isogeometric variational multiscale modeling of wall bounded turbulent flows with weakly enforced boundary conditions on unstretched meshes abstract in this work we combine i nurbs based isogeometric analysis ii residual driven turbulence modeling and iii weak imposition of no slip and no penetration dirichlet boundary conditions on unstretched meshes to compute wall bounded turbulent flows while the first two ingredients were shown to be successful for turbulence computations at medium to high reynolds number i akkerman y bazilevs v m calo t j r hughes s hulshoff the role of continuity in residual based variational multiscale modeling of turbulence comput mech 41 2008 371 378 y bazilevs v m calo j a cottrell t j r hughes a reali g scovazzi variational multiscale residual based turbulence modeling for large eddy simulation of incompressible flows comput methods appl mech engrg 197 2007 173 201 it is the weak imposition of no slip boundary conditions on coarse uniform meshes that maintains the good performance of the proposed methodology at higher reynolds number y bazilevs t j r hughes weak imposition of dirichlet boundary conditions in fluid mechanics comput fluids 36 2007 12 26 y bazilevs c michler v m calo t j r hughes weak dirichlet boundary conditions for wall bounded turbulent flows comput methods appl mech engrg 196 2007 4853 4862 these three ingredients form a basis of a possible practical strategy for computing engineering flows somewhere between rans and les in complexity we demonstrate this by solving two challenging incompressible turbulent benchmark problems channel flow at friction velocity reynolds number 2003 and flow in a planar asymmetric diffuser we observe good agreement between our calculations of mean flow quantities and both reference computations and experimental data this lends some credence to the proposed approach which we believe may become a viable engineering tool

numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media a numerical procedure for the determination of equivalent grid block permeability tensors for heterogeneous porous media is presented the method entails solution of the fine scale pressure equation subject to periodic boundary conditions to yield upon appropriate averaging of the fine scale velocity field the coarse scale or equivalent grid block permeability when the region over which this coarse scale permeability is computed constitutes a representative elementary volume rev the resulting equivalent permeability may be interpreted as the effective permeability of the region solution of the pressure equation on the fine scale is accomplished through the application of an accurate triangle based finite element numerical procedure which allows for the modeling of geometrically complex features the specification of periodic boundary conditions is shown to yield symmetric positive definite equivalent permeability tensors in all cases the method is verified through application to a periodic model problem and is then applied to the scale up of areal and cross sections with fractally generated permeability fields the applicability and limitations of the method for these more general heterogeneity fields are discussed

localization of the generalized sampling series and its numerical application we study the localization of the sampling series with general kernel and obtain its error estimates to achieve exponentially decaying accuracy for the series to approximate a band limited function and its derivatives a sufficient condition is given and a rigorous criterion is provided for the sampling to obtain any desired accuracy the result includes several known results as special cases and gives new results about wavelet sampling series

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

symmetric green s functions on complete manifolds on demontre d une facon constructive que l operateur de laplace admet une fonction de green symetrique g x y sur une variete de riemann complete

unified hybridization of discontinuous galerkin mixed and continuous galerkin methods for second order elliptic problems we introduce a unifying framework for hybridization of finite element methods for second order elliptic problems the methods fitting in the framework are a general class of mixed dual finite element methods including hybridized mixed continuous galerkin nonconforming and a new wide class of hybridizable discontinuous galerkin methods the distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements since the associated matrix is sparse symmetric and positive definite these methods can be efficiently implemented moreover the framework allows in a single implementation the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled finally the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom

multichannel interpolation of nonuniform samples with application to image recovery abstract the multichannel trigonometric reconstruction from uniform samples was proposed recently it not only makes use of multichannel information about the signal but is also capable to generate various kinds of interpolation formulas according to the types and amounts of the collected samples the paper presents the theory of multichannel interpolation from nonuniform samples two distinct models of nonuniform sampling patterns are considered namely recurrent and generic nonuniform sampling each model involves two types of samples nonuniform samples of the observed signal and its derivatives numerical examples and quantitative error analysis are provided to demonstrate the effectiveness of the proposed algorithms additionally the proposed algorithms for recovering highly corrupted images are also investigated in comparison with the median filter and the shepard interpolation the proposed algorithms produce superior results with lower errors

a comparison of pilot aided channel estimation methods for ofdm systems this paper deals with the estimation of the channel impulse response cir in orthogonal frequency division multiplexed ofdm systems in particular we focus on two pilot aided schemes the maximum likelihood estimator mle and the bayesian minimum mean square error estimator mmsee the advantage of the former is that it is simpler to implement as it needs no information on the channel statistics on the other hand the mmsee is expected to have better performance as it exploits prior information about the channel theoretical analysis and computer simulations are used in the comparisons at snr values of practical interest the two schemes are found to exhibit nearly equal performance provided that the number of pilot tones is sufficiently greater than the cirs length otherwise the mmsee is superior in any case the mmsee is more complex to implement

an adaptive gmsfem for high contrast flow problems in this paper we derive an a posteriori error indicator for the generalized multiscale finite element method gmsfem framework this error indicator is further used to develop an adaptive enrichment algorithm for the linear elliptic equation with multiscale high contrast coefficients the gmsfem which has recently been introduced in 13 allows solving multiscale parameter dependent problems at a reduced computational cost by constructing a reduced order representation of the solution on a coarse grid the main idea of the method consists of 1 the construction of snapshot space 2 the construction of the offline space and 3 the construction of the online space the latter for parameter dependent problems in 13 it was shown that the gmsfem provides a flexible tool to solve multiscale problems with a complex input space by generating appropriate snapshot offline and online spaces in this paper we study an adaptive enrichment procedure and derive an a posteriori error indicator which gives an estimate of the local error over coarse grid regions we consider two kinds of error indicators where one is based on the l 2 norm of the local residual and the other is based on the weighted h 1 norm of the local residual where the weight is related to the coefficient of the elliptic equation we show that the use of weighted h 1 norm residual gives a more robust error indicator which works well for cases with high contrast media the convergence analysis of the method is given in our analysis we do not consider the error due to the fine grid discretization of local problems and only study the errors due to the enrichment numerical results are presented that demonstrate the robustness of the proposed error indicators

nonuniform sampling of bandlimited signals with polynomial growth on the real axis we derive a sampling expansion for bandlimited signals with polynomial growth on the real axis the sampling expansion uses nonuniformly spaced sampling points but unlike other known sampling expansions for such signals ours converge uniformly to the signal on any compact set an estimate of the truncation error of such a series is also obtained

a generalized sampling theorem for locally compact abelian groups we present a sampling theorem for locally compact abelian groups the sampling sets are finite unions of cosets of a closed subgroup this generalizes the well known case of non equidistant but periodic sampling on the real line for non bandlimited functions an l1 type estimate for the aliasing error is given we discuss the application of the theorem to a class of sampling sets in r s give a general algorithm for computer implementation present a detailed description of the implementation for the s dimensional torus group and point out connections to lattice rules for numerical integration

a mixed multiscale finite element method for elliptic problems with oscillating coefficients the recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large scale structure of the solutions without resolving all the fine scale structures motivated by the numerical simulation of flow transport in highly heterogeneous porous media we propose a mixed multiscale finite element method with an over sampling technique for solving second order elliptic equations with rapidly oscillating coefficients the multiscale finite element bases are constructed by locally solving neumann boundary value problems we provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain the asymptotic structure of the solutions numerical experiments are carried out for flow transport in a porous medium with a random log normal relative permeability to demonstrate the efficiency and accuracy of the proposed method

an irregular sampling theorem for functions bandlimited in a generalized sense it is shown that a well known irregular sampling formula for ordinary bandlimited functions is also valid for functions bandlimited in a generalized sense except for some necessary modifications

a spectral multiscale hybridizable discontinuous galerkin method for second order elliptic problems abstract we design a multiscale model reduction framework within the hybridizable discontinuous galerkin finite element method our approach uses local snapshot spaces and local spectral decomposition following the concept of generalized multiscale finite element methods we propose several multiscale finite element spaces on the coarse edges that provide a reduced dimensional approximation for numerical traces within the hdg framework we provide a general framework for systematic construction of multiscale trace spaces using local snapshots we avoid high dimensional representation of trace spaces and use some local features of the solution space in constructing a low dimensional trace space we investigate the solvability and numerically study the performance of the proposed method on a representative number of numerical examples

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

sinc interpolation of nonuniform samples it is well known that a bandlimited signal can be uniquely recovered from nonuniformly spaced samples under certain conditions on the nonuniform grid and provided that the average sampling rate meets or exceeds the nyquist rate however reconstruction of the continuous time signal from nonuniform samples is typically more difficult to implement than from uniform samples motivated by the fact that sinc interpolation results in perfect reconstruction for uniform sampling we develop a class of approximate reconstruction methods from nonuniform samples based on the use of time invariant lowpass filtering i e sinc interpolation the methods discussed consist of four cases incorporated in a single framework the case of sub nyquist sampling is also discussed and nonuniform sampling is shown as a possible approach to mitigating the impact of aliasing

landau s necessary density conditions for lca groups we derive necessary conditions for sampling and interpolation of bandlimited functions on a locally compact abelian group in line with the classical results of h landau for bandlimited functions on rd our conditions are phrased as comparison principles involving a certain canonical lattice

a multiscale stabilized finite element method for the advection diffusion equation this paper presents a multiscale method that yields a stabilized finite element formulation for the advection diffusion equation the multiscale method arises from a decomposition of the scalar field into coarse resolved scale and fine unresolved scale the resulting stabilized formulation possesses superior properties like that of the supg and the gls methods a significant feature of the present method is that the definition of the stabilization term appears naturally and therefore the formulation is free of any user designed or user defined parameters another important ingredient is that since the method is residual based it satisfies consistency ab initio based on the proposed formulation a family of 2 d elements comprising 3 and 6 node triangles and 4 and 9 node quadrilaterals has been developed numerical results show the good performance of the method on uniform skewed as well as composite meshes and confirm convergence at optimal rates 2004 elsevier b v all rights reserved

multiscale finite element methods for high contrast problems using local spectral basis functions in this paper we study multiscale finite element methods msfems using spectral multiscale basis functions that are designed for high contrast problems multiscale basis functions are constructed using eigenvectors of a carefully selected local spectral problem this local spectral problem strongly depends on the choice of initial partition of unity functions the resulting space enriches the initial multiscale space using eigenvectors of local spectral problem the eigenvectors corresponding to small asymptotically vanishing eigenvalues detect important features of the solutions that are not captured by initial multiscale basis functions multiscale basis functions are constructed such that they span these eigenfunctions that correspond to small asymptotically vanishing eigenvalues we present a convergence study that shows that the convergence rate in energy norm is proportional to h 1 2 where is proportional to the minimum of the eigenvalues that the corresponding eigenvectors are not included in the coarse space thus we would like to reach to a larger eigenvalue with a smaller coarse space this is accomplished with a careful choice of initial multiscale basis functions and the setup of the eigenvalue problems numerical results are presented to back up our theoretical results and to show higher accuracy of msfems with spectral multiscale basis functions we also present a hierarchical construction of the eigenvectors that provides cpu savings

regularized sampling of multiband signals this paper presents a regularized sampling method for multiband signals that makes it possible to approach the landau limit while keeping the sensitivity to noise and perturbations at a low level the method is based on band limited windowing followed by trigonometric approximation in consecutive time intervals the key point is that the trigonometric approximation inherits the multiband property that is its coefficients are formed by bursts of elements corresponding to the multiband components it is shown that this method can be well combined with the recently proposed synchronous multirate sampling smrs scheme given that the resulting linear system is sparse and formed by ones and zeroes the proposed method allows one to trade sampling efficiency for noise sensitivity and is specially well suited for bounded signals with unbounded energy like those in communications navigation audio systems etc besides it is also applicable to finite energy signals and periodic band limited signals trigonometric polynomials the paper includes a subspace method for blindly estimating the support of the multiband signal as well as its components and the results are validated through several numerical examples

banach spaces related to integrable group representations and their atomic decompositions i abstract we present a general theory of banach spaces which are invariant under the action of an integrable group representation and give their atomic decompositions with respect to coherent states i e the atoms arise from a single element under the group action several well known decomposition theories are contained as special examples and are unified under the aspect of group theory

poisson equation on complete manifolds abstract we develop heat kernel and green s function estimates for manifolds with positive bottom spectrum the results are then used to establish existence and sharp estimates of the solution to the poisson equation on such manifolds with ricci curvature bounded below as an application we show that the curvature of a steady gradient ricci soliton must decay exponentially if it decays faster than linear and the potential function is bounded above

a multiscale hdg method for second order elliptic equations part i polynomial and homogenization based multiscale spaces we introduce a finite element method for numerical upscaling of second order elliptic equations with highly heterogeneous coefficients the method is based on a mixed formulation of the problem and the concepts of the domain decomposition and the hybrid discontinuous galerkin methods the method utilizes three different scales 1 the scale of the partition of the domain of the problem 2 the scale of partition of the boundaries of the subdomains related to the corresponding space of lagrange multipliers and 3 the fine grid scale that is assumed to resolve the scale of the heterogeneous variation of the coefficients our proposed method gives a flexible framework that 1 couples independently generated multiscale basis functions in each coarse patch 2 provides a stable global coupling independent of local discretization physical scales and contrast and 3 allows avoiding any constraints arbogast et al multiscale model simul 6 2007 pp 319 346 on coarse spaces in this paper we

error estimates for irregular sampling of band limited functions on a locally compact abelian group abstract band limited functions f can be recovered from their sampling values f x i by means of iterative methods if only the sampling density is high enough we present an error analysis for these methods treating the typical forms of errors i e jitter error truncation error aliasing error quantization error and their combinations the derived apply uniformly to whole families of spaces e g to weighted l p spaces over some locally compact abelian group with growth rate up to some given order in contrast to earlier papers we do not make use of any relative separation condition on the sampling sets furthermore we discard the assumption on polynomial growth of the weights that has been used over euclidean spaces consequently even for the case of regular sampling i e sampling along lattices in g the results are new in the given generality

the separation of zeros for entire functions of exponential type soit f une fonction entiere a valeur reelle sur l axe reel r de type exponentiel de carre integrable sur r si f a une infinite de zeros sur cet axe alors il existe au moins une paire de zeros consecutifs dont la distance est au moins

estimation of time limited channel spectra from nonuniform samples this paper deals with the estimation of a time invariant channel spectrum from its own nonuniform samples assuming there is a bound on the channel s delay spread except for this last assumption this is the basic estimation problem in systems providing channel spectral samples however as shown in the paper the delay spread bound leads us to view the spectrum as a band limited signal rather than the fourier transform of a tapped delay line tdl using this alternative model a linear estimator is presented that approximately minimizes the expected root mean square rms error for a deterministic channel its main advantage over the tdl is that it takes into account the spectrum s smoothness time width thus providing a performance improvement the proposed estimator is compared numerically with the maximum likelihood ml estimator based on a tdl model in pilot assisted channel estimation pace for ofdm

channel estimation in wireless ofdm systems with irregular pilot distribution this paper addresses pilot assisted channel estimation for wireless orthogonal frequency division multiplexing ofdm systems with irregular pilot arrangements using nonuniform sampling techniques we propose a novel channel estimator that is based on conjugate gradient iterations and features very low computational complexity we investigate the impact of the pilot arrangement on the channel estimation mean square error and we provide a heuristic stopping criterion for early termination of the conjugate gradient iterations one dimensional and two dimensional channel estimator implementations and applications of our method to multiple antenna and multiuser ofdm systems are discussed extensive numerical simulations corroborate that the proposed method performs similarly to computationally much more expensive minimum mean square error estimators

functionally weighted lagrange interpolation of band limited signals from nonuniform samples a modification of the conventional lagrange interpolator is proposed in this paper that allows one to approximate a band limited signal from its own nonuniform samples with high accuracy the modification consists in applying the lagrange method to the signal but pre multiplied by a fixed function and then solving for the desired signal value its efficiency lies in the fact that the fixed function is independent of the sampling instants it is shown in this paper that the function can be selected so that the interpolation error decreases exponentially with the number of samples for the case in which the sampling instants have a maximum deviation from a uniform grid this paper includes a low complexity recursive implementation of the method its accuracy is validated in the numerical examples by comparison with several interpolators in the literature and by deriving upper and lower bounds for its maximum error

on green functions of second order elliptic operators on riemannian manifolds the critical case abstract let p be a second order linear elliptic operator with real coefficients which is defined on a noncompact and connected riemannian manifold m it is well known that the equation p u 0 in m admits a positive supersolution which is not a solution if and only if p admits a unique positive minimal green function on m and in this case p is said to be subcritical in m if p does not admit a positive green function but admits a global positive super solution then such a solution is called a ground state of p in m and p is said to be critical in m we prove for a critical operator p in m the existence of a green function which is dominated above by the ground state of p away from the singularity moreover in a certain class of green functions such a green function is unique up to an addition of a product of the ground states of p and p under some further assumptions we describe the behavior at infinity of such a green function this result extends and sharpens the celebrated result of p li and l f tam concerning the existence of a symmetric green function for the laplace beltrami operator on a smooth and complete riemannian manifold m

fast reconstruction methods for bandlimited functions from periodic nonuniform sampling a well known generalization of shannon s sampling theorem states that a bandlimited function can be reconstructed from its periodic nonuniformly spaced samples if the effective sampling rate is at least the nyquist rate analogous to shannon s sampling theorem this generalization requires that an infinite number of samples be available which however is never the case in practice most existing reconstruction methods for periodic nonuniform sampling yield very low order often not even first order accuracy when only a finite number of samples is given in this paper we propose a fast numerically robust root exponential accurate reconstruction method the efficiency and accuracy of the algorithm is obtained by fully exploiting the sampling structure and utilizing localized fourier analysis we discuss applications in analog to digital conversion where nonuniform periodic sampling arises in various situations finally we demonstrate the performance of our algorithm by numerical examples

convergence of a nonconforming multiscale finite element method the multiscale finite element method msfem t y hou x h wu and z cai math comp 1998 to appear t y hou and x h wu j comput phys 134 1997 pp 169 189 has been introduced to capture the large scale solutions of elliptic equations with highly oscillatory coefficients this is accomplished by constructing the multiscale base functions from the local solutions of the elliptic operator our previous study reveals that the leading order error in this approach is caused by the resonant sampling which leads to large error when the mesh size is close to the small scale of the continuous problem similar difficulty also arises in numerical upscaling methods an oversampling technique has been introduced to alleviate this difficulty t y hou and x h wu j comput phys 134 1997 pp 169 189 a consequence of the oversampling method is that the resulting finite element method is no longer conforming here we give a detailed analysis of the nonconforming error our analysis also reveals a new cell resonance error which is caused by the mismatch between the mesh size and the wavelength of the small scale we show that the cell resonance error is of lower order our numerical experiments demonstrate that the cell resonance error is generically small and is difficult to observe in practice

on quasi isometry of threshold based sampling the problem of isometry for threshold based sampling such as integrate and fire or send on delta is addressed while for uniform sampling the parseval theorem provides isometry and makes the euclidean metric canonical there is no analogy for threshold based sampling the relaxation of the isometric postulate to quasi isometry however allows the discovery of the underlying metric structure of threshold based sampling this paper characterizes this metric structure making hermann weyl s discrepancy measure canonical for threshold based sampling the usefulness of the approach is demonstrated by means of a novel adaptive threshold based sampling scheme which applied on electrocardiography signals allows a substantial reduction of the number of samples while maintaining nearly the same snr after reconstruction

stochastic homogenization of multicontinuum heterogeneous flows abstract we consider a multicontinuum model in porous media applications which is described as a system of coupled flow equations the coupling between different continua depends on many factors and its modeling is important for porous media applications the coefficients depend on particle deposition that is described in terms of a stochastic process solution of an sde the stochastic process is considered to be faster than the flow motion and we introduce time space scales to model the problem our goal is to pass to the limit in time and space and to find an associated averaged system this is an averaging homogenization problem where the averages are computed in terms of the invariant measure associated to the fast motion and the spatial variable we use the techniques developed in our previous paper bessaih et al 2019 to model the interactions between the continua and derive the averaged model problem that can be used in many applications

generalized multiscale finite element methods gmsfem in this paper we propose a general approach called generalized multiscale finite element method gmsfem for performing multiscale simulations for problems without scale separation over a complex input space as in multiscale finite element methods msfems the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space in the proposed approach we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage the enrichment in the online stage is performed based on a spectral decomposition of the offline space in the online stage for any input parameter a multiscale space is constructed to solve the global problem on a coarse grid the online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues the computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be re used for solving the forward problem with any forcing and boundary condition compared with the other approaches where global snapshots are used the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarse grid level we present various examples in the paper and some numerical results to demonstrate the effectiveness of our method

variational multiscale analysis the fine scale green s function projection optimization localization and stabilized methods we derive an explicit formula for the fine scale green s function arising in variational multiscale analysis the formula is expressed in terms of the classical green s function and a projector which defines the decomposition of the solution into coarse and fine scales the theory is presented in an abstract operator format and subsequently specialized for the advection diffusion equation it is shown that different projectors lead to fine scale green s functions with very different properties for example in the advection dominated case the projector induced by the h 1 0 seminorm produces a fine scale green s function which is highly attenuated and localized these are very desirable properties in a multiscale method and ones that are not shared by the l 2 projector by design the coarse scale solution attains optimality in the norm associated with the projector this property combined with a localized fine scale green s function indicates the possibility of effective methods with local character for dominantly hyperbolic problems the constructs lead to a new class of stabilized methods and the relationship between h 1 0 optimality and the streamline upwind petrov galerkin supg method is described

fundamental distortion limits of analog to digital compression representing a continuous time signal by a set of samples is a classical problem in signal processing we study this problem under the additional constraint that the samples are quantized or compressed in a lossy manner under a limited bitrate budget to this end we consider a combined sampling and source coding problem in which an analog stationary gaussian signal is reconstructed from its encoded samples these samples are obtained by a set of bounded linear functionals of the continuous time path with a limitation on the average number of samples per unit time given in this setting we provide a full characterization of the minimal distortion in terms of the sampling frequency the bitrate and the signal s spectrum assuming that the signal s energy is not uniformly distributed over its spectral support we show that for each compression bitrate there exists a critical sampling frequency smaller than the nyquist rate such that the distortion in signal reconstruction when sampling at this frequency is minimal our results can be seen as an extension of the classical sampling theorem for bandlimited random processes in the sense that they describe the minimal amount of excess distortion in the reconstruction due to lossy compression of the samples and provide the minimal sampling frequency required in order to achieve this distortion finally we compare the fundamental limits in the combined source coding and sampling problem to the performance of pulse code modulation where each sample is quantized by a scalar quantizer using a fixed number of bits

design of barycentric interpolators for uniform and nonuniform sampling grids this paper proposes a method to convert a fixed instant interpolator for band limited signals into a variable instant one which has the form of a barycentric interpolator this interpolator makes it possible to approximate the signal and its derivatives with minimal complexity in a range that surrounds the initial instant and is applicable to uniform and nonuniform sampling grids the barycentric form of the interpolator is derived using two different procedures first by the repeated use of bernstein s inequality and taylor s theorem and second by truncating a lagrange type series these procedures show that the proposed method can be applied to existing fixed instant interpolators and that it can be accurate in long time intervals finally an evaluation procedure for barycentric interpolators and their derivatives is presented that minimizes the number of divisions solving at the same time the numerical problems associated with small denominators this paper includes several numerical examples

optimal reconstruction of bandlimited bounded signals a formula is presented for restoring bounded bandlimited signals from a finite number of samples the restoration is linear in the samples used and the error bound of the method is the theoretical minimum

on positive solutions of second order elliptic equations stability results and classification soit p un operateur uniformement elliptique d ordre 2 defini dans un domaine cr n soit p le cone convexe de toutes les solutions positives de l equation pu 0 dans on etudie les proprietes generales de p pour p et generaux

generalized multiscale finite element method symmetric interior penalty coupling motivated by applications to numerical simulations of flows in highly heterogeneous porous media we develop multiscale finite element methods for second order elliptic equations we discuss a multiscale model reduction technique in the framework of the discontinuous galerkin finite element method we propose two different finite element spaces on the coarse mesh the first space is based on a local eigenvalue problem that uses an interior weighted l 2 norm and a boundary weighted l 2 norm for computing the mass matrix the second choice is based on generation of a snapshot space and subsequent selection of a subspace of a reduced dimension the approximation with these multiscale spaces is based on the discontinuous galerkin finite element method framework we investigate the stability and derive error estimates for the methods and further experimentally study their performance on a representative number of numerical examples

irregular sampling theorems and series expansions of band limited functions abstract we present a new approach to the problem of irregular sampling of band limited functions that is based on the approximation and factorization of convolution operators a special case of our main result is the following theorem if r n is compact g l 1 rn a band limited function with g t 0 on and xi i i and yj j j are two discrete sets in rn which are dense enough then every band limited function f lwp 1 p our methods work a for a very general class of function spaces not only weighted lwp spaces b in all dimensions even in general locally compact abelian groups c to provide the correct behavior of the coefficients in contrast to the traditional estimates currently used by engineers furthermore the method is constructive and stable with respect to input errors such as round off errors truncation effors jitter errors or aliasing errors

metric based upscaling we consider divergence form elliptic operators in dimension n ge 2 with l coefficients although solutions of these operators are only holder continuous we show that they are differentiable c1 with respect to harmonic coordinates it follows that numerical homogenization can be extended to situations where the medium has no ergodicity at small scales and is characterized by a continuum of scales this new numerical homogenization method is based on the transfer of a new metric in addition to traditional averaged homogenized quantities from subgrid scales into computational scales error bounds can be given and this method can also be used as a compression tool for differential operators 2006 wiley periodicals inc

analysis of a two scale locally conservative subgrid upscaling for elliptic problems we present a two scale theoretical framework for approximating the solution of a second order elliptic problem the elliptic coefficient is assumed to vary on a scale that can be resolved on a fine numerical grid but limits on computational power require that computations be performed on a coarse grid we consider the elliptic problem in mixed variational form over w times bf v subset l 2 times h rm div we base our scale expansion on local mass conservation over the coarse grid it is used to define a direct sum decomposition of w times bf v into coarse and subgrid subspaces w c times bf v c and delta w times delta bf v such that 1 nabla cdot bf v c w c and nabla cdot delta bf v delta w and 2 the space delta bf v is locally supported over the coarse mesh we then explicitly decompose the variational problem into coarse and subgrid scale problems the subgrid problem gives a well defined operator taking w c times bf v c to delta w times delta bf v which is localized in space and it is used to upscale that is to remove the subgrid from the coarse scale problem using standard mixed finite element spaces two scale mixed spaces are defined a mixed approximation is defined which can be viewed as a type of variational multiscale method or a residual free bubble technique a numerical green s function approach is used to make the approximation to the subgrid operator efficient to compute a mixed method pi operator is defined for the two scale approximation spaces and used to show optimal order error estimates

a discontinuous galerkin method with lagrange multipliers for the solution of helmholtz problems in the mid frequency regime we present a discontinuous galerkin method dgm for the solution of the helmholtz equation in the mid frequency regime our approach is based on the discontinuous enrichment method in which the standard polynomial field is enriched within each finite element by a non conforming field that contains free space solutions of the homogeneous partial differential equation to be solved hence for the helmholtz equation the enrichment field is chosen here as the superposition of plane waves we enforce a weak continuity of these plane waves across the element interfaces by suitable lagrange multipliers preliminary results obtained for two dimensional model problems discretized by uniform meshes reveal that the proposed dgm enables the development of elements that are far more competitive than both the standard linear and the standard quadratic galerkin elements for the discretization of helmholtz problems

reconstruction algorithms in irregular sampling a constructive solution of the irregular sampling problem for band limited functions is given we show how a band limited function can be com pletely reconstructed from any random sampling set whose density is higher than the nyquist rate and give precise estimates for the speed of convergence of this iteration method variations of this algorithm allow for irregular sampling with derivatives reconstruction of band limited functions from local averages and irregular sampling of multivariate band limited functions in the irregular sampling problem one is asked whether and how a band limited function can be completely reconstructed from its irregularly sam pled values f xi this has many applications in signal and image processing seismology meteorology medical imaging etc finding constructive solutions of this problem has received considerable attention among mathematicians and engineers the mathematical literature provides several uniqueness results 1 2 17 18 19 it is now part of the folklore that for stable sampling the sampling rate must be at least the nyquist rate 18 these results as deep as they are have had little impact for the applied sciences because they were not constructive if the sampling set is just a perturbation of the regular oversampling then a reconstruction method has been obtained in a seminal paper by duffin and schaeffer 6 see also 29 if for some l 0 a 0 and o 0 the sampling points xk k e z satisfy a xk ok a k i then the norm equivalence a ir f x 2dx with w n o this norm equivalence implies that it is possible to reconstruct through an iterative procedure the so called frame method most of the later work on constructive methods consists of variations of this method 3 21 22 26 the above conditions on the sampling set exclude random irregular sampling sets e g sets with regions of higher sampling density a partial but undesirable remedy to handle highly irregular sampling sets would be to force the above conditions by throwing away information on part of the points and accept a very slow convergence of the iteration

efficient numerical methods in non uniform sampling theory summary we present a new second generation reconstruction algorithm for irregular sampling i e for the problem of recovering a band limited function from its non uniformly sampled values the efficient new method is a combination of the adaptive weights method which was developed by the two first named authors and the method of conjugate gradients for the solution of positive definite linear systems the choice of adaptive weights can be seen as a simple but very efficient method of preconditioning further substantial acceleration is achieved by utilizing the toeplitztype structure of the system matrix this new algorithm can handle problems of much larger dimension and condition number than have been accessible so far furthermore if some gaps between samples are large then the algorithm can still be used as a very efficient extrapolation method across the gaps

efficient sampling of band limited signals from sine wave crossings this correspondence presents an efficient method for reconstructing a band limited signal in the discrete domain from its crossings with a sine wave the method makes it possible to design analog to discrete a d converters that only deliver the crossing timings which are then used to interpolate the input signal at arbitrary instants potentially it may allow for reductions in power consumption and complexity in these converters the reconstruction in the discrete domain is based on a recently proposed modification of the lagrange interpolator which is readily implementable with linear complexity and efficiently given that it reuses known schemes for variable fractional delay vfd filters as a spin off the method allows one to perform spectral analysis from sine wave crossings with the complexity of the fft finally the results in the correspondence are validated in several numerical examples

localization of elliptic multiscale problems this paper constructs a local generalized finite element basis for elliptic problems with heterogeneous and highly varying coefficients the basis functions are solutions of local problems on vertex patches the error of the corresponding generalized finite element method decays exponentially with respect to the number of layers of elements in the patches hence on a uniform mesh of size h patches of diameter h log 1 h are sufficient to preserve a linear rate of convergence in h without pre asymptotic or resonance effects the analysis does not rely on regularity of the solution or scale separation in the coefficient this result motivates new and justifies old classes of variational multiscale methods see more at http www ams org journals mcom 2014 83 290 s0025 5718 2014 02868 8 sthash z2ccfxig dpuf

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

overview of the discontinuous enrichment method the ultra weak variational formulation and the partition of unity method for acoustic scattering in the medium frequency regime and performance comparisons

the discontinuous enrichment method for elastic wave propagation in the medium frequency regime the discontinuous enrichment method dem is specified and developed for the solution of two dimensional elastic wave propagation problems in the frequency domain the classical finite element polynomial approximation of the displacement field is enriched by the superposition of discontinuous pressure and shear wave functions the continuity of the solution across the element interfaces is weakly enforced by suitable lagrange multipliers higher order rectangular dem elements are constructed and benchmarked against the standard higher order polynomial galerkin elements for two dimensional problems in the medium frequency regime in general it is found that for such applications dem can achieve the same accuracy as the p finite element method using a similar computational complexity but with about 10 times fewer degrees of freedom this highlights the potential of this hybrid method for problems where the scale of vibrations is very small compared to the characteristic dimension of the physical medium copyright 2006 john wiley sons ltd

bandwidth estimation from multiple level crossings of stochastic signals motivated by a broad range of potential applications in event driven signal processing inference we examine a class of spectral moments estimators derived from multiple level crossings of a stationary gaussian stochastic signal as well as non gaussian signals being monotone transformations of the gaussian signal the established spectral moments estimators are used to obtain estimates of the stochastic signal bandwidth both the mean squared bandwidth and the absolute bandwidth are taken into account also the notion of the power bandwidth is introduced and its estimation is examined we prove that the proposed estimates are consistent as the length of the observed time interval is increasing the accuracy of the proposed bandwidth estimation methods from multiple level crossings is assessed in simulation studies

multi tiling sets riesz bases and sampling near the critical density in lca groups abstract we prove the existence of sampling sets and interpolation sets near the critical density in paley wiener spaces of a locally compact abelian lca group g this solves a problem left by grochenig kutyniok and seip 2008 7 to achieve this result we prove the existence of universal riesz bases of characters for l 2 provided that the relatively compact subset of the dual group g satisfies a multi tiling condition this last result generalizes fuglede s theorem and extends to lca groups setting recent constructions of riesz bases of exponentials in bounded sets of r d

residual driven online generalized multiscale finite element methods the construction of local reduced order models via multiscale basis functions has been an area of active research in this paper we propose online multiscale basis functions which are constructed using the offline space and the current residual online multiscale basis functions are constructed adaptively in some selected regions based on our error indicators we derive an error estimator which shows that one needs to have an offline space with certain properties to guarantee that additional online multiscale basis function will decrease the error this error decrease is independent of physical parameters such as the contrast and multiple scales in the problem the offline spaces are constructed using generalized multiscale finite element methods gmsfem we show that if one chooses a sufficient number of offline basis functions one can guarantee that additional online multiscale basis functions will reduce the error independent of contrast we note that the construction of online basis functions is motivated by the fact that the offline space construction does not take into account distant effects using the residual information we can incorporate the distant information provided the offline approximation satisfies certain properties in the paper theoretical and numerical results are presented our numerical results show that if the offline space is sufficiently large in terms of the dimension such that the coarse space contains all multiscale spectral basis functions that correspond to small eigenvalues then the error reduction by adding online multiscale basis function is independent of the contrast we discuss various ways computing online multiscale basis functions which include a use of small dimensional offline spaces

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

irregular sampling toeplitz matrices and the approximation of entire functions of exponential type in many applications one seeks to recover an entire function of exponential type from its non uniformly spaced samples whereas the mathematical theory usually addresses the question of when such a function in l 2 r can be recovered numerical methods operate with a finite dimensional model the numerical reconstruction or approximation of the original function amounts to the solution of a large linear system we show that the solutions of a particularly efficient discrete model in which the data are fit by trigonometric polynomials converge to the solution of the original infinite dimensional reconstruction problem this legitimatizes the numerical computations and explains why the algorithms employed produce reasonable results the main mathematical result is a new type of approximation theorem for entire functions of exponential type from a finite number of values from another point of view our approach provides a new method for proving sampling theorems

discrete irregular sampling with larger gaps abstract discrete irregular sampling is the problem of recovering a band limited discrete signal from its nonuniformly sampled values we introduce a new method for discrete irregular sampling it features a new sampling geometry where the sampling points are chosen to come in pairs a frequency domain formulation reduces the problem to a system of linear equations for which the matrix is of toeplitz type and is positive definite as soon as there are enough samples an explicit estimate of the condition number of that matrix is given under some condition on the maximum sampling gap this condition allows gaps larger than the nyquist gap we recombine the new method and the conjugate method to deduce an efficient iterative reconstruction scheme for band limited discrete signals numerical results are provided to illustrate the efficiency of the method