on differential games for infinite dimensional systems with nonlinear unbounded operators abstract we consider a two player zero sum differential game governed by an abstract nonlinear differential equation of accretive type in an infinite dimensional space we prove that the value function of the game is the unique viscosity solution of the corresponding hamilton jacobi isaacs equation in the sense of 12 we also discuss some properties of this notion of solution

nonsmooth lyapunov pairs for infinite dimensional first order differential inclusions the main objective of this paper is to provide new explicit criteria to characterize weak lower semicontinuous lyapunov pairs or functions associated to first order differential inclusions in hilbert spaces these inclusions are governed by a lipschitzian perturbation of a maximally monotone operator the dual criteria we give are expressed by means of the proximal and basic subdifferentials of the nominal functions while primal conditions are described in terms of the contingent directional derivative we also propose a unifying review of many other criteria given in the literature our approach is based on advanced tools of variational analysis and generalized differentiation

nonsmooth lyapunov pairs for differential inclusions governed by operators with nonempty interior domain the general theory of lyapunov stability of first order differential inclusions in hilbert spaces has been studied by the authors in the previous paper adly et al in nonlinear anal 75 3 985 1008 2012 this new contribution focuses on the case when the interior of the domain of the maximally monotone operator governing the given differential inclusion is nonempty this includes in a natural way the finite dimensional case the current setting leads to simplified more explicit criteria and permits some flexibility in the choice of the generalized subdifferentials some consequences of the viability of closed sets are given our analysis makes use of standard tools from convex and variational analysis

decay estimates for time fractional and other non local in time subdiffusion equations in mathbb r d we prove optimal estimates for the decay in time of solutions to a rather general class of non local in time subdiffusion equations in mathbb r d an important special case is the time fractional diffusion equation which has seen much interest during the last years mostly due to its applications in the modeling of anomalous diffusion processes we follow three different approaches and techniques to study this particular case a estimates based on the fundamental solution and young s inequality b fourier multiplier methods and c the energy method it turns out that the decay behaviour is markedly different from the heat equation case in particular there occurs a critical dimension phenomenon the general subdiffusion case is treated by method b and relies on a careful estimation of the underlying relaxation function several examples of kernels including the ultraslow diffusion case illustrate our results

probabilistic model identification of uncertainties in computational models for dynamical systems and experimental validation we present a methodology to perform the identification and validation of complex uncertain dynamical systems using experimental data for which uncertainties are taken into account by using the nonparametric probabilistic approach such a probabilistic model of uncertainties allows both model uncertainties and parameter uncertainties to be addressed by using only a small number of unknown identification parameters consequently the optimization problem which has to be solved in order to identify the unknown identification parameters from experiments is feasible two formulations are proposed the first one is the mean square method for which a usual differentiable objective function and an unusual non differentiable objective function are proposed the second one is the maximum likelihood method coupling with a statistical reduction which leads us to a considerable improvement of the method three applications with experimental validations are presented in the area of structural vibrations and vibroacoustics

maximum likelihood estimation of stochastic chaos representations from experimental data this paper deals with the identification of probabilistic models of the random coefficients in stochastic boundary value problems sbvp the data used in the identification correspond to measurements of the displacement field along the boundary of domains subjected to specified external forcing starting with a particular mathematical model for the mechanical behaviour of the specimen the unknown field to be identified is projected on an adapted functional basis such as that provided by a finite element discretization for each set of measurements of the displacement field along the boundary an inverse problem is formulated to calculate the corresponding optimal realization of the coefficients of the unknown random field on the adapted basis realizations of these coefficients are then used in conjunction with the maximum likelihood principle to set up and solve an optimization problem for the estimation of the coefficients in a polynomial chaos representation of the parameters of the sbvp

viscosity solutions for hamilton jacobi equations with unbounded nonlinear term a simplified approach abstract in a previous paper the author defined a notion for hamilton jacobi equations with an unbounded nonlinear term ax du and for such solutions proved existence and uniqueness results here we define an equivalent notion of solution for such equations which in addition has the advantage of being simpler and more intuitive

a deterministic filter for non gaussian bayesian estimation applications to dynamical system estimation with noisy measurements abstract we present a fully deterministic method to compute sequential updates for stochastic state estimates of dynamic models from noisy measurements it does not need any assumptions about the type of distribution for either data or measurement in particular it does not have to assume any of them as gaussian here the implementation is based on a polynomial chaos expansion pce of the stochastic variables of the model however any other orthogonal basis would do we use a minimum variance estimator that combines an a priori state estimate and noisy measurements in a bayesian way for computational purposes the update equation is projected onto a finite dimensional pce subspace the resulting kalman type update formula for the pce coefficients can be efficiently computed solely within the pce as it does not rely on sampling the method is deterministic robust and fast in this paper we discuss the theory and practical implementation of the method the original kalman filter is shown to be a low order special case in a first experiment we perform a bi modal identification using noisy measurements additionally we provide numerical experiments by applying it to the well known lorenz 84 model and compare it to a related method the ensemble kalman filter

maximum entropy approach for modeling random uncertainties in transient elastodynamics a new approach is presented for analyzing random uncertainties in dynamical systems this approach consists of modeling random uncertainties by a nonparametric model allowing transient responses of mechanical systems submitted to impulsive loads to be predicted in the context of linear structural dynamics the information used does not require the description of the local parameters of the mechanical model the probability model is deduced from the use of the entropy optimization principle whose available information is constituted of the algebraic properties related to the generalized mass damping and stiffness matrices which have to be positive definite symmetric matrices and the knowledge of these matrices for the mean reduced matrix model an explicit construction and representation of the probability model have been obtained and are very well suited to algebraic calculus and to monte carlo numerical simulation in order to compute the transient responses of structures submitted to impulsive loads the fundamental properties related to the convergence of the stochastic solution with respect to the dimension of the random reduced matrix model are analyzed finally an example is presented

a probabilistic construction of model validation we describe a procedure to assess the predictive accuracy of process models subject to approximation error and uncertainty the proposed approach is a functional analysis based probabilistic approach for which we represent random quantities using polynomial chaos expansions pces the approach permits the formulation of the uncertainty assessment in validation a significant component of the process as a problem of approximation theory it has two essential parts first a statistical procedure is implemented to calibrate uncertain parameters of the candidate model from experimental or model based measurements such a calibration technique employs pces to represent the inherent uncertainty of the model parameters based on the asymptotic behavior of the statistical parameter estimator the associated pce coefficients are then characterized as independent random quantities to represent epistemic uncertainty due to lack of information second a simple hypothesis test is implemented to explore the validation of the computational model assumed for the physics of the problem the above validation path is implemented for the case of dynamical system validation challenge exercise

polynomial chaos representation of databases on manifolds characterizing the polynomial chaos expansion pce of a vector valued random variable with probability distribution concentrated on a manifold is a relevant problem in data driven settings the probability distribution of such random vectors is multimodal in general leading to potentially very slow convergence of the pce in this paper we build on a recent development for estimating and sampling from probabilities concentrated on a diffusion manifold the proposed methodology constructs a pce of the random vector together with an associated generator that samples from the target probability distribution which is estimated from data concentrated in the neighborhood of the manifold the method is robust and remains efficient for high dimension and large datasets the resulting polynomial chaos construction on manifolds permits the adaptation of many uncertainty quantification and statistical tools to emerging questions motivated by data driven queries

on asymptotic properties of maximum likelihood estimators for parabolic stochastic pde s we investigate asymptotic properties of the maximum likelihood estimators for parameters occurring in parabolic spdes of the form du t x a 0 theta a 1 u t x dt dw t x wherea 0 anda 1 are partial differential operators andw is a cylindrical brownian motion we introduce a spectral method for computing mles based on finite dimensional approximations to solutions of such systems and establish criteria for consistency asymptotic normality and asymptotic efficiency as the dimension of the approximation goes to infinity we derive the asymptotic properties of the mle from a condition on the order of the operators in particular the mle is consistent if and only if ord a 1 1 2 ord a 0 a 1 d

probabilistic equivalence and stochastic model reduction in multiscale analysis this paper presents a probabilistic upscaling of mechanics models a reduced order probabilistic model is constructed as a coarse scale representation of a specified fine scale model whose probabilistic structure can be accurately determined equivalence of the fine and coarse scale representations is identified such that a reduction in the requisite degrees of freedom can be achieved while accuracy in certain quantities of interest is maintained a significant stochastic model reduction can a priori be expected if a separation of spatial and temporal scales exists between the fine and coarse scale representations the upscaling of probabilistic models is subsequently formulated as an optimization problem suitable for practical computations an illustration in stochastic structural dynamics is provided to demonstrate the proposed framework

dynamic data driven bayesian gmsfem abstract in this paper we propose a bayesian approach for multiscale problems with the availability of dynamic observational data our method selects important degrees of freedom probabilistically in a generalized multiscale finite element method framework due to scale disparity in many multiscale applications computational models cannot resolve all scales dominant modes in the generalized multiscale finite element method are used as permanent basis functions which we use to compute an inexpensive multiscale solution and the associated uncertainties through our bayesian framework we can model approximate solutions by selecting the unresolved scales probabilistically we consider parabolic equations in heterogeneous media the temporal domain is partitioned into subintervals using residual information and given dynamic data we design appropriate prior distribution for modeling missing subgrid information the likelihood is designed to minimize the residual in the underlying pde problem and the mismatch of observational data using the resultant posterior distribution the sampling process identifies important degrees of freedom beyond permanent basis functions the method adds important degrees of freedom in resolving subgrid information and ensuring the accuracy of the observations

ito sde based generator for a class of non gaussian vector valued random fields in uncertainty quantification this paper is concerned with the derivation of a generic sampling technique for a class of non gaussian vector valued random fields such an issue typically arises in uncertainty quantification for complex systems where the input coefficients associated with the elliptic operators must be identified by solving statistical inverse problems specifically we consider the case of non gaussian random fields with values in some arbitrary bounded or semi bounded subsets of r n the approach involves two main features the first one is the construction of a family of random fields converging at a user controlled rate towards the target random field each of these auxialiary random fields can be subsequently simulated by solving a family of ito stochastic differential equations the second ingredient is the definition of an adaptive discretization algorithm the latter allows refining the integration step on the fly and prevents the scheme from diverging the proposed strategy is finally exemplified on three examples each of which serving as a benchmark either for the adaptivity procedure or for the convergence of the diffusions

on the minimum time function and the minimum energy problem a nonlinear case abstract for a semilinear control system in general banach spaces we prove results on exact and approximate controllability regularity of the minimum time function connections between the minimum time function and the minimum energy

invariant sets and lyapunov pairs for differential inclusions with maximal monotone operators we give different conditions for the invariance of closed sets with respect to differential inclusions governed by a maximal monotone operator defined on hilbert spaces which is subject to a lipschitz continuous perturbation depending on the state these sets are not necessarily weakly closed as in 3 4 while the invariance criteria are still written by using only the data of the system so no need to the explicit knowledge of neither the solution of this differential inclusion nor the semi group generated by the maximal monotone operator these invariant viability results are next applied to derive explicit criteria for a lyapunov pairs of lower semi continuous not necessarily weakly lsc functions associated to these differential inclusions the lack of differentiability of the candidate lyapunov functions and the consideration of general invariant sets possibly not convex or smooth are carried out by using techniques from nonsmooth analysis

lyapunov pairs for continuous perturbations of nonlinear evolutions abstract this paper deals with the lyapunov pairs for a nonlinear evolution system in banach spaces applications on a priori estimates finite extinction time null controllability and regularity of the minimal time function are given

ito sde mcmc method for bayesian characterization of errors associated with data limitations in stochastic expansion methods for uncertainty quantification abstract this paper is concerned with the characterization and the propagation of errors associated with data limitations in polynomial chaos based stochastic methods for uncertainty quantification such an issue can arise in uncertainty quantification when only a limited amount of data is available when the available information does not suffice to accurately determine the probability distributions that must be assigned to the uncertain variables the bayesian method for assigning these probability distributions becomes attractive because it allows the stochastic model to account explicitly for insufficiency of the available information in previous work such applications of the bayesian method had already been implemented by using the metropolis hastings and gibbs markov chain monte carlo mcmc methods in this paper we present an alternative implementation which uses an alternative mcmc method built around an ito stochastic differential equation sde that is ergodic for the bayesian posterior we draw together from the mathematics literature a number of formal properties of this ito sde that lend support to its use in the implementation of the bayesian method and we describe its discretization including the choice of the free parameters by using the implicit euler method we demonstrate the proposed methodology on a problem of uncertainty quantification in a complex nonlinear engineering application relevant to metal forming

karhunen loeve expansion revisited for vector valued random fields scaling errors and optimal basis due to scaling effects when dealing with vector valued random fields the classical karhunen loeve expansion which is optimal with respect to the total mean square error tends to favorize the components of the random field that have the highest signal energy when these random fields are to be used in mechanical systems this phenomenon can introduce undesired biases for the results this paper presents therefore an adaptation of the karhunen loeve expansion that allows us to control these biases and to minimize them this original decomposition is first analyzed from a theoretical point of view and is then illustrated on a numerical example

stability of dynamical systems with competitive controls the degenerate case abstract we study the stability of a dynamical system with two competitive controls to a general closed set the stability is proven to be equivalent to the continuity on such set of the value function of a differential game with maximum cost functional the existence of a suitably defined and continuous lyapunov function is proven to imply the stability our lyapunov function satisfies a differential inequality in the sense of the theory of viscosity solutions

cal h _ infty control of nonlinear systems differential games and viscosity solutions dealing with disturbances is one of the most important questions for controlled systems cal h infty optimal control theory is a deterministic way to tackle the problem in the presence of unfavorable disturbances the theory of differential games and the study of the associated hamilton jacobi isaacs equation appear to be basic tools of the theory we consider a general nonlinear system and prove that the existence of a continuous local viscosity supersolution of the isaacs equation corresponding to the cal h infty control problem is sufficient for its solvability we also show that the existence of a lower semicontinuous viscosity supersolution is necessary

uncertainty quantification in computational stochastic multiscale analysis of nonlinear elastic materials abstract this paper is devoted to a computational stochastic multiscale analysis of nonlinear structures made up of heterogeneous hyperelastic materials at the microscale level the nonlinear constitutive equation of the material is characterized by a stochastic potential for which a polynomial chaos representation is used the geometry of the microstructure is random and characterized by a high number of random parameters the method is based on a deterministic non concurrent multiscale approach devoted to micro macro nonlinear mechanics which leads us to characterize the nonlinear constitutive equation with an explicit continuous form of the strain energy density function with respect to the large scale cauchy green strain states to overcome the curse of dimensionality due to the high number of involved random variables the problem is transformed into another one consisting in identifying the potential on a polynomial chaos expansion several strategies based on novel algorithms dedicated to high stochastic dimension are used and adapted for the class of multi modal random variables which may characterize the potential numerical examples at both small and large scales allow analyzing the efficiency of the approach through comparisons with classical methods

lyapunov functions for infinite dimensional systems abstract we study lyapunov functions for infinite dimensional dynamical systems governed by general maximal monotone operators we obtain a characterization of lyapunov pairs by means of the associated hamilton jacobi partial differential equations whose solutions are meant in the viscosity sense as evolved in works of tataru and crandall lions our approach also leads to a new sufficient condition for lyapunov pairs generalizing a classical result of pazy

large deviation for diffusions and hamilton jacobi equation in hilbert spaces large deviation for markov processes can be studied by hamilton jacobi equation techniques the method of proof involves three steps first we apply a nonlinear transform to generators of the markov processes and verify that limit of the transformed generators exists such limit induces a hamilton jacobi equation second we show that a strong form of uniqueness the comparison principle holds for the limit equation finally we verify an exponential compact containment estimate the large deviation principle then follows from the above three verifications this paper illustrates such a method applied to a class of hilbert space valued small diffusion processes the examples include stochastically perturbed allen cahn cahn hilliard pdes and a one dimensional quasilinear pde with a viscosity term we prove the comparison principle using a variant of the tataru method we also discuss different notions of viscosity solution in infinite dimensions in such context

on the bellman equation for the minimum time problem in infinite dimensions we consider the time optimal control problem for a semilinear parabolic control system where the target is the closed ball with center 0 and radius rgeq0 in a hilbert space x in particular we allow the origin of x to be the target using an appropriate kruzkov type transformation we give an existence and uniqueness result for the associated hamilton jacobi bellman equation even when the reachable set is not the whole space

differential games and nonlinear boldmath cal h_ infty control in infinite dimensions this paper studies the cal h infty control problem for a nonlinear unbounded infinite dimensional system with state constraints we characterize the solvability of the problem by means of a hamilton jacobi isaacs hji equation proving that the cal h infty problem can be solved if and only if the hji equation has a positive definite viscosity supersolution vanishing and continuous at the origin in order to do so the standard definition of the cal h infty problem has to be relaxed by using the theory of differential games we apply our results to the one phase stefan problem

identification of bayesian posteriors for coefficients of chaos expansions this article is concerned with the identification of probabilistic characterizations of random variables and fields from experimental data the data used for the identification consist of measurements of several realizations of the uncertain quantities that must be characterized the random variables and fields are approximated by a polynomial chaos expansion and the coefficients of this expansion are viewed as unknown parameters to be identified it is shown how the bayesian paradigm can be applied to formulate and solve the inverse problem the estimated polynomial chaos coefficients are hereby themselves characterized as random variables whose probability density function is the bayesian posterior this allows to quantify the impact of missing experimental information on the accuracy of the identified coefficients as well as on subsequent predictions an illustration in stochastic aeroelastic stability analysis is provided to demonstrate the proposed methodology

reduced model of macro scale stochastic plasticity identification by bayesian inference application to quasi brittle failure of concrete abstract in this paper we deal with a probability based scale bridging for concrete material when passing the detailed information at the meso scale the scale where the aggregate vs cement microstructure is visible with a voronoi cell based microstructure representation towards the chosen reduced model at the macro scale the scale where the concrete is a homogenized continuum with a stochastic plasticity model for localized failure this is accomplished by using bayesian inference providing the probability distributions of the macro scale model parameters expressed as random variables rv in order to compensate for the model reduction from the meso scale where parameters are expressed as random fields rf the original aspect of this approach is in the resulting macro scale stochastic plasticity model which can best quantify the uncertainty due to data loss in terms of the corresponding probability distribution of its parameters the proposed procedure is illustrated in detail for the concrete meso scale model presented in part i of this paper see karavelic et al 2019 both for the simple elastic response as well as for the plastic response with hardening in the fracture process zone fpz followed by a softening response in the localized failure phase this context of localized failure implies that the classical homogenization procedure no longer applies and should be replaced by a macro scale reduced model defined with respect to a quantity of interest qoi which is not necessarily the same for each particular response phase the complete set of results for the parameter identification is combined together at the macro scale in terms of a solid finite element with embedded discontinuity ed fem granting it very powerful predictive properties the particular choice of ed fem for the macro scale reduced model allows one to compute the qoi at an element level either as strain energy for the elastic response or as plastic dissipation for each failure mode thus the bayesian inference computation of the probability distributions of the rvs representing the macro scale model parameters performed by matching such qoi results computed for the macro scale and the meso scale models reduces to a homogenization type procedure within a probabilistic setting that can successfully handle any case where the separation of scales no longer applies

reduced wiener chaos representation of random fields via basis adaptation and projection abstract a new characterization of random fields appearing in physical models is presented that is based on their well known homogeneous chaos expansions we take advantage of the adaptation capabilities of these expansions where the core idea is to rotate the basis of the underlying gaussian hilbert space in order to achieve reduced functional representations that concentrate the induced probability measure in a lower dimensional subspace for a smooth family of rotations along the domain of interest the uncorrelated gaussian inputs are transformed into a gaussian process thus introducing a mesoscale that captures intermediate characteristics of the quantity of interest

sampling free linear bayesian update of polynomial chaos representations we present a fully deterministic approach to a probabilistic interpretation of inverse problems in which unknown quantities are represented by random fields or processes described by possibly non gaussian distributions the description of the introduced random fields is given in a white noise framework which enables us to solve the stochastic forward problem through galerkin projection onto polynomial chaos with the help of such a representation the probabilistic identification problem is cast in a polynomial chaos expansion setting and the baye s linear form of updating by introducing the hermite algebra this becomes a direct purely algebraic way of computing the posterior which is comparatively inexpensive to evaluate in addition we show that the well known kalman filter is the low order part of this update the proposed method is here tested on a stationary diffusion equation with prescribed source terms characterised by an uncertain conductivity parameter which is then identified from limited and noisy data obtained by a measurement of the diffusing quantity

kernel principal component analysis for stochastic input model generation stochastic analysis of random heterogeneous media provides useful information only if realistic input models of the material property variations are used these input models are often constructed from a set of experimental samples of the underlying random field to this end the karhunen loeve k l expansion also known as principal component analysis pca is the most popular model reduction method due to its uniform mean square convergence however it only projects the samples onto an optimal linear subspace which results in an unreasonable representation of the original data if they are non linearly related to each other in other words it only preserves the first order mean and second order statistics covariance of a random field which is insufficient for reproducing complex structures this paper applies kernel principal component analysis kpca to construct a reduced order stochastic input model for the material property variation in heterogeneous media kpca can be considered as a nonlinear version of pca through use of kernel functions kpca further enables the preservation of higher order statistics of the random field instead of just two point statistics as in the standard karhunen loeve k l expansion thus this method can model non gaussian non stationary random fields in this work we also propose a new approach to solve the pre image problem involved in kpca in addition polynomial chaos pc expansion is used to represent the random coefficients in kpca which provides a parametric stochastic input model thus realizations which are statistically consistent with the experimental data can be generated in an efficient way we showcase the methodology by constructing a low dimensional stochastic input model to represent channelized permeability in porous media

identification of polynomial chaos representations in high dimension from a set of realizations this paper deals with the identification in high dimensions of a polynomial chaos expansion of random vectors from a set of realizations due to numerical and memory constraints the usual polynomial chaos identification methods are based on a series of truncations that induce a numerical bias this bias becomes very detrimental to the convergence analysis of polynomial chaos identification in high dimensions this paper therefore proposes a new formulation of the usual polynomial chaos identification algorithms to avoid this numerical bias after a review of the polynomial chaos identification method the influence of the numerical bias on the identification accuracy is quantified the new formulation is then described in detail and illustrated using two examples

construction of probability distributions in high dimension using the maximum entropy principle applications to stochastic processes random fields and random matrices the construction of probabilistic models in computational mechanics requires the effective construction of probability distributions of random variables in high dimension this paper deals with the effective construction of the probability distribution in high dimension of a vector valued random variable using the maximum entropy principle the integrals in high dimension are then calculated in constructing the stationary solution of an ito stochastic differential equation associated with its invariant measure a random generator of independent realizations is explicitly constructed in this paper three fundamental applications are presented the first one is a new formulation of the stochastic inverse problem related to the construction of the probability distribution in high dimension of an unknown non stationary random time series random accelerograms for which the velocity response spectrum is given the second one is also a new formulation related to the construction of the probability distribution of positive definite band random matrices finally we present an extension of the theory when the support of the probability distribution is not all the space but is any part of the space the third application is then a new formulation related to the construction of the probability distribution of the karhunen loeve expansion of non gaussian positive valued random fields

parametrization of random vectors in polynomial chaos expansions via optimal transportation polynomial chaos pc expansions are used for the propagation of uncertainty through dynamical systems as an alternative to monte carlo methods model parameters in a given dynamical system are assumed to have known expansions which correspond to simple standard distributions and one is usually interested in the polynomial expansion of the system solution we are concerned with the problem of estimating the pc expansion of a parameter vector when only realizations from its distribution are given to this end we apply ideas from optimal transportation theory and network optimization

a space fractional stefan problem abstract we study a space fractional stefan problem where the non local diffusion flux is modeled by the caputo derivative we obtain the unique existence of classical solution to this problem

viscosity solutions of hamilton jacobi equations with unbounded nonlinear terms abstract the aim of this paper is to introduce a new definition of viscosity solutions for hamilton jacobi equations in infinite dimensions with unbounded nonlinear terms we prove existence and uniqueness results under usual assumptions regarding the bounded part of the hamiltonian

the minimum time function for semilinear evolutions we study the bellman equation for the minimum time problem associated with a semilinear evolution system from the point of view of contingent solutions for this purpose new contingent derivatives are defined which involve functions instead of vectors as directions our analysis is based on flow invariance results for appropriate differential inclusions

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

distributed order calculus and equations of ultraslow diffusion abstract we consider equations of the form d u t x u t x f t x t 0 x r n where d is a distributed order derivative that is d t 0 1 d t d d is the caputo dzhrbashyan fractional derivative of order is a positive weight function the above equation is used in physical literature for modeling diffusion with a logarithmic growth of the mean square displacement in this work we develop a mathematical theory of such equations study the derivatives and integrals of distributed order

a viscosity solution approach to the infinite dimensional hjb equation related to a boundary control problem in a transport equation the paper concerns the infinite dimensional hamilton jacobi bellman equation related to an optimal control problem regulated by a linear transport equation with boundary control a suitable viscosity solution approach is needed in view of the presence of the unbounded control related term in the state equation in the hilbert setting an existence and uniqueness result is obtained

on the construction and analysis of stochastic models characterization and propagation of the errors associated with limited data this paper investigates the predictive accuracy of stochastic models in particular a formulation is presented for the impact of data limitations associated with the calibration of parameters for these models on their overall predictive accuracy in the course of this development a new method for the characterization of stochastic processes from corresponding experimental observations is obtained specifically polynomial chaos representations of these processes are estimated that are consistent in some useful sense with the data the estimated polynomial chaos coefficients are themselves characterized as random variables with known probability density function thus permitting the analysis of the dependence of their values on further experimental evidence moreover the error in these coefficients associated with limited data is propagated through a physical system characterized by a stochastic partial differential equation spde this formalism permits the rational allocation of resources in view of studying the possibility of validating a particular predictive model a bayesian inference scheme is relied upon as the logic for parameter estimation with its computational engine provided by a metropolis hastings markov chain monte carlo procedure

almost sure stabilizability of controlled degenerate diffusions we develop a direct lyapunov method for the almost sure open loop stabilizability and asymptotic stabilizability of controlled degenerate diffusion processes the infinitesimal decrease condition for a lyapunov function is a new form of hamilton jacobi bellman partial differential inequality of second order we give local and global versions of the first and second lyapunov theorems assuming the existence of a lower semicontinuous lyapunov function satisfying such an inequality in the viscosity sense an explicit formula for a stabilizing feedback is provided for affine systems with smooth lyapunov function several examples illustrate the theory

a viscosity approach to infinite dimensional hamilton jacobi equations arising in optimal control with state constraints we consider nonlinear optimal control problems with state constraints and nonnegative cost in infinite dimensions where the constraint is a closed set possibly with empty interior for a class of systems with a maximal monotone operator and satisfying certain stability properties of the set of trajectories that allow the value function to be lower semicontinuous we prove that the value function is a viscosity solution of the bellman equation and is in fact the minimal nonnegative supersolution

stochastic spectral methods for efficient bayesian solution of inverse problems we present a reformulation of the bayesian approach to inverse problems that seeks to accelerate bayesian inference by using polynomial chaos pc expansions to represent random variables evaluation of integrals over the unknown parameter space is recast more efficiently as monte carlo sampling of the random variables underlying the pc expansion we evaluate the utility of this technique on a transient diffusion problem arising in contaminant source inversion the accuracy of posterior estimates is examined with respect to the order of the pc representation the choice of pc basis and the decomposition of the support of the prior the computational cost of the new scheme shows significant gains over direct sampling

the dynamic programming equation for the time optimal control problem in infinite dimensions the existence and uniqueness of a viscosity solution for the bellman equation associated with the time optimal control problem for a semilinear evolution equation in hilbert space is provided applications to time optimal control problems governed by parabolic equations are given

characterization of lyapunov pairs in the nonlinear case and applications abstract the paper presents a characterization of the lyapunov pairs for a general initial value problem with a possibly multivalued m accretive operator on an arbitrary banach space by means of the contingent derivative related to the operator the proof is based on tangency and flow invariance arguments combined with a priori estimates and approximation the abstract results are applied to obtain precise a priori estimates for the mild solutions they readily lead to the existence of global solutions and various controllability properties important lyapunov pairs are pointed out in connection with specific problems

dimensionality reduction and polynomial chaos acceleration of bayesian inference in inverse problems we consider a bayesian approach to nonlinear inverse problems in which the unknown quantity is a spatial or temporal field endowed with a hierarchical gaussian process prior computational challenges in this construction arise from the need for repeated evaluations of the forward model e g in the context of markov chain monte carlo and are compounded by high dimensionality of the posterior we address these challenges by introducing truncated karhunen loeve expansions based on the prior distribution to efficiently parameterize the unknown field and to specify a stochastic forward problem whose solution captures that of the deterministic forward model over the support of the prior we seek a solution of this problem using galerkin projection on a polynomial chaos basis and use the solution to construct a reduced dimensionality surrogate posterior density that is inexpensive to evaluate we demonstrate the formulation on a transient diffusion equation with prescribed source terms inferring the spatially varying diffusivity of the medium from limited and noisy data

a computational inverse method for identification of non gaussian random fields using the bayesian approach in very high dimension this paper is devoted to the identification of bayesian posteriors for the random coefficients of the high dimension polynomial chaos expansions of non gaussian tensor valued random fields using partial and limited experimental data the experimental data sets correspond to an observation vector which is the response of a stochastic boundary value problem depending on the tensor valued random field which has to be identified so an inverse stochastic problem must be solved to perform the identification of the random field a complete methodology is proposed to solve this very challenging problem in high dimension which consists in using the first four steps introduced in a previous paper followed by the identification of the posterior model the steps of the methodology are the following 1 introduction of a family of prior algebraic stochastic model pasm 2 identification of an optimal pasm in the constructed family using the partial experimental data 3 construction of a statistical reduced order optimal pasm 4 construction in high dimension of the polynomial chaos expansion with deterministic vector valued coefficients of the reduced order optimal pasm 5 substitution of these deterministic vector valued coefficients by random vector valued coefficients in order to extend the capability of the polynomial chaos expansion to represent the experimental data and for which the joint probability distribution must be identified 6 construction of the prior probability model of these random vector valued coefficients and finally 7 identification of the posterior probability model of these random vector valued coefficients using partial and limited experimental data through the stochastic boundary value problem two methods are proposed to carry out the identification of the posterior model the first one is based on the use of the classical bayesian method the second one is a new approach derived from the bayesian method which is more efficient in high dimension an application is presented for which several millions of random coefficients are identified

initial boundary value problems for multi term time fractional diffusion equations with positive constant coefficients in this paper we investigate the well posedness and the long time asymptotic behavior for initial boundary value problems for multi term time fractional diffusion equations the governing equation under consideration includes a linear combination of caputo derivatives in time with decreasing orders in 0 1 and positive constant coefficients by exploiting several important properties of multinomial mittag leffler functions various estimates follow from the explicit solutions in form of these special functions then we prove the uniqueness and continuous dependency on initial values and source terms from which we further verify the lipschitz continuous dependency of solutions with respect to coefficients and orders of fractional derivatives finally by a laplace transform argument it turns out that the decay rate of the solution as t is given by the minimum order of the time fractional derivatives

nested polynomial trends for the improvement of gaussian process based predictors the role of simulation has kept increasing for the sensitivity analysis and the uncertainty quantification of complex systems such numerical procedures are generally based on the processing of a huge amount of code evaluations when the computational cost associated with one particular evaluation of the code is high such direct approaches based on the computer code only can be not affordable surrogate models have therefore to be introduced to interpolate the information given by a fixed set of code evaluations to the whole input space when confronted to deterministic mappings the gaussian process based regression gpr or kriging presents a good compromise between complexity efficiency and error control such a method considers the quantity of interest of the system as a particular realization of a gaussian stochastic process which mean and covariance functions have to be identified from the available code evaluations in this context this work proposes an innovative parameterization of this mean function which is based on the composition of two polynomials this approach is particularly relevant for the approximation of strongly non linear quantities of interest from very little information after presenting the theoretical basis of this method this work compares its efficiency to alternative approaches on a series of examples

decay of solutions to parabolic type problem with distributed order caputo derivative abstract we consider the decay of solution to a fractional diffusion equation with distributed order caputo derivative we assume that the elliptic operator is time dependent and that the weight function contained in the definition of the distributed order caputo derivative is just integrable we establish the relation between behavior of the weight function near zero and the decay rate of solution

second order hamilton jacobi equations in hilbert spaces and stochastic boundary control the paper is concerned with fully nonlinear second order hamilton jacobi bellman isaacs equations of elliptic type in separable hilbert spaces which have unbounded first and second order terms the viscosity solution approach is adapted to the equations under consideration and the existence and uniqueness of viscosity solutions are proved a stochastic optimal control problem driven by a parabolic stochastic pde with control of dirichlet type on the boundary is considered it is proved that the value function of this problem is the unique viscosity solution of the associated hamilton jacobi bellman equation

maximum principle for the generalized time fractional diffusion equation abstract in the paper a maximum principle for the generalized time fractional diffusion equation over an open bounded domain g 0 t g r n is formulated and proved the proof of the maximum principle is based on an extremum principle for the caputo dzherbashyan fractional derivative that is given in the paper too the maximum principle is then applied to show that the initial boundary value problem for the generalized time fractional diffusion equation possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions

identification of high dimension polynomial chaos expansions with random coefficients for non gaussian tensor valued random fields using partial and limited experimental data this paper is devoted to the identification of high dimension polynomial chaos expansions with random coefficients for non gaussian tensor valued random fields using partial and limited experimental data the experimental data sets correspond to partial experimental data made up of an observation vector which is the response of a stochastic boundary value problem depending on the tensor valued random field which has to be identified so an inverse stochastic problem has to be solved to carry out the identification of the random field a complete methodology is proposed to solve this challenging problem and consists in introducing a family of prior probability models in identifying an optimal prior model in the constructed family using the experimental data in constructing a statistical reduced order optimal prior model in constructing the polynomial chaos expansion with deterministic vector valued coefficients of the reduced order optimal prior model and finally in constructing the probability distribution of random coefficients of the polynomial chaos expansion and in identifying the parameters using experimental data an application is presented for which several millions of random coefficients are identified solving an inverse stochastic problem c 2010 elsevier b v all rights reserved

viscosity solutions for a class of hamilton jacobi equations in hilbert spaces we study hamilton jacobi equations with an unbounded term in hilbert spaces we introduce a new variant of the notion of viscosity solution for a class of hamilton jacobi equations and obtain comparison and existence results for viscosity solutions we also examine the usefulness of the notion of viscosity solution in optimal control

path dependent hamilton jacobi equations in infinite dimensions abstract we propose notions of minimax and viscosity solutions for a class of fully nonlinear path dependent pdes with nonlinear monotone and coercive operators on hilbert space our main result is well posedness existence uniqueness and stability for minimax solutions a particular novelty is a suitable combination of minimax and viscosity solution techniques in the proof of the comparison principle one of the main difficulties the lack of compactness in infinite dimensional hilbert spaces is circumvented by working with suitable compact subsets of our path space as an application our theory makes it possible to employ the dynamic programming approach to study optimal control problems for a fairly general class of delay evolution equations in the variational framework furthermore differential games associated to such evolution equations can be investigated following the krasovski subbotin approach similarly as in finite dimensions

uniqueness in the inversion of distributed orders in ultraslow diffusion equations abstract this paper deals with the distributed order time fractional diffusion equations with non homogeneous dirichlet neumann boundary condition we first prove the wellposedness of the forward problem by means of eigenfunction expansion which ensures that the weak solution has the classical derivatives we next give a harnack type inequality of the solution in the frequency domain under the laplace transform finally we show a uniqueness result for the inverse problem in determining the weight function in the distributed order time derivative from one point observation

asymptotic sampling distribution for polynomial chaos representation from data a maximum entropy and fisher information approach a procedure is presented for characterizing the asymptotic sampling distribution of estimators of the polynomial chaos pc coefficients of a second order nonstationary and non gaussian random process by using a collection of observations the random process represents a physical quantity of interest and the observations made over a finite denumerable subset of the indexing set of the random process are considered to form a set of realizations of a random vector mathcal y representing a finite dimensional projection of the random process the karhunen loeve decomposition and a scaling transformation are employed to produce a reduced order model mathcal z of mathcal y the pc expansion of mathcal z is next determined by having recourse to the maximum entropy principle the metropolis hastings markov chain monte carlo algorithm and the rosenblatt transformation the resulting pc expansion has random coefficients where the random characteristics of the pc coefficients can be attributed to the limited data available from the experiment the estimators of the pc coefficients of mathcal y obtained from that of mathcal z are found to be maximum likelihood estimators as well as consistent and asymptotically efficient computation of the covariance matrix of the associated asymptotic normal distribution of estimators of the pc coefficients of mathcal y requires knowledge of the fisher information matrix fim the fim is evaluated here by using a numerical integration scheme as well as a sampling technique the resulting confidence interval on the pc coefficient estimators essentially reflects the effect of incomplete information due to data limitation on the characterization of the stochastic process this asymptotic distribution is significant as its characteristics can be propagated through predictive models for which the stochastic process in question describes uncertainty on some input parameters

lyapunov pairs for multi valued semi linear evolutions abstract we give here a characterization of a lyapunov pair for a multi valued semi linear evolution equation on a banach space by means of an appropriate lower contingent derivative the contingent derivative introduced in this paper is related to a new concept of tangency introduced recently in o c rj m necula i i vrabie necessary and sufficient conditions for viability for semilinear differential inclusions trans amer math soc 361 2009 343 390 as an application we give a controllability result and a lipschitz estimate of the corresponding minimum time function under a petrov like condition

reduced chaos decomposition with random coefficients of vector valued random variables and random fields we develop a stochastic functional representation that is adapted to problems involving various forms of epistemic uncertainties including modeling error and data paucity the new representation builds on the polynomial chaos decomposition and eventually yields a karhunen loeve expansion with random multiplicative coefficients in this expansion one set of uncertainty is captured in the usual manner as uncorrelated scalar random variables another component of the uncertainty statistically independent from the first is captured by constructing the usually deterministic functions in the kl expansion as random functions we think of the first set of uncertainties as associated with a coarse scale model and of the second set as associated with subscale fluctuations not captured in the coarse scale description