reconstructing hyperbolic cross trigonometric polynomials by sampling along rank 1 lattices with given fourier coefficients the evaluation of multivariate trigonometric polynomials at the nodes of a rank 1 lattice leads to a one dimensional discrete fourier transform in many applications one is also interested in the reconstruction of the fourier coefficients from samples in the spatial domain we present necessary and sufficient conditions on rank 1 lattices allowing a stable reconstruction of trigonometric polynomials supported on hyperbolic crosses in addition we suggest approaches for determining suitable rank 1 lattices using a component by component algorithm we present numerical results for reconstructing trigonometric polynomials up to spatial dimension 100

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

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

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

iterative strong coupling of dimensionally heterogeneous models in this article we address decomposition strategies especially tailored to perform strong coupling of dimensionally heterogeneous models under the hypothesis that one wants to solve each submodel separately and implement the interaction between subdomains by boundary conditions alone the novel methodology takes full advantage of the small number of interface unknowns in this kind of problems existing algorithms can be viewed as variants of the natural staggered algorithm in which each domain transfers function values to the other and receives fluxes or forces and vice versa this natural algorithm is known as dirichlet to neumann in the domain decomposition literature essentially we propose a framework in which this algorithm is equivalent to applying gauss seidel iterations to a suitably defined linear or nonlinear system of equations it is then immediate to switch to other iterative solvers such as gmres or other krylov based method which we assess through numerical experiments showing the significant gain that can be achieved indeed the benefit is that an extremely flexible automatic coupling strategy can be developed which in addition leads to iterative procedures that are parameter free and rapidly converging further in linear problems they have the finite termination property copyright 2009 john wiley sons ltd

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

supg finite element computation of compressible flows with the entropy and conservation variables formulations abstract supg stabilized finite element formulations of compressible euler equations based on the conservation and entropy variables are investigated and compared the formulation based on the conservation variables consists of the formulation introduced by tezduyar and hughes plus a shock capturing term the formulation based on the entropy variables is the same as the one by hughes franca and mallet which has a shock capturing term built in these formulations are tested on several subsonic transonic and supersonic compressible flow problems it is shown that the stabilized formulation based on the conservation variables gives solutions which are just as good as those obtained with the entropy variables furthermore the solutions obtained using the two formulations are very close and in some cases almost indistinguishable consequently it can be deduced that the relative merits of these two formulations will continue to remain under debate

the jackknife estimate of variance tukey s jackknife estimate of variance for a statistic s x 1 x 2 cdots x n which is a symmetric function of i i d random variables x i is investigated using an anova like decomposition of s it is shown that the jackknife variance estimate tends always to be biased upwards a theorem to this effect being proved for the natural jackknife estimate of operatorname var s x 1 x 2 cdots x n 1 based on x 1 x 2 cdots x n

a novel method to look for the hysteresis thresholds for the canny edge detector in the last few years several works have been proposed to solve the problem of determining the hysteresis thresholds in an unsupervised way in this paper a novel method to solve this problem is proposed given a set of candidates for hysteresis thresholds the basic idea of the proposed method is to combine gradient information with information obtained when the linking process is applied to all candidates using the same dataset and the same evaluation methodology already proposed by other works the results obtained by our method show a performance better than that of the previous methods the results obtained by the proposed method have been validated only for the canny edge detector but there are no restrictions on applying the proposed method to any other edge detector whose strategy is based on the hysteresis mechanism

quasi monte carlo methods for linear two stage stochastic programming problems quasi monte carlo qmc algorithms are studied for generating scenarios to solve two stage linear stochastic programming problems their integrands are piecewise linear quadratic but do not belong to the function spaces considered for qmc error analysis we show that under some weak geometric condition on the two stage model all terms of their anova decomposition except the one of highest order are continuously differentiable and second order mixed derivatives exist almost everywhere and belong to l 2 l2 this implies that randomly shifted lattice rules may achieve the optimal rate of convergence o n 1 delta o n 1 with delta in 0 frac 1 2 0 12 and a constant not depending on the dimension if the effective superposition dimension is less than or equal to two the geometric condition is shown to be satisfied for almost all covariance matrices if the underlying probability distribution is normal we discuss effective dimensions and techniques for dimension reduction numerical experiments for a production planning model with normal inputs show that indeed convergence rates close to the optimal rate are achieved when using randomly shifted lattice rules or scrambled sobol point sets accompanied with principal component analysis for dimension reduction

the role of frolov s cubature formula for functions with bounded mixed derivative we prove upper bounds on the order of convergence of frolov s cubature formula for numerical integration in function spaces of dominating mixed smoothness on the unit cube with homogeneous boundary condition more precisely we study worst case integration errors for besov mathbf b s p theta and triebel lizorkin spaces mathbf f s p 2 and our results treat the whole range of admissible parameters s geq 1 p in particular we obtain upper bounds for the difficult the case of small smoothness which is given for triebel lizorkin spaces mathbf f s p 2 in case 1 1 theta in the latter case the presented upper bounds are optimal i e they cannot be improved by any other cubature formula the optimality for small smoothness is open

a generalized discrepancy and quadrature error bound an error bound for multidimensional quadrature is derived that includes the koksma hlawka inequality as a special case this error bound takes the form of a product of two terms one term which depends only on the integrand is defined as a generalized variation the other term which depends only on the quadrature rule is defined as a generalized discrepancy the generalized discrepancy is a figure of merit for quadrature rules and includes as special cases the l p star discrepancy and p that arises in the study of lattice rules

algebraic discrete nonlocal dnl absorbing boundary condition for the ship wave resistance problem an absorbing boundary condition for the ship wave resistance problem is presented in contrast to the dawson like methods it avoids the use of numerical viscosities in the discretization so that a centered scheme can be used for the free surface operator the absorbing boundary condition is completely absorbing in the sense that the solution is independent of the position of the downstream boundary and is derived from straightforward analysis of the resulting constant coefficients difference equations assuming that the mesh is 1d structured in the longitudinal direction and requires the eigen decomposition of a matrix one dimension lower than the system matrix the use of a centered scheme for the free surface operator allows a full finite element discretization the drag is computed by a momentum flux balance this method is more accurate and guarantees positive resistances

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

absorbing boundary conditions for free surface wave simulations with a panel method abstract the numerical implementation and stability of first and second order absorbing boundary conditions for simulating free surface gravity waves are considered the free surface waves are solved with a panel method the stability of the boundary conditions is proved assuming certain properties of the integral operator arguments are given that support these assumptions the theoretical results are confirmed in the test cases and the boundary conditions give low reflections for plane waves

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

weighted aggregation operators based on minimization minimization based aggregation operators a g d are discussed special attention is paid to weighting function g based cases related to some fixed dissimilarity function d when d 2 x y x y 2 we recognize mixture operators and we recall some sufficient conditions for g ensuring the monotonicity of a g d for d 1 x y x y and non decreasing non increasing g a g d is shown to be the upper lower median whenever a g d is an aggregation operator

ordered directional monotonicity in the construction of edge detectors abstract in this paper we provide a specific construction method of ordered directionally monotone functions we show that the functions obtained with this construction method can be used to build edge detectors for grayscale images we compare the results of these detectors to those obtained with some other ones that are widely used in the literature finally we show how a consensus edge detector can be built improving the results obtained both by our proposal and by those in the literature when applied individually

optimal quadrature for haar wavelet spaces this article considers the error of the scrambled equidistribution quadrature rules in the worst case random case and average case settings the underlying space of integrands is a hilbert space of multidimensional haar wavelet series hwav the asymptotic orders of the errors are derived for the case of the scrambled t m s nets and t s sequences these rules are shown to have the best asymptotic convergence rates for any random quadrature rule for the space of integrands hwav

lattice rules in non periodic subspaces of sobolev spaces we investigate quasi monte carlo qmc integration over the s dimensional unit cube based on rank 1 lattice point sets in weighted non periodic sobolev spaces mathcal h k alpha varvec gamma s mathrm sob and their subspaces of high order smoothness alpha 1 where varvec gamma denotes a set of the weights a recent paper by dick nuyens and pillichshammer has studied qmc integration in half period cosine spaces with smoothness parameter alpha 1 2 consisting of non periodic smooth functions denoted by mathcal h k alpha varvec gamma s mathrm cos and also in the sum of half period cosine spaces and korobov spaces with common parameter alpha denoted by mathcal h k alpha varvec gamma s mathrm kor mathrm cos motivated by the results shown there we first study embeddings and norm equivalences on those function spaces in particular for an integer alpha we provide their corresponding norm equivalent subspaces of mathcal h k alpha varvec gamma s mathrm sob this implies that mathcal h k alpha varvec gamma s mathrm kor mathrm cos is strictly smaller than mathcal h k alpha varvec gamma s mathrm sob as sets for alpha ge 2 which solves an open problem by dick nuyens and pillichshammer then we study the worst case error of tent transformed lattice rules in mathcal h left k 2 varvec gamma s mathrm sob right and also the worst case error of symmetrized lattice rules in an intermediate space between mathcal h k alpha varvec gamma s mathrm kor mathrm cos and mathcal h k alpha varvec gamma s mathrm sob we show that the almost optimal rate of convergence can be achieved for both cases while a weak dependence of the worst case error bound on the dimension can be obtained for the former case

aggregation functions based on penalties this article studies a large class of averaging aggregation functions based on minimizing a distance from the vector of inputs or equivalently minimizing a penalty imposed for deviations of individual inputs from the aggregated value we provide a systematization of various types of penalty based aggregation functions and show how many special cases arise as the result we show how new aggregation functions can be constructed either analytically or numerically and provide many examples we establish connection with the maximum likelihood principle and present tools for averaging experimental noisy data with distinct noise distributions

boundary conditions at outflow for a problem with transport and diffusion abstract we present a numerical test of a class of approximate boundary conditions at an artificial boundary for the solution of time dependent problems in semi infinite spatial domains the conditions are constructed by use of a general method developed by the author in another work a problem modeling the transport and diffusion of soluble matter by a parallel flow is taken as the test case we demonstrate the effectiveness of the method in obtaining accurate results on small computational domains

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

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 monte carlo integration using digital nets with antithetics antithetic sampling which goes back to the classical work by hammersley and morton 1956 is one of the well known variance reduction techniques for monte carlo integration in this paper we investigate its application to digital nets over z b for quasi monte carlo qmc integration a deterministic counterpart of monte carlo of functions defined over the s dimensional unit cube by looking at antithetic sampling as a geometric technique in a compact totally disconnected abelian group we first generalize the notion of antithetic sampling from base 2 to an arbitrary base b 2 then we analyze the qmc integration error of digital nets over z b with b adic antithetics moreover for a prime b we prove the existence of good higher order polynomial lattice point sets with b adic antithetics for qmc integration of smooth functions in weighted sobolev spaces numerical experiments based on sobol point sets up to s 100 show that the rate of convergence can be improved for smooth integrands by using antithetic sampling technique which is quite encouraging beyond the reach of our theoretical result and motivates future work to address

on edge detection edge detection is the process that attempts to characterize the intensity changes in the image in terms of the physical processes that have originated them a critical intermediate goal of edge detection is the detection and characterization of significant intensity changes this paper discusses this part of the edge detection problem to characterize the types of intensity changes derivatives of different types and possibly different scales are needed thus we consider this part of edge detection as a problem in numerical differentiation we show that numerical differentiation of images is an ill posed problem in the sense of hadamard differentiation needs to be regularized by a regularizing filtering operation before differentiation this shows that this part of edge detection consists of two steps a filtering step and a differentiation step following this perspective the paper discusses in detail the following theoretical aspects of edge detection 1 the properties of different types of filters with minimal uncertainty with a bandpass spectrum and with limited support are derived minimal uncertainty filters optimize a tradeoff between computational efficiency and regularizing properties 2 relationships among several 2 d differential operators are established in particular we characterize the relation between the laplacian and the second directional derivative along the gradient zero crossings of the laplacian are not the only features computed in early vision 3 geometrical and topological properties of the zero crossings of differential operators are studied in terms of transversality and morse theory

improving the convergence rate of the petrov galerkin techniques for the solution of transonic and supersonic flows this paper report progress on a technique to accelerate the convergence to steady solutions when the streamline upwind petrov galerkin supg technique is used both the description of a supg formulation and the documentation of the development of a code for the finite element solution of transonic and supersonic flows are reported the aim of this work is to present a formulation to be able to treat domains of any configuration and to use the appropriate physical boundary conditions which are the major stumbling blocks of the finite difference schemes together with an appropriate convergence rate to the steady solution the implemented code has the following features the hughes supg type formulation with an oscillation free shock capturing operator adaptive refinement explicit integration with local time step and hourglassing control an automatic scheme for dealing with slip boundary conditions and a boundary augmented lumped mass matrix for speeding up convergence it is shown that the velocities at which the error is absorbed in and ejected from the domain that is damping and group velocities respectively are strongly affected by the time step used and that damping gives an o n2 algorithm contrasting with the o n one given by absorption at the boundaries nonetheless the absorbing effect is very low when very different eigenvalues are present such as in the transonic case because the stability condition imposes a too slow group velocity for the smaller eigenvalues to overcome this drawback we present a new mass matrix that provides us with a scheme having the highest group velocity attainable in all the components in section 1 we will describe briefly the theoretical background of the supg formulation in section 2 it is described how the foregoing formulation was used in the finite element code and which are the appropriate boundary conditions to be used finally in section 3 we will show some results obtained with this code

dynamic boundary conditions in computational fluid dynamics abstract the number and type of boundary conditions to be used in the numerical modeling of fluid mechanics problems is normally chosen according to a simplified analysis of the characteristics and also from the experience of the modeler the problem is harder at inflow outflow boundaries which are in most cases artificial boundaries so that a bad decision about the boundary conditions to be imposed may affect the precision and stability of the whole computation for inviscid flows the analysis of the sense of propagation in the normal direction to the boundaries gives the number of conditions to be imposed and in addition the conditions that are absorbing for the waves impinging normally to the boundary in practice it amounts to counting the number of positive and negative eigenvalues of the advective flux jacobian projected onto the normal the problem is even harder when the number of incoming characteristics varies during the computation and the correct treatment of these cases poses both mathematical and practical problems one example considered here is a compressible flow where the flow regime at a certain part of an inlet outlet boundary can change from subsonic to supersonic and the flow can revert in this work the technique for dynamically imposing the correct number of boundary conditions along the computation using lagrange multipliers and penalization is discussed and several numerical examples are presented

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

l_ 0 gradient projection minimizing l 0 gradient the number of the non zero gradients of an image together with a quadratic data fidelity to an input image has been recognized as a powerful edge preserving filtering method however the l 0 gradient minimization has an inherent difficulty a user given parameter controlling the degree of flatness does not have a physical meaning since the parameter just balances the relative importance of the l 0 gradient term to the quadratic data fidelity term as a result the setting of the parameter is a troublesome work in the l 0 gradient minimization to circumvent the difficulty we propose a new edge preserving filtering method with a novel use of the l 0 gradient our method is formulated as the minimization of the quadratic data fidelity subject to the hard constraint that the l 0 gradient is less than a user given parameter alpha this strategy is much more intuitive than the l 0 gradient minimization because the parameter alpha has a clear meaning the l 0 gradient value of the output image itself so that one can directly impose a desired degree of flatness by alpha we also provide an efficient algorithm based on the so called alternating direction method of multipliers for computing an approximate solution of the nonconvex problem where we decompose it into two subproblems and derive closed form solutions to them the advantages of our method are demonstrated through extensive experiments

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

quantitative error measures for edge detection how to measure the performance of an edge detection method is an open problem many performance measures have been presented in the literature however there seems to be no agreement on the best option and there exist no practical studies of the features of some of the most recent measures in this work we make a comprehensive overview of the different proposals followed by a practical comparison of the most representative measures on synthetic as well as natural edge images

fast algorithms for component by component construction of rank 1 lattice rules in shift invariant reproducing kernel hilbert spaces we reformulate the original component by component algorithm for rank 1 lattices in a matrix vector notation so as to highlight its structural properties for function spaces similar to a weighted korobov space we derive a technique which has construction cost o snlog n in contrast with the original algorithm which has construction cost o sn 2 herein s is the number of dimensions and n the number of points taken prime in contrast to other approaches to speed up construction our fast algorithm computes exactly the same quantity as the original algorithm the presented algorithm can also be used to construct randomly shifted lattice rules in weighted sobolev spaces

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

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

cyclic digital nets hyperplane nets and multivariate integration in sobolev spaces cyclic nets are a special case of digital nets and were recently introduced by niederreiter here we present a construction algorithm for such nets where we use the root mean square worst case error of a randomly digitally shifted point set in a weighted sobolev space as a selection criterion this yields a feasible construction algorithm since for a cyclic net with qm points with fixed bijections and fixed ground field there are qm possible choices our results here match the convergence rate and strong tractability results for polynomial lattice rules hence providing us with an alternative construction algorithm further we improve upon previous results by including constructions over arbitrary finite fields and an arbitrary choice of bijections

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

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 geometric approach to edge detection this paper describes edge detection as a composition of four steps conditioning feature extraction blending and scaling we examine the role of geometry in determining good features for edge detection and in setting parameters for functions to blend the features we find that 1 statistical features such as the range and standard deviation of window intensities can be as effective as more traditional features such as estimates of digital gradients 2 blending functions that are roughly concave near the origin of feature space ran provide visually better edge images than traditional choices such as the city block and euclidean norms 3 geometric considerations ran be used to specify the parameters of generalized logistic functions and takagi sugeno input output systems that yield a rich variety of edge images and 4 understanding the geometry of the feature extraction and blending functions is the key to using models based on computational learning algorithms such as neural networks and fuzzy systems for edge detection edge images derived from a digitized mammogram are given to illustrate various facets of our approach

stabilization and shock capturing parameters in supg formulation of compressible flows abstract the streamline upwind petrov galerkin supg formulation is one of the most widely used stabilized methods in finite element computation of compressible flows it includes a stabilization parameter that is known as typically the supg formulation is used in combination with a shock capturing term that provides additional stability near the shock fronts the definition of the shock capturing term includes a shock capturing parameter in this paper we describe for the finite element formulation of compressible flows based on conservation variables new ways for determining the and the shock capturing parameter the new definitions for the shock capturing parameter are much simpler than the one based on the entropy variables involve less operations in calculating the shock capturing term and yield better shock quality in the test computations

on the definition of penalty functions in data aggregation abstract in this paper we point out several problems in the different definitions and related results of penalty functions found in the literature then we propose a new standard definition of penalty functions that overcomes such problems some results related to averaging aggregation functions in terms of penalty functions are presented as the characterization of averaging aggregation functions based on penalty functions some examples are shown as the penalty functions based on spread measures which happen to be continuous we also discuss the definition of quasi penalty functions in order to deal with non monotonic or weakly directionally monotonic averaging functions

on candidates selection for hysteresis thresholds in edge detection manual determination of hysteresis thresholds is time consuming several methods approach the problem of unsupervised determination of edge detector parameters but they require human intervention to establish the initial range of values in which to detect the best parameter value and the result depends on the range of values initially used in this paper a method is proposed to determine candidates to hysteresis thresholds in an unsupervised manner the method provides a criterion to reduce in a significant way the number of initial values to be considered as threshold candidates the proposed method can be applied to any feature image provided by an edge detector upon which hysteresis must be implemented

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

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

component by component construction of good lattice rules this paper provides a novel approach to the construction of good lattice rules for the integration of korobov classes of periodic functions over the unit s dimensional cube theorems are proved which justify the construction of good lattice rules one component at a time that is the lattice rule for dimension s 1 is obtained from the rule for dimension s by searching over all possible choices of the s 1 th component while keeping all the existing components unchanged the construction which goes against accepted wisdom is illustrated by numerical examples the construction is particularly useful if the components of the integrand are ordered in the sense that the first component is more important than the second and so on

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

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

finite element computation of absorbing boundary conditions for time harmonic wave problems this paper proposes a new method in the frequency domain to define absorbing boundary conditions for general two dimensional problems the main feature of the method is that it can obtain boundary conditions from the discretized equations without much knowledge of the analytical behavior of the solutions and is thus very general it is based on the computation of waves in periodic structures and needs the dynamic stiffness matrix of only one period in the medium which can be obtained by standard finite element software boundary conditions at various orders of accuracy can be obtained in a simple way this is then applied to study some examples for which analytical or numerical results are available good agreements between the present results and analytical solutions allow to check the efficiency and the accuracy of the proposed method

walsh spaces containing smooth functions and quasi monte carlo rules of arbitrary high order we define a walsh space which contains all functions whose partial mixed derivatives up to order delta ge 1 exist and have finite variation in particular for a suitable choice of parameters this implies that certain sobolev spaces are contained in these walsh spaces for this walsh space we then show that quasi monte carlo rules based on digital t alpha s sequences achieve the optimal rate of convergence of the worst case error for numerical integration this rate of convergence is also optimal for the subspace of smooth functions explicit constructions of digital t alpha s sequences are given hence providing explicit quasi monte carlo rules which achieve the optimal rate of convergence of the integration error for arbitrarily smooth functions

finite element methods for first order hyperbolic systems with particular emphasis on the compressible euler equations a petrov galerkin finite element formulation is presented for first order hyperbolic systems of conservation laws with particular emphasis on the compressible euler equations applications of the methodology are made to one and two dimensional steady and unsteady flows with shocks results obtained suggest the potential of the type of methods developed

tight error bounds for rank 1 lattice sampling in spaces of hybrid mixed smoothness we consider the approximate recovery of multivariate periodic functions from a discrete set of function values taken on a rank 1 lattice moreover the main result is the fact that any non linear reconstruction algorithm taking function values on any integration lattice of size m has a dimension independent lower bound of 2 alpha 1 2 m alpha 2 when considering the optimal worst case error with respect to function spaces of hybrid mixed smoothness alpha 0 on the d torus we complement this lower bound with upper bounds that coincide up to logarithmic terms these upper bounds are obtained by a detailed analysis of a rank 1 lattice sampling strategy where the rank 1 lattices are constructed by a component by component method the lattice group structure allows for an efficient approximation of the underlying function from its sampled values using a single one dimensional fast fourier transform this is one reason why these algorithms keep attracting significant interest we compare our results to recent almost optimal methods based upon samples on sparse grids

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

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

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

optimized general sparse grid approximation spaces for operator equations this paper is concerned with the construction of optimized sparse grid approximation spaces for elliptic pseudodifferential operators of arbitrary order based on the framework of tensor product biorthogonal wavelet bases and stable subspace splittings we construct operator adapted subspaces with a dimension smaller than that of the standard full grid spaces but which have the same approximation order as the standard full grid spaces provided that certain additional regularity assumptions on the solution are fulfilled specifically for operators of positive order their dimension is o 2 j independent of the dimension n of the problem compared to o 2 jn for the full grid space also for operators of negative order the overall cost is significantly in favor of the new approximation spaces we give cost estimates for the case of continuous linear information we show these results in a constructive manner by proposing a galerkin method together with optimal preconditioning the theory covers elliptic boundary value problems as well as boundary integral equations

the dnl absorbing boundary condition applications to wave problems abstract a general methodology for developing absorbing boundary conditions is presented for planar surfaces it is based on a straightforward solution of the system of block difference equations that arise from partial discretization in the directions transversal to the artificial boundary followed by discretization on a constant step 1d grid in the direction normal to the boundary this leads to an eigenvalue problem of the size of the number of degrees of freedom in the lateral discretization the eigenvalues are classified as right or left going and the absorbing boundary condition consists in imposing a null value for the ingoing modes leaving free the outgoing ones whereas the classification is straightforward for operators with definite sign like the laplace operator a virtual dissipative mechanism has to be added in the mixed case usually associated with wave propagation phenomena like the helmholtz equation the main advantage of the method is that it can be implemented as a black box routine taking as input the coefficients of the linear system obtained from standard discretization fem or fdm packages and giving on output the absorption matrix we present the application of the dnl methodology to typical wave problems like helmholtz equations and potential flow with free surface the ship wave resistance and sea keeping problems