a mass redistributed finite element method mr fem for acoustic problems using triangular mesh the accuracy of numerical results using standard finite element method fem in acoustic problems will deteriorate with increasing frequency due to the dispersion error such dispersion error depends on the balance between the stiffness and mass of discretization equation systems this paper reports an improved finite element method fem for solving acoustic problems by re distributing the mass in the mass matrix to tune the balance aiming to minimize the dispersion errors this is done by shifting the integration point locations when computing the entries of the mass matrix while ensuring the mass conservation the new method is verified through the detailed numerical error analysis and a strategy is also proposed for the best mass redistribution in terms of minimizing dispersion error the relative dispersion error of present mass redistributed finite element method mr fem is found to be much smaller than the fem solution in both theoretical prediction and numerical examination the present mr fem works well by using the linear triangular elements that can be generated automatically which enables automation in computation and saving computational cost in mesh generation numerical examples demonstrate the advantages of mr fem in comparison with the standard fem using the same triangular meshes and quadrilateral meshes

good parameters for a class of node sets in quasi monte carlo integration for 2 s 12 we determine good parameters in a general construction of node sets for s dimensional quasi monte carlo integration recently introduced by the third author some of the parameters represent optimal choices in this construction and lead to improvements on node sets obtained by earlier techniques

strong tractability of quasi monte carlo quadrature using nets for certain banach spaces we consider multivariate integration in the weighted spaces of functions with mixed first derivatives bounded in l p norms and the weighted coefficients introduced via ell q norms where p q in 1 infty the integration domain may be bounded or unbounded the worst case error and randomized error are investigated for quasi monte carlo quadrature rules for the worst case setting the quadrature rule uses deterministic t u s sequences in base b and for the randomized setting the quadrature rule uses randomly scrambled digital t u m s nets in base b sufficient conditions are found under which multivariate integration is strongly tractable in the worst case and randomized settings respectively similar results hold for the banach spaces of finite order weights results presented in this article extend and improve upon those found previously

a reduced fast component by component construction of lattice points for integration in weighted spaces with fast decreasing weights lattice rules and polynomial lattice rules are quadrature rules for approximating integrals over the s dimensional unit cube since no explicit constructions of such quadrature methods are known for dimensions s 2 one usually has to resort to computer search algorithms the fast component by component approach is a useful algorithm for finding suitable quadrature rules we present a modification of the fast component by component algorithm which yields savings of the construction cost for polynomial lattice rules in weighted function spaces the idea is to reduce the size of the search space for coordinates which are associated with small weights and are therefore of less importance to the overall error compared to coordinates associated with large weights we analyze tractability conditions of the resulting qmc rules numerical results demonstrate the effectiveness of our method

wavelet based ilu preconditioners for the numerical solution by pufem of high frequency elastic wave scattering daubechies family of wavelets combined to the incomplete lower upper ilu factorization are considered as preconditioners for a block sparse linear system arising from the approximation of the time harmonic elastic wave equations by the partition of unity finite element method pufem after applying the discrete wavelet transform dwt to each dense block in the final matrix and the known right hand side due to the local enrichment by pressure p and shear s plane waves the resulting linear system is solved by the restarted generalized minimum residual method gmres with ilu preconditioners allowing fill in elements in the l and u matrix factors a reordering algorithm of the vertices based on gibbs method is also introduced it leads to a significant reduction of the bandwidth of the wave based finite element fe matrix and enables the ilu preconditioners to be more effective to study the performance of the proposed preconditioners a problem of a horizontal s plane wave scattered by a rigid circular body in an infinite elastic medium is considered the numerical tests show the good performance of the dwt based ilu preconditioners in improving the rate of convergence of gmres for high numbers of approximating p and s plane waves on coarse mesh grids containing multi wavelength sized elements moreover the haar dwt enhances the scalability with respect to the problem size when the number of the nodal points increases of the ilu preconditioner which uses the threshold strategy in the control of the fill in elements despite the high level of the conditioning a good accuracy may be achieved for a discretization level of around 1 9 degrees of freedom per s wavelength which is far below the rule of thumb of 10 nodal points per wavelength adopted for the conventional fe

the asymptotic efficiency of randomized nets for quadrature an l 2 type discrepancy arises in the average and worst case error analyses for multidimensional quadrature rules this discrepancy is uniquely defined by k x y which serves as the covariance kernel for the space of random functions in the average case analysis and a reproducing kernel for the space of functions in the worst case analysis this article investigates the asymptotic order of the root mean square discrepancy for randomized 0 m s nets in base b for moderately smooth k x y the discrepancy is o n 1 log n s 1 2 and for k x y with greater smoothness the discrepancy is o n 3 2 log n s 1 2 where n b m is the number of points in the net numerical experiments indicate that the t m s nets of faure niederreiter and sobol do not necessarily attain the higher order of decay for sufficiently smooth kernels however niederreiter nets may attain the higher order for kernels corresponding to spaces of periodic functions

a new computational method for transient dynamics including the low and the medium frequency ranges this paper deals with a new computational method for transient dynamic analysis which enables one to cover both the low and medium frequency ranges this is a frequency approach in which the low frequency part is obtained through a classical technique while the medium frequency part is handled through the variational theory of complex rays vtcr initially introduced for vibrations preliminary examples are shown

quasi monte carlo numerical integration on mathbb r s digital nets and worst case error quasi monte carlo rules are equal weight quadrature rules defined over the domain 0 1 s here we introduce quasi monte carlo type rules for numerical integration of functions defined on mathbb r s these rules are obtained by way of some transformation of digital nets such that locally one obtains quasi monte carlo rules but at the same time globally one also has the required distribution we prove that these rules are optimal for numerical integration in spaces of bounded fractional variation the analysis is based on certain tilings of the walsh phase plane as a proof of concept some numerical examples are included for dimensions between 3 and 10 we compare our method with quasi monte carlo rules transformed to mathbb r s using the inverse cumulative distribution function in these examples the new method significantly improves upon both methods for dimensions up to 5 no improvement is seen for dimension 10

brief control parametrization enhancing technique for optimal discrete valued control problems in this paper we consider a class of optimal discrete valued control problems since the range set of the control function is a discrete set and hence not convex these problems are in fact nonlinear combinatorial optimization problems using the novel idea of the control parametrization enhancing technique it is shown that optimal discrete valued control problems are equivalent to optimal control problems involving a new control function which is piecewise constant with pre fixed switching points the transformed problems are essentially optimal parameter selection problems and can hence be readily solved by various existing algorithms a practical numerical example is solved using the proposed method

representation of functions as walsh series to different bases and an application to the numerical integration of high dimensional walsh series we will prove the following theorem on walsh series and we will derive from this theorem an effective and constructive method for the numerical integration of walsh series by number theoretic methods further concrete computer calculations are given theorem for base b 2 dimension s 1 and o 1 c 0 b s n c a e r let i e c be the class of all functions f 0 l s c which are representable by absolutely convergent walsh series to base b with walsh coefficients w h hs with the following property w h hs 2 provided that a 1 s where

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

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

response variance prediction in the statistical energy analysis of built up systems in the statistical energy analysis sea of high frequency noise and vibration a complex engineering structure is represented as an assembly of subsystems the response of the system to external excitation is expressed in terms of the vibrational energy of each subsystem and these energies are found by employing the principle of power balance strictly the computed energy is an average taken over an ensemble of random structures and for many years there has been interest in extending the sea prediction to the variance of the energy a variance prediction method for a general built up structure is presented here closed form expressions for the variance are obtained in terms of the standard sea parameters and an additional set of parameters alpha k that describe the nature of the power input to each subsystem k and alpha ks that describe the nature of the coupling between subsystems k and s the theory is validated by comparison with monte carlo simulations of plate networks and structural acoustic systems

induction heating processes optimization a general optimal control approach a general automatic optimization procedure coupled to a finite element induction heating process simulation has been developed the mathematical model and the numerical methods are presented along with results validating the model the first part of this paper presents the direct induction heating mathematical model the related main numerical choices and especially the ultra weak coupling procedure the general optimization problem is then presented with the full detailed transposition of the ultra weak coupling procedure to the adjoint problem numerical results provided at the end prove the efficiency and robustness of the adjoint model in optimizing induction heating processes

monte carlo variance of scrambled net quadrature 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 this paper studies the variance of one such hybrid scrambled nets by applying a multidimensional multiresolution wavelet analysis to the integrand the integrand is assumed to be measurable and square integrable but not necessarily of bounded variation in simple monte carlo every nonconstant term of the multiresolution contributes to the variance of the estimated integral for scrambled nets certain low dimensional and coarse terms do not contribute to the variance for any integrand in l2 the sampling variance tends to zero faster under scrambled net quadrature than under monte carlo sampling as the number of function evaluations n tends to infinity some finite n results bound the variance under scrambled net quadrature by a small constant multiple of the monte carlo variance uniformly over all integrands f latin hypercube sampling is a special case of scrambled net quadrature

dpg method with optimal test functions for a transmission problem we propose and analyze a numerical method to solve an elliptic transmission problem in full space the method consists of a variational formulation involving standard boundary integral operators on the coupling interface and an ultra weak formulation in the interior to guarantee the discrete inf sup condition the system is discretized by the dpg method with optimal test functions we prove that the principal unknowns are approximated quasi optimally numerical experiments for problems with smooth and singular solutions confirm optimal convergence orders

a monte carlo method for integration of multivariate smooth functions we study a monte carlo algorithm that is based on a specific randomly shifted and dilated lattice point set the main result of this paper is that the mean squared error for a given compactly supported square integrable function is bounded by n 1 2 times the l 2 norm of the fourier transform outside a region around the origin where n is the expected number of function evaluations as corollaries we obtain the optimal order of convergence for functions from the sobolev spaces h s p with isotropic anisotropic or mixed smoothness with given compact support for all values of the parameters if the region of integration is the unit cube we obtain the same optimal orders for functions without boundary conditions this proves in particular that the optimal order of convergence in the latter case is n s 1 2 for p ge2 which is in contrast to the case of deterministic algorithms independent of the dimension this shows that monte carlo algorithms can improve the order by more than n

extension of the variational theory of complex rays to shells for medium frequency vibrations a new approach to the calculation of vibrations of weakly damped elastic structures in the medium frequency range called the variational theory of complex rays vtcr is being developed here the extension of this theory to shells of relatively small curvature is considered numerical examples of structures made of plates and shells demonstrate the capabilities of the vtcr

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

a relaxation procedure for domain decomposition methods using finite elements we present the convergence analysis of a new domain decomposition technique for finite element approximations this technique was introduced in 11 and is based on an iterative procedure among subdomains in which transmission conditions at interfaces are taken into account partly in one subdomain and partly in its adjacent no global preconditioner is needed in the practice but simply single domain finite element solvers are required an optimal strategy for an automatic selection of a relaxation parameter to be used at interface subdomains is indicated applications are given to both elliptic equations and incompressible stokes equations

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 phase based interior penalty discontinuous galerkin method for the helmholtz equation with spatially varying wavenumber this paper is concerned with an interior penalty discontinuous galerkin ipdg method based on a flexible type of non polynomial local approximation space for the helmholtz equation with varying wavenumber the local approximation space consists of multiple polynomial modulated phase functions which can be chosen according to the phase information of the solution we obtain some approximation properties for this space and a priori l2l2 error estimates for the h convergence of the ipdg method using duality argument we also provide ample numerical examples to show that building phase information into the local spaces often gives more accurate results comparing to using the standard polynomial spaces

discrete valued control of linear time invariant systems by sum of absolute values optimization in this paper we propose a new design method of discrete valued control for continuous time linear time invariant systems based on sum of absolute values soav optimization we first formulate the discrete valued control design as a finite horizon soav optimal control which is an extended version of mathbb l 1 optimal control we then give simple conditions that guarantee the existence discreteness and uniqueness of the soav optimal control also we show that the value function is continuous by which we prove the stability of infinite horizon model predictive soav control systems we provide a fast algorithm for the soav optimization based on the alternating direction method of multipliers admm which has an important advantage in real time control computation a simulation result shows the effectiveness of the proposed method

directional sparsity in optimal control of partial differential equations we study optimal control problems in which controls with certain sparsity patterns are preferred for time dependent problems the approach can be used to find locations for control devices that allow controlling the system in an optimal way over the entire time interval the approach uses a nondifferentiable cost functional to implement the sparsity requirements additionally bound constraints for the optimal controls can be included we study the resulting problem in appropriate function spaces and present two solution methods of newton type based on different formulations of the optimality system using elliptic and parabolic test problems we research the sparsity properties of the optimal controls and analyze the behavior of the proposed solution algorithms

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

an a posteriori error estimate for the generalized finite element method for transient heat diffusion problems summary we propose the study of a posteriori error estimates for time dependent generalized finite element simulations of heat transfer problems a residual estimate is shown to provide reliable and practically useful upper bounds for the numerical errors independent of the heuristically chosen enrichment functions two sets of numerical experiments are presented first the error estimate is shown to capture the decrease in the error as the number of enrichment functions is increased or the time discretization refined second the estimate is used to predict the behaviour of the error where no exact solution is available it also reflects the errors incurred in the poorly conditioned systems typically encountered in generalised finite element methods finally we study local error indicators in individual time steps and elements of the mesh this creates a basis towards the adaptive selection and refinement of the enrichment functions this article is protected by copyright all rights reserved

a least squares method for the helmholtz equation abstract we investigate the use of least squares methods to approximate the helmholtz equation the basis used in the discrete method consists of solutions of the helmholtz equation either consisting of plane waves or bessel functions on each element of a finite element grid unlike previous methods of this type we do not use polynomial based finite elements the use of small elements and relatively few basis functions per element allows us to prove convergence theorems for the method and to some extent control the conditioning of the resulting linear sy stem numerical results show the efficiency of the new method and suggest that it may be possible to obtain accurate results with a coarser grid than is usual for standard finite element methods

an ultra weak method for acoustic fluid solid interaction we introduce the ultra weak variational formulation uwvf for fluid solid vibration problems in particular we consider the scattering of time harmonic acoustic pressure waves from solid elastic objects the problem is modeled using a coupled system of the helmholtz and navier equations the transmission conditions on the fluid solid interface are represented in an impedance type form after which we can employ the well known ultra weak formulations for the helmholtz and navier equations the uwvf approximation for both equations is computed using a superposition of propagating plane waves a condition number based criterion is used to define the plane wave basis dimension for each element as a model problem we investigate the scattering of sound from an infinite elastic cylinder immersed in a fluid a comparison of the uwvf approximation with the analytical solution shows that the method provides a means for solving wave problems on relatively coarse meshes however particular care is needed when the method is used for problems at frequencies near the resonance frequencies of the fluid solid system

application of an ultra weak variational formulation of elliptic pdes to the two dimensional helmholtz problem a new technique to solve elliptic linear pdes called ultra weak variational formulation uwvf in this paper is introduced in b despres c r acad sci paris 318 1994 pp 939 944 this paper is devoted to an evaluation of the potentialities of this technique it is applied to a model wave problem the two dimensional helmholtz problem the new method is presented in three parts following the same style of presentation as the classical one of the finite elements method even though they are definitely conceptually different methods the first part is committed to the variational formulation and to the continuous problem the second part defines the discretization process using a galerkin procedure the third part actually studies the efficiency of the technique from the order of convergence point of view this is achieved using theoretical proofs and a series of numerical experiments in particular it is proven and shown the order of convergence is lower bounded by a linear function of the number of degrees of freedom an application to scattering problems is presented in a fourth part

a multivariate fast discrete walsh transform with an application to function interpolation for high dimensional problems such as approximation and integration one cannot afford to sample on a grid because of the curse of dimensionality an attractive alternative is to sample on a low discrepancy set such as an integration lattice or a digital net this article introduces a multivariate fast discrete walsh transform for data sampled on a digital net that requires only o n log n operations where n is the number of data points this algorithm and its inverse are digital analogs of multivariate fast fourier transforms this fast discrete walsh transform and its inverse may be used to approximate the walsh coefficients of a function and then construct a spline interpolant of the function this interpolant may then be used to estimate the function s effective dimension an important concept in the theory of numerical multivariate integration numerical results for various functions are presented

the partition of unity finite element method for the simulation of waves in air and poroelastic media recently chazot et al j sound vib 332 1918 1929 2013 applied the partition of unity finite element method for the analysis of interior sound fields with absorbing materials the method was shown to allow a substantial reduction of the number of degrees of freedom compared to the standard finite element method the work is however restricted to a certain class of absorbing materials that react like an equivalent fluid this paper presents an extension of the method to the numerical simulation of biot s waves in poroelastic materials the technique relies mainly on expanding the elastic displacement as well as the fluid phase pressure using sets of plane waves which are solutions to the governing partial differential equations to show the interest of the method for tackling problems of practical interests poroelastic acoustic coupling conditions as well as fixed or sliding edge conditions are presented and numerically tested it is shown that the technique is a good candidate for solving noise control problems at medium and high frequency

efficient weighted lattice rules with applications to finance good lattice rules are an important type of quasi monte carlo algorithm they are known to have good theoretical properties in the sense that they can achieve an error bound or optimal error bound that is independent of the dimension for weighted function spaces with suitably decaying weights to use the theory of weighted function spaces for practical applications one has to determine what weights should be used after explaining that lattice rules based on figures of merit with classical weights classical lattice rules may not give good results for high dimensional problems we propose a matching strategy which chooses the weights in such a way that the typical functions of a weighted korobov space are similar in the sense of similar relative sensitivity indices to those of the given function the matching strategy relates the weights of the spaces to the sensitivity indices of the given function we apply the korobov construction and the component by component construction of lattice rules with the suitably chosen weights to the valuation of high dimensional financial derivative securities and find that such weighted lattice rules improve dramatically on the results for the classical lattice rules moreover they are competitive with other types of low discrepancy sequences we also demonstrate that lattice rules combined with variance reduction techniques and dimension reduction techniques increase the efficiency significantly we show that an acceptance rejection approach to the construction of lattice rules can reduce the construction cost with almost no loss of accuracy

residual free bubbles for the helmholtz equation the galerkin method enriched with residual free bubbles is considered for approximating the solution of the helmholtz equation two dimensional tests demonstrate the improvement over the standard galerkin method and the galerkin least squares method using piecewise bilinear interpolations

approximation of elliptic control problems in measure spaces with sparse solutions optimal control problems in measure spaces governed by elliptic equations are considered for distributed and neumann boundary control which are known to promote sparse solutions optimality conditions are derived and some of the structural properties of their solutions in particular sparsity are discussed a framework for their approximation is proposed which is efficient for numerical computations and for which we prove convergence and provide error estimates

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

is the helmholtz equation really sign indefinite the usual variational or weak formulations of the helmholtz equation are sign indefinite in the sense that the bilinear forms cannot be bounded below by a positive multiple of the appropriate norm squared this is often for a good reason since in bounded domains under certain boundary conditions the solution of the helmholtz equation is not unique at wavenumbers that correspond to eigenvalues of the laplacian and thus the variational problem cannot be sign definite however even in cases where the solution is unique for all wavenumbers the standard variational formulations of the helmholtz equation are still indefinite when the wavenumber is large this indefiniteness has implications for both the analysis and the practical implementation of finite element methods in this paper we introduce new sign definite also called coercive or elliptic formulations of the helmholtz equation posed in either the interior of a star shaped domain with impedance boundary conditions or the exterior of a star shape

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

randomized polynomial lattice rules for multivariate integration and simulation lattice rules are among the best methods to estimate integrals in a large number of dimensions they are part of the quasi monte carlo set of tools a theoretical framework for a class of lattice rules defined in a space of polynomials with coefficients in a finite field is developed in this paper a randomized version is studied implementations and criteria for selecting the parameters are discussed and examples of its use as a variance reduction tool in stochastic simulation are provided certain types of digital net constructions as well as point sets constructed by taking all vectors of successive output values produced by a tausworthe random number generator are special cases of this method

a multi level fast multipole bem for 3 d elastodynamics in the frequency domain to reduce computational complexity and memory requirement for 3 d elastodynamics using the boundary element method bem a multi level fast multipole bem fm bem is proposed the diagonal form for the expansion of the elastodynamic fundamental solution is used with a truncation parameter adjusted to the subdivision level a feature necessary for achieving optimal computational efficiency both the single level and mul ti level forms of the elastodynamic fm bem are considered with emphasis on the latter crucial implementation issues including the truncation of the multipole expansion the optimal number of levels the direct and inverse extrapolation steps are examined in detail with the backing of numerical experiments a complexity analysis for both the single level and multi level versions is conducted the correctness and computational performances of the proposed elastodynamic fmm are demonstrated on numerical examples featuring up to o 10 6 dofs run on a single processor pc and including the diffraction of an incident p plane wave by a semi spherical or semi ellipsoidal canyon representative of topographic site effects

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

the fourier version of the variational theory of complex rays for medium frequency acoustics the variational theory of complex rays vtcr is a wave based computational approach dedicated to the resolution of medium frequency problems it uses a variational formulation of the problem which enables one to use any type of shape function within the substructures provided that it satisfies the governing equation thus the solution can be approximated using plane waves which is very interesting in the medium frequency vibration domain and also leads to a strong convergence of the method in the previous works this was shown in the case of acoustic problems in which the amplitudes of the plane waves were calculated as wavebands in this paper we propose a new approximation of these amplitudes based on fourier series we show that this approach increases the robustness of the method makes it more efficient numerically and extends its applicability to somewhat higher frequencies

a first order system least squares method for the helmholtz equation we present a first order system least squares fosls method for the helmholtz equation at high wave number k which always leads to a hermitian positive definite algebraic system by utilizing a non trivial solution decomposition to the dual fosls problem which is quite different from that of the standard finite element methods we give an error analysis to the h p version of the fosls method where the dependence on the mesh size h the approximation order p and the wave number k is given explicitly in particular under some assumption of the boundary of the domain the l 2 norm error estimate of the scalar solution from the fosls method is shown to be quasi optimal under the condition that k h p is sufficiently small and the polynomial degree p is at least o log k numerical experiments are given to verify the theoretical results

on the asymptotic distribution of scrambled net quadrature recently in a series of articles owen proposed the use of scrambled t m s nets and t s sequences in high dimensional numerical integration these scrambled nets and sequences achieve the superior accuracy of equidistribution methods while allowing for the simpler error estimation techniques of monte carlo methods the main aim of this article is to use stein s method to study the asymptotic distribution of the scrambled 0 m s net integral estimate in particular it is shown that for suitably smooth integrands on the s dimensional unit hypercube the estimate has an asymptotic normal distribution

a comparison of nrbcs for pufem in 2d helmholtz problems at high wave numbers in this work exact and approximate non reflecting boundary conditions nrbcs are implemented with the partition of unity finite element method pufem to solve short wave scattering problems governed by the helmholtz equation in two dimensions by short wave problems we mean situations in which the wavelength is a small fraction of the characteristic dimension of the scatterer various nrbcs are implemented and a comparison of their performance is carried out based on the accuracy of the results ease of implementation and computational cost the aim is to accurately model such problems in a reduced computational domain around the scatterer with fewer elements and without refining the mesh at each wave number

figures of merit for digital multistep pseudorandom numbers the statistical independence properties of s successive digital multistep pseudorandom numbers are governed by the figure of merit p s i which depends on s and the characteristic polynomial f of the recursion used in the generation procedure we extend previous work for s 2 and describe how to obtain large figures of merit for s 2 thus arriving at digital multistep pseudorandom numbers with attractive statistical independence properties tables of figures of merit for s 3 4 5 and degrees 32 are included

the ultra weak variational formulation for elastic wave problems the ultra weak variational formulation has been used effectively to solve time harmonic acoustic and electromagnetic wave propagation in inhomogeneous media we develop the ultra weak variational formulation for elastic wave propagation in two space dimensions in order to improve the accuracy and stability of the method we find it necessary to approximate the s and p wave components of the solution in a balanced way some preliminary analysis is provided and numerical evidence is presented for the efficiency of the scheme in comparison to piecewise linear finite elements

the variational theory of complex rays an answer to the resolution of mid frequency 3d engineering problems the variational theory of complex rays vtrc is an approach to the simulation of mid frequency phenomena whose wavelengths are relatively small compared to the dimensions of the domain this is a wave based computational technique which involves a nonclassical variational formulation this paper focuses on the development of the approach for 3d engineering problems and shows that it is a mature technique illustrations are given to show the capabilities of this method

value function in maximum hands off control for linear systems in this brief paper we study the value function in maximum hands off control maximum hands off control also known as sparse control is the l 0 optimal control among the feasible controls although the l 0 measure is discontinuous and non convex we prove that the value function or the minimum l 0 norm of the control is a continuous and strictly convex function of the initial state in the reachable set under an assumption on the controlled plant model we then extend the finite horizon maximum hands off control to model predictive control mpc and prove the recursive feasibility and the stability by using the continuity and convexity properties of the value function

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

construction algorithms for digital nets with low weighted star discrepancy we introduce a new construction method for digital nets which yield point sets in the s dimensional unit cube with low star discrepancy the digital nets are constructed using polynomials over finite fields it has long been known that there exist polynomials which yield point sets with low unweighted star discrepancy this result was obtained by niederreiter by the means of averaging over all polynomials hence concrete examples of good polynomials were not known in many cases here we show that good polynomials can be found by computer search the search algorithm introduced in this paper is based on minimizing a quantity closely related to the star discrepancy it has been pointed out that many integration problems can be modeled by weighted function spaces and it has been shown that in this case point sets with low weighted discrepancy are required hence it is particularly useful to be able to adjust a point set to some given weights we are able to extend our results from the unweighted case to show that this can be done using our construction algorithms this way we can find point sets with low weighted star discrepancy making such point sets especially useful for many applications

well posedness and generalized plane waves simulations of a 2d mode conversion model certain types of electro magnetic waves propagating in a plasma can undergo a mode conversion process in magnetic confinement fusion this phenomenon is very useful to heat the plasma since it permits to transfer the heat at or near the plasma center this work focuses on a mathematical model of wave propagation around the mode conversion region from both theoretical and numerical points of view it aims at developing for a well posed equation specific basis functions to study a wave mode conversion process these basis functions called generalized plane waves are intrinsically based on variable coefficients as such they are particularly adapted to the mode conversion problem the design of generalized plane waves for the proposed model is described in detail their implementation within a discontinuous galerkin method then provides numerical simulations of the process these first 2d simulations for this model agree with qualitative aspects studied in previous works

the construction of extensible polynomial lattice rules with small weighted star discrepancy in this paper we introduce a construction algorithm for extensible polynomial lattice rules and we prove that the construction algorithm yields generating vectors of polynomials which are optimal for a range of moduli chosen in advance the construction algorithm uses a sieve where the generating vectors are extended by one coefficient in each component at each step and where one keeps a certain number of good ones and discards the rest we also show that the construction can be done component by component

the discontinuous enrichment method for multiscale analysis computation naturally separates scales of a problem according to the mesh size a variety of improved numerical methods are described by multiscale considerations differing in the treatment of the unresolved fine scales the discontinuous enrichment method provides a unique multiscale approach to computation by employing fine scales that contain solutions of the homogeneous partial differential equation in a discontinuous framework the method thus combines relative ease of implementation with improved numerical performance these properties are demonstrated for both multiscale wave and transport problems pointing to the potential of considerable savings in computational resources

optimal polynomials for t m s nets and numerical integration of multivariate walsh series existence theorems for good parameters in a construction of low discrepancy point sets based on rational functions over finite fields are proved a special class of these point sets is introduced and good parameters in the construction of these special point sets are tabulated in this way point sets in the s dimensional unit cube for 3 leq s leq 15 are obtained with very small discrepancy these special point sets yield excellent numerical results when applied to the quasi monte carlo integration of multivariate walsh series

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

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

on the numerical integration of walsh series by number theoretic methods in analogy to the theory of good lattice points for the numerical integration of rapidly converging fourier series a theory for the fast numerical integration of walsh series is developed the basis for this theory is provided by a class of very well distributed point sets in the s dimensional unit cube the so called t m s nets

the generalized finite element method an example of its implementation and illustration of its performance the generalized finite element method gfem was introduced in reference 1 as a combination of the standard fem and the partition of unity method the standard mapped polynomial finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data the geometry of the domain the loads and the boundary conditions the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and are pasted to the existing finite element basis to construct a conforming approximation the essential boundary conditions can be imposed exactly as in the standard fem adaptive numerical quadrature is used to ensure that the errors in integration do not affect the accuracy of the approximation this paper gives an example of how the gfem can be developed for the laplacian in domains with multiple elliptical voids and illustrates implementation issues and the superior accuracy of the gfem versus the standard fem copyright 2000 john wiley sons ltd

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

improvement of pufem for the numerical solution of high frequency elastic wave scattering on unstructured triangular mesh grids the purpose of this paper which builds on previous work int j numer meth engng 2009 77 1646 1669 is to improve a numerical scheme based on the partition of unity finite element method pufem for the solution of the time harmonic elastic wave equations the approach consists to approximate the displacement field by the standard finite element shape functions enriched locally by superimposing pressure p and shear s plane waves the aim is to accurately model two dimensional elastic wave problems on relatively coarse mesh grids capable of containing many wavelengths per nodal spacing for wide ranges of frequencies this allows us to relax the traditional requirement of about 10 nodal points per s wavelength in this work an exact integration scheme for the linear triangular finite element is developed to evaluate the oscillatory integrals arising from the use of the pufem the main contribution here consists in developing an explicit closed form solution for two dimensional wave based integrals when the phase variation is linear in the local coordinate element system the evaluation of the element mass matrix is performed from appropriate edge integrals all other element matrices obtained by adequate splitting of the element stress tensor matrix are simply deduced from the element mass matrix entries the results show clearly that the proposed integration scheme evaluates accurately the entries of the global matrix with drastic reduction of the computational time numerical tests dealing with the scattering of s elastic plane waves by a circular rigid body show that for the same discretization level it is possible to improve the accuracy by using large elements associated with high numbers of approximating plane waves rather than using small elements with less plane waves however this increases the conditioning and the fill in of the global matrix at high frequency it is even possible to push the number of degrees of freedom per s wavelength under 2 and still achieve good accuracy finally some remarks on the choice of the numbers of p and s plane waves leading to better accuracy and conditioning are discussed copyright 2010 john wiley sons ltd

clot norm minimization for continuous hands off control abstract in this paper we propose optimal control that is both sparse and continuous unlike previously proposed alternatives to maximum hands off control the maximum hands off control is the l 0 optimal or sparsest control among all feasible controls that are bounded by a specified value and transfer the state from a given initial state to the origin within a fixed time duration the proposed control is obtained via minimization of the clot combined l one and two norm of the control input along with the constraints on the state variable the constraints on the state variable ensures that the states are not blown up while achieving the optimal control by using the non smooth maximum principle we prove that the clot norm optimal control is unique and it is continuous in the time variable we show by numerical simulations that the clot control is continuous unlike l 0 optimal control or maximum hands off control and much sparser i e has longer time duration on which the control equals 0 than the conventional en elastic net control which is a convex combination of l 1 and squared l 2 norms

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

the generalized finite element method for helmholtz equation theory computation and open problems in this paper we address the generalized finite element method for the helmholtz equation we obtain our method by employing the finite element method on cartesian meshes which may overlap the boundaries of the problem domain and by enriching the approximation by plane waves pasted into the finite element basis at each mesh vertex by the partition of unity method here we address the q convergence of the method where q is the number of plane waves added at each vertex for the class of smooth analytic solutions for which we get better than exponential convergence for sufficiently small h depending on p an important observation is that we can monitor the accuracy in any computed solution quantity of interest at negligible cost by using q extrapolation our results assume exact integration of all the employed integrals further studies are needed to analyze the effects of the numerical integrations and also the effect of the roundoff errors

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

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

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

the construction of good extensible rank 1 lattices it has been shown by hickernell and niederreiter that there exist generating vectors for integration lattices which yield small integration errors for n p p 2 for all integers p 2 this paper provides algorithms for the construction of generating vectors which are finitely extensible for n p p 2 for all integers p 2 the proofs which show that our algorithms yield good extensible rank 1 lattices are based on a sieve principle particularly fast algorithms are obtained by using the fast component by component construction of nuyens and cools analogous results are presented for generating vectors with small weighted star discrepancy

a generalized finite element method for solving the helmholtz equation in two dimensions with minimal pollution when using the galerkin fem for solving the helmholtz equation in two dimensions the error of the corresponding solution differs substantially from the error of the best approximation and this effect increases with higher wave number k in this paper we will design a generalized finite element method gfem for the helmholtz equation such that the pollution effect is minimal

adaptive numerical solution of a discontinuous galerkin method for a helmholtz problem in low frequency regime we develop an a posteriori error analysis for helmholtz problem using the local discontinuous galerkin ldg for short approach for the sake of completeness we give a description of the main a priori results of this method indeed under some assumptions on regularity of the solution of an adjoint problem we prove that a the corresponding indefinite discrete scheme is well posed b the approach is convergent with the expected convergence rates as long as the meshsize h is small enough we give precise information on how small h has to be in terms of the size of the wave number and its distance to the set of eigenvalues for the same boundary value problem for the laplacian after that we present a reliable and efficient a posteriori error estimator with detailed information on the dependence of the constants on the wave number we finish presenting extensive numerical experiments which illustrate the theoretical results proven in this paper and suggest that stability and convergence may occur under less restrictive assumptions than those taken in the present work we derive an a posteriori error estimator that is reliable and locally quasi efficient the efficiency and reliability constants have been characterized with respect to the physical parameter we include numerical examples that validate our theoretical results recognizing singularities and inner layers for moderate values of

richardson s iteration for nonsymmetric matrices abstract to solve the linear n n system 1 ax a for any nonsingular matrix a richardson s iteration 2 x j 1 x j j ax j a j 1 2 n which is applied in a cyclic manner with cycle length n is investigated where the j are free parameters the objective is to minimize the error x n 1 x where x is the solution of 1 if the spectrum of a is known to lie in a compact set s one is led to the chebyshev type approximation problem 3 min p 1 v n max z s p z where v n is the linear span of z z 2 z n if p solves 3 then the reciprocals of the zeros of p are optimal iteration parameters j it is shown that for a real problem 1 the iteration 2 can be carried out with real arithmetic alone even when there are complex j the stationary case n 1 is solved completely i e for all compact sets s the problem 3 is solved explicitly as a consequence the converging stationary iteration processes can be characterized for arbitrary closed disks s the problem 3 can be solved for all n n and a simple proof is provided the lemniscates associated with s are introduced they appear as an important tool for studying the stability of the iteration 2

three dimensional discontinuous galerkin elements with plane waves and lagrange multipliers for the solution of mid frequency helmholtz problems recently a discontinuous galerkin finite element method with plane wave basis functions and lagrange multiplier degrees of freedom was proposed for the efficient solution in two dimensions of helmholtz problems in the mid frequency regime in this paper this method is extended to three dimensions and several new elements are proposed computational results obtained for several wave guide and acoustic scattering model problems demonstrate one to two orders of magnitude solution time improvement over the higher order galerkin method

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

strong tractability of integration using scrambled niederreiter points we study the randomized worst case error and the randomized error of scrambled quasi monte carlo qmc quadrature as proposed by owen the function spaces considered in this article are the weighted hilbert spaces generated by haar like wavelets and the weighted sobolev hilbert spaces conditions are found under which multivariate integration is strongly tractable in the randomized worst case setting and the randomized setting respectively the e exponents of strong tractability are found for the scrambled niederreiter nets and sequences the sufficient conditions for strong tractability for sobolev spaces are more lenient for scrambled qmc quadratures than those for deterministic qmc net quadratures

the variational theory of complex rays a predictive tool for medium frequency vibrations abstract today the main numerical modeling techniques for the analysis of medium frequency vibrations are all based on finite element or boundary element approaches in order to represent small wavelength phenomena in complex structures such as car chassis satellites or ships these techniques require a huge number of degrees of freedom at least seven elements per wavelength are required to represent oscillating solutions in addition the solution obtained is highly sensitive to material properties and boundary conditions the use of high frequency approaches such as the statistical energy analysis sea or any of its improvements does not appear to be suitable for medium frequency vibrations the vibrational behavior becomes too smooth and in general the coupling loss factor cannot be calculated in a predictive way in this paper a new and multiscale approach to the calculation of the vibrations of elastic structures in the medium frequency range called the variational theory of complex rays is presented the feasibility and effectiveness of the method are demonstrated for structures made of plate assemblies several 3d examples are analyzed and compared to the results from a classical finite element approach and from a sea code this comparison shows that our method is able to predict the effective quantities at a very low numerical cost

why are high dimensional finance problems often of low effective dimension many problems in mathematical finance can be formulated as high dimensional integrals where the large number of dimensions arises from small time steps in time discretization and or a large number of state variables quasi monte carlo qmc methods have been successfully used for approximating such integrals to understand this success this paper focuses on investigating the special features of some typical high dimensional finance problems namely option pricing and bond valuation we provide new insight into the connection between the effective dimension and the efficiency of qmc and we present methods to analyze the dimension structure of a function we confirm the observation of caflisch morokoff and owen that functions from finance are often of low effective dimension in the sense that they can be well approximated by their low order anova analysis of variance terms usually just the order 1 and order 2 terms we explore why the effective dimension is small for many integrals from finance by deriving explicit forms of the anova terms in simple cases we find that the importance of each dimension is naturally weighted by certain hidden weights these weights characterize the relative importance of different variables or groups of variables and limit the importance of the higher order anova terms we study the variance ratios captured by low order anova terms and their asymptotic properties as the dimension tends to infinity and we show that with the increase of dimension the lower order terms continue to play a significant role and the higher order terms tend to be negligible this provides some insight into high dimensional problems from finance and explains why qmc algorithms are efficient for problems of this kind

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

the discontinuous enrichment method we propose a finite element based discretization method in which the standard polynomial field is enriched within each element by a nonconforming field that is added to it the enrichment contains free space solutions of the homogeneous differential equation that are not represented by the underlying polynomial field continuity of the enrichment across element interfaces is enforced weakly by lagrange multipliers in this manner we expect to attain high coarse mesh accuracy without significant degradation of conditioning assuring good performance of the computation at any mesh resolution examples of application to acoustics and advection diffusion are presented

element free galerkin solutions for helmholtz problems fomulation and numerical assessment of the pollution effect abstract the element free galerkin method efgm a particular case of the meshless methods is examined in its application to acoustic wave propagation addressed by the helmholtz equation dispersion and pollution effects two problems encountered by the classical numerical methods are reviewed numerical tests on two dimensional problems focus on the parameters governing the formulation of the efgm they also demonstrate that the efgm is affected by dispersion and pollution effects as well as fem but these effects are rather low showing that the efgm is a promising method