characterization of discontinuities in high dimensional stochastic problems on adaptive sparse grids in this paper we present a set of efficient algorithms for detection and identification of discontinuities in high dimensional space the method is based on extension of polynomial annihilation for discontinuity detection in low dimensions compared to the earlier work the present method poses significant improvements for high dimensional problems the core of the algorithms relies on adaptive refinement of sparse grids it is demonstrated that in the commonly encountered cases where a discontinuity resides on a small subset of the dimensions the present method becomes optimal in the sense that the total number of points required for function evaluations depends linearly on the dimensionality of the space the details of the algorithms will be presented and various numerical examples are utilized to demonstrate the efficacy of the method

adaptive smolyak pseudospectral approximations polynomial approximations of computationally intensive models are central to uncertainty quantification this paper describes an adaptive method for nonintrusive pseudospectral approximation based on smolyak s algorithm with generalized sparse grids we rigorously analyze and extend the nonadaptive method proposed in p g constantine m s eldred and e t phipps comput methods appl mech engrg 229 232 2012 pp 1 12 and compare it to a common alternative approach for using sparse grids to construct polynomial approximations direct quadrature analysis of direct quadrature shows that mathcal o 1 errors are an intrinsic property of some configurations of the method as a consequence of internal aliasing we provide precise conditions based on the chosen polynomial basis and quadrature rules under which this aliasing error occurs we then establish theoretical results on the accuracy of smolyak pseudospectral approximation and show that the smolyak approximation avoids internal alia

stochastic convergence acceleration through basis enrichment of polynomial chaos expansions given their mathematical structure methods for computational stochastic analysis based on orthogonal approximations and projection schemes are well positioned to draw on developments from deterministic approximation theory this is demonstrated in the present paper by extending basis enrichment from deterministic analysis to stochastic procedures involving the polynomial chaos decomposition this enrichment is observed to have a significant effect on the efficiency and performance of these stochastic approximations in the presence of non continuous dependence of the solution on the stochastic parameters in particular given the polynomial structure of these approximations the severe degradation in performance observed in the neighbourhood of such discontinuities is effectively mitigated an enrichment of the polynomial chaos decomposition is proposed in this paper that can capture the behaviour of such non smooth functions by integrating a priori knowledge about their behaviour the proposed enrichment scheme is applied to a random eigenvalue problem where the smoothness of the functional dependence between the random eigenvalues and the random system parameters is controlled by the spacing between the eigenvalues it is observed that through judicious selection of enrichment functions the spectrum of such a random system can be more efficiently characterized even for systems with closely spaced eigenvalues copyright 2007 john wiley sons ltd

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

symmetric dual multiobjective programming abstract in this paper a pair of symmetric dual multiobjective programming problems is formulated and the duality theorems are established for pseudo convex pseudo concave functions

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

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

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

minimax mixed integer symmetric duality for multiobjective variational problems abstract a mond weir type multiobjective variational mixed integer symmetric dual program over arbitrary cones is formulated applying the separability and generalized f convexity on the functions involved weak strong and converse duality theorems are established self duality theorem is proved a close relationship between these variational problems and static symmetric dual minimax mixed integer multiobjective programming problems is also presented

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

stochastic collocation methods on unstructured grids in high dimensions via interpolation in this paper we propose a method for conducting stochastic collocation on arbitrary sets of nodes to accomplish this we present the framework of least orthogonal interpolation which allows one to construct interpolation polynomials based on arbitrarily located grids in arbitrary dimensions these interpolation polynomials are constructed as a subspace of the family of orthogonal polynomials corresponding to the probability distribution function on stochastic space this feature enables one to conduct stochastic collocation simulations in practical problems where one cannot adopt some popular node selections such as sparse grids or cubature nodes we present in detail both the mathematical properties of the least orthogonal interpolation and its practical implementation algorithm numerical benchmark problems are also presented to demonstrate the efficacy of the method

uncertainty quantification in chemical systems summary we demonstrate the use of multiwavelet spectral polynomial chaos techniques for uncertainty quantification in non isothermal ignition of a methane air system we employ bayesian inference for identifying the probabilistic representation of the uncertain parameters and propagate this uncertainty through the ignition process we analyze the time evolution of moments and probability density functions of the solution we also examine the role and significance of dependence among the uncertain parameters we finish with a discussion of the role of non linearity and the performance of the algorithm copyright q 2009 john wiley sons ltd

transverse vibrations of elastically connected double string complex system part i free vibrations this paper analyzes the forced transverse vibrations of an elastically connected double string complex continuous system the general solutions of forced vibrations of strings subjected to arbitrarily distributed continuous loads are found by using the method of expansion in a series of the mode shape functions different cases of exciting loadings are analyzed the action of stationary harmonic loads and moving concentrated forces is considered vibrations caused by the harmonic exciting forces are discussed and conditions of resonance and dynamic vibration absorption are formulated thus the string type dynamic absorber can be used to suppress the excessive vibrations of corresponding string systems the dynamic string type damper is a new concept of a continuous dynamic vibration absorber cdva it is shown that a corresponding two degree of freedom discrete system is an analogue of an elastically connected double string complex system theoretical analysis is illustrated by a numerical example

uncertainty quantification given discontinuous model response and a limited number of model runs we outline a methodology for forward uncertainty quantification in systems with uncertain parameters discontinuous model response and a limited number of model runs our approach involves two stages first we detect the discontinuity with bayesian inference thus obtaining a probabilistic representation of the discontinuity curve for arbitrarily distributed input parameters then employing the rosenblatt transform we construct spectral representations of the uncertain model output using polynomial chaos pc expansions on either side of the discontinuity curve leading to an averaged pc representation of the forward model response that allows efficient uncertainty quantification we obtain pc modes by either orthogonal projection or bayesian inference and argue for a hybrid approach that targets a balance between the accuracy provided by the orthogonal projection and the flexibility provided by the bayesian inference the uncertain model output is then computed by taking an ensemble average over pc expansions corresponding to sampled realizations of the discontinuity curve the methodology is demonstrated on synthetic examples of discontinuous model response with adjustable sharpness and structure

multiobjective mixed symmetric duality involving cones a pair of multiobjective mixed symmetric dual programs is formulated over arbitrary cones weak strong converse and self duality theorems are proved for these programs under k preinvexity and k pseudoinvexity assumptions this mixed symmetric dual formulation unifies the symmetric dual formulations of suneja et al 2002 14 and khurana 2005 15

wave synthesis in a non linear stretched string with time varying length or tension abstract this paper is concerned with the numerical tone synthesis of a non linear stretched string the length or tension of which varies with time 1 at a constant rate 2 at a cosine function rate and 3 at a rate determined by a handwritten input function listening to the results for the analytical waveforms transformed by a digital to analog converter indicates that the case of the constant rate time varying length corresponds closely to a guitar string positive or negative feedback compensation for the analytically obtained frequency brings the analytical waveforms close to the numerically calculated waveforms a simple method is proposed with lower order polynomials for smoothly connecting two discontinuous waveforms into continuous waveforms a fourier series synthesizer procedure is described to aid rapid real time synthesis of tones of desired timbre

bounded input bounded output stability of a damped nonlinear string in this paper we prove that the displacement of a damped nonlinear string under bounded distributed forces is bounded the proof is achieved by showing that an energy like lyapunov function corresponding to the string is bounded

discontinuity detection in multivariate space for stochastic simulations edge detection has traditionally been associated with detecting physical space jump discontinuities in one dimension e g seismic signals and two dimensions e g digital images hence most of the research on edge detection algorithms is restricted to these contexts high dimension edge detection can be of significant importance however for instance stochastic variants of classical differential equations not only have variables in space time dimensions but additional dimensions are often introduced to the problem by the nature of the random inputs the stochastic solutions to such problems sometimes contain discontinuities in the corresponding random space and a prior knowledge of jump locations can be very helpful in increasing the accuracy of the final solution traditional edge detection methods typically require uniform grid point distribution they also often involve the computation of gradients and or laplacians which can become very complicated to compute as the number of dimensions increases the polynomial annihilation edge detection method on the other hand is more flexible in terms of its geometric specifications and is furthermore relatively easy to apply this paper discusses the numerical implementation of the polynomial annihilation edge detection method to high dimensional functions that arise when solving stochastic partial differential equations

pade legendre approximants for uncertainty analysis with discontinuous response surfaces a novel uncertainty propagation method for problems characterized by highly non linear or discontinuous system responses is presented the approach is based on a pade legendre pl formalism which does not require modifications to existing computational tools non intrusive approach and it is a global method the paper presents a novel pl method for problems in multiple dimensions which is non trivial in the pade literature in addition a filtering procedure is developed in order to minimize the errors introduced in the approximation close to the discontinuities the numerical examples include fluid dynamic problems characterized by shock waves a simple dual throat nozzle problem with uncertain initial state and the turbulent transonic flow over a transonic airfoil where the flight conditions are assumed to be uncertain results are presented in terms of statistics of both shock position and strength and are compared to monte carlo simulations

a continuous analogue of the tensor train decomposition abstract we develop new approximation algorithms and data structures for representing and computing with multivariate functions using the functional tensor train ft a continuous extension of the tensor train tt decomposition the ft represents functions using a tensor train ansatz by replacing the three dimensional tt cores with univariate matrix valued functions the main contribution of this paper is a framework to compute the ft that employs adaptive approximations of univariate fibers and that is not tied to any tensorized discretization the algorithm can be coupled with any univariate linear or nonlinear approximation procedure we demonstrate that this approach can generate multivariate function approximations that are several orders of magnitude more accurate for the same cost than those based on the conventional approach of compressing the coefficient tensor of a tensor product basis our approach is in the spirit of other continuous computation packages such as chebfun and yields an algorithm which requires the computation of continuous matrix factorizations such as the lu and qr decompositions of vector valued functions to support these developments we describe continuous versions of an approximate maximum volume cross approximation algorithm and of a rounding algorithm that re approximates an ft by one of lower ranks we demonstrate that our technique improves accuracy and robustness compared to tt and quantics tt approaches with fixed parameterizations of high dimensional integration differentiation and approximation of functions with local features such as discontinuities and other nonlinearities

bayesian inference with optimal maps we present a new approach to bayesian inference that entirely avoids markov chain simulation by constructing a map that pushes forward the prior measure to the posterior measure existence and uniqueness of a suitable measure preserving map is established by formulating the problem in the context of optimal transport theory we discuss various means of explicitly parameterizing the map and computing it efficiently through solution of an optimization problem exploiting gradient information from the forward model when possible the resulting algorithm overcomes many of the computational bottlenecks associated with markov chain monte carlo advantages of a map based representation of the posterior include analytical expressions for posterior moments and the ability to generate arbitrary numbers of independent posterior samples without additional likelihood evaluations or forward solves the optimization approach also provides clear convergence criteria for posterior approximation and facilitates model selection through automatic evaluation of the marginal likelihood we demonstrate the accuracy and efficiency of the approach on nonlinear inverse problems of varying dimension involving the inference of parameters appearing in ordinary and partial differential equations

multi output local gaussian process regression applications to uncertainty quantification we develop an efficient bayesian uncertainty quantification framework using a novel treed gaussian process model the tree is adaptively constructed using information conveyed by the observed data about the length scales of the underlying process on each leaf of the tree we utilize bayesian experimental design techniques in order to learn a multi output gaussian process the constructed surrogate can provide analytical point estimates as well as error bars for the statistics of interest we numerically demonstrate the effectiveness of the suggested framework in identifying discontinuities local features and unimportant dimensions in the solution of stochastic differential equations

technical note on symmetric duality in multiobjective programming some remarks on recent results abstract certain shortcomings are pointed out in some recent works of das and nanda european journal of operational research 97 1997 167 171 and kim et al european journal of operational research 107 1998 686 691 which show that these studies are highly restricted

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

efficient localization of discontinuities in complex computational simulations surrogate models for computational simulations are input output approximations that allow computationally intensive analyses such as uncertainty propagation and inference to be performed efficiently when a simulation output does not depend smoothly on its inputs the error and convergence rate of many approximation methods deteriorate substantially this paper details a method for efficiently localizing discontinuities in the input parameter domain so that the model output can be approximated as a piecewise smooth function the approach comprises an initialization phase which uses polynomial annihilation to assign function values to different regions and thus seed an automated labeling procedure followed by a refinement phase that adaptively updates a kernel support vector machine representation of the separating surface via active learning the overall approach avoids structured grids and exploits any available simplicity in the geometry of the separating surface thus reducing the number of model

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

some nonlinear degenerate wave equations on etudie la resolubilite locale du probleme aux valeurs limites et initiales suivant 2 u t 2 m gradu x t 2 dx u 0 dans 0 u 0 sur w 0 u 0 0 u t 0 dans ou m r est une fonction c 1 non negative sur 0 un domaine borne de r n a frontiere lisse w

duality and reciprocity for vector programming abstract starting from the well known result that the lagrange multipliers measure the sensitivity of the optimum value function with respect to the constraint parameters we develop a generalized duality theory for vector programming problems then we consider the connection stated by the complementarity property of the optimum value function with the saturation of the constraints and introduce reciprocal problems with many constraints and a vector valued objective function for such problems we derive some monotonicity properties significantly related to the active constraints and apply these results to the sensitivity analysis of economic problems

a parametrically driven harmonic analysis of a non linear stretched string with time varying length abstract this paper is concerned with the first second and third harmonic lateral vibrations of a string having a forced periodic small length variation account is taken of the non linear term arising from the correction to the tension of the string due to local elongation it is found that increasing the amplitude of the length variation of the string results in a narrowing of the frequency response range of each of the harmonic oscillation components both numerical and analytical calculations of the harmonic frequency responses have been carried out the relationships for initial values between the original differential equation and the approximate differential equations for the harmonic responses are discussed in detail