of unstable mathbb a 1 homotopy sheaves of the general linear group using this identification we establish suslin s conjecture in degree 5 for infinite fields having characteristic unequal to 2 or 3 we do this by linking the relevant unstable mathbb a 1 homotopy sheaf of the general linear group to the stable mathbb a 1 homotopy of motivic spheres

global optimization in hilbert space we propose a complete search algorithm for solving a class of non convex possibly infinite dimensional optimization problems to global optimality we assume that the optimization variables are in a bounded subset of a hilbert space and we determine worst case run time bounds for the algorithm under certain regularity conditions of the cost functional and the constraint set because these run time bounds are independent of the number of optimization variables and in particular are valid for optimization problems with infinitely many optimization variables we prove that the algorithm converges to an varepsilon suboptimal global solution within finite run time for any given termination tolerance varepsilon 0 finally we illustrate these results for a problem of calculus of variations

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

reachability analysis of nonlinear systems using matrix measures matrix measures also known as logarithmic norms have historically been used to provide bounds on the divergence of trajectories of a system of ordinary differential equations in this technical note we use them to compute guaranteed overapproximations of reachable sets for nonlinear continuous time systems using numerically simulated trajectories and to bound the accumulation of numerical simulation errors along simulation traces our method employs a user supplied bound on the matrix measure of the system s jacobian matrix to compute bounds on the behavior of nearby trajectories leading to efficient computation of reachable sets when such bounds are available we demonstrate that the proposed technique scales well to systems with a large number of states

comparison of various formulations of three phase flow in porous media various formulations of the governing equations that describe three phase e g water oil and gas flow in porous media including phase global and pseudo global pressure saturation formulations are discussed in this paper comparisons of these differential formulations are theoretically and numerically presented for the first time it is shown that the global pressure saturation formulation is the most efficient one from the computational point of view in the case where the three phase relative permeability and capillary pressure curves satisfy a so called total differential condition the pseudo global formulation is useful when the fractional flow functions of the water and gas phases are close to their respective mean values and the phase formulation can be applied generally

geometric models for higher grothendieck witt groups in mathbb a 1 homotopy theory we show that the higher grothendieck witt groups a k a algebraic hermitian k groups are represented by an infinite orthogonal grassmannian in the mathbb a 1 homotopy category of smooth schemes over a regular base for which 2 is a unit in the ring of regular functions we also give geometric models for various mathbb p 1 and s 1 loop spaces of hermitian k theory

a comprehensive method for reachability analysis of uncertain nonlinear hybrid systems reachability analysis of nonlinear uncertain hybrid systems i e continuous discrete dynamical systems whose continuous dynamics guard sets and reset functions are defined by nonlinear functions can be decomposed in three algorithmic steps computing the reachable set when the system is in a given operation mode computing the discrete transitions i e detecting and localizing when and where the continuous flowpipe intersects the guard sets and aggregating the multiple trajectories that result from an uncertain transition once the whole flow pipe has transitioned so that the algorithm can resume this paper proposes a comprehensive method that provides a nicely integrated solution to the hybrid reachability problem at the core of the method is the concept of mspb i e geometrical object obtained as the minkowski sum of a parallelotope and an axes aligned box mspb are a way to control the over approximation of the taylor s interval integration method as they happen to be a specific type of zonotope they articulate perfectly with the zonotope bounding method that we propose to enclose in an optimal way the set of flowpipe trajectories generated by the transition process the method is evaluated both theoretically by analyzing its complexity and empirically by applying it to well chosen hybrid nonlinear examples

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