infinite dimensional integration and the multivariate decomposition method abstract we further develop the multivariate decomposition method mdm for the lebesgue integration of functions of infinitely many variables x 1 x 2 x 3 with respect to a corresponding product of a one dimensional probability measure the method is designed for functions that admit a dominantly convergent decomposition f u f u where u runs over all finite subsets of positive integers and for each u i 1 i k the function f u depends only on x i 1 x i k although a number of concepts of infinite dimensional integrals have been used in the literature questions of uniqueness and compatibility have mostly not been studied we show that under appropriate convergence conditions the lebesgue integral equals the anchored integral independently of the anchor for approximating the integral the mdm assumes that point values of f u are available for important subsets u at some known cost in this paper we introduce a new setting in which it is assumed that each f u belongs to a normed space f u and that bounds b u on f u f u are known this contrasts with the assumption in many papers that weights u appearing in the norm of the infinite dimensional function space are somehow known often such weights u were determined by minimizing an error bound depending on the b u the u and the chosen algorithm resulting in weights that depend on the algorithm in contrast in this paper only the bounds b u are assumed to be known we give two examples in which we specialize the mdm in the first case f u is the u fold tensor product of an anchored reproducing kernel hilbert space in the second case it is a particular non hilbert space for integration over an unbounded domain

structure properties of min max systems and existence of global cycle time studies min max systems which are dynamic systems including three operations min max with unknown or stochastic parameters some sufficient conditions are given for the existence of global cycle times our results are based on structure properties of min max systems

on the equivalence of dynamically orthogonal and bi orthogonal methods theory and numerical simulations abstract the karhunen loeve kl decomposition provides a low dimensional representation for random fields as it is optimal in the mean square sense although for many stochastic systems of practical interest described by stochastic partial differential equations spdes solutions possess this low dimensional character they also have a strongly time dependent form and to this end a fixed in time basis may not describe the solution in an efficient way motivated by this limitation of standard kl expansion sapsis and lermusiaux 2009 26 developed the dynamically orthogonal do field equations which allow for the simultaneous evolution of both the spatial basis where uncertainty lives but also the stochastic characteristics of uncertainty recently cheng et al 2013 28 introduced an alternative approach the bi orthogonal bo method which performs the exact same tasks i e it evolves the spatial basis and the stochastic characteristics of uncertainty in the current work we examine the relation of the two approaches and we prove theoretically and illustrate numerically their equivalence in the sense that one method is an exact reformulation of the other we show this by deriving a linear and invertible transformation matrix described by a matrix differential equation that connects the bo and the do solutions we also examine a pathology of the bo equations that occurs when two eigenvalues of the solution cross resulting in an instantaneous infinite speed internal rotation of the computed spatial basis we demonstrate that despite the instantaneous duration of the singularity this has important implications on the numerical performance of the bo approach on the other hand it is observed that the bo is more stable in nonlinear problems involving a relatively large number of modes several examples linear and nonlinear are presented to illustrate the do and bo methods as well as their equivalence

construction algorithms for polynomial lattice rules for multivariate integration we introduce a new construction algorithm for digital nets for integration in certain weighted tensor product hilbert spaces the first weighted hilbert space we consider is based on walsh functions dick and pillichshammer calculated the worst case error for integration using digital nets for this space here we extend this result to a special construction method for digital nets based on polynomials over finite fields this result allows us to find polynomials which yield a small worst case error by computer search we prove an upper bound on the worst case error for digital nets obtained by such a search algorithm which shows that the convergence rate is best possible and that strong tractability holds under some condition on the weights we extend the results for the weighted hilbert space based on walsh functions to weighted sobolev spaces in this case we use randomly digitally shifted digital nets the construction principle is the same as before only the worst case error is slightly different again digital nets obtained from our search algorithm yield a worst case error achieving the optimal rate of convergence and as before strong tractability holds under some condition on the weights these results show that such a construction of digital nets yields the until now best known results of this kind and that our construction methods are comparable to the construction methods known for lattice rules we conclude the article with numerical results comparing the expected worst case error for randomly digitally shifted digital nets with those for randomly shifted lattice rules

a fast monte carlo method with a reduced basis of control variates applied to uncertainty propagation and bayesian estimation abstract the reduced basis control variate monte carlo method was introduced recently in s boyaval t lelievre a variance reduction method for parametrized stochastic differential equations using the reduced basis paradigm commun math sci 8 2010 735 762 special issue mathematical issues on complex fluids as an improved monte carlo method for the fast estimation of many parametrized expected values at many parameter values we provide here a more complete analysis of the method including precise error estimates and convergence results we also numerically demonstrate that it can be useful to some parametric frameworks in uncertainty quantification in particular i the case where the parametrized expectation is a scalar output of the solution to a partial differential equation pde with stochastic coefficients an uncertainty propagation problem and ii the case where the parametrized expectation is the bayesian estimator of a scalar output in a similar pde context moreover in each case a pde has to be solved many times for many values of its coefficients this is costly and we also use a reduced basis of pde solutions like in s boyaval c le bris y maday n nguyen a patera a reduced basis approach for variational problems with stochastic parameters application to heat conduction with variable robin coefficient comput methods appl mech eng 198 2009 3187 3206 to our knowledge this is the first combination of various reduced basis ideas with a view to reducing as much as possible the computational cost of a simple versatile monte carlo approach to uncertainty quantification

an efficient ensemble algorithm for numerical approximation of stochastic stokes darcy equations abstract we propose and analyze an efficient ensemble algorithm for fast computation of multiple realizations of the stochastic stokes darcy model with a random hydraulic conductivity tensor the algorithm results in a common coefficient matrix for all realizations at each time step making solving the linear systems much less expensive while maintaining comparable accuracy to traditional methods that compute each realization separately moreover it decouples the stokes darcy system into two smaller sub physics problems which reduces the size of the linear systems and allows parallel computation of the two sub physics problems we prove the ensemble method is long time stable and first order in time convergent under a time step condition and two parameter conditions numerical examples are presented to support the theoretical results and illustrate the application of the algorithm

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

quasi monte carlo integration for affine parametric elliptic pdes local supports and product weights we analyze convergence rates of first order quasi monte carlo qmc integration with randomly shifted lattice rules and for higher order interlaced polynomial lattice rules for a class of countably parametric integrands that result from linear functionals of solutions of linear elliptic diffusion equations with affine parametric uncertain coefficient function a x y bar a x sum j geq 1 y j psi j x in a bounded domain d subset mathbb r d extending the result in f y kuo c schwab and i h sloan siam j numer anal 50 2012 pp 3351 3374 where psi j was assumed to have global support in the domain d we assume in the present paper that supp psi j is localized in d and that we have control on the overlaps of these supports under these conditions we prove dimension independent convergence rates in 1 2 1 of randomly shifted lattice rules with product weights and corresponding higher order convergence rates by higher order interlaced polynomial lattice rule

dynamically orthogonal field equations for continuous stochastic dynamical systems abstract in this work we derive an exact closed set of evolution equations for general continuous stochastic fields described by a stochastic partial differential equation spde by hypothesizing a decomposition of the solution field into a mean and stochastic dynamical component we derive a system of field equations consisting of a partial differential equation pde for the mean field a family of pdes for the orthonormal basis that describe the stochastic subspace where the stochasticity lives as well as a system of stochastic differential equations that defines how the stochasticity evolves in the time varying stochastic subspace these new evolution equations are derived directly from the original spde using nothing more than a dynamically orthogonal condition on the representation of the solution if additional restrictions are assumed on the form of the representation we recover both the proper orthogonal decomposition equations and the generalized polynomial chaos equations we apply this novel methodology to two cases of two dimensional viscous fluid flows described by the navier stokes equations and we compare our results with monte carlo simulations

recursive parameter estimation using incomplete data abstract stochastic approximation procedures are considered for the estimation of parameters using incomplete data one procedure is stated and illustrated which often leads to asymptotically efficient estimators others are developed which although possibly not optimal in the above sense will be very much easier to apply this will be particularly advantageous when quick recursive estimates are required examples are given and a link is made between one of the sub optimal methods and the em algorithm author

estimating mean dimensionality of analysis of variance decompositions analysis of variance anova is now often applied to functions defined on the unit cube where it serves as a tool for the exploratory analysis of functions the mean dimension of a function defined as a natural weighted combination of its anova mean squares provides one measure of how hard or easy it is to integrate the function by quasi monte carlo sampling this article presents some new identities relating the mean dimension and some analogously defined higher moments to the variance importance measures of i m sobol as a result we are able to measure the mean dimension of certain functions arising in computational finance we produce an unbiased and nonnegative estimate of the variance contribution of the highest order interaction that avoids the cancellation problems of previous estimates in an application to extreme value theory we find that among other things the minimum of d independent u 0 1 random variables has a mean dimension of 2 d 1 d 3

a construction of polynomial lattice rules with small gain coefficients in this paper we construct polynomial lattice rules which have in some sense small gain coefficients using a component by component approach the gain coefficients as introduced by owen indicate to what degree the method improves upon monte carlo we show that the variance of an estimator based on a scrambled polynomial lattice rule constructed component by component decays at a rate of n 2 1 for all 0 assuming that the function under consideration has bounded variation of order for some 0 1 and where n denotes the number of quadrature points an analogous result is obtained for korobov polynomial lattice rules it is also established that these rules are almost optimal for the function space considered in this paper furthermore we discuss the implementation of the component by component approach and show how to reduce the computational cost associated with it finally we present numerical results comparing scrambled polynomial lattice rules and scrambled digital nets

multilevel monte carlo path simulation we show that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using monte carlo path simulations in the simplest case of a lipschitz payoff and a euler discretisation the computational cost to achieve an accuracy of o e is reduced from o e 3 to o e 2 log e 2 the analysis is supported by numerical results showing significant computational savings

attractor local dimensionality nonlinear energy transfers and finite time instabilities in unstable dynamical systems with applications to two dimensional fluid flows we examine the geometry of the finite dimensional attractor associated with fluid flows described by navier stokes equations and relate its nonlinear dimensionality to energy exchanges between dynamical components modes of the flow specifically we use a stochastic framework based on the dynamically orthogonal equations to perform efficient order reduction and describe the stochastic attractor in the reduced order phase space in terms of the associated probability measure we introduce the notion of local fractal dimensionality to describe the geometry of the attractor and we establish a connection with the number of positive finite time lyapunov exponents subsequently we illustrate in specific fluid flows that the low dimensionality of the stochastic attractor is caused by the synergistic activity of linearly unstable and stable modes as well as the action of the quadratic terms in particular we illustrate the connection of the low dimensionality of the attractor with the circulation of energy i from the mean flow to the unstable modes due to their linearly unstable character ii from the unstable modes to the stable ones due to a nonlinear energy transfer mechanism and iii from the stable modes back to the mean due to the linearly stable character of these modes

the balance problem of min max systems is co np hard abstract min max systems are timed discrete event dynamic systems containing min max and plus operations a min max system is balanced if it admits a global cycle time generalized eigenvalue for every possible assignment of parameters this paper proves that the testing of the balance property for min max systems is co np hard by showing that the test of inseparability is co np hard the latter result is established on a new complexity result namely that the testing of existence of non trivial fixed points for monotone boolean functions is np complete

analysis of quasi monte carlo methods for elliptic eigenvalue problems with stochastic coefficients we consider the forward problem of uncertainty quantification for the generalised dirichlet eigenvalue problem for a coercive second order partial differential operator with random coefficients motivated by problems in structural mechanics photonic crystals and neutron diffusion the pde coefficients are assumed to be uniformly bounded random fields represented as infinite series parametrised by uniformly distributed i i d random variables the expectation of the fundamental eigenvalue of this problem is computed by a truncating the infinite series which define the coefficients b approximating the resulting truncated problem using lowest order conforming finite elements and a sparse matrix eigenvalue solver and c approximating the resulting finite but high dimensional integral by a randomly shifted quasi monte carlo lattice rule with specially chosen generating vector we prove error estimates for the combined error which depend on the truncation dimension s the finite element mesh diameter h and the number of quasi monte carlo samples n under suitable regularity assumptions our bounds are of the particular form mathcal o h 2 n 1 delta where delta 0 is arbitrary and the hidden constant is independent of the truncation dimension which needs to grow as h rightarrow 0 and n rightarrow infty as for the analogous pde source problem the conditions under which our error bounds hold depend on a parameter p in 0 1 representing the summability of the terms in the series expansions of the coefficients although the eigenvalue problem is nonlinear which means it is generally considered harder than the source problem in almost all cases p ne 1 we obtain error bounds that converge at the same rate as the corresponding rate for the source problem the proof involves a detailed study of the regularity of the fundamental eigenvalue as a function of the random parameters as a key intermediate result in the analysis we prove that the spectral gap between the fundamental and the second eigenvalues is uniformly positive over all realisations of the random problem

qmc integration for lognormal parametric elliptic pdes local supports and product weights we analyze convergence rates of quasi monte carlo qmc quadratures for countably parametric solutions of linear elliptic partial differential equations pde in divergence form with log gaussian diffusion coefficient based on the error bounds in nichols and kuo j complex 30 4 444 468 2014 https doi org 10 1016 j jco 2014 02 004 we prove for representations of the gaussian random field pde input with locally supported basis functions and for continuous piecewise polynomial finite element discretizations in the physical domain novel qmc error bounds in weighted spaces with product weights that exploit localization of supports of the basis elements representing the input gaussian random field in this case the cost of the fast component by component algorithm for constructing the qmc points scales linearly in terms of the integration dimension the qmc convergence rate mathcal o n 1 delta independent of the parameter space dimension s is achieved under weak summability conditions on the expansion coefficients

sparse approximation of multilinear problems with applications to kernel based methods in uq we provide a framework for the sparse approximation of multilinear problems and show that several problems in uncertainty quantification fit within this framework in these problems the value of a multilinear map has to be approximated using approximations of different accuracy and computational work of the arguments of this map we propose and analyze a generalized version of smolyak s algorithm which provides sparse approximation formulas with convergence rates that mitigate the curse of dimension that appears in multilinear approximation problems with a large number of arguments we apply the general framework to response surface approximation and optimization under uncertainty for parametric partial differential equations using kernel based approximation the theoretical results are supplemented by numerical experiments

the optimal multilevel monte carlo approximation of the stochastic drift diffusion poisson system abstract existence and local uniqueness theorems for weak solutions of a system consisting of the drift diffusion poisson equations and the poisson boltzmann equation all with stochastic coefficients are presented for the numerical approximation of the expected value of the solution of the system we develop a multi level monte carlo mlmc finite element method fem and we analyze its rate of convergence and its computational complexity this allows to find the optimal choice of discretization parameters finally numerical results show the efficiency of the method applications are among others noise and fluctuations in nanoscale transistors in field effect bio and gas sensors and in nanopores

gaussian processes with built in dimensionality reduction uncertainty quantification uq tasks such as model calibration uncertainty propagation and optimization under uncertainty typically require several thousand evaluations of the underlying computer codes to cope with the cost of simulations one replaces the real response surface with a cheap surrogate based e g on polynomial chaos expansions neural networks support vector machines or gaussian processes gp however the number of simulations required to learn a generic multivariate response grows exponentially as the input dimension increases this curse of dimensionality can only be addressed if the response exhibits some special structure that can be discovered and exploited a wide range of physical responses exhibit a special structure known as an active subspace as an as is a linear manifold of the stochastic space characterized by maximal response variation the idea is that one should first identify this low dimensional manifold project the high dimensional input onto it and then link the projection to the output if the dimensionality of the as is low enough then learning the link function is a much easier problem than the original problem of learning a high dimensional function the classic approach to discovering the as requires gradient information a fact that severely limits its applicability furthermore and partly because of its reliance to gradients it is not able to handle noisy observations the latter is an essential trait if one wants to be able to propagate uncertainty through stochastic simulators e g through molecular dynamics codes in this work we develop a probabilistic version of as which is gradient free and robust to observational noise our approach relies on a novel gaussian process regression with built in dimensionality reduction in particular the as is represented as an orthogonal projection matrix that serves as yet another covariance function hyper parameter to be estimated from the data to train the model we design a two step maximum likelihood optimization procedure that ensures the orthogonality of the projection matrix by exploiting recent results on the stiefel manifold i e the manifold of matrices with orthogonal columns the additional benefit of our probabilistic formulation is that it allows us to select the dimensionality of the as via the bayesian information criterion we validate our approach by showing that it can discover the right as in synthetic examples without gradient information using both noiseless and noisy observations we demonstrate that our method is able to discover the same as as the classical approach in a challenging one hundred dimensional problem involving an elliptic stochastic partial differential equation with random conductivity finally we use our approach to study the effect of geometric and material uncertainties in the propagation of solitary waves in a one dimensional granular system

projection approximation subspace tracking subspace estimation plays an important role in a variety of modern signal processing applications we present a new approach for tracking the signal subspace recursively it is based on a novel interpretation of the signal subspace as the solution of a projection like unconstrained minimization problem we show that recursive least squares techniques can be applied to solve this problem by making an appropriate projection approximation the resulting algorithms have a computational complexity of o nr where n is the input vector dimension and r is the number of desired eigencomponents simulation results demonstrate that the tracking capability of these algorithms is similar to and in some cases more robust than the computationally expensive batch eigenvalue decomposition relations of the new algorithms to other subspace tracking methods and numerical issues are also discussed

the smoothing effect of integration in r d and the anova decomposition this paper studies the anova decomposition of a d variate function f dened on the whole of r d where f is the maximum of a smooth function and zero or f could be the absolute value of a smooth function our study is motivated by option pricing problems we show that under suitable conditions all terms of the anova decomposition except the one of highest order can have unlimited smoothness in particular this is the case for arithmetic asian options with both the standard and brownian bridge constructions of the brownian motion 2010 mathematics subject classication primary 41a63 41a99 secondary 65d30

time separations of cyclic event rule systems with min max timing constraints the analysis of the time separations of events is a fundamental problem in the design and evaluation of discrete event systems important progresses have been made based on the event rule system model in the last decade the existing results for event rule systems with min and max constraints can be summarized briefly as the exact evaluation of time separations for acyclic systems is np complete for cyclic systems the structural condition of being tightly coupled is sufficient for long term time separations of events to be bounded in this paper we establish a necessary and sufficient structural boundedness condition uniformity for cyclic event rule systems with both min and max constraints tightly coupled systems are shown to be a special class of uniform systems the well known cas algorithm for finding bounds on long term time separations is adapted to find finite bounds for uniform systems our results are obtained by exploring the algebraic structures guiding the evolution of the systems

technical communique on stabilization of min max systems this note studies min max systems which are dynamic systems involving three operations min max we present a feedback stabilization policy for a class of min max systems such that the closed loop systems have a global cycle time which is the same as the maximal lyapunov exponent of the open loop systems our results are based on structural properties of min max systems

stability analysis of hierarchical tensor methods for time dependent pdes abstract we address the question of whether it is possible to stably integrate time dependent high dimensional linear pdes with hierarchical tensor methods and explicit time stepping schemes to this end we develop sufficient conditions for stability and convergence of tensor solutions evolving on tensor manifolds with constant rank we also argue that the applicability of pde solvers with explicit time stepping may be limited by time step restriction dependent on the dimension of the problem numerical applications are presented and discussed for variable coefficients linear hyperbolic and parabolic pdes

multi level monte carlo weak galerkin method for elliptic equations with stochastic jump coefficients in this paper we present a multi level monte carlo weak galerkin method for solving elliptic equations with stochastic jump coefficients the multi level monte carlo technique balances the spatial approximation error and the sampling error the weak galerkin technique is a stable and high order accurate method which can easily handle deterministic partial differential equations with complex geometries or jump coefficients given by each sample and this method is also able to capture highly complex solutions exhibiting discontinuities or oscillations with high resolution comparing with the standard monte carlo method by using the multi level monte carlo weak galerkin method the computational cost can be sharply reduced to log linear complexity with respect to the degree of freedom in spatial direction the numerical experiments verify the efficiency of our algorithms

solving boolean equations using rosop forms boolean equations are important tools in digital logic previous algorithms for solving boolean equations are based on the boolean algebra of disjoint sop forms in this paper we develop a new boolean algebra with more efficient boolean operation algorithms called the reduced ordered sop rosop forms which are canonical representations rosops are closely related to the well known obdd data structure the results here also show the algebraic structure of obdds

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

higher order quasi monte carlo integration for bayesian pde inversion abstract we analyze combined quasi monte carlo quadrature and finite element approximations in bayesian estimation of solutions to countably parametric operator equations with holomorphic dependence on the parameters as considered in schillings and schwab 2014 such problems arise in numerical uncertainty quantification and in bayesian inversion of operator equations with distributed uncertain inputs such as uncertain coefficients uncertain domains or uncertain source terms and boundary data we show that the parametric bayesian posterior densities belong to a class of weighted bochner spaces of functions of countably many variables with a particular structure of the qmc quadrature weights up to a problem dependent and possibly large finite dimension s product weights can be used and beyond this dimension weighted spaces with so called spod weights recently introduced in dick et al 2014 are used to describe the solution regularity we establish error bounds for higher order quasi monte carlo quadrature for the bayesian estimation based on dick et al 2016 it implies in particular regularity of the parametric solution and of the countably parametric bayesian posterior density in spod smoothness driven product and order dependent weighted spaces of integrand functions this in turn implies that the quasi monte carlo quadrature methods in dick et al 2014 are applicable to these problem classes with dimension independent convergence rates o n 1 p of n point hoqmc approximated bayesian estimates where 0 p 1 depends only on the sparsity class of the uncertain input in the bayesian estimation fast component by component cbc for short construction gantner and schwab 2016 allow efficient deterministic bayesian estimation with up to 1 0 4 parameters

a high performance computing and sensitivity analysis algorithm for stochastic many particle wave scattering we present a high performance computing framework for quantifying uncertainties in the propagation of acoustic waves through a stochastic media comprising a large number of three dimensional particles we subsequently describe an efficient postprocessing approach using our framework to statistically quantify the sensitivity of the uncertainties with respect to input parameters that govern the stochasticity in the model the stochasticity arises through the random positions and orientations of the component particles in the media simulation even for a single deterministic three dimensional configuration is inherently difficult because of the large number of particles the stochasticity leads to a larger dimensional model involving three spatial variables and additional stochastic variables and accounting for uncertainty in key parameters of the input probability distributions leads to prohibitive computational complexity in the first part of our paper we describe a high performance computing framework f

optimal randomized multilevel algorithms for infinite dimensional integration on function spaces with anova type decomposition in this paper we consider the infinite dimensional integration problem on weighted reproducing kernel hilbert spaces with norms induced by an underlying function space decomposition of analysis of variance type the weights model the relative importance of different groups of variables we present new randomized multilevel algorithms to tackle this integration problem and prove upper bounds for their randomized error furthermore we provide in this setting the first nontrivial lower error bounds for general randomized algorithms which in particular may be adaptive or nonlinear these lower bounds show that our multilevel algorithms are optimal our analysis refines and extends the analysis provided in f j hickernell t muller gronbach b niu and k ritter j complexity 26 2010 pp 229 254 and our error bounds improve substantially on the error bounds presented there as an illustrative example we discuss the unanchored sobolev space and employ randomized quasi monte carlo multilevel a

multi level monte carlo finite element method for elliptic pdes with stochastic coefficients in monte carlo methods quadrupling the sample size halves the error in simulations of stochastic partial differential equations spdes the total work is the sample size times the solution cost of an instance of the partial differential equation a multi level monte carlo method is introduced which allows in certain cases to reduce the overall work to that of the discretization of one instance of the deterministic pde the model problem is an elliptic equation with stochastic coefficients multi level monte carlo errors and work estimates are given both for the mean of the solutions and for higher moments the overall complexity of computing mean fields as well as k point correlations of the random solution is proved to be of log linear complexity in the number of unknowns of a single multi level solve of the deterministic elliptic problem numerical examples complete the theoretical analysis

infinite dimensional integration on weighted hilbert spaces we study the numerical integration problem for functions with infinitely many variables the functions we want to integrate are from a reproducing kernel hilbert space which is endowed with a weighted norm we study the worst case e complexity which is defined as the minimal cost among all algorithms whose worst case error over the hilbert space unit ball is at most e here we assume that the cost of evaluating a function depends polynomially on the number of active variables the infinite dimensional integration problem is polynomially tractable if the e complexity is bounded by a constant times a power of 1 e the smallest such power is called the exponent of tractability first we study finite order weights we provide improved lower bounds for the exponent of tractability for general finite order weights and improved upper bounds for three newly defined classes of finite order weights the constructive upper bounds are obtained by multilevel algorithms that use for each level quasimonte carlo integration points whose projections onto specific sets of coordinates exhibit a small discrepancy the newly defined finite intersection weights model the situation where each group of variables interacts with at most other groups of variables where is some fixed number for these weights we obtain a sharp upper bound this is the first class of weights for which the exact exponent of tractability is known for any possible decay of the weights and for any polynomial degree of the cost function for the other two classes of finite order weights our upper bounds are sharp if e g the decay of the weights is fast or slow enough we extend our analysis to the case of arbitrary weights in particular from our results for finite order weights we conclude a lower bound on the exponent of tractability for arbitrary weights and a constructive upper bound for product weights

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

the error bounds and tractability of quasi monte carlo algorithms in infinite dimension dimensionally unbounded problems are frequently encountered in practice such as in simulations of stochastic processes in particle and light transport problems and in the problems of mathematical finance this paper considers quasi monte carlo integration algorithms for weighted classes of functions of infinitely many variables in which the dependence of functions on successive variables is increasingly limited the dependence is modeled by a sequence of weights the integrands belong to rather general reproducing kernel hilbert spaces that can be decomposed as the direct sum of a series of their subspaces each subspace containing functions of only a finite number of variables the theory of reproducing kernels is used to derive a quadrature error bound which is the product of two terms the generalized discrepancy and the generalized variation tractability means that the minimal number of function evaluations needed to reduce the initial integration error by a factor s is bounded by ce p for some exponent p and some positive constant c the e exponent of tractability is defined as the smallest power of e 1 in these bounds it is shown by using monte carlo quadrature that the e exponent is no greater than 2 for these weighted classes of integrands under a somewhat stronger assumption on the weights and for a popular choice of the reproducing kernel it is shown constructively using the halton sequence that the e exponent of tractability is 1 which implies that infinite dimensional integration is no harder than one dimensional integration

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

good interlaced polynomial lattice rules for numerical integration in weighted walsh spaces quadrature rules using higher order digital nets and sequences are known to exploit the smoothness of a function for numerical integration and to achieve an improved rate of convergence as compared to classical digital nets and sequences for smooth functions a construction principle of higher order digital nets and sequences based on a digit interlacing function was introduced in dick 2007 which interlaces classical digital nets or sequences whose number of components is a multiple of the dimension in this paper we study the use of polynomial lattice point sets for interlaced components we call quadrature rules using such point sets interlaced polynomial lattice rules we consider weighted walsh spaces containing smooth functions and derive two upper bounds on the worst case error for interlaced polynomial lattice rules both of which can be employed as a quality criterion for the construction of interlaced polynomial lattice rules we investigate the component by component construction and the korobov construction as a means of explicit constructions of good interlaced polynomial lattice rules that achieve the optimal rate of the worst case error through this approach we are able to obtain a good dependence of the worst case error bounds on the dimension under certain conditions on the weights while significantly reducing the construction cost as compared to higher order polynomial lattice rules

recursive k distribution parameter estimation recursive estimation of the parameter of the k distribution is studied and tested the probability density function pdf of the k distribution is seen as a mixture pdf allowing the application of titterington s recursive expectation maximization em technique under mild conditions the technique produces estimates that are strongly consistent and asymptotically normal for the k distribution the complete data information matrix required by the recursive em has an explicit form making the algorithm easy to implement the algorithm is tested using k distributed data with both constant and time varying parameter for the constant parameter case recursive em estimates are compared to numerical maximization of the likelihood for the time varying parameter case recursive em estimates are compared to estimates obtained using a fast routine from the literature and implemented by sliding a rectangular window over the observations

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

generalisation of euler lagrange equations to find min max optimal solution of uncertain systems a major problem in optimal control theory is existing uncertainty in initial state cost function or in state equations in this paper a generalised euler lagrange approach to find min max optimal solution of uncertain systems with uncertain or certain cost function using calculus of variation is considered the work is based on an admissible system path or a trajectory which minimises the maximum value of the functional over all uncertainty first of all a new form of euler lagrange conditions for uncertain systems is presented then several cases are indicated where final condition can be specified or free also necessary conditions are introduced to existence of min max optimal solution of the uncertain systems also the method is generalised when the uncertainties are bounded with using pontryagin s minimum principle finally efficiency of the proposed methods is verified through some examples

recursive maximum likelihood algorithm for dependent observations a recursive maximum likelihood algorithm rml is proposed that can be used when both the observations and the hidden data have continuous values and are statistically dependent between different time samples the algorithm recursively approximates the probability density functions of the observed and hidden data by analytically computing the integrals with respect to the state variables where the parameters are updated using gradient steps a full convergence proof is given based on the ordinary differential equation approach which shows that the algorithm converges to a local minimum of the kullback leibler divergence between the true and the estimated parametric probability density functions a result that is useful even for a miss specified parametric model compared to other rml algorithms proposed in the literature this contribution extends the state space model and provides a theoretical analysis in a nontrivial statistical model that was not analyzed so far we further extend the rml analysis to constrained parameter estimation problems two examples including nonlinear state space models are given to highlight this contribution

space alternating generalized expectation maximization algorithm the expectation maximization em method can facilitate maximizing likelihood functions that arise in statistical estimation problems in the classical em paradigm one iteratively maximizes the conditional log likelihood of a single unobservable complete data space rather than maximizing the intractable likelihood function for the measured or incomplete data em algorithms update all parameters simultaneously which has two drawbacks 1 slow convergence and 2 difficult maximization steps due to coupling when smoothness penalties are used the paper describes the space alternating generalized em sage method which updates the parameters sequentially by alternating between several small hidden data spaces defined by the algorithm designer the authors prove that the sequence of estimates monotonically increases the penalized likelihood objective derive asymptotic convergence rates and provide sufficient conditions for monotone convergence in norm two signal processing applications illustrate the method estimation of superimposed signals in gaussian noise and image reconstruction from poisson measurements in both applications the sage algorithms easily accommodate smoothness penalties and converge faster than the em algorithms

recursive expectation maximization em algorithms for time varying parameters with applications to multiple target tracking we investigate the application of expectation maximization em algorithms to the classical problem of multiple target tracking mtt for a known number of targets conventional algorithms which deal with this problem have a computational complexity that depends exponentially on the number of targets and usually divide the problem into a localization stage and a tracking stage the new algorithms achieve a linear dependency and integrate these two stages three optimization criteria are proposed using deterministic and stochastic dynamic models for the targets

a recursive sparse grid collocation method for differential equations with white noise we consider a sparse grid collocation method in conjunction with a time discretiza tion of the differential equations for computing expectations of functionals of solutions to differential equations perturbed by time dependent white noise we first analyze the error of smolyak s sparse grid collocation used to evaluate expectations of functionals of solutions to stochastic differential equations discretized by the euler scheme we show theoretically and numerically that this algo rithm can have satisfactory accuracy for small noise magnitude or small integration time however it does not converge either with decrease of the euler scheme s time step size or with increase of smolyak s sparse grid level subsequently we use this method as a building block for proposing a new algorithm by combining sparse grid collocation with a recursive procedure this approach allows us to numerically integrate linear stochastic partial differential equations over longer times which is illustrated in numerical tests on a stochastic advection diffusion equation

a remark on inseparability of min max systems min max systems or min max plus systems are algebraic models of discrete event dynamic systems they are nonlinear extensions of the well known linear max plus system models the structural property inseparability of min max systems was proposed by us in a recent paper it turns out that inseparability is equivalent to the property of irreducibility proposed recently by van der woude and subiono although the two properties appear in very different forms we will prove this fact here

fast qmc matrix vector multiplication quasi monte carlo qmc rules 1 n sum n 0 n 1 f boldsymbol y n a can be used to approximate integrals of the form int 0 1 s f boldsymbol y a mathrm d boldsymbol y where a is a matrix and boldsymbol y is a row vector this type of integral arises for example from the simulation of a normal distribution with a general covariance matrix from the approximation of the expectation value of solutions of pdes with random coefficients or from applications from statistics in this paper we design qmc quadrature points boldsymbol y 0 ldots boldsymbol y n 1 in 0 1 s such that for the matrix y boldsymbol y 0 top ldots boldsymbol y n 1 top top whose rows are the quadrature points one can use the fast fourier transform to compute the matrix vector product y boldsymbol a top boldsymbol a in mathbb r s in mathcal o n log n operations and at most 2 s 1 extra additions the proposed method can be applied to lattice rules polynom

explicit constructions of quasi monte carlo rules for the numerical integration of high dimensional periodic functions in this paper we give explicit constructions of point sets in the s dimensional unit cube yielding quasi monte carlo algorithms which achieve the optimal rate of convergence of the worst case error for numerically integrating high dimensional periodic functions in the classical measure p alpha of the worst case error introduced by korobov the convergence for every even integer alpha ge 1 is of cal o n min alpha d log n s alpha 2 where d is a parameter of the construction which can be chosen arbitrarily large and n is the number of quadrature points this convergence rate is known to be the best possible up to some log n factors we prove the result for the deterministic and also a randomized setting the construction is based on a suitable extension of digital t m s nets over the finite field mathbb z b

on line expectation maximization algorithm for latent data models in this contribution we propose a generic online also sometimes called adaptive or recursive version of the expectation maximisation em algorithm applicable to latent variable models of independent observations compared to the algorithm of titterington 1984 this approach is more directly connected to the usual em algorithm and does not rely on integration with respect to the complete data distribution the resulting algorithm is usually simpler and is shown to achieve convergence to the stationary points of the kullback leibler divergence between the marginal distribution of the observation and the model distribution at the optimal rate i e that of the maximum likelihood estimator in addition the proposed approach is also suitable for conditional or regression models as illustrated in the case of the mixture of linear regressions model

recursive em and sage inspired algorithms with application to doa estimation this paper is concerned with recursive estimation using augmented data we study two recursive procedures closely linked with the well known expectation and maximization em and space alternating generalized em sage algorithms unlike iterative methods the recursive em and sage inspired algorithms give a quick update on estimates given new data under mild conditions estimates generated by these procedures are strongly consistent and asymptotically normally distributed these mathematical properties are valid for a broad class of problems when applied to direction of arrival doa estimation the recursive em and sage inspired algorithms lead to a very simple and fast implementation of the maximum likelihood ml method the most complicated computation in each recursion is inversion of the augmented information matrix through data augmentation this matrix is diagonal and easy to invert more importantly there is no search in such recursive procedures consequently the computational time is much less than that associated with existing numerical methods for finding ml estimates this feature greatly increases the potential of the ml approach in real time processing numerical experiments show that both algorithms provide good results with low computational cost

comparative convergence analysis of em and sage algorithms in doa estimation in this work the convergence rates of direction of arrival doa estimates using the expectation maximization em and space alternating generalized em sage algorithms are investigated the em algorithm is a well known iterative method for locating modes of a likelihood function and is characterized by simple implementation and stability unfortunately the slow convergence associated with em makes it less attractive for practical applications the sage algorithm proposed by fessler and hero 1994 based on the same idea of data augmentation has the potential to speed up convergence and preserves the advantage of simple implementation we study both algorithms within the framework of array processing theoretical analysis shows that sage has faster convergence speed than em under certain conditions on observed and augmented information matrices the analytical results are supported by numerical simulations carried out over a wide range of signal to noise ratios snrs and various source locations

real time reduced order modeling of stochastic partial differential equations via time dependent subspaces abstract we present a new methodology for the real time reduced order modeling of stochastic partial differential equations called the dynamically bi orthonormal dbo decomposition in this method the stochastic fields are approximated by a low rank decomposition to spatial and stochastic subspaces each of these subspaces is represented by a set of orthonormal time dependent modes we derive exact evolution equations of these time dependent modes and the evolution of the factorization of the reduced covariance matrix we show that dbo is equivalent to the dynamically orthogonal do 1 and bi orthogonal bo 2 decompositions via linear and invertible transformation matrices that connect dbo to do and bo however dbo shows several improvements compared to do and bo i dbo performs better than do and bo for cases with ill conditioned covariance matrix ii in contrast to bo the issue of eigenvalue crossing is not present in the dbo formulation iii in contrast to do the stochastic modes are orthonormal which leads to more accurate representation of the stochastic subspace we study the convergence properties of the method and compare it to the do and bo methods for demonstration we consider three cases i stochastic linear advection equation ii stochastic burgers equation and iii stochastic incompressible flow over a bump in a channel overall we observe improvements in the numerical accuracy of dbo compared against do and bo

music maximum likelihood and cramer rao bound the performance of the music and ml methods is studied and their statistical efficiency is analyzed the cramer rao bound crb for the estimation problems is derived and some useful properties of the crb covariance matrix are established the relationship between the music and ml estimators is investigated as well a numerical study is reported of the statistical efficiency of the music estimator for the problem of finding the directions of two plane waves using a uniform linear array an exact description of the results is included

performance study of conditional and unconditional direction of arrival estimation a numerical and analytical study of conditional and unconditional direction of arrival doa estimation is presented explicit expressions for the unconditional cramer rao bounds on the doa estimation accuracy and the covariance matrix of the conditional maximum likelihood method are given it is shown that many doa estimation methods have the same asymptotic statistical properties under conditional and unconditional models the situation of two narrowband plane signals impinging on a uniformly spaced linear array is discussed

on decompositions of multivariate functions we present formulas that allow us to decompose a function f of d variables into a sum of 2 d terms fu indexed by subsets u of f1 dg where each term fu depends only on the variables with indices in u the decomposition depends on the choice of d commuting projections fpjg d 1 where pj f does not depend on the variable xj we present an explicit formula for fu which is new even for the anova and anchored decompositions both are special cases of the general decomposition we show that the decomposition is minimal in the following sense if f is expressible as a sum in which there is no term that depends on all of the variables indexed by the subset z then for every choice of fpjg d 1 the terms fu 0 for all subsets u containing z furthermore in a reproducing kernel hilbert space setting we give su cient conditions for the terms fu to be mutually orthogonal

numerical schemes for dynamically orthogonal equations of stochastic fluid and ocean flows the quantification of uncertainties is critical when systems are nonlinear and have uncertain terms in their governing equations or are constrained by limited knowledge of initial and boundary conditions such situations are common in multiscale intermittent and non homogeneous fluid and ocean flows the dynamically orthogonal do field equations provide an adaptive methodology to predict the probability density functions of such flows the present work derives efficient computational schemes for the do methodology applied to unsteady stochastic navier stokes and boussinesq equations and illustrates and studies the numerical aspects of these schemes semi implicit projection methods are developed for the mean and for the do modes and time marching schemes of first to fourth order are used for the stochastic coefficients conservative second order finite volumes are employed in physical space with new advection schemes based on total variation diminishing methods other results include i the definition of pseudo stochastic pressures to obtain a number of pressure equations that is linear in the subspace size instead of quadratic ii symmetric advection schemes for the stochastic velocities iii the use of generalized inversion to deal with singular subspace covariances or deterministic modes and iv schemes to maintain orthonormal modes at the numerical level to verify our implementation and study the properties of our schemes and their variations a set of stochastic flow benchmarks are defined including asymmetric dirac and symmetric lock exchange flows lid driven cavity flows and flows past objects in a confined channel different reynolds number and grashof number regimes are employed to illustrate robustness optimal convergence under both time and space refinements is shown as well as the convergence of the probability density functions with the number of stochastic realizations

a domain mapping approach for elliptic equations posed on random bulk and surface domains in this article we analyse the domain mapping method approach to approximate statistical moments of solutions to linear elliptic partial differential equations posed over random geometries including smooth surfaces and bulk surface systems in particular we present the necessary geometric analysis required by the domain mapping method to reformulate elliptic equations on random surfaces onto a fixed deterministic surface using a prescribed stochastic parametrisation of the random domain an abstract analysis of a finite element discretisation coupled with a monte carlo sampling is presented for the resulting elliptic equations with random coefficients posed over the fixed curved reference domain and optimal error estimates are derived the results from the abstract framework are applied to a model elliptic problem on a random surface and a coupled elliptic bulk surface system and the theoretical convergence rates are confirmed by numerical experiments

acceleration of stochastic approximation by averaging a new recursive algorithm of stochastic approximation type with the averaging of trajectories is investigated convergence with probability one is proved for a variety of classical optimization and identification problems it is also demonstrated for these problems that the proposed algorithm achieves the highest possible rate of convergence

a multilevel sparse kernel based stochastic collocation finite element method for elliptic problems with random coefficients abstract a new stochastic collocation finite element method is proposed for the numerical solution of elliptic boundary value problems bvp with random coefficients assuming that the randomness is well approximated by a finite number of random variables with given probability distributions the proposed method consists of a finite element approximation in physical space along with a stochastic collocation quadrature approach utilizing the recent multilevel sparse kernel based interpolation musik technique georgoulis et al 2013 musik is based on a multilevel sparse grid type algorithm with the basis functions consisting of directionally anisotropic gaussian radial basis functions kernels placed at directionally uniform grid points we prove that musik is interpolatory at these nodes and therefore can be naturally used to define a quadrature scheme numerical examples are also presented assessing the performance of the new algorithm in the context of high dimensional stochastic collocation finite element methods

quasi monte carlo finite element methods for elliptic pdes with lognormal random coefficients in this paper we analyze the numerical approximation of diffusion problems over polyhedral domains in mathbb r d rd d 1 2 3 d 1 2 3 with diffusion coefficient a varvec x omega a x given as a lognormal random field i e a varvec x omega exp z varvec x omega a x exp z x where varvec x x is the spatial variable and z varvec x cdot z x is a gaussian random field the analysis presents particular challenges since the corresponding bilinear form is not uniformly bounded away from 0 0 or infty over all possible realizations of a a focusing on the problem of computing the expected value of linear functionals of the solution of the diffusion problem we give a rigorous error analysis for methods constructed from 1 standard continuous and piecewise linear finite element approximation in physical space 2 truncated karhunen loeve expansion for computing realizations of a a leading to a possibly high dimensional parametrized deterministic diffusion problem and 3 lattice based quasi monte carlo qmc quadrature rules for computing integrals over parameter space which define the expected values the paper contains novel error analysis which accounts for the effect of all three types of approximation the qmc analysis is based on a recent result on randomly shifted lattice rules for high dimensional integrals over the unbounded domain of euclidean space which shows that under suitable conditions the quadrature error decays with mathcal o n 1 delta o n 1 with respect to the number of quadrature points n n where delta 0 0 is arbitrarily small and where the implied constant in the asymptotic error bound is independent of the dimension of the domain of integration

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

an offline online algorithm for a class of stochastic multiple obstacle scattering configurations in the half plane efficient computational models are required to understand quantities of interest qoi such as the intensity of sound radiated by non deterministic configurations comprising multiple obstacles in the half plane the stochastic nature of the configurations requires tens of thousands of realizations to compute the expected value and standard deviation of the random qoi the acoustic wave scattering configuration usually comprises multiple deterministic structures whose locations in the half plane are non deterministic thus understanding the nature of the deterministic structures is intrinsically an offline process whilst understanding the nature of the resulting qoi is an online process the online process includes interactions between the structures and the half plane boundary in accordance with the physical offline online process we develop a computational t matrix based offline online reduced order model for large realizations of stochastic acoustic wave scattering configurations in the half plane we present results with more than 60 000 monte carlo mc quasi mc qmc and generalized polynomial chaos gpc simulations which are evaluated using our efficient online scheme for the acoustic scattering half plane model our offline online deterministic and stochastic framework can be used in conjunction with any numerical method for the offline simulations of scattering from a single obstacle in free space

dynamically orthogonal tensor methods for high dimensional nonlinear pdes abstract we develop new dynamically orthogonal tensor methods to approximate multivariate functions and the solution of high dimensional time dependent nonlinear partial differential equations pdes the key idea relies on a hierarchical decomposition of the approximation space obtained by splitting the independent variables of the problem into disjoint subsets this process which can be conveniently visualized in terms of binary trees yields series expansions analogous to the classical tensor train and hierarchical tucker tensor formats by enforcing dynamic orthogonality conditions at each level of the binary tree we obtain coupled evolution equations for the modes spanning each subspace within the hierarchical decomposition this allows us to effectively compute the solution to high dimensional time dependent nonlinear pdes on tensor manifolds of constant rank with no need for rank reduction methods we also propose new algorithms for dynamic addition and removal of modes within each subspace numerical examples are presented and discussed for high dimensional hyperbolic and parabolic pdes in bounded domains

multi index monte carlo when sparsity meets sampling we propose and analyze a novel multi index monte carlo mimc method for weak approximation of stochastic models that are described in terms of differential equations either driven by random measures or with random coefficients the mimc method is both a stochastic version of the combination technique introduced by zenger griebel and collaborators and an extension of the multilevel monte carlo mlmc method first described by heinrich and giles inspired by giles s seminal work we use in mimc high order mixed differences instead of using first order differences as in mlmc to reduce the variance of the hierarchical differences dramatically this in turn yields new and improved complexity results which are natural generalizations of giles s mlmc analysis and which increase the domain of the problem parameters for which we achieve the optimal convergence mathcal o mathrm tol 2 o tol 2 moreover in mimc the rate of increase of required memory with respect to mathrm tol tol is independent of the number of directions up to a logarithmic term which allows far more accurate solutions to be calculated for higher dimensions than what is possible when using mlmc we motivate the setting of mimc by first focusing on a simple full tensor index set we then propose a systematic construction of optimal sets of indices for mimc based on properly defined profits that in turn depend on the average cost per sample and the corresponding weak error and variance under standard assumptions on the convergence rates of the weak error variance and work per sample the optimal index set turns out to be the total degree type in some cases using optimal index sets mimc achieves a better rate for the computational complexity than the corresponding rate when using full tensor index sets we also show the asymptotic normality of the statistical error in the resulting mimc estimator and justify in this way our error estimate which allows both the required accuracy and the confidence level in our computational results to be prescribed finally we include numerical experiments involving a partial differential equation posed in three spatial dimensions and with random coefficients to substantiate the analysis and illustrate the corresponding computational savings of mimc

constructing lattice points for numerical integration by a reduced fast successive coordinate search algorithm abstract in this paper we study an efficient algorithm for constructing node sets of high quality quasi monte carlo integration rules for weighted korobov walsh and sobolev spaces the algorithm presented is a reduced fast successive coordinate search scs algorithm which is adapted to situations where the weights in the function space show a sufficiently fast decay the new scs algorithm is designed to work for the construction of lattice points and in a modified version for polynomial lattice points and the corresponding integration rules can be used to treat functions in different kinds of function spaces we show that the integration rules constructed by our algorithms satisfy error bounds of optimal convergence order furthermore we give details on efficient implementation such that we obtain a considerable speed up of previously known scs algorithms this improvement is illustrated by numerical results the speed up obtained by our results may be of particular interest in the context of qmc for pdes with random coefficients where both the dimension and the required number of points are usually very large furthermore our main theorems yield previously unknown generalizations of earlier results

sparse grid reduced basis bayesian inversion abstract we analyze reduced basis rb for short short acceleration of recently proposed sparse bayesian inversion algorithms for partial differential equations with uncertain distributed parameter for observation data subject to additive gaussian observation noise specifically bayesian inversion of affine parametric linear operator families on possibly high dimensional parameter spaces we consider high fidelity petrov galerkin pg discretizations of these countably parametric operator families we allow general families of inf sup stable pg finite element methods covering most conforming primal and mixed finite element discretizations of standard problems in mechanics rb acceleration of the high dimensional parametric forward response maps which need to be numerically solved numerous times in bayesian inversion is proposed and convergence rate bounds for the error in the bayesian estimate incurred by the use of rb are derived as consequence of recent theoretical results on dimension independent sparsity of parametric responses and preservation of sparsity for holomorphic parametric problems we establish new convergence rates of greedy rb compressions for both the parametric forward maps as well as for the countably parametric posterior densities which arise in bayesian inversion we show that the convergence rates for the rb compressions of the parametric forward maps as well as of the countably parametric sparse bayesian posterior densities are free from the curse of dimensionality and depend only on the sparsity of the uncertain input data in particular we establish the quadratic convergence of the rb compression for the posterior densities with respect to that for the parametric forward maps numerical experiments for linear elliptic affine parametric model problems in two space dimensions with hundreds of parameters are reported which confirm that the proposed adaptive sparse grid reduced basis algorithms indeed exploit sparsity of both the parametric forward maps as well as the bayesian posterior density

good lattice rules in weighted korobov spaces with general weights we study the problem of multivariate integration and the construction of good lattice rules in weighted korobov spaces with general weights these spaces are not necessarily tensor products of spaces of univariate functions sufficient conditions for tractability and strong tractability of multivariate integration in such weighted function spaces are found these conditions are also necessary if the weights are such that the reproducing kernel of the weighted korobov space is pointwise non negative the existence of a lattice rule which achieves the nearly optimal convergence order is proven a component by component cbc algorithm that constructs good lattice rules is presented the resulting lattice rules achieve tractability or strong tractability error bounds and achieve nearly optimal convergence order for suitably decaying weights we also study special weights such as finite order and order dependent weights for these special weights the cost of the cbc algorithm is polynomial numerical computations show that the lattice rules constructed by the cbc algorithm give much smaller worst case errors than the mean worst case errors over all quasi monte carlo rules or over all lattice rules and generally smaller worst case errors than the best korobov lattice rules in dimensions up to hundreds numerical results are provided to illustrate the efficiency of cbc lattice rules and korobov lattice rules with suitably chosen weights in particular for high dimensional finance problems

quasi monte carlo finite element methods for a class of elliptic partial differential equations with random coefficients in this paper quasi monte carlo qmc methods are applied to a class of elliptic partial differential equations pdes with random coefficients where the random coefficient is parametrized by a co