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

a pearson random walk with steps of uniform orientation and dirichlet distributed lengths a constrained diffusive random walk of n steps in d and a random flight in d which are equivalent were investigated independently in recent papers j stat phys 127 813 2007 j theor probab 20 769 2007 and j stat phys 131 1039 2008 the n steps of the walk are independent and identically distributed random vectors of exponential length and uniform orientation conditioned on the sum of their lengths being equal to a given value l closed form expressions for the distribution of the endpoint of the walk were obtained altogether for any n for d 1 2 4 uniform distributions of the endpoint inside a ball of radius l were evidenced for a walk of three steps in 2d and of two steps in 4d

isotropic random motion at finite speed with k erlang distributed direction alternations we study uniformly distributed direction of motion at finite speed where the direction alternations occur according to the renewal epochs of a k erlang pdf at first sight our generalizations of previous markovian results appears to be a small step however it must be seen as an important non markovian case where we have found closed form expressions for the pdf and the conditional characteristic function of this semi markov transport process we present detailed calculations of a three dimensional example for the 2 erlang case which is important not only from physical applications point of view but also to understand more general models for instance in principle the example of the 2 erlang case can be extended to a k erlang case k 3 4 but some of the mathematical expressions may be cumbersome

a new family of solvable pearson dirichlet random walks an n step pearson gamma random walk in d starts at the origin and consists of n independent steps with gamma distributed lengths and uniform orientations the gamma distribution of each step length has a shape parameter q 0 constrained random walks of n steps in d are obtained from the latter walks by imposing that the sum of the step lengths is equal to a fixed value simple closed form expressions were obtained in particular for the distribution of the endpoint of such constrained walks for any d d0 and any n 2 when q is either q frac d 2 1 d0 3 or q d 1 d0 2 le caer in j stat phys 140 728 751 2010 when the total walk length is chosen without loss of generality to be equal to 1 then the constrained step lengths have a dirichlet distribution whose parameters are all equal to q and the associated walk is thus named a pearson dirichlet random walk the density of the endpoint position of a n step planar walk of this type n 2 with q d 2 was shown recently to be a weighted mixture of 1 floor n 2 endpoint densities of planar pearson dirichlet walks with q 1 beghin and orsingher in stochastics 82 201 229 2010 the previous result is generalized to any walk space dimension and any number of steps n 2 when the parameter of the pearson dirichlet random walk is q d 1 we rely on the connection between an unconstrained random walk and a constrained one which have both the same n and the same q d to obtain a closed form expression of the endpoint density the latter is a weighted mixture of 1 floor n 2 densities with simple forms equivalently expressed as a product of a power and a gauss hypergeometric function the weights are products of factors which depends both on d and n and bessel numbers independent of d

random evolutions are driven by the hyperparabolic operators we resolve the long standing problem of describing the multidimensional random evolutions by means of the telegraph equations this problem was posed by mark kac more than 50 years ago and has become the subject of intense discussion among researchers on whether the multidimensional random flights could be described by the telegraph equations similarly to the one dimensional case we give the exhaustive answer to this question and show that the multidimensional random evolutions are driven by the hyperparabolic operators composed of the telegraph operators and their integer powers the only exception is the 2d random flight whose transition density is the fundamental solution to the two dimensional telegraph equation the reason of the exceptionality of the 2d case is explained we also show that under the standard kac s condition the governing hyperparabolic operator turns into the generator of the brownian motion

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

dirichlet problem for semilinear hyperbolic equation multidimensional case abstract we discuss the solvability of the dirichlet problem for the semilinear equation of the vibrating string x t t t y x t y f t y x t y 0 in higher dimensions with sides length being irrational numbers and superlinear nonlinearity to this effect we derive a new dual variational method

scrambled net variance for integrals of smooth functions hybrids of equidistribution and monte carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter one version randomized t m s nets has the property that the integral estimates are unbiased and that the variance is o 1 n for any square integrable integrand stronger assumptions on the integrand allow one to find rates of convergence this paper shows that for smooth integrands over s dimensions the variance is of order n 3 log n 3 1 compared to n 1 for ordinary monte carlo thus the integration errors are of order n 3 2 log n s 1 2 in probability this compares favorably with the rate n 1 log n s 1 for unrandomized t m s nets

on korobov lattice rules in weighted spaces this paper studies the error bounds of multivariate integration in weighted function spaces using lattice rules of the korobov form in which the generating vector for an n point rule with n prime has the form 1 a ad 1 mod n with the parameter a chosen optimally we establish new error bounds for korobov lattice rules in weighted korobov spaces in particular we prove that if the weights decay sufficiently fast the optimal korobov lattice rule has an error bound of order o n alpha 2 delta for arbitrary delta 0 with the implied constant depending at worst polynomially on the dimension here alpha 1 is the smoothness parameter of the weighted korobov spaces we generalize the construction to the case where n is a product of arbitrary distinct prime numbers with the purpose of reducing the construction cost without sacrificing much of the quality of the lattice rules a corresponding result is deduced for weighted sobolev spaces of nonperiodic functions using randomly shifted optimal korobov lattice rules a comparison of the worst case errors for korobov lattice rules and the recent component by component constructions is presented the investigation establishes the usefulness of shifted optimal korobov lattice rules for integration even in high dimensions if the weights which characterize the weighted spaces are suitably chosen

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

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

the tent transformation can improve the convergence rate of quasi monte carlo algorithms using digital nets in this paper we investigate multivariate integration in reproducing kernel sobolev spaces for which the second partial derivatives are square integrable as quadrature points for our quasi monte carlo algorithm we use digital t m s nets over mathbb z 2 which are randomly digitally shifted and then folded using the tent transformation for this qmc algorithm we show that the root mean square worst case error converges with order 2 m 2 varepsilon for any 0 where 2m is the number of points a similar result for lattice rules has previously been shown by hickernell

random motions at finite speed in higher dimensions we present a general method of studying the transport process bold x t t 0 in the euclidean space m m 2 based on the analysis of the integral transforms of its distributions we show that the joint characteristic functions of bold x t are connected with each other by a convolution type recurrent relation this enables us to prove that the characteristic function fourier transform of bold x t in any dimension m 2 satisfies a convolution type volterra integral equation of second kind we give its solution and obtain the characteristic function of bold x t in terms of the multiple convolutions of the kernel of the equation with itself an explicit form of the laplace transform of the characteristic function in any dimension is given the complete solution of the problem of finding the initial conditions for the governing partial differential equations is given

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

dirichlet problem for semilinear hyperbolic equation abstract we discuss the solvability of the dirichlet problem for the semilinear equation of the vibrating string x tt t y x yy t y f t y x t y 0 in rectangle with sides length being irrational numbers and superlinear nonlinearity to this effect we derive new dual variational method

exact probability distributions for noncorrelated random walk models a stochastic model for the idealized locomotion of cells is studied the cell is assumed to cover a polygonal line in n the times between turns are exponentially distributed and independent of the directions and the density of thenth directione does not depend on the n 1 th directione the resulting markov process x t d t for position and direction of the motion at timet is studied by using the integrodifferential equation for the transition function for example the joint distribution of x t d t is derived in closed form ifn 2 orn 3 and all chosen directions including the initial one are uniformly distributed for higher dimensions the combined fourier laplace transform ofx t is given the case of a fixed initial direction is also considered

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

approximation of quasi monte carlo worst case error in weighted spaces of infinitely times smooth functions abstract in this paper we consider quasi monte carlo qmc worst case error of weighted smooth function classes in c 0 1 s by a digital net over f 2 we show that the ratio of the worst case error to the qmc integration error of an exponential function is bounded above and below by constants this result provides us with a simple interpretation that a digital net with small qmc integration error for an exponential function also gives the small integration error for any function in this function space

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

construction of interlaced polynomial lattice rules for infinitely differentiable functions we study multivariate integration over the s dimensional unit cube in a weighted space of infinitely differentiable functions it is known from a recent result by suzuki that there exists a good quasi monte carlo qmc rule which achieves a super polynomial convergence of the worst case error in this function space and moreover that this convergence behavior is independent of the dimension under a certain condition on the weights in this paper we provide a constructive approach to finding a good qmc rule achieving such a dimension independent super polynomial convergence of the worst case error specifically we prove that interlaced polynomial lattice rules with an interlacing factor chosen properly depending on the number of points n and the weights can be constructed using a fast component by component algorithm in at most o sn log n 2 arithmetic operations to achieve a dimension independent super polynomial convergence the key idea for the proof of the worst case error bound is to use a variant of jensen s inequality with a purposely designed concave function

random motion with uniformly distributed directions and random velocity in this paper we deal with uniformly distributed direction of motion or isotropic motion at random speed or velocity where the direction alternations occur according to the renewal epochs of a general distribution we derive the renewal equation for the characteristic function of the transition density of the multidimensional motion then by using the renewal equation we study the behavior of the transition density near the sphere of its singularity for one two three and four dimensional cases to illustrate our solution methodology we present detailed calculations of many solvable examples

quasi monte carlo point sets with small t values and wafom the t value of a t m s net is an important criterion of point sets for quasi monte carlo integration and many point sets are constructed in terms of the t values as this leads to small integration error bounds recently matsumoto saito and matoba proposed the walsh figure of merit wafom as a quickly computable criterion of point sets that ensures higher order convergence for function classes of very high smoothness in this paper we consider a search algorithm for point sets whose t value and wafom are both small so as to be effective for a wider range of function classes for this we fix digital t m s nets with small t values e g sobol or niederreiter xing nets in advance apply random linear scrambling and select scrambled digital t m s nets in terms of wafom experiments show that the resulting point sets improve the rates of convergence for smooth functions and are robust for non smooth functions

quasi periodic solutions of hamiltonian perturbations of 2d linear schrodinger equations the general problem discussed here is the persistency of quasi periodic solutions of linear or integrable equations after hamiltonian perturbation this subject is closely related to the well known kam theory of invariant tori in smooth dynamical systems the main source of new difficulties is the fact that one considers here finite dimensional tori in an infinite dimensional phase space for instance already in the study of time periodic solutions the fact that the phase space is infinite dimensional leads to small divisor problems now the standard kam technology may be reworked in a form applicable to treat persistency problems in space dimension 1d in the case of dirichlet boundary conditions say where in particular the normal frequencies of the problem are well separated results along this line were obtained by s kuksin ki and c e wayne w among other works we mentioned ki as the main expository reference for the kam approach considering pde s in id with periodic boundary conditions and especially in space dimension 1 a significant new problem arises due to the presence of clusters of normal frequencies

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

optimal quasi monte carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions we investigate quasi monte carlo rules for the numerical integration of multivariate periodic functions from besov spaces s r p q b mathbb t d sp qrb td with dominating mixed smoothness 1 p

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

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

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

asymptotic relation for the transition density of the three dimensional markov random flight on small time intervals we consider the markov random flight varvec x t t 0 in the three dimensional euclidean space mathbb r 3 with constant finite speed c 0 and the uniform choice of the initial and each new direction at random time instants that form a homogeneous poisson flow of rate lambda 0 series representations for the conditional characteristic functions of varvec x t corresponding to two and three changes of direction are obtained based on these results an asymptotic formula as t rightarrow 0 for the unconditional characteristic function of varvec x t is derived by inverting it we obtain an asymptotic relation for the transition density of the process we show that the error in this formula has the order o t 3 and therefore it gives a good approximation on small time intervals whose lengths depend on lambda an asymptotic formula as t rightarrow 0 for the probability of being in a three dimensional ball of radius r ct is also derived estimate of the accuracy of the approximation is analysed

an explicit construction of optimal order quasi monte carlo rules for smooth integrands in a recent paper by the authors it is shown that there exists a quasi monte carlo qmc rule which achieves the best possible rate of convergence for numerical integration in a reproducing kernel hilbert space consisting of smooth functions in this paper we provide an explicit construction of such an optimal order qmc rule our approach is to exploit both the decay and the sparsity of the walsh coefficients of the reproducing kernel simultaneously this can be done by applying digit interlacing composition due to dick to digital nets with large minimum hamming and niederreiter rosenbloom tsfasman metrics due to chen and skriganov to our best knowledge our construction gives the first qmc rule which achieves the best possible convergence in this function space

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

duality theory and propagation rules for generalized digital nets digital nets are used in quasi monte carlo algorithms for approximating high dimensional integrals over the unit cube hence one wants to have explicit constructions of digital nets of high quality in this paper we consider the so called propagation rules for digital nets which state how one can obtain a new digital net of different size from existing digital nets this way one often can generate digital nets of higher quality than were previously known here we generalize existing propagation rules for classical digital nets to generalized digital nets as introduced by dick

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

two step dirichlet random walks abstract random walks of n steps taken into independent uniformly random directions in a d dimensional euclidean space d 2 which are characterized by a sum of step lengths which is fixed and taken to be 1 without loss of generality are named dirichlet when this constraint is realized via a dirichlet law of step lengths the latter continuous multivariate distribution which depends on n positive parameters generalizes the beta distribution n 2 it is simply obtained from n independent gamma random variables with identical scale factors previous literature studies of these random walks dealt with symmetric dirichlet distributions whose parameters are all equal to a value q which takes half integer or integer values in the present work the probability density function of the distance from the endpoint to the origin is first made explicit for a symmetric dirichlet random walk of two steps it is valid for any positive value of q and for all d 2 the latter pdf is used in turn to express the related density of a random walk of two steps whose step length is distributed according to an asymmetric beta distribution which depends on two parameters namely q and q s where s is a positive integer

higher order qmc petrov galerkin discretization for affine parametric operator equations with random field inputs we construct quasi monte carlo methods to approximate the expected values of linear functionals of petrov galerkin discretizations of parametric operator equations which depend on a possibly infinite sequence of parameters such problems arise in the numerical solution of differential and integral equations with random field inputs we analyze the regularity of the solutions with respect to the parameters in terms of the rate of decay of the fluctuations of the input field if p in 0 1 denotes the summability exponent corresponding to the fluctuations in affine parametric families of operators then we prove that deterministic interlaced polynomial lattice rules of order alpha lfloor 1 p rfloor 1 in s dimensions with n points can be constructed using a fast component by component algorithm in mathcal o alpha s n log n alpha 2 s 2 n operations to achieve a convergence rate of mathcal o n 1 p with the implied constant independent of s this dimension independen

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

multiplicity of periodic solutions of nonlinear wave equations abstract we prove multiplicity of small amplitude periodic solutions with fixed frequency of completely resonant wave equations with general nonlinearities as 1 the number n of 2 periodic solutions u1 un un tends to the minimal period of the nth solution un is 2 n the proofs are based on the variational lyapunov schmidt reduction comm math phys to appear and minimax arguments 1

some two and three dimensional persistent random walks abstract we formulate non markovian versions of the persistent random walk in two and three dimensions and in continuous time these models can be regarded as being generalizations of the original pearson random walk and of the freely jointed chain which has been of some importance in polymer physics solutions for the probability density of the displacement of the random walker can be furnished for a restricted essentially markovian set of these models it is shown that in two dimensions the solution to one of these models is equivalent to the solution to an inhomogeneous telegrapher s equation it does not appear to be possible starting from a similar model in three dumensions to find any form of the telegrapher s equation that follows from our solution of the equations in the fourier laplace transform domains

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

the mean square discrepancy of scrambled t s sequences the discrepancy arises in the worst case error analysis for quasi monte carlo quadrature rules low discrepancy sets yield good quadrature rules this article considers the mean square discrepancies for scrambled lam t m s nets and t s sequences in base b it is found that the mean square discrepancy for scrambled nets and sequences is never more than a constant multiple of that under simple monte carlo sampling if the reproducing kernel defining the discrepancy satisfies a lipschitz condition with respect to one of its variables separately then the asymptotic order of the root mean square discrepancy is o n 1 log n s 1 2 for scrambled nets if the reproducing kernel satisfies a lipschitz condition with respect to both of its variables then the asymptotic order of the root mean square discrepancy is o n 3 2 log n s 1 2 for scrambled nets for an arbitrary number of points taken from a t s sequence the root mean square discrepancy appears to be no better than o n 1 log n s 1 2 regardless of the smoothness of the reproducing kernel

diophantine approximation bessel functions and radially symmetric periodic solutions of semilinear wave equations in a ball the aim of this paper is to consider the radially symmetric periodic dirichlet problem on 0 t x b n a for the equation u tt deltau f t x u where delta is the classical laplacian operator and b n a denotes the open ball of center 0 and radius a in r n when alpha a t is a sufficiently large irrational with bounded partial quotients we combine some number theory techniques with the asymptotic properties of the bessel functions to show that 0 is not an accumulation point of the spectrum of the linear part this result is used to obtain existence conditions for the nonlinear problem

constructing randomly shifted lattice rules in weighted sobolev spaces shifted rank 1 lattice rules a special class of quasi monte carlo methods have recently been proposed by the present authors for the integration of functions belonging to certain weighted sobolev spaces the shifts in these rules were generated in a deterministic manner in contrast in this paper we generate these shifts randomly this allows probabilistic estimates for the error in a given integral it also reduces the number of operations required to find the generating vectors for the underlying lattice rules component by component the rules thus constructed achieve a worst case strong tractability error bound in an average or probabilistic sense

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

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

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

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

the exact probability distribution of a two dimensional random walk a calculation is made of the exact probability distribution of the two dimensional displacement of a particle at timet that starts at the origin moves in straight line paths at constant speed and changes its direction after exponentially distributed time intervals where the lengths of the straight line paths and the turn angles are independent the angles being uniformly distributed this random walk is the simplest model for the locomotion of microorganisms on surfaces its weak convergence to a wiener process is also shown

lattice rules for nonperiodic smooth integrands the aim of this paper is to show that one can achieve convergence rates of n alpha delta n for alpha 1 2 1 2 and for delta 0 0 arbitrarily small for nonperiodic alpha smooth cosine series using lattice rules without random shifting the smoothness of the functions can be measured by the decay rate of the cosine coefficients for a specific choice of the parameters the cosine series space coincides with the unanchored sobolev space of smoothness 1 we study the embeddings of various reproducing kernel hilbert spaces and numerical integration in the cosine series function space and show that by applying the so called tent transformation to a lattice rule one can achieve the almost optimal rate of convergence of the integration error the same holds true for symmetrized lattice rules for the tensor product of the direct sum of the korobov space and cosine series space but with a stronger dependence on the dimension in this case

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

on the step by step construction of quasi monte carlo integration rules that achieve strong tractability error bounds in weighted sobolev spaces we develop and justify an algorithm for the construction of quasi monte carlo qmc rules for integration in weighted sobolev spaces the rules so constructed are shifted rank 1 lattice rules the parameters characterising the shifted lattice rule are found component by component the d 1 th component of the generator vector and the shift are obtained by successive 1 dimensional searches with the previous d components kept unchanged the rules constructed in this way are shown to achieve a strong tractability error bound in weighted sobolev spaces a search for n point rules with n prime and all dimensions 1 to d requires a total cost of o n3d2 operations this may be reduced to o n3d operations at the expense of o n2 storage numerical values of parameters and worst ease errors are given for dimensions up to 40 and n up to a few thousand the worst case errors for these rules are found to be much smaller than the theoretical bounds

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

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