partial period crosscorrelations of geometric sequences for a binary pseudorandom sequence s sub i with period n the partial period autocorrelation function a sub s spl tau k d is defined by correlating the portion of the sequence within a window of size d and start position k with the portion in another window of the same size but starting spl tau steps later in the sequence a distribution of possible partial period autocorrelation values is obtained by allowing the start position k to vary over all possible values o spl les k

stochastic mixed multiscale finite element methods and their applications in random porous media we present a framework for stochastic mixed multiscale finite element methods mixed msfems for elliptic equations with heterogeneous random coefficients the use of some global information is necessary in multiscale simulations when there is no scale separation for the heterogeneity the methods in the proposed framework for the stochastic mixed msfems use some global information the media properties in a stochastic environment drastically vary among realizations and thus many global fields are needed for multiscale simulation the computations of these global fields on a fine grid can be very expensive one can utilize upscaling methods to compute the global information on an intermediate coarse grid that reduces the computational cost we investigate two approaches of stochastic mixed msfems in the framework first approach entails no stochastic interpolation and the second approach uses stochastic interpolation if the random media have deterministic features that play significant roles in the flow we can use the deterministic features of the random media as the global information this reduces the computational cost of the simulations we make convergence analysis of the stochastic mixed msfems and investigate their applications to incompressible two phase flows in random porous media the numerical results demonstrate the effectiveness of the proposed methods and confirm the convergence

wavelet like bases for the fast solutions of second kind integral equations a class of vector space bases is introduced for the sparse representation of discretizations of integral operators an operator with a smooth nonoscillatory kernel possessing a finite number of singularities in each row or column is represented in these bases as a sparse matrix to high precision a method is presented that employs these bases for the numerical solution of second kind integral equations in time bounded by o n log 2 n where n is the number of points in the discretization numerical results are given which demonstrate the effectiveness of the approach and several generalizations and applications of the method are discussed

what is the optimal shape of a pipe we consider an incompressible fluid in a three dimensional pipe following the navier stokes system with classical boundary conditions we are interested in the following question is there any optimal shape for the criterion energy dissipated by the fluid moreover is the cylinder the optimal shape we prove that there exists an optimal shape in a reasonable class of admissible domains but the cylinder is not optimal for that purpose we explicit the first order optimality condition thanks to adjoint state and we prove that it is impossible that the adjoint state be a solution of this over determined system when the domain is the cylinder at last we show some numerical simulations for that problem

efficient inversion of the galerkin matrix of general second order elliptic operators with nonsmooth coefficients this article deals with the efficient approximate inversion of finite element stiffness matrices of general second order elliptic operators with l coefficients it will be shown that the inverse stiffness matrix can be approximated by hierarchical matrices h matrices furthermore numerical results will demonstrate that it is possible to compute an approximate inverse with almost linear complexity

the merit factor of legendre sequences corresp turyn has shown by numerical computation that the merit factor of long legendre sequences offset by a quarter of their length has in all likelihood the asymptotic value 6 the ergodicity postulate is used in this correspondence to calculate that the merit factor f of a legendre sequence offset by a fraction f of its length has an asymptotic value given by 1 f 2 3 4 f 8f 2 f leq 1 2 which gives f 6 for f 1 4

subdivision surfaces a new paradigm for thin shell finite element analysis we develop a new paradigm for thin shell finite element analysis based on the use of subdivision surfaces for i describing the geometry of the shell in its undeformed configuration and ii generating smooth interpolated displacement fields possessing bounded energy within the strict framework of the kirchhoff love theory of thin shells the particular subdivision strategy adopted here is loop s scheme with extensions such as required to account for creases and displacement boundary conditions the displacement fields obtained by subdivision are h2 and consequently have a finite kirchhoff love energy the resulting finite elements contain three nodes and element integrals are computed by a one point quadrature the displacement field of the shell is interpolated from nodal displacements only in particular no nodal rotations are used in the interpolation the interpolation scheme induced by subdivision is non local i e the displacement field over one element depend on the nodal displacements of the element nodes and all nodes of immediately neighbouring elements however the use of subdivision surfaces ensures that all the local displacement fields thus constructed combine conformingly to define one single limit surface numerical tests including the belytschko et al 10 obstacle course of benchmark problems demonstrate the high accuracy and optimal convergence of the method

numerical homogenization of the acoustic wave equations with a continuum of scales in this paper we consider numerical homogenization of acoustic wave equations with heterogeneous coefficients namely when the bulk modulus and the density of the medium are only bounded we show that under a cordes type condition the second order derivatives of the solution with respect to harmonic coordinates are l 2 instead h 1 with respect to euclidean coordinates and the solution itself is in l 0 t h 2 instead of l 0 t h 1 with respect to euclidean coordinates then we propose an implicit time stepping method to solve the resulted linear system on coarse spatial scales and present error estimates of the method it follows that by pre computing the associated harmonic coordinates it is possible to numerically homogenize the wave equation without assumptions of scale separation or ergodicity

the peak sidelobe level of families of binary sequences a numerical investigation is presented for the peak sidelobe level psl of legendre sequences maximal length shift register sequences m sequences and rudin shapiro sequences the psl gives an alternative to the merit factor for measuring the collective smallness of the aperiodic autocorrelations of a binary sequence the growth of the psl of these infinite families of binary sequences is tested against the desired growth rate o spl radic nlnn for sequence length n the claim that the psl of m sequences grows like o spl radic n which appears frequently in the radar literature is concluded to be unproven and not currently supported by data notable similarities are uncovered between the psl and merit factor behavior under cyclic rotations of the sequences

wavelet based numerical homogenization a numerical homogenization procedure for elliptic differential equations is presented it is based on wavelet decompositions of discrete operators in fine and coarse scale components followed by the elimination of the fine scale contributions if the operator is in divergence form this is preserved by the homogenization procedure for periodic problems results similar to classical effective coefficient theory are proved the procedure can be applied to problems that are not cell periodic

rudin shapiro like sequences with maximum asymptotic merit factor borwein and mossinghoff investigated the rudin shapiro like sequences which are infinite families of binary sequences usually represented as polynomials each family of rudin shapiro like sequences is obtained from a starting sequence which we call the seed by a recursive construction that doubles the length of the sequence at each step and many sequences produced in this manner have exceptionally low aperiodic autocorrelation borwein and mossinghoff showed that the asymptotic autocorrelation merit factor for any such family is at most 3 and found the seeds of length 40 or less that produce the maximum asymptotic merit factor of 3 the definition of rudin shapiro like sequences was generalized by katz lee and trunov to include sequences with arbitrary complex coefficients among which are families of low autocorrelation polyphase sequences katz lee and trunov proved that the maximum asymptotic merit factor is also 3 for this larger class here we show that a family of such rudin shapiro like sequences achieves asymptotic merit factor 3 if and only if the seed is either of length 1 or is the interleaving of a pair of golay complementary sequences for small seed lengths where this is not possible the optimal seeds are interleavings of pairs that are as close as possible to being complementary pairs and the idea of an almost complementary pair makes sense of remarkable patterns in previously unexplained data on optimal seeds for binary rudin shapiro like sequences

sparse tensor finite elements for elastic wave equation with multiple scales we consider an elastic wave equation whose moduli depend on n separable microscopic scales and are periodic with respect to each scale in a domain d in r d by using multiscale convergence we deduce an n 1 d dimensional problem that contains all the macroscopic and microscopic information on the multiscale equation the full tensor product finite element approach is natural to solve this high dimensional multiscale homogenized equation but is too expensive practically we develop a sparse tensor finite element approach that achieves essentially equal accuracy as the full tensor product finite elements but requires an essentially equal number of degrees of freedom as for solving a problem in r d restricting our consideration to the case of zero initial wave displacement we deduce a corrector with a homogenization error estimate in the case of two scales from this numerical correctors are constructed from the finite element solutions of the high dimensional multiscale homogenized problem with an error of the order of the sum of the homogenization and the finite element errors for the general multiscale problems we construct numerical correctors from the finite element solutions but without an explicit error numerical examples on two and three scale elastic wave equations in two dimensions confirm our analysis

some remarks on a multigrid preconditioner this paper is devoted to a class of multilevel preconditioners developed in math comp 55 1990 pp 1 22 by bramble pasciak and xu and in the ph d thesis of xu at cornell university ithaca ny 1989 a simple proof is given for optimal estimation of the conditioning a derivation of matrix representations and a description of efficient implementation techniques are also given some modified preconditioners are proposed combining the hierarchical basis method numerical examples are also given that compare various preconditioners discussed in this paper

weighted extended b spline approximation of dirichlet problems we describe a new finite element method which uses weighted extended b splines on a regular grid as basis functions for solving dirichlet problems on bounded domains in arbitrary dimensions this web method does not require any grid generation and can be implemented very efficiently it yields smooth high order accurate approximations with relatively low dimensional subspaces

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

the merit factor of binary sequences related to difference sets long binary sequences related to cyclic difference sets are investigated among all known constructions of cyclic difference sets it is shown that only sequences constructed from hadamard difference sets can have an asymptotic nonzero merit factor maximal length shift register sequences legendre and twin prime sequences are all constructed from hadamard difference sets the authors prove that the asymptotic merit factor of any maximal length shift register sequence is three for twin prime sequences it is shown that the best asymptotic merit factor is six this value is obtained by shifting the twin prime sequence one quarter of its length it turns out that legendre sequences and twin prime sequences have similar behavior jacobi sequences are investigated on the basis of the jacobi symbol the best asymptotic merit factor is shown to be six through the introduction of product sequences it is argued that the maximal merit factor among all sequences of length n is at least six when n is large the authors also demonstrate that it is fairly easy to construct sequences of moderate composite length with a merit factor close to six

change of sign of the corrector s determinant for homogenization in three dimensional conductivity in this paper we study the positivity of the determinant of the local electric field in a conducting composite we know by 1 that the positivity holds true in two dimensions for any periodic structure using a different approach from 11 we prove that is also the case for a laminate microstructure in any dimension however and this is the main result of the paper we provide an example of a two phase three dimensional periodic composite for which the determinant changes sign

numerical methods for multiscale elliptic problems we present an overview of the recent development on numerical methods for elliptic problems with multiscale coefficients we carry out a thorough study of two representative techniques the heterogeneous multiscale method hmm and the multiscale finite element method msfem for problems with scale separation but without specific assumptions on the particular form of the coefficients analytical and numerical results show that hmm gives comparable accuracy as msfem with much less cost for problems without scale separation our numerical results suggest that hmm performs at least as well as msfem in terms of accuracy and cost even though in this case both methods may fail to converge since the cost of msfem is comparable to that of solving the full fine scale problem one might expect that it does not need scale separation and still retains some accuracy we show that this is not the case specifically we give an example showing that if there exists an intermediate scale comparable to h the size of the macroscale mesh then msfem commits a finite error even with overlapping

homogenization of parabolic equations with a continuum of space and time scales this paper addresses the issue of the homogenization of linear divergence form parabolic operators in situations where no ergodicity and no scale separation in time or space are available namely we consider divergence form linear parabolic operators in omega subset mathbb r n with l infty omega times 0 t coefficients it appears that the inverse operator maps the unit ball of l 2 omega times 0 t into a space of functions which at small time and space scales are close in h 1 norm to a functional space of dimension n it follows that once one has solved these equations at least n times it is possible to homogenize them both in space and in time reducing the number of operation counts necessary to obtain further solutions in practice we show under a cordes type condition that the first order time derivatives and second order space derivatives of the solution of these operators with respect to caloric coordinates are in l 2 instead of h 1 with euclidean coordinates if the medium is time independent then it is sufficient to solve n times the associated elliptic equation in order to homogenize the parabolic equation

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

an upscaling method using coefficient splitting and its applications to elliptic pdes in this paper we develop an upscaling method using coefficient splitting techniques green s function is constructed using the differential operator associated with the first part of the splitting an effective upscaling coefficient is recursively calculated by green s function the computation of the upscaling process involves some independent steps combining the proposed upscaling method with the stochastic collocation method we present a stochastic space reduction collocation method where the stochastic collocation method is performed on a lower dimension stochastic space than the full dimension stochastic space we thoroughly analyze the convergence of the proposed upscaling method for both deterministic and stochastic elliptic pdes computation complexity is also addressed for the stochastic upscaling method a number of numerical tests are presented to confirm the convergence analysis

spatio spectral concentration of convolutions differential equations may possess coefficients that vary on a spectrum of scales because coefficients are typically multiplicative in real space they turn into convolution operators in spectral space mixing all wavenumbers however in many applications only the largest scales of the solution are of interest and so the question turns to whether it is possible to build effective coarse scale models of the coefficients in such a manner that the large scales of the solution are left intact here we apply the method of numerical homogenisation to deterministic linear equations to generate sub grid scale models of coefficients at desired frequency cutoffs we use the fourier basis to project filter and compute correctors for the coefficients the method is tested in 1d and 2d scenarios and found to reproduce the coarse scales of the solution to varying degrees of accuracy depending on the cutoff we relate this method to mode elimination renormalisation group rg and discuss the connection between accuracy and the cutoff wavenumber the tradeoff is governed by a form of the uncertainty principle for convolutions which states that as the convolution operator is squeezed in the spectral domain it broadens in real space as a consequence basis sparsity is a high virtue and the choice of the basis can be critical

the sub cauchy stokes problem solvability issues and lagrange multiplier methods with artificial boundary conditions in contrast to the conventional cauchy stokes problems in which the velocity and the stress force data are given on the accessible boundary in the present paper we reduce the accessible boundary data information and we consider a problem which deals only with shear stress data we refer to this problem as a sub cauchy stokes problem this problem is ill posed because of severe instability and even uniqueness is unknown we first address the uniqueness issues associated with this problem resorting to the domain decomposition techniques together with the duplication process of vogelius 1 we propose new lagrange multiplier methods to solve the sub cauchy stokes problem these methods consist in recasting the problem in terms of interfacial equations by equalizing two solutions of the sub cauchy stokes problem using matching conditions defined on the inaccessible boundary the matching is based on second order conditions and depends on the equations used to match the values of the unknowns on the inaccessible boundary the underlying interfacial problems are then solved by iterative procedures in which coefficients can be optimized to improve convergence rates a complete analysis of the methods is presented and various numerical results illustrate the effectiveness and the performance of the proposed approaches

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

multiscale homogenization with bounded ratios and anomalous slow diffusion we show that the effective diffusivity matrix d v n for the heat operator t 1 2 rv n r in a periodic potential v n d p nd0 uk x rk obtained as a superposition of hlder continuous periodic potentials uk of period t d vd r d z d d 2 n uk 0 d 0 decays exponentially fast with the number of scales when the scale ratios rkc1 rk are bounded above and below from this we de duce the anomalous slow behavior for a brownian motion in a potential obtained as a superposition of an innite number of scales dyt d d t rv 1 yt dt

a stochastic collocation method for the second order wave equation with a discontinuous random speed in this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions the speed is piecewise smooth in the physical space and depends on a finite number of random variables the numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space this approach leads to the solution of uncoupled deterministic problems as in the monte carlo method we consider both full and sparse tensor product spaces of orthogonal polynomials we provide a rigorous convergence analysis and demonstrate different types of convergence of the probability error with respect to the number of collocation points for full and sparse tensor product spaces and under some regularity assumptions on the data in particular we show that unlike in elliptic and parabolic problems the solution to hyperbolic problems is not in general analytic with respect to the random variables therefore the rate of convergence may only be algebraic an exponential fast rate of convergence is still possible for some quantities of interest and for the wave solution with particular types of data we present numerical examples which confirm the analysis and show that the collocation method is a valid alternative to the more traditional monte carlo method for this class of problems

ensemble level multiscale finite element and preconditioner for channelized systems and applications we consider multiscale flow in porous media we assume that we can characterize the ensemble of all possible flow scenarios that is we can describe all possible permeability configurations needed for the simulations we construct coarse basis functions that can provide inexpensive coarse approximations that are 1 adequate for all possible flow scenarios in the given ensemble 2 robust with respect to the small scales and high variations in each flow scenario the coarse approximations developed here can be used as a multiscale finite element method or as the coarse solver in a two level domain decomposition iterative method the methods presented here extend to the ensemble case some of the results in j galvis y efendiev domain decomposition preconditioners for multiscale flows in high contrast media reduced dimension coarse spaces siam multiscale model simul 8 5 2010 1621 1644 and y efendiev j galvis x h wu multiscale finite element methods for high contrast problems using local spectral basis functions j comput phys 230 4 2011 937 955 specifically ensembles of permeability fields with high contrast channels and inclusions are considered our main objective here is to construct special multiscale basis functions for the whole ensemble of flow scenarios the coarse basis functions are pre computed for selected or constructed permeability fields with certain topological properties this procedure is a preprocessing step and it avoids constructing basis functions or computing upscaling parameters for each permeability realization then for any permeability the solution of elliptic equation can be project to the space spanned by these pre computed basis functions we apply this coarse multiscale solver to the design of two level domain decomposition preconditioner numerical experiments show that the ensemble level multiscale finite element method converges to the reference solution numerical experiments also show that the ensemble level domain decomposition preconditioner condition number is independent of the high contrast in the coefficient

generalized multiscale finite element method gmsfem for elastic wave propagation in heterogeneous anisotropic media it is important to develop fast yet accurate numerical methods for seismic wave propagation to characterize complex geological structures and oil and gas reservoirs however the computational cost of conventional numerical modeling methods such as finite difference method and finite element method becomes prohibitively expensive when applied to very large models we propose a generalized multiscale finite element method gmsfem for elastic wave propagation in heterogeneous anisotropic media where we construct basis functions from multiple local problems for both the boundaries and interior of a coarse node support or coarse element the application of multiscale basis functions can capture the fine scale medium property variations and allows us to greatly reduce the degrees of freedom that are required to implement the modeling compared with conventional finite element method for wave equation while restricting the error to low values we formulate the continuous galerkin and discontinuous galerkin formulation of the multiscale method both of which have pros and cons applications of the multiscale method to three heterogeneous models show that our multiscale method can effectively model the elastic wave propagation in anisotropic media with a significant reduction in the degrees of freedom in the modeling system

autocorrelation properties of a class of infinite binary sequences corresp a class of infinite pm 1 sequences is presented whose autocorrelation function is zero for all nonzero shifts

special finite element methods for a class of second order elliptic problems with rough coefficients in this paper the approximate solution of a class of second order elliptic equations with rough coefficients is considered problems of the type considered arise in the analysis of unidirectional composites where the coefficients represent the properties of the material several methods for this class of problems are presented and it is shown that they have the same accuracy as usual methods have for problems with smooth coefficients the methods are referred to as special finite element methods because they are of finite element type but employ special shape functions chosen to accurately model the unknown solution

spectral multiscale finite element for nonlinear flows in highly heterogeneous media a reduced basis approach in this paper we study multiscale finite element methods for richards equation a mathematical model to describe fluid flow in unsaturated and highly heterogeneous porous media in order to compute solutions of richard s equation one can use numerical homogenization or multiscale methods that use two grid procedures a fine grid that resolves the heterogeneities and a coarse grid where computations are done the idea is that the coarse solution procedure captures the fine grid variations of the solution since the media has complicated variations inside of coarse grid blocks a large error can be generated during the computation of coarse scale solutions in this paper we consider the case of highly varying coefficients where variations can occur within coarse regions we develop accurate multiscale methods in order to obtain accurate coarse scale numerical solutions for richards equation we design an effective multiscale method that is able to capture the multiscale features of the solution without discarding the small scale details with a careful choice of the coarse basis functions for multiscale finite element methods we can significantly reduce errors we use coarse basis functions construction that combines local spectral problems and a reduced basis rb approach this is an extension to the nonlinear case of the method proposed by efendiev et al 2012 that combines spectral constructions of coarse spaces with rb procedures to efficiently solve linear parameter dependent flow problems the construction of coarse spaces begins with an initial choice of multiscale basis functions supported in coarse regions these basis functions are complemented using a local parameter dependent weighted eigenvalue problem the obtained basis functions can capture the small scale features of the solution within a coarse grid block and give us an accurate coarse scale solution the rb procedures are used to efficiently solve for all possible flow scenarios encountered in every single iteration of a fixed point iterative method we present representative numerical experiments

a numerical homogenization method for heterogeneous anisotropic elastic media based on multiscale theory the development of reliable methods for upscaling finescale models of elastic media has long been an important topic for rock physics and applied seismology several effective medium theories have been developed to provide elastic parameters for materials such as finely layered media or randomly oriented or aligned fractures in such cases the analytic solutions for upscaled properties can be used for accurate prediction of wave propagation however such theories cannot be applied directly to homogenize elastic media with more complex arbitrary spatial heterogeneity therefore we have proposed a numerical homogenization algorithm based on multiscale finite element methods for simulating elastic wave propagation in heterogeneous anisotropic elastic media specifically our method used multiscale basis functions obtained from a local linear elasticity problem with appropriately defined boundary conditions homogenized effective medium parameters were then computed using these basis functions and the approach applied a numerical discretization that was similar to the rotated staggered grid finite difference scheme comparisons of the results from our method and from conventional analytical approaches for finely layered media showed that the homogenization reliably estimated elastic parameters for this simple geometry additional tests examined anisotropic models with arbitrary spatial heterogeneity in which the average size of the heterogeneities ranged from several centimeters to several meters and the ratio between the dominant wavelength and the average size of the arbitrary heterogeneities ranged from 10 to 100 comparisons to finite difference simulations proved that the numerical homogenization was equally accurate for these complex cases

anomalous slow diffusion from perpetual homogenization this paper is concerned with the asymptotic behavior of solutions of stochastic differential equations dy t d omega t nabla v y t dt y 0 0 when d 1 and v is not periodic but obtained as a superposition of an infinite number of periodic potentials with geometrically increasing periods v x sum k 0 infty u k x r k where u k are smooth functions of period 1 u k 0 0 and r k grows exponentially fast with k we can show that y t has an anomalous slow behavior and we obtain quantitative estimates on the anomaly using and developing the tools of homogenization pointwise estimates are based on a new analytical inequality for subharmonic functions when d geq 1 and v is periodic quantitative estimates are obtained on the heat kernel of y t showing the rate at which homogenization takes place the latter result proves davies conjecture and is based on a quantitative estimate for the laplace transform of martingales that can be used to obtain similar results for periodic elliptic generators

harmonic calculus on fractals a measure geometric approach ii differentiation of functions w r t finite atomless measures with compact support on the real line is introduced the related harmonic calculus is similar to that of the classical lebesgue case as an application we obtain the weyl exponent for the spectral asymptotics of the laplacians w r t linear cantor type measures with arbitrary weights

multiscale enrichment based on partition of unity a new multiscale enrichment method based on the partition of unity mepu method is presented it is a synthesis of mathematical homogenization theory and the partition of unity method its primary objective is to extend the range of applicability of mathematical homogenization theory to problems where scale separation may not be possible mepu is perfectly suited for enriching the coarse scale continuum descriptions pdes with fine scale features and the quasi continuum formulations with relevant atomistic data numerical results show that it provides a considerable improvement over classical mathematical homogenization theory and quasi continuum formulations copyright 2005 john wiley sons ltd

existence of ℋ matrix approximants to the inverse fe matrix of elliptic operators with l coefficients this article deals with the existence of blockwise low rank approximants so called matrices to inverses of fem matrices in the case of uniformly elliptic operators with l coefficients unlike operators arising from boundary element methods for which the matrix theory has been extensively developed the inverses of these operators do not benefit from the smoothness of the kernel function however it will be shown that the corresponding green functions can be approximated by degenerate functions giving rise to the existence of blockwise low rank approximants of fem inverses numerical examples confirm the correctness of our estimates as a side product we analyse the matrix property of the inverse of the fe mass matrix

on second order shape optimization methods for electrical impedance tomography this paper is devoted to the analysis of a second order method for recovering the a priori unknown shape of an inclusion omega inside a body omega from boundary measurement this inverse problem known as electrical impedance tomography has many important practical applications and hence has been the focus of much attention during the past few years however to the best of our knowledge no work has yet considered a second order approach for this problem this paper aims to fill that void we investigate the existence of second order derivative of the state u with respect to perturbations of the shape of the interface partial omega then we choose a cost function in order to recover the geometry of partial omega and derive the expression of the derivatives needed to implement the corresponding newton method we then investigate the stability of the process and explain why this inverse problem is severely ill posed by proving the compactness of the hessian at the global minimizer

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

local global multiscale model reduction for flows in high contrast heterogeneous media in this paper we study model reduction for multiscale problems in heterogeneous high contrast media our objective is to combine local model reduction techniques that are based on recently introduced spectral multiscale finite element methods see 19 with global model reduction methods such as balanced truncation approaches implemented on a coarse grid local multiscale methods considered in this paper use special eigenvalue problems in a local domain to systematically identify important features of the solution in particular our local approaches are capable of homogenizing localized features and representing them with one basis function per coarse node that are used in constructing a weight function for the local eigenvalue problem global model reduction based on balanced truncation methods is used to identify important global coarse scale modes this provides a substantial cpu savings as lyapunov equations are solved for the coarse system typical local multiscale methods are designed to find an approximation of the solution for any given coarse level inputs in many practical applications a goal is to find a reduced basis when the input space belongs to a smaller dimensional subspace of coarse level inputs the proposed approaches provide efficient model reduction tools in this direction our numerical results show that only with a careful choice of the number of degrees of freedom for local multiscale spaces and global modes one can achieve a balanced and optimal result

fast construction of hierarchical matrix representation from matrix vector multiplication we develop a hierarchical matrix construction algorithm using matrix vector multiplications based on the randomized singular value decomposition of low rank matrices the algorithm uses o logn applications of the matrix on structured random test vectors and o nlogn extra computational cost where n is the dimension of the unknown matrix numerical examples on constructing green s functions for elliptic operators in two dimensions show efficiency and accuracy of the proposed algorithm

characterization of sub gaussian heat kernel estimates on strongly recurrent graphs sub gaussian estimates for random walks are typical of fractal graphs we characterize them in the strongly recurrent case in terms of resistance estimates only without assuming elliptic harnack inequalities

a domain decomposition method for the acoustic wave equation with discontinuous coefficients and grid change a domain decomposition technique is proposed for the computation of the acoustic wave equation in which the bulk modulus and density fields are allowed to be discontinuous at the interfaces inside each subdomain the method presented coincides with the second order finite difference schemes traditionally used in geophysical modeling however the possibility of assigning to each subdomain its own space step makes numerical simulations much less expensive another interest of the method lies in the fact that its hybrid variational formulation naturally leads to exact equations for gridpoints on the interfaces transposing babuska brezzi s formalism on mixed and hybrid finite elements provides a suitable functional framework for this domain decomposition formulation and shows that the inf sup condition remains the basic requirement for convergence to occur

an energy minimizing interpolation for robust multigrid methods we propose a robust interpolation for multigrid based on the concepts of energy minimization and approximation it can handle pde coefficients of various types on structured or unstructured grids under one framework the formulation is general it can be applied to any dimension we demonstrate numerically the effectiveness of the multigrid method in two dimensions by applying it to a discontinuous coefficient problem an oscillatory coefficient problem and an anisotropic problem empirically the convergence rate is independent of the coefficients of the underlying pde in addition to being independent of the mesh size the proposed method is primarily designed for second order elliptic pdes with possible extensions to other classes of problems such as integral equations

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

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

a multiscale method for linear elasticity reducing poisson locking we propose a generalized finite element method for linear elasticity equations with highly varying and oscillating coefficients the method is formulated in the framework of localized orthogonal decomposition techniques introduced by malqvist and peterseim 2014 assuming only l infinity coefficients we prove linear convergence in the h 1 norm also for materials with large lame parameter a the theoretical a priori error estimate is confirmed by numerical examples

mixed gmsfem for the simulation of waves in highly heterogeneous media numerical simulations of waves in highly heterogeneous media have important applications but direct computations are prohibitively expensive in this paper we develop a new generalized multiscale finite element method with the aim of simulating waves at a much lower cost our method is based on a mixed galerkin type method with carefully designed basis functions that can capture various scales in the solution the basis functions are constructed based on some local snapshot spaces and local spectral problems defined on them the spectral problems give a natural ordering of the basis functions in the snapshot space and allow systematically enrichment of basis functions in addition by using a staggered coarse mesh our method is energy conserving and has block diagonal mass matrix which are desirable properties for wave propagation we will prove that our method has spectral convergence and present numerical results to show the performance of the method

flux norm approach to finite dimensional homogenization approximations with non separated scales and high contrast we consider linear divergence form scalar elliptic equations and vectorial equations for elasticity with rough l r d coefficients a x that in particular model media with non separated scales and high contrast in material properties while the homogenization of pdes with periodic or ergodic coefficients and well separated scales is now well understood we consider here the most general case of arbitrary bounded coefficients for such problems we introduce explicit and optimal finite dimensional approximations of solutions that can be viewed as a theoretical galerkin method with controlled error estimates analogous to classical homogenization approximations in particular this approach allows one to analyze a given medium directly without introducing the mathematical concept of an e family of media as in classical homogenization we define the flux norm as the l 2 norm of the potential part of the fluxes of solutions which is equivalent to the usual h 1 norm we show that in the flux norm the error associated with approximating in a properly defined finite dimensional space the set of solutions of the aforementioned pdes with rough coefficients is equal to the error associated with approximating the set of solutions of the same type of pdes with smooth coefficients in a standard space for example piecewise polynomial we refer to this property as the transfer property a simple application of this property is the construction of finite dimensional approximation spaces with errors independent of the regularity and contrast of the coefficients and with optimal and explicit convergence rates this transfer property also provides an alternative to the global harmonic change of coordinates for the homogenization of elliptic operators that can be extended to elasticity equations the proofs of these homogenization results are based on a new class of elliptic inequalities these inequalities play the same role in our approach as the div curl lemma in classical homogenization

gamblets for opening the complexity bottleneck of implicit schemes for hyperbolic and parabolic odes pdes with rough coefficients implicit schemes are popular methods for the integration of time dependent pdes such as hyperbolic and parabolic pdes however the necessity to solve corresponding linear systems at each time step constitutes a complexity bottleneck in their application to pdes with rough coefficients we present a generalization of gamblets introduced in 62 enabling the resolution of these implicit systems in near linear complexity and provide rigorous a priori error bounds on the resulting numerical approximations of hyperbolic and parabolic pdes these generalized gamblets induce a multiresolution decomposition of the solution space that is adapted to both the underlying hyperbolic and parabolic pde and the system of odes resulting from space discretization and to the time steps of the numerical scheme

a new multiscale finite element method for high contrast elliptic interface problems we introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes and without resolving the interfaces a typical application would be the modelling of flow in a porous medium containing a number of inclusions of low or high permeability embedded in a matrix of high respectively low permeability our method is h 1 conforming with degrees of freedom at the nodes of a triangular mesh and requiring the solution of subgrid problems for the basis functions on elements which straddle the coefficient interface but which use standard linear approximation otherwise a key point is the introduction of novel coefficientdependent boundary conditions for the subgrid problems under moderate assumptions we prove that our methods have optimal convergence rate of o h in the energy norm and o h 2 in the l 2 norm where h is the coarse mesh diameter and the hidden constants in these estimates are independent of the contrast i e ratio of largest to smallest value of the pde coefficient for standard elements the best estimate in the energy norm would be o h 1 2 e with a hidden constant which in general depends on the contrast the new interior boundary conditions depend not only on the contrast of the coefficients but also on the angles of intersection of the interface with the element edges

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

instability of an inverse problem for the stationary navier stokes equations this paper provides a theoretical study of the detection of an object immersed in a fluid when the fluid motion is governed by the stationary navier stokes equations with nonhomogeneous dirichlet boundary conditions to solve this inverse problem we make a boundary measurement on a part of the exterior boundary first we present an identifiability result we then use a shape optimization method in order to identify the obstacle we minimize a nonlinear least squares thus we prove the existence of the first order shape derivative of the state characterize it and deduce the gradient of the least squares functional finally we study the stability of this setting doing a shape sensitivity analysis of order two hence we prove the existence of the second order shape derivatives and we give the expression of the shape hessian at possible solutions of the original inverse problem then the compactness of the riesz operator corresponding to this shape hessian is shown and the ill posedness of the ident

an analysis of the nlmc upscaling method for high contrast problems abstract in this paper we propose simple multiscale basis functions with constraint energy minimization to solve elliptic problems with high contrast medium our methodology is based on the recently developed non local multicontinuum method nlmc the main ingredient of the method is the construction of suitable local basis functions with the capability of capturing multiscale features and non local effects in our method each coarse block is decomposed into various regions according to the contrast ratio and we require that the contrast ratio should be relatively small within each region the basis functions are constructed by solving a local problem defined on the oversampling domains and they have mean value one on the chosen region and zero mean otherwise numerical analysis shows that the resulting basis functions can be localizable and have a decay property the convergence of the multiscale solution is also proved finally some numerical experiments are carried out to illustrate the performances of the proposed method they show that the proposed method can solve problem with high contrast medium efficiently in particular if the oversampling size is large enough then we can achieve the desired error

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

aperiodic correlations and the merit factor of a class of binary sequences corresp a class of binary sequences of length n 2 m is considered and it is shown that their aperiodic autocorrelations can be calculated recursively in a simple way based on this the merit factor of the sequences is calculated and it is shown that the asymptotic value is 3 finally it is proved that the magnitude of the maximal aperiodic autocorrelation is bounded by n 0 9

multi scale finite volume method for elliptic problems in subsurface flow simulation in this paper we present a multi scale finite volume msfv method to solve elliptic problems with many spatial scales arising from flow in porous media the method efficiently captures the effects of small scales on a coarse grid is conservative and treats tensor permeabilities correctly the underlying idea is to construct transmissibilities that capture the local properties of the differential operator this leads to a multi point discretization scheme for the finite volume solution algorithm transmissibilities for the msfv have to be constructed only once as a preprocessing step and can be computed locally therefore this step is perfectly suited for massively parallel computers furthermore a conservative fine scale velocity field can be constructed from the coarse scale pressure solution two sets of locally computed basis functions are employed the first set of basis functions captures the small scale heterogeneity of the underlying permeability field and it is computed in order to construct the effective coarse scale transmissibilities a second set of basis functions is required to construct a conservative fine scale velocity field the accuracy and efficiency of our method is demonstrated by various numerical experiments

harmonic calculus on limits of networks and its application to dendrites abstract the main purpose of this paper is to give a general framework to analysis on fractals including finitely ramified self similar sets random fractals canter sets dendrites and the recurrent case of infinitely ramified self similar sets we study a limit of a sequence of electrical networks satisfying certain compatibility which is a generalization of the decimation invariance of random walks playing an important role in constructing diffusion processes on self similar sets as a limit we propose notions of finite resistance forms and effective resistance metrics also it is shown that a finite resistance form induces an effective resistance metric and vice versa a finite resistance form becomes a regular dirichlet form if the domain is dense in l 2 space in such a case the effective resistance metric is thought of as the intrinsic metric for the dirichlet form in the latter half of the paper we apply the general framework to dendrites which are topological spaces containing no loops a shortest path metric where all arcs are geodesics on a dendrite is shown to be an effective resistance metric and the corresponding finite resistance form turns out to be a local regular dirichlet form furthermore we establish a simple relation between the hausdorff dimensions and the spectral dimensions of dendrites with shortest path metrics

function spaces on fractals abstract we construct function spaces analogs of holder zygmund besov and sobolev spaces on a class of post critically finite self similar fractals in general and the sierpinski gasket in particular based on the laplacian and effective resistance metric of kigami this theory is unrelated to the usual embeddings of these fractals in euclidean space and so our spaces are distinct from the function spaces of jonsson and wallin although there are some coincidences for small orders of smoothness we show that the laplacian acts as one would expect an elliptic pseudodifferential operator of order d 1 on a space of dimension d to act where d is determined by the growth rate of the measure of metric balls we establish some sobolev embedding theorems and some results on complex interpolation on these spaces

multiscale modeling of acoustic wave propagation in 2d media abstractconventional finite difference methods produce accurate solutions to the acoustic and elastic wave equation for many applications but they face significant challenges when material properties vary significantly over distances less than the grid size this challenge is likely to occur in reservoir characterization studies because important reservoir heterogeneity can be present on scales of several meters to ten meters here we describe a new multiscale finite element method for simulating acoustic wave propagation in heterogeneous media that addresses this problem by coupling fine and coarse scale grids the wave equation is solved on a coarse grid but it uses basis functions that are generated from the fine grid and allow the representation of the fine scale variation of the wavefield on the coarser grid time stepping also takes place on the coarse grid providing further speed gains another important property of the method is that the basis functions are only computed once and time savin

a multiscale finite element method for the helmholtz equation it is well known that when the standard galerkin method is applied to the helmholtz equation it exhibits an error in the wavenumber and the solution does not therefore preserve the phase characteristics of the exact solution improvements on the galerkin method including galerkin least squares gls methods have been proposed however these approaches rely on a dispersion analysis of the underlying difference stencils in order to reduce error in the solution in this paper we propose a multiscale finite element for the helmholtz equation the method employs a multiscale variational formulation which leads to a subgrid model in which subgrid scales are incorporated analytically through appropriate green s functions it is shown that entirely new and accurate methods emerge and that gls methods can be obtained as special cases of the more general subgrid model

a priori error analysis of the finite element heterogeneous multiscale method for the wave equation over long time a fully discrete a priori analysis of the finite element heterogeneous multiscale method introduced in a abdulle m grote and c stohrer multiscale model simul 12 2014 pp 1135 1166 for the wave equation with highly oscillatory coefficients over long time is presented a sharp a priori convergence rate for the numerical method is derived for long time intervals the effective model over long time is a boussinesq type equation that has been shown to approximate the one dimensional multiscale wave equation with epsilon periodic coefficients up to time cal o epsilon 2 in a lamacz math models methods appl sci 21 2011 pp 1871 1899 in this paper we also revisit this result by deriving and analyzing a family of effective boussinesq type equations for the approximation of the multiscale wave equation that depends on the normalization chosen for certain micro functions used to define the macroscopic models

stationary stokes oseen and navier stokes equations with singular data the concept of very weak solution introduced by giga math z 178 287 329 1981 for the stokes equations has hardly been studied in recent years for either the navier stokes equations or the navier stokes type equations we treat the stationary stokes oseen and navier stokes systems in the case of a bounded open set connected of class mathcal c 1 1 of mathbb r 3 taking up once again the duality method introduced by lions and magenes problemes aus limites non homogenes et applications vols 1 2 dunod paris 1968 and giga math z 178 287 329 1981 for open sets of class mathcal c infty see also chapter 4 of necas les methodes directes en theorie des equations elliptiques french masson et cie ed paris academia editeurs prague 1967 which considers the hilbertian case p 2 for general elliptic operators we give a simpler proof of the existence of a very weak solution for stationary oseen and navier stokes equations when data are not regular enough based on density arguments and a functional framework adequate for defining more rigourously the traces of non regular vector fields in the stationary navier stokes case the results will be valid for external forces not necessarily small which lets us extend the uniqueness class of solutions for these equations considering more regular data regularity results in fractional sobolev spaces will also be discussed for the three systems all these results can be extended to other dimensions

high dimensional finite elements for two scale maxwell wave equations abstract we develop an essentially optimal numerical method for solving two scale maxwell wave equations in a domain d r d the problems depend on two scales one macroscopic scale and one microscopic scale solving the macroscopic two scale homogenized problem we obtain the desired macroscopic and microscopic information this problem depends on two variables in r d one for each scale that the original two scale equation depends on and is thus posed in a high dimensional tensorized domain the straightforward full tensor product finite element fe method is exceedingly expensive we develop the sparse tensor product fes that solve this two scale homogenized problem with essentially optimal number of degrees of freedom i e the number of degrees of freedom differs by only a logarithmic multiplying factor from that required for solving a macroscopic problem in a domain in r d only for obtaining a required level of accuracy numerical correctors are constructed from the fe solution we derive a rate of convergence for the numerical corrector in terms of the microscopic scale and the fe mesh width numerical examples confirm our analysis

one to one piecewise linear mappings over triangulations we call a piecewise linear mapping from a planar triangulation to the plane a convex combination mapping if the image of every interior vertex is a convex combination of the images of its neighbouring vertices such mappings satisfy a discrete maximum principle and we show that they are one to one if they map the boundary of the triangulation homeomorphically to a convex polygon this result can be viewed as a discrete version of the rado kneser choquet theorem for harmonic mappings but is also closely related to tutte s theorem on barycentric mappings of planar graphs