spherical harmonics and integration in superspace the study of spherical harmonics in superspace introduced in de bie and sommen 2007 j phys a math theor 40 7193 is further elaborated a detailed description of spherical harmonics of degree k is given in terms of bosonic and fermionic pieces which also determines the irreducible pieces under the action of so m x sp 2n in the second part of the paper this decomposition is used to describe all possible integrations over the supersphere it is then shown that only one possibility yields the orthogonality of spherical harmonics of different degrees this is the so called pizzetti integral of which it was shown in de bie and sommen 2007 j phys a math theor 40 7193 that it leads to the berezin integral

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

uncertainty propagation using wiener haar expansions an uncertainty quantification scheme is constructed based on generalized polynomial chaos pc representations two such representations are considered based on the orthogonal projection of uncertain data and solution variables using either a haar or a legendre basis governing equations for the unknown coefficients in the resulting representations are derived using a galerkin procedure and then integrated in order to determine the behavior of the stochastic process the schemes are applied to a model problem involving a simplified dynamical system and to the classical problem of rayleigh benard instability for situations involving random parameters close to a critical point the computational implementations show that the wiener haar wha representation provides more robust predictions that those based on a wiener legendre wle decomposition however when the solution depends smoothly on the random data the wle scheme exhibits superior convergence suggestions regarding future extensions are finally drawn based on these experiences

minimal multi element stochastic collocation for uncertainty quantification of discontinuous functions we propose a multi element stochastic collocation method that can be applied in high dimensional parameter space for functions with discontinuities lying along manifolds of general geometries the key feature of the method is that the parameter space is decomposed into multiple elements defined by the discontinuities and thus only the minimal number of elements are utilized on each of the resulting elements the function is smooth and can be approximated using high order methods with fast convergence properties the decomposition strategy is in direct contrast to the traditional multi element approaches which define the sub domains by repeated splitting of the axes in the parameter space such methods are more prone to the curse of dimensionality because of the fast growth of the number of elements caused by the axis based splitting the present method is a two step approach firstly a discontinuity detector is used to partition parameter space into disjoint elements in each of which the function is smooth the detector uses an efficient combination of the high order polynomial annihilation technique along with adaptive sparse grids and this allows resolution of general discontinuities with a smaller number of points when the discontinuity manifold is low dimensional after partitioning an adaptive technique based on the least orthogonal interpolant is used to construct a generalized polynomial chaos surrogate on each element the adaptive technique reuses all information from the partitioning and is variance suppressing we present numerous numerical examples that illustrate the accuracy efficiency and generality of the method when compared against standard locally adaptive sparse grid methods the present method uses many fewer number of collocation samples and is more accurate

recent developments on the stability of systems with aperiodic sampling an overview this article presents basic concepts and recent research directions about the stability of sampled data systems with aperiodic sampling we focus mainly on the stability problem for systems with arbitrary time varying sampling intervals which has been addressed in several areas of research in control theory systems with aperiodic sampling can be seen as time delay systems hybrid systems input output interconnections discrete time systems with time varying parameters etc the goal of the article is to provide a structural overview of the progress made on the stability analysis problem without being exhaustive which would be neither possible nor useful we try to bring together results from diverse communities and present them in a unified manner for each of the existing approaches the basic concepts fundamental results converse stability theorems when available and relations with the other approaches are discussed in detail results concerning extensions of lyapunov and frequency domain methods for systems with aperiodic sampling are recalled as they allow to derive constructive stability conditions furthermore numerical criteria are presented while indicating the sources of conservatism the problems that remain open and the possible directions of improvement at last some emerging research directions such as the design of stabilizing sampling sequences are briefly discussed

fundamental solutions for the super laplace and dirac operators and all their natural powers the fundamental solutions of the super dirac and laplace operators and their natural powers are determined within the framework of clifford analysis

cauchy green type formulae in clifford analysis a cauchy integral formula is constructed for solutions to the polynomial dirac equation dk ekjbmdmn f 0 where each bm is a complex number d is the dirac operator in rn and f is defined on a domain in rn and takes values in a complex clifford algebra some basic properties for the solutions to this equation arising from the integral formula are described including an approximation theorem we also introduce a bergman kernel for square integrable solutions to d a f 0 over bounded domains with piecewise c or lipschitz boundary introduction it is well understood 1 3 4 that solutions to the dirac equation df 0 in rn can be described by a cauchy integral formula here d stands for the homogeneous dirac operator z7 ej 0 oxj and the ej s are the generators of a real clifford algebra an as d2 a the negative laplacian over rn it may easily be shown see 8 that green s formula for harmonic functions can be modified via clifford algebras to more closely resemble a cauchy integral formula in 12 x zhenyuan shows that solutions to the inhomogeneous dirac equation d a f 0 with a e c also possess a cauchy integral formula with a cauchy kernel e a x y a similar kemel for this equation is also produced for the case n 3 by gurlebeck and sprossig 4 chapter 4 and is used to describe boundary value problems for helmholtz equation ah 2h using the kernel ea produced in 12 these results automatically generalize to the cases n 3 clifford algebras have also been used by mitrea 6 to study boundary value problems and associated hp spaces for the helmholtz equation over nonsmooth domains in 11 sommen and x zhenyuan construct a cauchy kernel for d a as a power series amgm x where am e r and gm x is a fundamental solution to the operator dm in this paper we show that this method of construction generalizes to allow us to obtain cauchy kernels and cauchygreen type integral formulae for solutions to each polynomial equation dk ek4 bmdm h 0 with bm e c it automatically follows that many existing results in clifford analysis extend to the context described here for instance the approximation theorems described in 1 chapter 3 automatically received by the editors october 14 1993 and in revised form december 13 1993 originally communicated to the proceedings of the ams by palle e t jorgensen 1991 mathematics subject classification primary 30g35 1995 american mathematical society 0002 9947 95 1 00 25 per page

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

fast methods for determining the evolution of uncertain parameters in reaction diffusion equations abstract an important problem in mathematical modeling in science and engineering is the determination of the effects of uncertainty or variation in parameters and data on the output of a deterministic nonlinear operator for example such variations may describe the effect of experimental error in measured parameter values or may arise as part of a sensitivity analysis of the model the monte carlo method is a widely used tool for determining such effects it employs random sampling of the input space in order to produce a pointwise representation of the output it is a robust and easily implemented tool with relatively low dependence on the number of parameters unfortunately it generally requires sampling the operator very many times at a significant cost especially when the model is expensive to evaluate moreover standard analysis provides only asymptotic or distributional information about the error computed from a particular realization in this paper we present an alternative approach for ascertaining the effects of variations and uncertainty in parameters in a reaction diffusion equation on the solution the approach is based on techniques borrowed from a posteriori error analysis for finite element methods the generalized green s function solving the adjoint problem is used to efficiently compute the gradient of a quantity of interest with respect to parameters at sample points in the parameter space this derivative information is used in turn to produce an error estimate for the information computed from the sample points the error estimate provides the basis for both deterministic and probabilistic adaptive sampling algorithms the deterministic adaptive sampling method can be orders of magnitude faster than monte carlo sampling in case of a moderate number of parameters the gradient can also be used to compute useful information that cannot be obtained easily from a monte carlo sample for example the adaptive algorithm yields a natural dimensional reduction in the parameter space where applicable after describing the new methodology in detail we apply the new method to analyze the parameter sensitivity of a predator prey model with a holling ii functional response that has six parameters

the euler approximation in state constrained optimal control we analyze the euler approximation to a state constrained control problem we show that if the active constraints satisfy an independence condition and the lagrangian satisfies a coercivity condition then locally there exists a solution to the euler discretization and the error is bounded by a constant times the mesh size the proof couples recent stability results for state constrained control problems with results established here on discrete time regularity the analysis utilizes mappings of the discrete variables into continuous spaces where classical finite element estimates can be invoked

invariant integration on orthosymplectic and unitary supergroups the orthosymplectic supergroup osp m 2n and unitary supergroup u p q are studied following a new approach that starts from harish chandra pairs and links the sheaf theoretical supermanifold approach of berezin and others with the differential geometry approach of rogers and others the matrix elements of the fundamental representation of the lie supergroup are expressed in terms of functions on the product supermanifold with the underlying lie group and n the odd dimension of this product supermanifold is isomorphic to the supermanifold of this leads to a new expression for the standard generators of the corresponding lie superalgebra as invariant derivations on using these results a new and transparent formula for the invariant integrals on osp m 2n and u p q is obtained

pizzetti and cauchy formulae for higher dimensional surfaces a distributional approach abstract in this paper we study pizzetti type formulas for stiefel manifolds and cauchy type formulas for the tangential dirac operator from a distributional perspective first we illustrate a general distributional method for integration over manifolds in r m defined by means of k equations 1 x k x 0 next we apply this method to derive an alternative proof of the pizzetti formulae for the real stiefel manifolds so m so m k besides a distributional interpretation of invariant oriented integration is provided in particular we obtain a distributional cauchy theorem for the tangential dirac operator on an embedded m k dimensional smooth surface

finite elements for elliptic problems with stochastic coefficients abstract we describe a deterministic finite element fe solution algorithm for a stochastic elliptic boundary value problem sbvp whose coefficients are assumed to be random fields with finite second moments and known piecewise smooth two point spatial correlation function separation of random and deterministic variables parametrization of the uncertainty is achieved via a karhunen loeve kl expansion an o n log n algorithm for the computation of the kl eigenvalues is presented based on a kernel independent fast multipole method fmm truncation of the kl expansion gives an m 1 wiener polynomial chaos pc expansion of the stochastic coefficient and is shown to lead to a high dimensional deterministic boundary value problem dbvp analyticity of its solution in the stochastic variables with sharp bounds for the domain of analyticity are used to prescribe variable stochastic polynomial degree r r1 rm in an m r wiener pc expansion for the approximate solution pointwise error bounds for the fem approximations of kl eigenpairs the truncation of the kl expansion and the fe solution to the dbvp are given numerical examples show that m depends on the spatial correlation length of the random diffusion coefficient the variable polynomial degree r in pc stochastic galerkin fem allows to handle kl expansions with m up to 30 and r1 up to 10 in moderate time

galerkin methods for linear and nonlinear elliptic stochastic partial differential equations abstract stationary systems modelled by elliptic partial differential equations linear as well as nonlinear with stochastic coefficients random fields are considered the mathematical setting as a variational problem existence theorems and possible discretisations in particular with respect to the stochastic part are given and investigated with regard to stability different and increasingly sophisticated computational approaches involving both wiener s polynomial chaos as well as the karhunen loeve expansion are addressed in conjunction with stochastic galerkin procedures and stability within the galerkin framework is established new and effective algorithms to compute the mean and covariance of the solution are proposed the similarities and differences with better known monte carlo methods are exhibited as well as alternatives to integration in high dimensional spaces hints are given regarding the numerical implementation and parallelisation numerical examples serve as illustration

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

parametric models for samples of random functions a new class of parametric models referred to as sample parametric models is developed for random elements that match sample rather than the first two moments and or other global properties of these elements the models can be used to characterize e g material properties at small scale in which case their samples represent microstructures of material specimens selected at random from a population the samples of the proposed models are elements of finite dimensional vector spaces spanned by samples eigenfunctions of karhunen loeve kl representations or modes of singular value decompositions svds the implementation of sample parametric models requires knowledge of the probability laws of target random elements numerical examples including stochastic processes and random fields are used to demonstrate the construction of sample parametric models assess their accuracy and illustrate how these models can be used to solve efficiently stochastic equations

the harmonic transvector algebra in two vector variables the decomposition of polynomials of one vector variable into irreducible modules for the orthogonal group is a crucial result in harmonic analysis which makes use of the howe duality theorem and leads to the study of spherical harmonics the aim of the present paper is to describe a decomposition of polynomials in two vector variables and to obtain projection operators on each of the irreducible components to do so a particular transvector algebra will be used as a new dual partner for the orthogonal group leading to a generalisation of the classical howe duality the results are subsequently used to obtain explicit projection operators and formulas for integration of polynomials over the associated stiefel manifold

dirac operators schrodinger type operators in cn and huygens principle abstract a detailed study of the holomorphic continuations of primitives of iterates of the dirac operator d in c n is given this analysis is used to study holomorphic continuations of solutions to shrodinger type equations also analytic and continuous extensions of solutions to homogeneous dirac equations are studied and it is shown that for each cell of harmonicity considered here there exist solutions to each iterate of the dirac operator which may not be extended beyond any point of the closure of the cell

graded manifolds supermanifolds and infinite dimensional grassmann algebras a new approach to superdifferentiable functions of grassmann variables is developed which avoids ambiguities in odd derivatives this is used to give an improved definition of supermanifold over a finite dimensional grassmann algebra a natural embedding of super manifolds over grassmann algebras with increasing number l of generators is developed and thus a limit asl tends to infinity is possible a correspondence between graded manifolds and supermanifolds is constructed extending results of 5 and 8

discretization methods for optimal control problems with state constraints we consider an optimal control problem described by nonlinear ordinary differential equations with control and state constraints including pointwise state constraints since this problem may have no classical solutions it is also formulated in relaxed form the classical control problem is then discretized by using the implicit midpoint scheme while the controls are approximated by not necessarily continuous piecewise linear classical controls we first study the behavior in the limit of properties of discrete optimality and of discrete admissibility and extremality we then apply a penalized gradient projection method to each discrete classical problem and also a corresponding progressively refining combined discretization optimization method to the continuous classical problem thus reducing computing time and memory we prove that accumulation points of sequences generated by these methods are admissible and extremal in some sense for the corresponding discrete or continuous classical or relaxed problem for nonconvex problems whose solutions are nonclassical we show that we can apply the above methods to the problem formulated in gamkrelidze relaxed form finally numerical examples are given

q deformed harmonic and clifford analysis and the q hermite and laguerre polynomials we define a q deformation of the dirac operator inspired by the one dimensional q derivative this implies a q deformation of the partial derivatives by taking the square of this dirac operator we find a q deformation of the laplace operator this allows us to construct q deformed schrodinger equations in higher dimensions the equivalence of these schrodinger equations with those defined on q euclidean space in quantum variables is shown we also define the m dimensional q clifford hermite polynomials and show their connection with the q laguerre polynomials these polynomials are orthogonal with respect to an m dimensional q integration which is related to integration on q euclidean space the q laguerre polynomials are the eigenvectors of an suq 1 1 representation

on the time discretization of control systems this paper develops an approach for obtaining discrete approximations to nonlinear affine control systems that are of higher than first order of accuracy with respect to the discretization step h the approach consists of two parts first the set cal u of measurable admissible controls is replaced by an appropriate finite dimensional subset cal u n then the differential equations corresponding to controls from cal u n which are in a reasonable sense regular are discretized by single step discretization methods the main result estimates the accuracy in the first part measured in terms of a prescribed collection of performance indexes the result can be interpreted both in the context of approximation of optimal control problems and in the context of approximation of the reachable set in the first case accuracy o h2 is proven for appropriate runge kutta type discretization methods without explicitly or implicitly requiring any regularity of the optimal solutions in the case of a convex reachable set we obtain o h2 approximation with respect to the hausdorff distance and o h1 5 accuracy in the nonconvex case an application to the time aggregation of discrete time control systems is also presented

hermite and gegenbauer polynomials in superspace using clifford analysis the clifford hermite and the clifford gegenbauer polynomials of standard clifford analysis are generalized to the new framework of clifford analysis in superspace in a merely symbolic way this means that one does not a priori need an integration theory in superspace furthermore a lot of basic properties such as orthogonality relations differential equations and recursion formulae are proven finally an interesting physical application of the super clifford hermite polynomials is discussed thus giving an interpretation to the super dimension

realizing the berezin integral as a superspace contour integral integration on supermanifolds using contours and the covariant differential forms of kostant b kostant graded manifolds graded lie theory and prequantisation in lecture notes in mathematics vol 570 springer berlin 1977 is described the good properties that these integrals naturally acquire are considered it is then shown that the formal process of integration over even and odd variables introduced by berezin f a berezin the method of second quantization academic new york 1966 using partly covariant and partly contravariant forms can be regarded as a special case of these contour integrals over covariant forms

brief paper continuous time controller redesign for digital implementation a trajectory based approach given a continuous time nonlinear closed loop system we investigate sampled data feedback laws for which the trajectories of the sampled data closed loop system converge to the continuous time trajectories with a prescribed rate of convergence as the length of the sampling interval tends to zero we derive necessary and sufficient conditions for the existence of such sampled data feedback laws and in case of existence provide explicit redesign formulas and algorithms for these controllers

segal bargmann transforms associated with finite coxeter groups using a polarization of a suitable restriction map and heat kernel analysis we construct a generalized segal bargmann transform associated with every finite coxeter group g on n we find the integral representation of this transform and we prove its unitarity to define the segal bargmann transform we introduce a hilbert space open image in new window of holomorphic functions on open image in new window with reproducing kernel equal to the dunkl kernel the definition and properties of open image in new window extend naturally those of the well known classical fock space the generalized segal bargmann transform allows to exhibit some relationships between the dunkl theory in the schrodinger model and in the fock model further we prove a branching decomposition of open image in new window as a unitary open image in new window module and a general version of hecke s formula for the dunkl transform

orthosymplectically invariant functions in superspace the notion of spherically symmetric superfunctions as functions invariant under the orthosymplectic group is introduced this leads to dimensional reduction theorems for differentiation and integration in superspace these spherically symmetric functions can be used to solve orthosymplectically invariant schrodinger equations in superspace such as the an harmonic oscillator or the kepler problem finally the obtained machinery is used to prove the funk hecke theorem and bochner s relations in superspace

adaptive mesh strategies for the spectral element method abstract an adaptive spectral element method has been developed for the efficient solution of time dependent partial differential equations adaptive mesh strategies that include resolution refinement and coarsening by three different methods are illustrated on solutions to the one dimensional viscous burgers equation and the two dimensional navier stokes equations for driven flow in a cavity sharp gradients singularities and regions of poor resolution are resolved optimally as they develop in time using error estimators which indicate the choice of refinement to be used the adaptive formulation presents significant increases in efficiency flexibility and general capabilities for high order spectral methods

integration of grassmann variables over invariant functions on flat superspaces we study integration over functions on superspaces these functions are invariant under a transformation which maps the whole superspace onto the part of the superspace which only comprises purely commuting variables we get a compact expression for the differential operator with respect to the commuting variables which results from berezin integration over all grassmann variables also we derive cauchy like integral theorems for invariant functions on supervectors and symmetric supermatrices this extends theorems partly derived by other authors as a physical application we calculate the generating function of the one point correlation function in random matrix theory furthermore we give another derivation of supermatrix bessel functions for u k1 k2

ingredients for a general purpose stochastic finite elements implementation abstract the stochastic finite element method has seen a number of incarnations over the past few years ranging from perturbation based methods and neumann expansion based methods to methods for evaluating bounds on the response variability and more recent functional expansions combined with error minimization techniques of these only the last class of methods provides an explicit expression to the solution of the problem as opposed to evaluating a global norm of this solution and can thus be viewed as a rational extension of concepts developed for the deterministic finite element method the main obstacle in developing a substantial user community of various stochastic finite element methods has been the lack of a general purpose formalism for addressing issues of general engineering interest this paper presents the main ingredients for developing a general purpose version of the spectral stochastic finite element method

hilbert space for quantum mechanics on superspace in superspace a realization of sl2 is generated by the super laplace operator and the generalized norm squared in this paper an inner product on superspace for which this representation is skew symmetric is considered this inner product was already defined for spaces of weighted polynomials see k coulembier h de bie and f sommen orthogonality of hermite polynomials in superspace and mehler type formulae proc london math soc accepted arxiv 1002 1118 in this article it is proven that this inner product can be extended to the super schwartz space but not to the space of square integrable functions subsequently the correct hilbert space corresponding to this inner product is defined and studied a complete basis of eigenfunctions for general orthosymplectically invariant quantum problems is constructed for this hilbert space then the integrability of the sl2 representation is proven finally the heisenberg uncertainty principle for the super fourier transform is constructed

a robust bi orthogonal dynamically orthogonal method using the covariance pseudo inverse with application to stochastic flow problems abstract we develop a new robust methodology for the stochastic navier stokes equations based on the dynamically orthogonal do and bi orthogonal bo methods 1 2 3 both approaches are variants of a generalized karhunen loeve kl expansion in which both the stochastic coefficients and the spatial basis evolve according to system dynamics hence capturing the low dimensional structure of the solution the do and bo formulations are mathematically equivalent 3 but they exhibit computationally complimentary properties specifically the bo formulation may fail due to crossing of the eigenvalues of the covariance matrix while both bo and do become unstable when there is a high condition number of the covariance matrix or zero eigenvalues to this end we combine the two methods into a robust hybrid framework and in addition we employ a pseudo inverse technique to invert the covariance matrix the robustness of the proposed method stems from addressing the following issues in the do bo formulation i eigenvalue crossing we resolve the issue of eigenvalue crossing in the bo formulation by switching to the do near eigenvalue crossing using the equivalence theorem and switching back to bo when the distance between eigenvalues is larger than a threshold value ii ill conditioned covariance matrix we utilize a pseudo inverse strategy to invert the covariance matrix iii adaptivity we utilize an adaptive strategy to add remove modes to resolve the covariance matrix up to a threshold value in particular we introduce a soft threshold criterion to allow the system to adapt to the newly added removed mode and therefore avoid repetitive and unnecessary mode addition removal when the total variance approaches zero we show that the do bo formulation becomes equivalent to the evolution equation of the optimally time dependent modes 4 we demonstrate the capability of the proposed methodology with several numerical examples namely i stochastic burgers equation we analyze the performance of the method in the presence of eigenvalue crossing and zero eigenvalues ii stochastic kovasznay flow we examine the method in the presence of a singular covariance matrix and iii we examine the adaptivity of the method for an incompressible flow over a cylinder where for large stochastic forcing thirteen do bo modes are active

finite dimensional models for random functions abstract truncated karhunen loeve kl representations are used to construct finite dimensional fd models for non gaussian functions with finite variances the second moment specification of the random coefficients of these representations are enhanced to full probabilistic characterization by using translation polynomial chaos and translated polynomial chaos models referred to as t pc and pct models following theoretical considerations on kl representations and t pc and pct models three numerical examples are presented to illustrate the implementation and performance of these models the pct models inherit the desirable features of both t and pc models it approximates accurately all quantities of interest considered in these examples

an anisotropic sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes an anisotropic sparse grid stochastic collocation method for solving partial differential equations with random coefficients and forcing terms input data of the model the method consists of a galerkin approximation in the space variables and a collocation in probability space on sparse tensor product grids utilizing either clenshaw curtis or gaussian knots even in the presence of nonlinearities the collocation approach leads to the solution of uncoupled deterministic problems just as in the monte carlo method this work includes a priori and a posteriori procedures to adapt the anisotropy of the sparse grids to each given problem these procedures seem to be very effective for the problems under study the proposed method combines the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation the first approach is effective for problems depending on random variables which weigh approximately equally in the solution while the benefits of the latter approach become apparent when solving highly anisotropic problems depending on a relatively small number of random variables as in the case where input random variables are karhunen loeve truncations of smooth random fields this work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates sub exponential convergence in the asymptotic regime and algebraic convergence in the preasymptotic regime with respect to the total number of collocation points it also shows that the anisotropic approximation breaks the curse of dimensionality for a wide set of problems numerical examples illustrate the theoretical results and are used to compare this approach with several others including the standard monte carlo in particular for moderately large dimensional problems the sparse grid approach with a properly chosen anisotropy seems to be very efficient and superior to all examined methods

the berezin integral as a contour integral abstract in superspace formulations of supersymmetric field theories use is made of the berezin integral a linear map which assigns a real number to each polynomial function of the grasmann coordinates however this map has not been shown to be an integral in the mathematical sense of a limit of sums or an inverse to differentiation consequently its mathematical basis is unclear and its generalization to arbituary curved supermanifolds is problematic i will show that certain normalized contour integrals in the space of possess the properties of berezin integrals including translation invariance and correct transformation properties under changes of coordinates these contour integrals are mathematically well defined and may generalize to arbitrary supermanifolds alice rogers definition of super space is used

on the discretization of switched linear systems abstract a bilinear single control system which can be viewed as a control formulation of a linear switched system is considered the control is restricted to take values either i in 0 1 switched system or ii in 0 1 relaxed system in more practical considerations the control is often allowed to change only at the points of a given time net with a step length h the paper investigates what is the approximation error in terms of the reachable set in the two cases i and ii the error estimates that follow directly from known results are of order h and h respectively in the present paper estimations of order h and h 1 5 are proved in a constructive way the second one makes use of the effect of non accumulation of errors established earlier by the second author

distributions and integration in superspace distributions in superspace constitute a very useful tool for establishing an integration theory in particular distributions have been used to obtain a suitable extension of the cauchy formula to superspace and to define integration over the superball and the supersphere through the heaviside and dirac distributions respectively in this paper we extend the distributional approach to integration over more general domains and surfaces in superspace the notions of domain and surface in superspace are defined by smooth bosonic phase functions g this allows us to define domain integrals and oriented as well as non oriented surface integrals in terms of the heaviside and dirac distributions of the superfunction g it will be shown that the presented definition for the integrals does not depend on the choice of the phase function g defining the corresponding domain or surface in addition some examples of integration over a super paraboloid and a super hyperboloid will be presented finally a new distributional cauchy pompeiu formula will be obtained which generalizes and unifies the previously known approaches distributions in superspace constitute a very useful tool for establishing an integration theory in particular distributions have been used to obtain a suitable extension of the cauchy formula to superspace and to define integration over the superball and the supersphere through the heaviside and dirac distributions respectively in this paper we extend the distributional approach to integration over more general domains and surfaces in superspace the notions of domain and surface in superspace are defined by smooth bosonic phase functions g this allows us to define domain integrals and oriented as well as non oriented surface integrals in terms of the heaviside and dirac distributions of the superfunction g it will be shown that the presented definition for the integrals does not depend on the choice of the phase function g defining the corresponding domain or surface in addition some examples of integration over a super paraboloid and a super hyperboloid will be presented finally a new dist

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

fourier transform and related integral transforms in superspace abstract in this paper extensions of the classical fourier fractional fourier and radon transforms to superspace are studied previously a fourier transform in superspace was already studied but with a different kernel in this work the fermionic part of the fourier kernel has a natural symplectic structure derived using a clifford analysis approach several basic properties of these three transforms are studied using suitable generalizations of the hermite polynomials to superspace see h de bie f sommen hermite and gegenbauer polynomials in superspace using clifford analysis j phys a 40 2007 10441 10456 an eigenfunction basis for the fourier transform is constructed

higher order numerical schemes for affinely controlled nonlinear systems a systematic method for the derivation of high order schemes for affinely controlled nonlinear systems is developed using an adaptation of the stochastic taylor expansion for control systems we construct taylor schemes of arbitrary high order and indicate how derivative free runge kutta type schemes can be obtained furthermore an approximation technique for the multiple control integrals appearing in the schemes is proposed

long term behavior of polynomial chaos in stochastic flow simulations in this paper we focus on the long term behavior of generalized polynomial chaos gpc and multi element generalized polynomial chaos me gpc for partial differential equations with stochastic coefficients first we consider the one dimensional advection equation with a uniform random transport velocity and derive error estimates for gpc and me gpc discretizations subsequently we extend these results to other random distributions and high dimensional random inputs with numerical verification using the algebraic convergence rate of me gpc finally we apply our results to noisy flow past a stationary circular cylinder simulation results demonstrate that me gpc is effective in improving the accuracy of gpc for a long term integration whereas high order gpc cannot capture the correct asymptotic behavior

karhunen loeve approximation of random fields by generalized fast multipole methods kl approximation of a possibly instationary random field a x l2 dp l d subject to prescribed meanfield ea x a x dp and covariance va x x a x ea x a x ea x dp in a polyhedral domain d rd is analyzed we show how for stationary covariances va x x ga x x with ga z analytic outside of z 0 an m term approximate kl expansion am x of a x can be computed in log linear complexity the approach applies in arbitrary domains d and for nonseparable covariances ca it involves galerkin approximation of the kl eigenvalue problem by discontinuous finite elements of degree p 0 on a quasiuniform possibly unstructured mesh of width h in d plus a generalized fast multipole accelerated krylov eigensolver the approximate kl expansion am x of a x has accuracy o exp bm1 d if ga is analytic at z 0 and accuracy o m k d if ga is ck at zero it is obtained in o mn logn b operations where n o h d

an adaptive multi element generalized polynomial chaos method for stochastic differential equations we formulate a multi element generalized polynomial chaos me gpc method to deal with long term integration and discontinuities in stochastic differential equations we first present this method for legendre chaos corresponding to uniform random inputs and subsequently we generalize it to other random inputs the main idea of me gpc is to decompose the space of random inputs when the relative error in variance becomes greater than a threshold value in each subdomain or random element we then employ a generalized polynomial chaos expansion we develop a criterion to perform such a decomposition adaptively and demonstrate its effectiveness for odes including the kraichnan orszag three mode problem as well as advection diffusion problems the new method is similar to spectral element method for deterministic problems but with h p discretization of the random space

sparse tensor discretization of elliptic spdes we propose and analyze sparse deterministic stochastic tensor galerkin finite element methods sparse sgfems for the numerical solution of elliptic partial differential equations pdes with random coefficients in a physical domain d subset mathbb r d in tensor product sgfems the variational solution to the boundary value problem is approximated in tensor product finite element spaces v gamma otimes v d where v gamma and v d denote suitable finite dimensional subspaces of the stochastic and deterministic function spaces respectively these approaches lead to sgfem algorithms of complexity o n gamma n d where n gamma dim v gamma and n d dim v d in this work we use hierarchic sequences v gamma 1 subset v gamma 2 subset ldots and v d 1 subset v d 2 subset ldots of finite dimensional spaces to approximate the law of the random solution the hierarchies of approximation spaces allow us to define sparse tensor product spaces v gamma ell hat otimes v d ell ell 1 2 dots yielding algorithms of o n gamma log n d n d log n gamma work and memory we estimate the convergence rate of sgfem for algebraic decay of the input random field karhunen loeve coefficients we give an algorithm for an input adapted a priori selection of deterministic and stochastic discretization spaces the convergence rate in terms of the total number of degrees of freedom of the proposed method is superior to monte carlo approximations numerical examples illustrate the theoretical results and demonstrate superiority of the sparse tensor product discretization proposed here versus the full tensor product approach

convergence analysis for approximations of optimal control problems subject to higher index differential algebraic equations and mixed control state constraints this paper establishes a convergence result for implicit euler discretizations of optimal control problems with differential algebraic equations of higher index and mixed control state constraints

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

probabilistic collocation for period 1 limit cycle oscillations in this paper probabilistic collocation for limit cycle oscillations pclco is proposed probabilistic collocation pc is a non intrusive approach to compute the polynomial chaos description of uncertainty numerically polynomial chaos can require impractical high orders to approximate long term time integration problems due to the fast increase of required polynomial chaos order with time pclco is a pc formulation for modeling the long term stochastic behavior of dynamical systems exhibiting a periodic response i e a limit cycle oscillation lco in the pc method deterministic time series are computed at collocation points in probability space in pclco pc is applied to a time independent parametrization of the periodic response of the deterministic solves instead of to the time dependent functions themselves due to the time independent parametrization the accuracy of pclco is independent of time the approach is applied to period 1 oscillations with one main frequency subject to a random parameter numerical results are presented for the harmonic oscillator a two dof airfoil flutter model and the fluid structure interaction of an elastically mounted cylinder

generalized spectral decomposition for stochastic nonlinear problems we present an extension of the generalized spectral decomposition method for the resolution of nonlinear stochastic problems the method consists in the construction of a reduced basis approximation of the galerkin solution and is independent of the stochastic discretization selected polynomial chaos stochastic multi element or multi wavelets two algorithms are proposed for the sequential construction of the successive generalized spectral modes they involve decoupled resolutions of a series of deterministic and low dimensional stochastic problems compared to the classical galerkin method the algorithms allow for significant computational savings and require minor adaptations of the deterministic codes the methodology is detailed and tested on two model problems the one dimensional steady viscous burgers equation and a two dimensional nonlinear diffusion problem these examples demonstrate the effectiveness of the proposed algorithms which exhibit convergence rates with the number of modes essentially dependent on the spectrum of the stochastic solution but independent of the dimension of the stochastic approximation space

generalized green s functions and the effective domain of influence one well known approach to a posteriori analysis of finite element solutions of elliptic problems estimates the error in a quantity of interest in terms of residuals and a generalized green s function the generalized green s function solves the adjoint problem with data related to a quantity of interest and measures the effects of stability including any decay of influence characteristic of elliptic problems we show that consideration of the generalized green s function can be used to improve the efficiency of the solution process when the goal is to compute multiple quantities of interest and or to compute quantities of interest that involve globally supported information such as average values and norms in the latter case we introduce a solution decomposition in which we solve a set of problems involving localized information and then recover the desired information by combining the local solutions by treating each computation of a quantity of interest independently the maximum number of elements required to achieve the desired accuracy can be decreased significantly

the multi element probabilistic collocation method me pcm error analysis and applications stochastic spectral methods are numerical techniques for approximating solutions to partial differential equations with random parameters in this work we present and examine the multi element probabilistic collocation method me pcm which is a generalized form of the probabilistic collocation method in the me pcm the parametric space is discretized and a collocation cubature grid is prescribed on each element both full and sparse tensor product grids based on gauss and clenshaw curtis quadrature rules are considered we prove analytically and observe in numerical tests that as the parameter space mesh is refined the convergence rate of the solution depends on the quadrature rule of each element only through its degree of exactness in addition the l 2 error of the tensor product interpolant is examined and an adaptivity algorithm is provided numerical examples demonstrating adaptive me pcm are shown including low regularity problems and long time integration we test the me pcm on two dimensional navier stokes examples and a stochastic diffusion problem with various random input distributions and up to 50 dimensions while the convergence rate of me pcm deteriorates in 50 dimensions the error in the mean and variance is two orders of magnitude lower than the error obtained with the monte carlo method using only a small number of samples e g 100 the computational cost of me pcm is found to be favorable when compared to the cost of other methods including stochastic galerkin monte carlo and quasi random sequence methods

solving optimization constrained differential equations with discontinuity points with application to atmospheric chemistry ordinary differential equations are coupled with mixed constrained optimization problems when modeling the thermodynamic equilibrium of a system evolving with time a particular application arises in the modeling of atmospheric particles discontinuity points are created by the activation deactivation of inequality constraints a numerical method for the solution of optimization constrained differential equations is proposed by coupling an implicit runge kutta method radau5 with numerical techniques for the detection of the events activation and deactivation of constraints the computation of the events is based on dense output formulas continuation techniques and geometric arguments numerical results are presented for the simulation of the time dependent equilibrium of organic atmospheric aerosol particles and show the efficiency and accuracy of the approach

a method for solving stochastic equations by reduced order models and local approximations a method is proposed for solving equations with random entries referred to as stochastic equations ses the method is based on two recent developments the first approximates the response surface giving the solution of a stochastic equation as a function of its random parameters by a finite set of hyperplanes tangent to it at expansion points selected by geometrical arguments the second approximates the vector of random parameters in the definition of a stochastic equation by a simple random vector referred to as stochastic reduced order model srom and uses it to construct a srom for the solution of this equation the proposed method is a direct extension of these two methods it uses sroms to select expansion points rather than selecting these points by geometrical considerations and represents the solution by linear and or higher order local approximations the implementation and the performance of the method are illustrated by numerical examples involving random eigenvalue problems and stochastic algebraic differential equations the method is conceptually simple non intrusive efficient relative to classical monte carlo simulation accurate and guaranteed to converge to the exact solution

finite elements for stochastic media problems we consider discretization methods for stochastic partial differential equations which are used to model media with stochastic properties the wick product formulation of the stochastic pde is used and it is shown how it may be discretized both spatially and in the stochastic part a stochastic variational principle serves as the guideline for the discretization procedure it is shown that both the projection onto the wiener chaos and the hermite transform lead to the same fully discrete equations the application examples show that there are other advantageous bases besides the karhunen loeve expansion and that the methods may be used in a variety of settings

linear random vibration by stochastic reduced order models a practical method is developed for calculating statistics of the states of linear dynamic systems with deterministic properties subjected to non gaussian noise and systems with uncertain properties subjected to gaussian and non gaussian noise these classes of problems are relevant as most systems have uncertain properties physical noise is rarely gaussian and the classical theory of linear random vibration applies to deterministic systems and can only deliver the first two moments of a system state if the noise is non gaussian the method 1 is based on approximate representations of all or some of the random elements in the definition of linear random vibration problems by stochastic reduced order models sroms that is simple random elements having a finite number of outcomes of unequal probabilities 2 can be used to calculate statistics of a system state beyond its first two moments and 3 establishes bounds on the discrepancy between exact and srom based solutions of linear random vibration problems the implementation of the method has required to integrate existing and new numerical algorithms examples are presented to illustrate the application of the proposed method and assess its accuracy copyright 2009 john wiley sons ltd

efficient iterative algorithms for the stochastic finite element method with application to acoustic scattering in this study we describe the algebraic computations required to implement the stochastic finite element method for solving problems in which uncertainty is restricted to right hand side data coming from forcing functions or boundary conditions we show that the solution can be represented in a compact outer product form which leads to efficiencies in both work and storage and we demonstrate that block iterative methods for algebraic systems with multiple right hand sides can be used to advantage to compute this solution we also show how to generate a variety of statistical quantities from the computed solution finally we examine the behavior of these statistical quantities in one setting derived from a model of acoustic scattering

dynamic refinement algorithms for spectral element methods abstract a numerical method is presented that combines high order domain decomposition and adaptive mesh refinement the spatial discretization is based on a nonconforming spectral element method meshes are generated by adaptive refinement of a quadtree using either local solution gradients or the local polynomial spectrum as a refinement criteria the method is applied to solve scalar test problems and to simulate the flow in a lid driven cavity at moderate reynolds numbers

inexact restoration for runge kutta discretization of optimal control problems a numerical method is presented for runge kutta discretization of unconstrained optimal control problems first general runge kutta discretization is carried out to obtain a finite dimensional approximation of the continous time optimal control problem then a recent optimization technique the inexact restoration ir method due to martinez and coworkers e g birgin and j m martinez j optim theory appl 127 2005 pp 229 247 j m martinez and e a pilotta j optim theory appl 104 2000 pp 135 163 j m martinez j optim theory appl 111 2001 pp 39 58 is applied to the discretized problem to find an approximate solution it is proved that for optimal control problems a key sufficiency condition for convergence of the ir method is readily satisfied under reasonable assumptions the ir method for optimal control problems is shown to converge to a solution of the discretized problem convergence of a solution of the discretized problem to a solution of the continuous time problem is also shown it turns out that optimality phase equations of the ir method emanate from an associated hamiltonian system and so general runge kutta discretization induces a symplectic partitioned runge kutta scheme a computational algorithm is described and numerical experiments are made to demonstrate the working of the method for optimal control of the van der pol system employing the three stage order 6 gauss legendre discretization

roe solver with entropy corrector for uncertain hyperbolic systems this paper deals with intrusive galerkin projection methods with a roe type solver for treating uncertain hyperbolic systems using a finite volume discretization in physical space and a piecewise continuous representation at the stochastic level the aim of this paper is to design a cost effective adaptation of the deterministic dubois and mehlman corrector to avoid entropy violating shocks in the presence of sonic points the adaptation relies on an estimate of the eigenvalues and eigenvectors of the galerkin jacobian matrix of the deterministic system of the stochastic modes of the solution and on a correspondence between these approximate eigenvalues and eigenvectors for the intermediate states considered at the interface we derive some indicators that can be used to decide where a correction is needed thereby reducing the computational costs considerably the effectiveness of the proposed corrector is assessed on the burgers and euler equations including sonic points

on the efetov wegner terms by diagonalizing a hermitian supermatrix the diagonalization of hermitian supermatrices is studied such a change of coordinates is inevitable if one wishes to find certain structures in random matrix theory however it still poses serious problems since until now the calculation of all rothstein contributions known as efetov wegner terms in physics was quite cumbersome we derive the supermatrix bessel function with all efetov wegner terms for an arbitrary rotation invariant probability density function as applications we consider representations of generating functions for hermitian random matrices with and without an external field as integrals over eigenvalues of hermitian supermatrices all results are obtained with all efetov wegner terms which were previously unknown in such an explicit and compact representation

long duration response evaluation of linear structural system with random system properties using time dependent polynomial chaos abstract polynomial chaos pc is one of the popular methods for approximate evaluation of responses of stochastic structural mechanics problems in the case of a structural system under dynamic loading the pc method is combined with a time integration scheme the pc method with galerkin projection converts the stochastic pde to a set of simultaneous deterministic pde which can be solved using any standard time integration scheme however the convergence of responses is known to fail for long duration time integration problem due to the development of unacceptable error in calculated response the probability density functions of the responses are not of constant type as the time progresses however in the case of generalized pc gpc these are approximated using initially considered gpc thus loses its efficiency time dependent general polynomial chaos tdgpc can be considered to overcome these difficulties where polynomials are generated on the fly to match with the pdf of the responses the present study considered an investigation of structural mechanics responses with both gaussian and non gaussian random system properties under dynamic loading using tdgpc an adaptive updation scheme based on the rms value of the coefficients of pc expansion is investigated for a proper updation of pc expansion the curse of dimensionality of pc expansion is addressed by considering only the dominant components of the responses evaluated using kl expansion further starting with lower order pc the order of pc expansion is increased only after the first updation thus reducing computational complexities

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

large sample properties of simulations using latin hypercube sampling latin hypercube sampling mckay conover and beckman 1979 is a method of sampling that can be used to produce input values for estimation of expectations of functions of output variables the asymptotic variance of such an estimate is obtained the estimate is also shown to be asymptotically normal asymptotically the variance is less than that obtained using simple random sampling with the degree of variance reduction depending on the degree of additivity in the function being integrated a method for producing latin hypercube samples when the components of the input variables are statistically dependent is also described these techniques are applied to a simulation of the performance of a printer actuator

solving elliptic problems with non gaussian spatially dependent random coefficients we propose a simple and effective numerical procedure for solving elliptic problems with non gaussian random coefficients assuming that samples of the non gaussian random inputs are available from a statistical model given a correlation function the karhunen loeve k l expansion is employed to reduce the dimensionality of random inputs using the kernel density estimation technique we obtain the marginal probability density functions pdfs of the random variables in the k l expansion based on which we define an auxiliary joint pdf we then implement the generalized polynomial chaos gpc method via a collocation projection according to the auxiliary joint pdf based on the observation that the solution has an analytic extension in the parametric space we ensure that the polynomial interpolation achieves point wise convergence in the parametric space regardless of the pdf where the energy norm is employed in the physical space hence we can sample the gpc solution using the joint pdf instead of the auxiliary one to obtain the correct statistics we also implement monte carlo methods to further refine the statistics using the gpc solution for variance reduction numerical results are presented to demonstrate the efficiency of the proposed approach

induced measures in the space of mixed quantum states we analyse several product measures in the space of mixed quantum states in particular we study measures induced by the operation of partial tracing the natural rotationally invariant measure on the set of all pure states of a n k composite system induces a unique measure in the space of n n mixed states or in the space of k k mixed states if the reduction takes place with respect to the first subsystem for k n the induced measure is equal to the hilbert schmidt measure which is shown to coincide with the measure induced by singular values of non hermitian random gaussian matrices pertaining to the ginibre ensemble we compute several averages with respect to this measure and show that the mean entanglement of n n pure states behaves as lnn 1 2

monogenic differential calculus in this paper we study differential forms satisfying a dirac type equation and taking values in a clifford algebra for them we establish a cauchy representation formula and we compute winding numbers for pairs of nonintersecting cycles in r m as residues of special differential forms next we prove that the cohomology spaces for the complex of monogenic differential forms split as direct sums of de rham cohomology spaces we also study duals of spaces of monogenic differential forms leading to a general residue theory in euclidean space our theory includes the one established in our paper 11 and is strongly related to certain differential forms introduced by habetha in 4

probabilistic models for stochastic elliptic partial differential equations mathematical requirements that the random coefficients of stochastic elliptical partial differential equations must satisfy such that they have unique solutions have been studied extensively yet additional constraints that these coefficients must satisfy to provide realistic representations for physical quantities referred to as physical requirements have not been examined systematically it is shown that current models for random coefficients constructed solely by mathematical considerations can violate physical constraints and consequently be of limited practical use we develop alternative models for the random coefficients of stochastic differential equations that satisfy both mathematical and physical constraints theoretical arguments are presented to show potential limitations of current models and establish properties of the models developed in this study numerical examples are used to illustrate the construction of the proposed models assess the performance of these models and demonstrate the sensitivity of the solutions of stochastic differential equations to probabilistic characteristics of their random coefficients

fast and reliable methods for determining the evolution of uncertain parameters in differential equations a very common problem in science and engineering is the determination of the effects of uncertainty or variation in parameters and data on the output of a deterministic nonlinear operator for example such variations may describe the effect of experimental error or may arise as part of a sensitivity analysis of the model the monte carlo method is a justifiably popular tool for analyzing such effects it does however require many evaluations of the operator and it is difficult to extract precise information about the accuracy of any particular result in this paper we borrow techniques from a posteriori error analysis for finite element methods to compute information about the derivative of an operator with respect to its parameters these techniques employ the generalized green s function to describe how variation propagates into the solution around localized points in the parameter space we show how this derivative information can be used either to create a higher order method or produce an error estimate for information computed from a given representation in the latter case this provides the basis for adaptive sampling according to the variation in the output values both the higher order method and the adaptive sampling method are generally orders of magnitude faster than monte carlo methods in the case that the parameter space is not too high dimensional

runge kutta methods in optimal control and the transformed adjoint system the convergence rate is determined for runge kutta discretizations of nonlinear control problems the analysis utilizes a connection between the kuhn tucker multipliers for the discrete problem and the adjoint variables associated with the continuous minimum principle this connection can also be exploited in numerical solution techniques that require the gradient of the discrete cost function

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

berezin integral as a limit of riemann sum berezin integration of functions of anticommuting grassmann variables is usually seen as a formal operation sometimes even defined via differentiation using the formalism of geometric algebra and geometric calculus in which the grassmann numbers are endowed with a second associative product coming from a clifford algebra structure we show how berezin integrals can be realized in the high dimensional limit as integrals in the sense of geometric calculus we then show how the concepts of spinors and superspace transform into this framework

modeling uncertainty in flow simulations via generalized polynomial chaos we present a new algorithm to model the input uncertainty and its propagation in incompressible flow simulations the stochastic input is represented spectrally by employing orthogonal polynomial functionals from the askey scheme as trial basis to represent the random space a standard galerkin projection is applied in the random dimension to obtain the equations in the weak form the resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode this approach can be considered as a generalization of the original polynomial chaos expansion first introduced by wiener am j math 60 1938 897 the original method employs the hermite polynomials one of the 13 members of the askey scheme as the basis in random space the algorithm is applied to micro channel flows with random wall boundary conditions and to external flows with random freestream efficiency and convergence are studied by comparing with exact solutions as well as numerical solutions obtained by monte carlo simulations it is shown that the generalized polynomial chaos method promises a substantial speed up compared with the monte carlo method the utilization of different type orthogonal polynomials from the askey scheme also provides a more efficient way to represent general non gaussian processes compared with the original wiener hermite expansions

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

lie symmetries of a generalized kuznetsov zabolotskaya khokhlov equation abstract we consider a class of generalized kuznetsov zabolotskaya khokhlov gkzk equations and determine its equivalence group which is then used to give a complete symmetry classification of this class the infinite dimensional symmetry is used to reduce such equations to 1 1 dimensional pdes special attention is paid to group theoretical properties of a class of generalized dispersionless kp gdkp or zabolotskaya khokhlov equations as a subclass of gkzk equations the conditions are determined under which a gdkp equation is invariant under a lie algebra containing the virasoro algebra as a subalgebra this occurs if and only if this equation is completely integrable a similar connection is shown to hold for generalized kp equations

sparse grid collocation schemes for stochastic natural convection problems in recent years there has been an interest in analyzing and quantifying the effects of random inputs in the solution of partial differential equations that describe thermal and fluid flow problems spectral stochastic methods and monte carlo based sampling methods are two approaches that have been used to analyze these problems as the complexity of the problem or the number of random variables involved in describing the input uncertainties increases these approaches become highly impractical from implementation and convergence points of view this is especially true in the context of realistic thermal flow problems where uncertainties in the topology of the boundary domain boundary flux conditions and heterogeneous physical properties usually require high dimensional random descriptors the sparse grid collocation method based on the smolyak algorithm offers a viable alternate method for solving high dimensional stochastic partial differential equations an extension of the collocation approach to include adaptive refinement in important stochastic dimensions is utilized to further reduce the numerical effort necessary for simulation we show case the collocation based approach to efficiently solve natural convection problems involving large stochastic dimensions equilibrium jumps occurring due to surface roughness and heterogeneous porosity are captured comparison of the present method with the generalized polynomial chaos expansion and monte carlo methods are made

second order runge kutta approximations in control constrained optimal control in this paper we analyze second order runge kutta approximations to a nonlinear optimal control problem with control constraints if the optimal control has a derivative of bounded variation and a coercivity condition holds we show that for a special class of runge kutta schemes the error in the discrete approximating control is o h2 where h is the mesh spacing

fischer decomposition for polynomials on superspace recently the fischer decomposition for polynomials on superspace m 2n that is polynomials in m commuting and 2n anti commuting variables has been obtained unless the superdimension m m 2n is even and non positive in this case it turns out that the fischer decomposition of polynomials into spherical harmonics is quite analogous as in m and it is an irreducible decomposition under the natural action of lie superalgebra m 2n in this paper we describe explicitly the fischer decomposition in the exceptional case when m 2 0 in particular we show that under the action of m 2n the fischer decomposition is not in general a decomposition into irreducible but just indecomposable pieces

new interpolatory quadrature formulae with chebyshev abscissae abstract we study interpolatory quadrature formulae relative to the legendre weight function on 1 1 having as nodes the zeros of any one of the four chebyshev polynomials of degree n plus one of the points 1 or 1 in particular we derive explicit formulae for the weights and examine their positivity we determine the precise degree of exactness we obtain asymptotically optimal error bounds and we examine the definiteness f these quadrature formulae in addition we establish their convergence for riemann integrable functions on 1 1 as well as for functions having a monotonic singularity at 1 or 1

the integer approximation error in mixed integer optimal control we extend recent work on nonlinear optimal control problems with integer restrictions on some of the control functions mixed integer optimal control problems miocp we improve a theorem sager et al in math program 118 1 109 149 2009 that states that the solution of a relaxed and convexified problem can be approximated with arbitrary precision by a solution fulfilling the integer requirements unlike in previous publications the new proof avoids the usage of the krein milman theorem which is undesirable as it only states the existence of a solution that may switch infinitely often we present a constructive way to obtain an integer solution with a guaranteed bound on the performance loss in polynomial time we prove that this bound depends linearly on the control discretization grid a numerical benchmark example illustrates the procedure as a byproduct we obtain an estimate of the hausdorff distance between reachable sets we improve the approximation order to linear grid size h instead of the previously known result with order sqrt h hackl in reachable sets control sets and their computation augsburger mathematisch naturwissenschaftliche schriften dr bernd wisner augsburg 1996 we are able to include a special ordered set condition which will allow for a transfer of the results to a more general multi dimensional and nonlinear case compared to the theorems in pietrus and veliov in syst control lett 58 395 399 2009

integration in superspace using distribution theory in this paper a new class of cauchy integral formulae in superspace is obtained using formal expansions of distributions this allows us to solve five open problems in the study of harmonic and clifford analysis in superspace