a family of lower and higher order transversal linearization techniques in non linear stochastic engineering dynamics sample pathwise numerical integration of noise driven engineering dynamical systems cannot generally be performed beyond a limited level of accuracy especially when the noise processes are modelled using filtered white noises recently a locally transversal linearization ltl strategy has been proposed by the author proc roy soc london a 2001 457 539 566 for direct integration of deterministic and stochastic non linear dynamical systems the present effort is focussed on a host of extensions along with detailed theoretical error analyses of the linearization approach especially as applied to problems in non linear stochastic engineering dynamics thus to begin with estimates of local and global error orders in the basic ltl scheme are obtained separately for the displacement and velocity vectors when the system is driven either by a set of additive noises or by an arbitrary combination of independently evolving additive and multiplicative noises following this a new family of higher order ltl schemes is proposed in order to improve upon the basic ltl method and the associated error orders are established a stepwise implementation of the lower and higher order versions of the ltl method along with certain computational aspects is also outlined the proposed schemes are numerically illustrated to a limited extent for a single degree of freedom sdof and a two degree of freedom tdof non linear engineering systems under additive and or multiplicative white noise excitations

spectrum of ornstein uhlenbeck operators in lp spaces with respect to invariant measures abstract let a i j 1 n q ij d ij i j 1 n b ij x j d i be a possibly degenerate ornstein uhlenbeck operator in r n and assume that the associated markov semigroup has an invariant measure we compute the spectrum of a in l p for 1 p

on level set regularization for highly ill posed distributed parameter estimation problems the recovery of a distributed parameter function with discontinuities from inverse problems with elliptic forward pdes is fraught with theoretical and practical difficulties better results are obtained for problems where the solution may take on at each point only one of two values thus yielding a shape recovery problem this article considers level set regularization for such problems however rather than explicitly integrating a time embedded pde to steady state which typically requires thousands of iterations methods based on gauss newton are applied more directly one of these can be viewed as damped gauss newton utilized to approximate the steady state equations which in turn are viewed as the necessary conditions of a tikhonov type regularization with a sharpening sub step at each iteration in practice this method is eclipsed however by a special finite time or levenberg marquardt type method which we call dynamic regularization applied to the output least squares formulation our stopping criterion for the iteration does not involve knowledge of the true solution the regularization functional is applied to the smooth level set function rather than to the discontinuous function to be recovered and the second focus of this article is on selecting this functional typical choices may lead to flat level sets that in turn cause ill conditioning we propose a new quartic non local regularization term that penalizes flatness and produces a smooth level set evolution and compare its performance to more usual choices two numerical test cases are considered a potential problem and the classical eit dc resistivity problem

ray holder continuity for fractional sobolev spaces in infinite dimensions and applications we prove holder continuity on rays in the direction of vectors in the generalized cameron martin space for functions in sobolev spaces in lp of fractional order 1 over infinite dimensional linear spaces the underlying measures are required to satisfy some easy standard structural assumptions only apart from wiener measure they include gibbs measures on a lattice and euclidean interacting quantum fields in infinite volume a number of applications e g to the two dimensional polymer measure are presented in particular irreducibility of the dirichlet form associated with the latter measure is proved without restrictions on the coupling constant

liouville theorems for non local operators abstract the paper characterizes some classes of pseudo differential operators for which there are or there are not non constant bounded harmonic functions non local perturbations of ornstein uhlenbeck operators and operators with dissipative coefficients are considered the methods used are probabilistic and based on the concept of absorption function and on a new extension of the bismut elworthy li formula the probabilistic interpretation of the liouville theorem by means of absorption functions for general markov processes is given as well

on effective methods for implicit piecewise smooth surface recovery this paper considers the problem of reconstructing a piecewise smooth model function from given measured data the data are compared to a field which is given as a possibly nonlinear function of the model a regularization functional is added which incorporates the a priori knowledge that the model function is piecewise smooth and may contain jump discontinuities regularization operators related to total variation tv are therefore employed two popular methods are modified tv and huber s function both contain a parameter which must be selected the huber variant provides a more natural approach for selecting its parameter and we use this to propose a scheme for both methods our selected parameter depends both on the resolution and on the model average roughness thus it is determined adaptively its variation from one iteration to the next yields additional information about the progress of the regularization process the modified tv operator has a smoother generating function nonetheless we obtain a huber variant with comparable and occasionally better performance for large problems e g high resolution the resulting reconstruction algorithms can be tediously slow we propose two mechanisms to improve efficiency the first is a multilevel continuation approach aimed mainly at obtaining a cheap yet good estimate for the regularization parameter and the solution the second is a special multigrid preconditioner for the conjugate gradient algorithm used to solve the linearized systems encountered in the procedures for recovering the model function

a new numeric analytical principle for nonlinear deterministic and stochastic dynamical systems a new implicit local linearization method termed as the locally transversal linearization ltl is developed in the present paper for a numeric analytical integration of nonlinear dynamical systems under deterministic and or stochastic excitations the ltl procedure is developed in such a way that at a chosen time instant the ltl solution manifold transversally intersects the nonlinear solution manifold at that particular point or cross section in the state space where the solution vector needs to be obtained in general construction of the ltl equation is non unique and thus it is generally needed to substitute the conditionally linear ltl based solution into the given nonlinear ordinary differential equation ode at a specified time instant to ensure a transversal intersection in the present study however it is demonstrated that a transversal intersection at a given time instant may as well be ensured by just choosing a few specific forms of the ltl equation thereby eliminating the need for substitution this new variant of ltl is particularly suitable for discretization of stochastic differential equations sdes in that one simultaneously avoids dealing with formal derivatives of wiener processes while satisfying the nonlinear sde at discrete time points the stochastic ltl procedure is adaptable for obtaining strong pathwise solutions of a general nonlinear dynamical system with continuous but not necessarily differentiable vector fields under additive and or multiplicative excitations a specific advantage of the ltl method is that for a class of response regime herein referred to as the phase independent regime it can obtain an accurate description of response irrespective of the choice of the time step and without a propagation of local errors presently the method has been numerically illustrated for strong solutions of the hardening duffing oscillator under deterministic sinusoidal or additive stochastic white noise excitations or their combination limited numerical results on periodic solutions chaos with or without noise and stochastic resonance are included in this study the results presented are indicative of the sample path accuracy consistency stochastic numerical stability and a partial insensitivity to time step sizes of the proposed method

skew convolution semigroups and affine markov processes a general affine markov semigroup is formulated as the convolution of a homogeneous one with a skew convolution semigroup we provide some sufficient conditions for the regularities of the homogeneous affine semigroup and the skew convolution semigroup the corresponding affine markov process is constructed as the strong solution of a system of stochastic equations with non lipschitz coefficients and poisson type integrals over some random sets based on this characterization it is proved that the affine process arises naturally in a limit theorem for the difference of a pair of reactant processes in a catalytic branching system with immigration

absolutely continuous flows generated by sobolev class vector fields in finite and infinite dimensions abstract we prove the existence of the global flow ut generated by a vector field a from a sobolev class w1 1 on a finite or infinite dimensional space x with a measure provided is sufficiently smooth and that a a and a where a is the divergence with respect to are exponentially integrable in addition the measure is shown to be quasi invariant under ut in the case x r n and p dx where p w1 1loc r n is a locally uniformly positive probability density a sufficient condition is exp c a exp c a p p l1 for all c in the infinite dimensional case we get analogous results for measures differentiable along sufficiently many directions examples of measures which fit our framework important for applications are symmetric invariant measures of infinite dimensional diffusions and gibbs measures typically in both cases such measures are essentially non gaussian our result in infinite dimensions significantly extends previously studied cases where was a gaussian measure finally we study flows generated by vector fields whose values are not necessarily in the cameron martin space

a parametrix for the neumann problem on pseudoconvex domains of finite type abstract we construct a parametrix for the neumann problem on any pseudoconvex domain of finite type which reduces completely the study of isotropic lp sobolev and holder estimates to those for b and b we also establish the maximal nonisotropic estimates in the case of d q domains and provide a precise comparison of the bergman and szego projections and of the canonical solutions to with respect to these two operators the argument is substantially simpler than the known ones for the cases of strongly pseudoconvex domains and pseudoconvex domains of finite type in c 2

higher order weak linearizations of stochastically driven nonlinear oscillators we present derivative free weak and strong solutions of stochastically driven nonlinear oscillators of engineering interest using higher order forms of the locally transversal linearization ltl method unlike strong solutions weak stochastic solutions attempt to predict only mathematical expectations of functions of the true solution and are obtainable with much less computational effort the linearized equations corresponding to the higher order implicit ltl schemes are arrived at using backward euler maruyama and newmark expansions in order to conditionally replace nonlinear drift and multiplicative diffusion vector fields we also briefly describe alterations through which explicit forms of such higher order linearizations are obtained in the weak forms the gaussian stochastic integrals appearing in the linearized solutions are replaced by random variables with simpler and discrete probability distributions the resulting local approximations to the true solution are of significantly higher formal order of accuracy as reflected through local error estimates numerical illustrations are presently provided for the duffing and van der pol oscillators driven by additive and multiplicative noises which are indicative of the numerical accuracy computational speed and algorithmic simplicity

a pseudo dynamical systems approach to a class of inverse problems in engineering a pseudo dynamical approach for a class of inverse problems involving static measurements is proposed and explored following linearization of the minimizing functional associated with the underlying optimization problem the new strategy results in a system of linearized ordinary differential equations odes whose steady state solutions yield the desired reconstruction we consider some explicit and implicit schemes for integrating the odes and thus establish a deterministic reconstruction strategy without an explicit use of regularization a stochastic reconstruction strategy is then developed making use of an ensemble kalman filter wherein these odes serve as the measurement model finally we assess the numerical efficacy of the developed tools against a few linear and nonlinear inverse problems of engineering interest

a sequential importance sampling filter with a new proposal distribution for state and parameter estimation of nonlinear dynamical systems the problem of estimating parameters of nonlinear dynamical systems based on incomplete noisy measurements is considered within the framework of bayesian filtering using monte carlo simulations the measurement noise and unmodelled dynamics are represented through additive and or multiplicative gaussian white noise processes truncated ito taylor expansions are used to discretize these equations leading to discrete maps containing a set of multiple stochastic integrals these integrals in general constitute a set of non gaussian random variables the system parameters to be determined are declared as additional state variables the parameter identification problem is solved through a new sequential importance sampling filter this involves ito taylor expansions of nonlinear terms in the measurement equation and the development of an ideal proposal density function while accounting for the non gaussian terms appearing in the governing equations numerical illustrations on parameter identification of a few nonlinear oscillators and a geometrically nonlinear euler bernoulli beam reveal a remarkably improved performance of the proposed methods over one of the best known algorithms i e the unscented particle filter

self adjointness of some infinite dimensional elliptic operators and application to stochastic quantization we consider an operator k l in a hilbert space h where l is an ornstein uhlenbeck operator u w 1 4 h and is the invariant measure associated with l we show that k is essentially self adjoint in the space l 2 h where is the gibbs measure dx z 1 e 2u x dx an application to stochastic quantization is given

new computational approaches for wrinkled and slack membranes the static response of thin wrinkled membranes is studied using both a tension field approximation based on plane stress conditions and a 3d nonlinear elasticity formulation discretized through 8 noded cosserat point elements while the tension field approach only obtains the wrinkled slack regions and at best a measure of the extent of wrinkliness the 3d elasticity solution provides in principle the deformed shape of a wrinkled slack membrane however since membranes barely resist compression the discretized and linearized system equations via both the approaches are ill conditioned and solutions could thus be sensitive to discretizations errors as well as other sources of noises imperfections we propose a regularized pseudo dynamical recursion scheme that provides a sequence of updates which are almost insensitive to the regularizing term as well as the time step size used for integrating the pseudo dynamical form this is borne out through several numerical examples wherein the relative performance of the proposed recursion scheme vis a vis a regularized newton strategy is compared the pseudo time marching strategy when implemented using 3d cosserat point elements also provides a computationally cheaper numerically accurate and simpler alternative to that using geometrically exact shell theories for computing large deformations of membranes in the presence of wrinkles

nonsymmetric ornstein uhlenbeck semigroups in banach spaces letebe a separable real banach space and letq l e e be positive and symmetric lets s t t 0be ac0 semigroup onewe study the relations between the reproducing kernel hilbert spaces associated with the operatorsqt t0 s s qs s ds under the assumption that these operators are the covariances of centered gaussian measures tone we also study equivalence t sfor different values of s and t and we calculate their radon nikodym derivatives

harnack and functional inequalities for generalized mehler semigroups abstract harnack inequalities are established for a class of generalized mehler semigroups which in particular imply upper bound estimates for the transition density moreover poincare and log sobolev inequalities are proved in terms of estimates for the square field operators furthermore under a condition well known in the gaussian case we prove that generalized mehler semigroups are strong feller the results are illustrated by concrete examples in particular we show that a generalized mehler semigroup with an stable part is not hyperbounded but exponentially ergodic and that the log sobolev constant obtained by our method in the special gaussian case can be sharper than the one following from the usual curvature condition moreover a harnack inequality is established for the generalized mehler semigroup associated with the dirichlet heat semigroup on 0 1 we also prove that this semigroup is not hyperbounded

quadratic transformation formulas for basic hypergeometric series starting with some of the known transformation formulas for well poised 2 1 and very well poised 8 7 basic hypergeometric series we obtain q analogues of 36 quadratic transformation formulas given in 2 11 of higher transcendental functions vol 1 edited by erdelyi et al we also derive some new quadratic transformation formulas that give rise to identities connecting very well poised but unbalanced 10 9 series in base q with very well poised and balanced 12 11 series in base q 2 a rogers ramanujan type identity is also found as a limiting case

structural properties of semilinear spdes driven by cylindrical stable processes we consider a class of semilinear stochastic evolution equations driven by an additive cylindrical stable noise we investigate structural properties of the solutions like markov irreducibility stochastic continuity feller and strong feller properties and study integrability of trajectories the obtained results are applied to semilinear stochastic heat equations with dirichlet boundary conditions and bounded and lipschitz nonlinearities

generalized mehler semigroups and applications we construct and study generalized mehler semigroups p t t 0 and their associated markov processesm the construction methods for p t t 0 are based on some new purely functional analytic results implying in particular that any strongly continuous semigroup on a hilbert spaceh can be extended to some larger hilbert spacee with the embeddingh e being hilbert schmidt the same analytic extension results are applied to construct strong solutions to stochastic differential equations of typedx t c dw t ax t dt with possibly unbounded linear operatorsa andc onh on a suitably chosen larger spacee for gaussian generalized mehler semigroups p t t 0 with corresponding markov processm the associated non symmetric dirichlet forms ed e are explicitly calculated and a necessary and sufficient condition for path regularity ofm in terms of e d e is proved then using dirichlet form methods it is shown thatm weakly solves the above stochastic differential equation if the state spacee is chosen appropriately finally we discuss the differences between these two methods yielding strong resp weak solutions

on filtering equations in infinite dimensions abstract we consider a nonlinear filtering problem for a state processxin a hilbert spaceh given a finite dimensional observation processy we look for solutions of the corresponding filtering equations having densities with respect to a gaussian reference measure inh we derive the equations for the conditional densities and prove that they have solutions we can allow correlated noise in the state and observation equations provided additional assumptions hold

continuity and generators of dynamical semigroups for infinite bose systems abstract for a class of quasifree quantum dynamical semigroups on the algebra of the canonical commutation relations ccr we give sufficient conditions for these semigroups to extend to ultraweakly continuous semigroups of normal operators on the von neumann algebra associated with a representation of the ccr then the explicit form of the generators of the extended semigroups is calculated

generalized ornstein uhlenbeck semigroups littlewood paley stein inequalities and the p a meyer equivalence of norms abstract let be a nondegenerate gaussian measure on a hilbert space h for an arbitrary selfadjoint nonnegative operator a we consider the semigroup e tl e ta on l p where stands for the second quantization operator we provide an explicit characterization of the domains of i l m 2 in l p in terms of gaussian sobolev spaces thus extending the p a meyer result on equivalence of norms the main tools are the littlewood paley stein inequalities which are proved under minimal assumptions by a purely analytic method following e stein 1970 topics in harmonic analysis princeton univ press princeton nj

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

a differential geometric approach to nonlinear filtering the projection filter this paper presents a new and systematic method of approximating exact nonlinear filters with finite dimensional filters using the differential geometric approach to statistics the projection filter is defined rigorously in the case of exponential families a convenient exponential family is proposed which allows one to simplify the projection filter equation and to define an a posteriori measure of the local error of the projection filter approximation finally simulation results are discussed for the cubic sensor problem

constitutive error based parameter estimation technique for plate structures using free vibration signatures abstract in this paper a variant of constitutive equation error based material parameter estimation procedure for linear elastic plates is developed from partially measured free vibration sig natures it has been reported in many research articles that the mode shape curvatures are much more sensitive compared to mode shape themselves to localize inhomogeneity complying with this idea an identification procedure is framed as an optimization problem where the proposed cost function measures the error in constitutive relation due to incompatible curvature strain and moment stress fields unlike standard constitutive equation error based procedure wherein a solution of a couple system is unavoidable in each iteration we generate these incompatible fields via two linear solves a simple yet effective penalty based approach is followed to incorporate measured data the penalization parameter not only helps in incorporating corrupted measurement data weakly but also acts as a regularizer against the ill posedness of the inverse problem explicit linear update formulas are then developed for anisotropic linear elastic material numerical examples are provided to show the applicability of the proposed technique finally an experimental validation is also provided

self regularized pseudo time marching schemes for structural system identification with static measurements we explore the application of pseudo time marching schemes involving either deterministic integration or stochastic filtering to solve the inverse problem of parameter identification of large dimensional structural systems from partial and noisy measurements of strictly static response solutions of such non linear inverse problems could provide useful local stiffness variations and do not have to confront modeling uncertainties in damping an important yet inadequately understood aspect in dynamic system identification problems the usual method of least square solution is through a regularized gauss newton method gnm whose results are known to be sensitively dependent on the regularization parameter and data noise intensity finite time recursive integration of the pseudo dynamical gnm pd gnm update equation addresses the major numerical difficulty associated with the near zero singular values of the linearized operator and gives results that are not sensitive to the time step of integration therefore we also propose a pseudo dynamic stochastic filtering approach for the same problem using a parsimonious representation of states and specifically solve the linearized filtering equations through a pseudo dynamic ensemble kalman filter pd enkf for multiple sets of measurements involving various load cases we expedite the speed of thepd enkf by proposing an inner iteration within every time step results using the pseudo dynamic strategy obtained through pd enkf and recursive integration are compared with those from the conventional gnm which prove that the pd enkf is the best performer showing little sensitivity to process noise covariance and yielding reconstructions with less artifacts even when the ensemble size is small

on the cauchy problem for non local ornstein uhlenbeck operators abstract we study the cauchy problem involving non local ornstein uhlenbeck operators in finite and infinite dimensions we prove classical solvability without requiring that the levy measure corresponding to the large jumps part has a first finite moment moreover we determine a core of regular functions which is invariant for the associated transition markov semigroup such a core allows to characterize the marginal laws of the ornstein uhlenbeck stochastic process as unique solutions to fokker planck kolmogorov equations for measures

parameter identification in explicit structural dynamics performance of the extended kalman filter abstract an extended kalman filter ekf approach is adopted in this paper for structural systems subject to dynamic loadings to simultaneously estimate the state and calibrate constitutive parameters to pursue this aim after space and time discretizations a joint system state vector is introduced gathering nodal displacements and constitutive parameters to be identified because of the linearization of the discretized equations governing filter updates the ekf can lead to inaccurate model calibration it is shown that even though the state of the system is always followed with a high level of accuracy unsatisfactory parameter estimations can be obtained in the case of degrading strength of the structure that is in the case of softening both single degree of freedom dof and multi dof dynamic systems are analyzed to detect the possible sources of this inaccuracy

a kalman filter based time domain analysis for structural damage diagnosis with noisy signals abstract in this paper a procedure is presented for the time domain analysis of noise contaminated vibration signals for global structural damage diagnosis it extends from a previously established acceleration response only time domain auto regressive with exogenous input arx model where the process is defined such that the acceleration response at a given degree of freedom dof is regarded as the input while the accelerations at other dofs are the state with which the measurements are associated the novel idea in the present procedure is to retrieve the intrinsic input output set from noisy signals by using the kalman filter so that the underlying physical system is best presented to the subsequent diagnosis operation the theoretical basis of representing the system by pairing the raw measured input and the filtered response through the kalman filter is discussed when such raw input and filtered response signals are fed into the reference arx model the error feature becomes indicative of the change of the physical system by analyzing the residual error the damage status of the structure can be diagnosed applications to numerical and experimental examples demonstrate that the approach is effective in tackling the noises and both the occurrence and relative extent of damage can be assessed with an appropriate damage feature

on the continuity of ornstein uhlenbeck processes in infinite dimensions existence and continuity of ornstein uhlenbeck processes in banach and hilbert spaces are investigated under various assumptions

phase space linearization for non linear oscillators deterministic and stochastic systems abstract a new and efficient semi analytical phase space linearization psl scheme for a class of non linear oscillators is developed in this paper the method is based on replacement of the non linear vector field by a set of linear ones each valid over a short segment of the evolving trajectory or over sufficiently small interval of time based on this concept a few explicit and implicit integration schemes are first proposed and applied to a class of low dimensional non linear dynamical systems to accurately determine their response trajectories this approach of local linearization is further extended to non linear oscillators excited by formal derivatives of one or a combination of gauss markov processes since the present methodology reduces the non linear operator by a set of linear operators it is also demonstrated that the principles of linear random vibration may be suitable exploited to arrive at a faster and semi analytical monte carlo scheme for computing the response statistics both in stationary and non stationary regimes limited examples are presented and compared with exact solutions whenever available to illustrate the efficiency and versatility of the proposed schemes

co evolutionary algorithm for structural damage identification using minimal physical testing the problem of damage identification using minimum test data is studied in this work data sparsity in damage identification applications commonly results in inverse problems that are mathematically ill posed e g non unique solutions although solution non uniqueness may be addressed by performing multiple tests on a structure it is not trivial to decide which tests to carry out given that actual physical testing is costly this problem is addressed in this work through a new co evolutionary algorithm that interactively searches for damage scenarios and optimum physical tests the algorithm is composed of two stages the estimation phase which searches for damage scenarios that can predict current physical tests and the exploration phase which searches for tests that increase the level of information about the damaged system the feasibility of the methodology is demonstrated using numerical examples copyright 2006 john wiley sons ltd

a product formula for the continuous q jacobi polynomials abstract using a set of transformations on a series form product formula for the q wilson polynomials pn x a b c d q and the nassrallah rahman integral representation of an 8 7 series a product formula in the integral form for the continuous q jacobi polynomials pn x b bq c cq q2 with a positive kernel is obtained in the case 0 b

traces of harmonic functions and a new path space for the free quantum field the infinite dimensional ornstein uhlenbeck process associated with the free field of quantum field theory on rd is reconstructed as the traces of random harmonic functions these harmonic functions are the solutions of a generalised dirichlet problem with a random schwartz distribution as boundary data a detailed analysis of the associated trace space which is a subspace of a scaled sobolev space of negative fractional order leads to a new path space for the free field as well as to a convenient regularisation method because of the particular form of the trace space it is possible to give an explicit formula for the transition function of the process furthermore the dependence of the path space on the dimension d is discussed and a formulation of our result in terms of abstract wiener spaces is given

on the q analogues of some transformations of nearly poised hypergeometric series a number of transformation formulas for very well poised basic hypergeometric series have been obtained which in the limit q 1 approach the known transformation formulas for nearly poised ordinary hypergeometric series