multi resolution analysis scheme for uncertainty quantification in chemical systems this paper presents a multi resolution approach for the propagation of parametric uncertainty in chemical systems it is motivated by previous studies where galerkin formulations of wiener hermite expansions were found to fail in the presence of steep dependences of the species concentrations with regard to the reaction rates the multi resolution scheme is based on representation of the uncertain concentration in terms of compact polynomial multi wavelets allowing for the control of the convergence in terms of polynomial order and resolution level the resulting representation is shown to greatly improve the robustness of the galerkin procedure in presence of steep dependences however this improvement comes with a higher computational cost which drastically increases with the number of uncertain reaction rates to overcome this drawback an adaptive strategy is proposed to control locally in the parameter space and in time the resolution level the efficiency of the method is demonstrated for an uncertain chemical system having eight random parameters

dynamic stability of spinning viscoelastic cylinders at finite deformation the study of spinning axisymmetric cylinders undergoing finite deformation is a classic problem in several industrial settings the tire industry in particular we present a stability analysis of spinning elastic and viscoelastic cylinders using arpack to compute eigenvalues and eigenfunctions of finite element discretizations of the linearized evolution operator we show that the eigenmodes correspond to n peak standing or traveling waves for the linearized problem with an additional index describing the number of oscillations in the radial direction we find a second hierarchy of bifurcations to standing waves where these eigenvalues cross zero and confirm numerically the existence of finite amplitude standing waves for the nonlinear problem on one of the new branches in the viscoelastic case this analysis permits us to study the validity of two popular models of finite viscoelasticity we show that a commonly used finite deformation linear convolution model results in non physical energy growth and finite time blow up when the system is perturbed in a linearly unstable direction and followed nonlinearly in time on the other hand sidoroff style viscoelastic models are seen to be linearly and nonlinearly stable as is physically required 2014 elsevier ltd all rights reserved

multi element generalized polynomial chaos for arbitrary probability measures we develop a multi element generalized polynomial chaos me gpc method for arbitrary probability measures and apply it to solve ordinary and partial differential equations with stochastic inputs given a stochastic input with an arbitrary probability measure its random space is decomposed into smaller elements subsequently in each element a new random variable with respect to a conditional probability density function pdf is defined and a set of orthogonal polynomials in terms of this random variable is constructed numerically then the generalized polynomial chaos gpc method is implemented element by element numerical experiments show that the cost for the construction of orthogonal polynomials is negligible compared to the total time cost efficiency and convergence of me gpc are studied numerically by considering some commonly used random variables me gpc provides an efficient and flexible approach to solving differential equations with random inputs especially for problems related to long term integration large perturbation and stochastic discontinuities

multi resolution analysis of wiener type uncertainty propagation schemes a multi resolution analysis mra is applied to an uncertainty propagation scheme based on a generalized polynomial chaos pc representation the mra relies on an orthogonal projection of uncertain data and solution variables onto a multi wavelet basis consisting of compact piecewise smooth polynomial functions the coefficients of the expansion are computed through a galerkin procedure the mra scheme is applied to the simulation of the lorenz system having a single random parameter the convergence of the solution with respect to the resolution level and expansion order is investigated in particular results are compared to two monte carlo sampling strategies demonstrating the superiority of the mra for more complex problems however the mra approach may require excessive cpu times adaptive methods are consequently developed in order to overcome this drawback two approaches are explored the first is based on adaptive refinement of the multi wavelet basis while the second is based on adaptive block partitioning of the space of random variables computational tests indicate that the latter approach is better suited for large problems leading to a more efficient flexible and parallelizable scheme

intrusive galerkin methods with upwinding for uncertain nonlinear hyperbolic systems this paper deals with stochastic spectral methods for uncertainty propagation and quantification in nonlinear hyperbolic systems of conservation laws we consider problems with parametric uncertainty in initial conditions and model coefficients whose solutions exhibit discontinuities in the spatial as well as in the stochastic variables the stochastic spectral method relies on multi resolution schemes where the stochastic domain is discretized using tensor product stochastic elements supporting local polynomial bases a galerkin projection is used to derive a system of deterministic equations for the stochastic modes of the solution hyperbolicity of the resulting galerkin system is analyzed a finite volume scheme with a roe type solver is used for discretization of the spatial and time variables an original technique is introduced for the fast evaluation of approximate upwind matrices which is particularly well adapted to local polynomial bases efficiency and robustness of the overall method are assessed on the burgers and euler equations with shocks

newton krylov method for computing the cyclic steady states of evolution problems in nonlinear mechanics this work is focused on the newton krylov technique for computing the steady cyclic states of evolution problems in non linear mechanics with space time periodicity conditions this kind of problems can be faced for instance in the modeling of a rolling tyre with a periodic tread pattern where the cyclic state satisfies rolling periodicity condition including shifts both in time and space the newton krylov method is a combination of a newton nonlinear solver with a krylov linear solver looking for the initial state which provides the space time periodic solution the convergence of the krylov iterations is proved to hold in presence of an adequate preconditioner after preconditioning the newton krylov method can be also considered as an observer controller method correcting the transient solution of the initial value problem after each period using information stored while computing the residual the krylov solver computation time becomes negligible with respect to the residual computation time the method has been analyzed and tested on academic applications and compared to the standard evolution fixed point method finally it has been implemented into the michelin industrial code applied to a full 3d rolling tyre model

lie idempotent algebras abstract besides the inner product inherited from each symmetric group algebra ksn on the direct sum ks nksn a second convolution product arises from the hopf algebra structure on ks for any subalgebra a of ks for both the inner and the convolution product it is shown that the lie idempotent algebra l a of a generated by all lie idempotents in a with respect to the convolution product is also closed under inner products its nth homogeneous component l n a is thus a subalgebra of ksn for all n provided that a contains a lie idempotent in ksk for all k n solomon s descent algebra d n is a subalgebra of l n a in this case a number of results on the structure of l n a is derived including a description of its jacobson radical its principal indecomposables and its cartan invariants furthermore solomon s epimorphism from d n onto the ring of class functions clk sn of sn extends to an epimorphism of algebras c n l n a cl k s n additional results are obtained concerning a bialgebra structure of l a in the particular case of a n d n this amounts to a new approach to several well known results on d n

generalized solution of time dependent traveling load problem via moving finite element scheme abstract a finite element methodology is described which can handle the transient response of moving structures which are themselves subject to time dependent traveling load fields the generality of the approach is such that it can handle the overall transient response due to moving loads whose magnitude zone of application and trajectory are time dependent in addition to developing the overall transient solution methodology attention is also given to establishing the overall eigenvalue properties of the linear problem as well as developing a solution for the steady vibrating case to benchmark the overall procedure the results of several numerical experiments are included in the presentation

high order statistics in global sensitivity analysis decomposition and model reduction abstract analysis of variance anova is a common technique for computing a ranking of the input parameters in terms of their contribution to the output variance nevertheless the variance is not a universal criterion for ranking variables since non symmetric outputs could require higher order statistics for their description and analysis in this work we illustrate how third and fourth order moments i e skewness and kurtosis respectively can be decomposed mimicking the anova approach it is also shown how this decomposition is correlated to a polynomial chaos pc expansion leading to a simple strategy to compute each term new sensitivity indices based on the contribution to the skewness and kurtosis are proposed the outcome of the proposed analysis is depicted by considering several test functions moreover the ranking of the sensitivity indices is shown to vary according to their statistics order furthermore the problem of formulating a truncated polynomial representation of the original function is treated both the reduction of the number of dimensions and the reduction of the order of interaction between parameters are considered in both cases the impact on the reduction is assessed in terms of statistics namely the probability density function feasibility of the proposed analysis in a real case is then demonstrated by presenting the sensitivity analysis of the performances of a turbine cascade in an organic rankine cycles orcs in the presence of complex thermodynamic models and multiple sources of uncertainty

cylindric skew schur functions cylindric skew schur functions which are a generalisation of skew schur functions arise naturally in the study of p partitions also recent work of a postnikov shows they have a strong connection with a problem of considerable current interest that of finding a combinatorial proof of the non negativity of the 3 point gromov witten invariants after explaining these motivations we study cylindric skew schur functions from the point of view of schur positivity using a result of i gessel and c krattenthaler we generalise a formula of a bertram i ciocan fontanine and w fulton thus giving an expansion of an arbitrary cylindric skew schur function in terms of skew schur functions while we show that no non trivial cylindric skew schur functions are schur positive we conjecture that this can be reconciled using the new concept of cylindric schur positivity

new identities in dendriform algebras abstract dendriform structures arise naturally in algebraic combinatorics where they allow for example the splitting of the shuffle product into two pieces and through rota baxter algebra structures the latter appear among others in differential systems and in the renormalization process of pqft we prove new combinatorial identities in dendriform algebras that appear to be strongly related to classical phenomena such as the combinatorics of lyndon words rewriting rules in lie algebras or the fine structure of the malvenuto reutenauer algebra one of these identities is an abstract noncommutative dendriform generalization of the bohnenblust spitzer identity and of an identity involving iterated chen integrals due to c s lam

lyndon words and shuffle algebras for generating the coloured multiple zeta values relations tables we study here the coloured multiple zeta values obtained by extending the usual notion of the multiple zeta values by adding roots of unity we state their main combinatorial properties and we give as a result a maple algorithm which generates the relations table between these values

solution of stochastic partial differential equations using galerkin finite element techniques this paper presents a framework for the construction of galerkin approximations of elliptic boundary value problems with stochastic input data a variational formulation is developed which allows among others numerical treatment by the finite element method a theory of a posteriori error estimation and corresponding adaptive approaches based on practical experience can be utilized the paper develops a foundation for treating stochastic partial differential equations pdes which can be further developed in many directions

uncertainty quantification for systems of conservation laws uncertainty quantification through stochastic spectral methods has been recently applied to several kinds of non linear stochastic pdes in this paper we introduce a formalism based on kinetic theory to tackle uncertain hyperbolic systems of conservation laws with polynomial chaos pc methods the idea is to introduce a new variable the entropic variable in bijection with our vector of unknowns which we develop on the polynomial basis by performing a galerkin projection we obtain a deterministic system of conservation laws we state several properties of this deterministic system in the case of a general uncertain system of conservation laws we then apply the method to the case of the inviscid burgers equation with random initial conditions and we present some preliminary results for the euler system we systematically compare results from our new approach to results from the stochastic galerkin method in the vicinity of discontinuities the new method bounds the oscillations due to gibbs phenomenon to a certain range through the entropy of the system without the use of any adaptative random space discretizations it is found to be more precise than the stochastic galerkin method for smooth cases but above all for discontinuous cases

hypernormal form theory foundations and algorithms abstract hypernormal forms unique normal forms simplest normal forms are investigated both from the standpoint of foundational theory and algorithms suitable for use with computer algebra the baider theory of the campbell hausdorff group is refined by a study of its subgroups to determine the smallest substages into which the hypernormalization process can be divided this leads to a linear algebra algorithm to compute the generators needed for each substage with the least amount of work a concrete interpretation of jan sanders spectral sequence for hypernormal forms is presented examples are given and a proof is given for a little known theorem of belitskii expressing the hypernormal form space in the inner product style as the kernel of a higher order differential operator

excitation of rotating circumferentially periodic structures abstract a circumferentially periodic structure consists of a finite number of identical substructures coupled together in identical ways to form a closed ring type structure the vibrational behaviour of such structures when rotating about the axis of structural symmetry is considered especially the forced response due to forces fixed in space it is known that for a circumferentially symmetric structure rotating about its axis of structural symmetry and excited by a force fixed in space resonance for the n nodal diameters mode is obtained when the corresponding natural frequency is n n where is the angular velocity of rotation the deformation at these resonances consists of waves travelling backwards in relation to the direction of the structure rotation this resonance condition also holds for a circumferentially periodic structure but then additional resonance possibilities exist given by n kn n where n is the number of sub structures and k 0 1 2 these results are known to hold true for a structure fixed in space and acted upon by a rotating force in this paper these results are shown to hold true also for a rotating structure acted upon by a force distribution fixed in space both backward and forward travelling waves can be excited in those cases when the rotating structure exhibits coriolis accelerations the natural frequencies for backward 1n and forward 2n rotating mode shapes are different this well known property has a pronounced effect on the forced response due to forces fixed in space in the absence of coriolis effects the magnification factor is essentially 1 n 2 2 where is the forcing frequency due to rotation but in the presence of coriolis effects the magnification factor has the characteristics 1 1n 2n

subcell resolution in simplex stochastic collocation for spatial discontinuities subcell resolution has been used in the finite volume method fvm to obtain accurate approximations of discontinuities in the physical space stochastic methods are usually based on local adaptivity for resolving discontinuities in the stochastic dimensions however the adaptive refinement in the probability space is ineffective in the non intrusive uncertainty quantification framework if the stochastic discontinuity is caused by a discontinuity in the physical space with a random location the dependence of the discontinuity location in the probability space on the spatial coordinates then results in a staircase approximation of the statistics which leads to first order error convergence and an underprediction of the maximum standard deviation to avoid these problems we introduce subcell resolution into the simplex stochastic collocation ssc method for obtaining a truly discontinuous representation of random spatial discontinuities in the interior of the cells discretizing the probability space the presented ssc sr method is based on resolving the discontinuity location in the probability space explicitly as function of the spatial coordinates and extending the stochastic response surface approximations up to the predicted discontinuity location the applications to a linear advection problem the inviscid burgers equation a shock tube problem and the transonic flow over the rae 2822 airfoil show that ssc sr resolves random spatial discontinuities with multiple stochastic and spatial dimensions accurately using a minimal number of samples

the campbell hausdorff near ring over n a topological view abstract the famous baker campbell hausdorff series defines a group composition on the set of the so called grouplike elements of a completed free lie algebra as we can find in p cartier s paper on the campbell hausdorff formula in 1956 according to a baider and r c churchill we call this group the campbell hausdorff group over the alphabet x of the free lie algebra by a certain coproduct and certain bialgebra endomorphisms of the free associative algebra over a set x which should be at most countably infinite we get an alternative realisation of this campbell hausdorff group this realisation is much more comfortable to deal with than the classical one the usual composition of endomorphisms gives a near ring structure to this group in this paper we consider a metric on the campbell hausdorff near ring over the alphabet n and the compatibility of this metric with the near ring compositions and we make some topological remarks

the algebra of quasi symmetric functions is free over the integers let z denote the leibniz hopf algebra which also turns up as the solomon descent algebra and the algebra of noncommutative symmetric functions as an algebra z z z1 z2 the free associative algebra over the integers in countably many indeterminates the coalgebra structure is given by zn ni 0 zi zn i z0 1 let m be the graded dual of z this is the algebra of quasi symmetric functions the ditters conjecture says that this algebra is a free commutative algebra over the integers in this paper the ditters conjecture is proved

algebraic structures on grothendieck groups of a tower of algebras abstract the grothendieck group of the tower of symmetric group algebras has a self dual graded hopf algebra structure inspired by this we introduce by way of axioms a general notion of a tower of algebras and study two grothendieck groups on this tower linked by a natural paring using representation theory we show that our axioms give a structure of graded hopf algebras on each grothendieck groups and these structures are dual to each other we give some examples to indicate why these axioms are necessary we also give auxiliary results that are helpful to verify the axioms we conclude with some remarks on generalized towers of algebras leading to a structure of generalized bialgebras in the sense of loday on their grothendieck groups

duality between quasi symmetrical functions and the solomon descent algebra abstract the ring qsym of quasi symmetric functions is naturally the dual of the solomon descent algebra the product and the two coproducts of the first extending those of the symmetric functions correspond to a coproduct and two products of the second which are defined by restriction from the symmetric group algebra a consequence is that qsym is a free commutative algebra

a generating set of solomon s descent algebra abstract solomon s descent algebra is generated by sums of descent classes corresponding to certain hook shapes this particularly implies that the ring of class functions of any finite symmetric group s n is generated by the irreducible characters corresponding to certain hook partitions of n as another consequence a second generating set of solomon s descent algebra and of the ring of class functions of s n is obtained related to the major index of permutations

a hyperbolicity preserving discontinuous stochastic galerkin scheme for uncertain hyperbolic systems of equations abstract intrusive uncertainty quantification methods such as stochastic galerkin are gaining popularity whereas the classical stochastic galerkin approach is not ensured to preserve hyperbolicity of the underlying hyperbolic system we apply a modification of this method that uses a slope limiter to retain admissible solutions of the system while providing high order approximations in the physical and stochastic space this is done using a spatial discontinuous galerkin scheme and a multi element stochastic galerkin ansatz in the random space we analyze the convergence of the resulting scheme and apply it to the compressible euler equations with various uncertain initial states in one and two spatial domains with up to three uncertainties the performance in multiple stochastic dimensions is compared to the non intrusive stochastic collocation method the numerical results underline the strength of our method especially if discontinuities are present in the uncertainty of the solution

the peak algebra of the symmetric group revisited abstract the linear span p n of the sums of all permutations in the symmetric group sn with a given set of peaks is a sub algebra of the symmetric group algebra due to nyman this peak algebra is a left ideal of the descent algebra d n and the direct sum p of all p n is a hopf sub algebra of the direct sum d of all d n dual to the stembridge algebra of peak functions in our self contained approach peak counterparts of several results on the descent algebra are established including a simple combinatorial characterization of the algebra p n an algebraic characterization of p n based on the action on the poincare birkhoff witt basis of the free associative algebra the display of peak variants of the classical lie idempotents an eulerian type sub algebra of p n a description of the jacobson radical of p n and its nil potency index of the principal indecomposable and irreducible p n modules and of the cartan matrix of p n furthermore it is shown that the primitive lie algebra of p is free and that p is its enveloping algebra

adaptive anisotropic spectral stochastic methods for uncertain scalar conservation laws this paper deals with the design of adaptive anisotropic discretization schemes for conservation laws with stochastic parameters a finite volume scheme is used for the deterministic discretization while a piecewise polynomial representation is used at the stochastic level the methodology is designed in the context of intrusive galerkin projection methods with a roe type solver the adaptation aims at selecting the stochastic resolution level based on the local smoothness of the solution in the stochastic domain in addition the stochastic features of the solution greatly vary in space and time so that the constructed stochastic approximation space depends on space and time the dynamically evolving stochastic discretization uses a tree structure representation that allows for the efficient implementation of the various operators needed to perform anisotropic multiresolution analysis efficiency of the overall adaptive scheme is assessed on a stochastic nonlinear conservation law with uncertain initial

uncertainty quantification of geochemical and mechanical compaction in layered sedimentary basins abstract in this work we propose an uncertainty quantification methodology for sedimentary basins evolution under mechanical and geochemical compaction processes which we model as a coupled time dependent non linear monodimensional depth only system of pdes with uncertain parameters while in previous works formaggia et al 2013 porta et al 2014 we assumed a simplified depositional history with only one material in this work we consider multi layered basins in which each layer is characterized by a different material and hence by different properties this setting requires several improvements with respect to our earlier works both concerning the deterministic solver and the stochastic discretization on the deterministic side we replace the previous fixed point iterative solver with a more efficient newton solver at each step of the time discretization on the stochastic side the multi layered structure gives rise to discontinuities in the dependence of the state variables on the uncertain parameters that need an appropriate treatment for surrogate modeling techniques such as sparse grids to be effective we propose an innovative methodology to this end which relies on a change of coordinate system to align the discontinuities of the target function within the random parameter space the reference coordinate system is built upon exploiting physical features of the problem at hand we employ the fact that the locations of the material interfaces display a smooth dependence on the random parameters and are therefore amenable to sparse grid polynomial approximations we showcase the capabilities of our numerical methodologies through two synthetic test cases in particular we show that our methodology reproduces with high accuracy multi modal probability density functions displayed by target state variables e g porosity

the dynamic analysis of a flat plate under a moving load by the finite element method summary the object of this paper is to analyse by the finite element method the dynamic responses of a flat plate subjected to various moving loads first of all the actual continuous flat plate was replaced by a discrete system composed of isoparametric rectangular plate elements and then the elementary and overall stiffness and mass matrices were determined and the natural frequencies and mode shapes of the flat plate were solved next the newmark direct integration method was used to find the dynamic responses of the flat plate the effects of eccentricity acceleration and initial velocity of the moving load and the effect of span length were the key points of study the dynamic behaviour of a multi span flat plate supported by the beam members of rigid plane frames and subjected to the action of a series of moving loads in identical or opposite directions were also investigated

multiresolution algorithms for the numerical solution of hyperbolic conservation laws given any scheme in conservation form and an appropriate uniform grid for the numerical solution of the initial value problem for one dimensional hyperbolic conservation laws we describe a multiresolution algorithm that approximates this numerical solution to a prescribed tolerance in an efficient manner to do so we consider the grid averages of the numerical solution for a hierarchy of nested diadic grids in which the given grid is the finest and introduce an equivalent multiresolution representation the multiresolution representation of the numerical solution consists of its grid averages for the coarsest grid and the set of errors in predicting the grid averages of each level of resolution in this hierarchy from those of the next coarser one once the numerical solution is resolved to our satisfaction in a certain locality of some grid then the prediction errors there are small for this particular grid and all finer ones this enables us to compress data by setting to zero small components of the representation which fall below a prescribed tolerance therefore instead of computing the time evolution of the numerical solution on the given grid we compute the time evolution of its compressed multiresolution representation algorithmically this amounts to computing the numerical fluxes of the given scheme at the points of the given grid by a hierarchical algorithm which starts with the computation of these numerical fluxes at the points of the coarsest grid and then proceeds through diadic refinements to the given grid at each step of refinement we add the values of the numerical flux at the center of the coarser cells the information in the multiresolution representation of the numerical solution is used to determine whether the solution is locally well resolved when this is the case we replace the costly exact value of the numerical flux with an accurate enough approximate value which is obtained by an inexpensive interpolation from the coarser grid the computational efficiency of this multiresolution algorithm is proportional to the rate of data compression for a prescribed level of tolerance that can be achieved for the numerical solution of the given scheme