discretization on non orthogonal quadrilateral grids for inhomogeneous anisotropic media two classes of discretization methods are proposed for control volume formulations on quadrilateral grids in two space dimensions curvilinear grids are considered as a special case the methods are applicable for any system of conservation laws where the flux is defined by a gradient law like darcy s law for porous media flow a strong feature of the methods is the ability to handle media inhomogeneities in combination with full tensor anisotropy and or non orthogonality of the grid further properties of the methods will be discussed and examples of their use will be presented

error estimates on the approximate finite volume solution of convection diffusion equations with general boundary conditions we study here the convergence of a finite volume scheme for a diffusion convection equation on an open bounded set of r dim dim 2 or 3 for which we consider dirichlet neumann or robin boundary conditions we consider unstructured meshes which include voronoi or triangular meshes we use for the diffusion term an s points where s is the number of sides of each cell finite volume scheme and for the convection term an upstream finite volume scheme assuming the exact solution at least in h2 we prove error estimates in a discrete h 1 0 norm of order the size of the mesh discrete poincare inequalities then allow one to prove error estimates in the l2 norm

a small stencil and extremum preserving scheme for anisotropic diffusion problems on arbitrary 2d and 3d meshes in this paper a nonlinear extremum preserving scheme for the heterogeneous and anisotropic diffusion problems is proposed on general 2d and 3d meshes through a certain linearity preserving approach the so called harmonic averaging points located at the interface of heterogeneity are employed to define the auxiliary unknowns this new scheme is locally conservative has only cell centered unknowns and possesses a small stencil which is five point on the structured quadrilateral meshes and seven point on the structured hexahedral meshes the stability result in h 1 norm is obtained under quite general assumptions numerical results show that our scheme is robust and extremum preserving and the optimal convergence rates are verified on general distorted meshes in case that the diffusion tensor is taken to be anisotropic at times discontinuous

a new cement to glue nonconforming grids with robin interface conditions the finite element case robin interface conditions in domain decomposition methods enable the use of non overlapping subdomains and a speed up in the convergence non conforming grids make the grid generation much easier and faster since it is then a parallel task the goal of this paper is to propose and analyze a new discretization scheme which allows to combine the use of robin interface conditions with non matching grids we consider both a symmetric definite positive operator and the convection diffusion equation discretized by finite volume schemes numerical results are shown

two dimensional quantum random walk we analyze several families of two dimensional quantum random walks the feasible region the region where probabilities do not decay exponentially with time grows linearly with time as is the case with one dimensional qrw the limiting shape of the feasible region is however quite different the limit region turns out to be an algebraic set which we characterize as the rational image of a compact algebraic variety we also compute the probability profile within the limit region which is essentially a negative power of the gaussian curvature of the same algebraic variety our methods are based on analysis of the space time generating function following the methods of pemantle and wilson j comb theory ser a 97 1 129 161 2002

on convergence of block centered finite differences for elliptic problems we consider linear selfadjoint elliptic problems with neumann boundary conditions in rectangular domains we demonstrate that with sufficiently smooth data the discrete l 2 norm errors for tensor product block centered finite differences in both the approximate solution and its first derivatives are second order for all nonuniform grids extensions to nonselfadjoint and parabolic problems are discussed

a method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

an explicit a posteriori error estimations for the cell functional minimization scheme of elliptic problems abstract this paper presents an explicit form of a posteriori error estimator for the cell functional minimization scheme of elliptic problems it is shown that this estimator is reliable with respect to an energy type error numerical experiments are conducted to demonstrate the performance of the error estimator in the adaptive mesh refinement algorithm

a non overlapping domain decomposition method with non matching grids for modeling large finite antenna arrays a non overlapping domain decomposition method ddm is proposed herein to solve maxwell equations in r 3 in this work the maxwell equations are discretized using a vector finite element method with hierarchical h curl vector basis functions there are two major ingredients in the proposed non overlapping ddm a a proper 1st order transmission condition to enforce field continuity across domain boundaries and b a cement technique to allow non matching grids for neighboring domains moreover a detail fourier analysis of the transmission condition for a canonical half space example is presented the analysis provides significant insights into the convergence behavior of the proposed non overlapping ddm for solving electromagnetic radiation problems such as the large finite antenna arrays particularly for the antenna arrays the proposed non overlapping ddm is extremely efficient since the formulation can easily incorporate geometrical repetitions exponentially tapered notch vivaldi antenna arrays with size up to 100x100 elements are solved on a common pc to validate the proposed non overlapping ddm

coined quantum walks on weighted graphs we define a discrete time coined quantum walk on weighted graphs that is inspired by szegedy s quantum walk using this we prove that many lackadaisical quantum walks where each vertex has l integer self loops can be generalized to a quantum walk where each vertex has a single self loop of real valued weight l we apply this real valued lackadaisical quantum walk to two problems first we analyze it on the line or one dimensional lattice showing that it is exactly equivalent to a continuous deformation of the three state grover walk with faster ballistic dispersion second we generalize grover s algorithm or search on the complete graph to have a weighted self loop at each vertex yielding an improved success probability when

optimized schwarz methods without overlap for the helmholtz equation the classical schwarz method is a domain decomposition method to solve elliptic partial differential equations in parallel convergence is achieved through overlap of the subdomains we study in this paper a variant of the schwarz method which converges without overlap for the helmholtz equation we show that the key ingredients for such an algorithm are the transmission conditions we derive optimal transmission conditions which lead to convergence of the algorithm in a finite number of steps these conditions are however nonlocal in nature and we introduce local approximations which we optimize for performance of the schwarz method this leads to an algorithm in the class of optimized schwarz methods we present an asymptotic analysis of the optimized schwarz method for two types of transmission conditions robin conditions and transmission conditions with second order tangential derivatives numerical results illustrate the effectiveness of the optimized schwarz method on a model problem and on a problem from industry

the multiscale hybrid mixed method in general polygonal meshes this work extends the general form of the multiscale hybrid mixed mhm method for the second order laplace darcy equation to general non conforming polygonal meshes the main properties of the mhm method i e stability optimal convergence and local conservation are proven independently of the geometry of the elements used for the first level mesh more precisely it is proven that piecewise polynomials of degree k and k 1 k ge 0 for the lagrange multipliers flux along with continuous piecewise polynomial interpolations of degree k 1 posed on second level sub meshes are stable if the latter is fine enough with respect to the mesh for the lagrange multiplier we provide an explicit sufficient condition for this restriction also we prove that the error converges with order k 1 and k 2 in the broken h 1 and l 2 norms respectively under usual regularity assumptions and that such estimates also hold for non convex or even non simply connected elements numerical results confirm the theoretical findings and illustrate the gain that the use of multiscale functions provides

one dimensional three state quantum walk we study a generalized hadamard walk in one dimension with three inner states the particle governed by the three state quantum walk moves in superposition both to the left and to the right according to the inner state in addition to these two degrees of freedom it is allowed to stay at the same position we calculate rigorously the wave function of the particle starting from the origin for any initial qubit state and show the spatial distribution of probability of finding the particle in contrast with the hadamard walk with two inner states on a line the probability of finding the particle at the origin does not converge to zero even after infinite time steps except special initial states this implies that the particle is trapped near the origin after a long time with high probability

superconvergence of the velocity in mimetic finite difference methods on quadrilaterals superconvergence of the velocity is established for mimetic finite difference approximations of second order elliptic problems over h 2 uniform quadrilateral meshes the superconvergence result holds for a full tensor coefficient the analysis exploits the relation between mimetic finite differences and mixed finite element methods via a special quadrature rule for computing the scalar product in the velocity space the theoretical results are confirmed by numerical experiments

spectral properties of quantum walks on rooted binary trees we define coined quantum walks on the infinite rooted binary tree given by unitary operators u c on an associated infinite dimensional hilbert space depending on a unitary coin matrix c in u 3 and study their spectral properties for circulant unitary coin matrices c we derive an equation for the caratheodory function associated to the spectral measure of a cyclic vector for u c this allows us to show that for all circulant unitary coin matrices the spectrum of the quantum walk has no singular continuous component furthermore for coin matrices c which are orthogonal circulant matrices we show that the spectrum of the quantum walk is absolutely continuous except for four coin matrices for which the spectrum of u c is pure point

non conforming curved finite element schemes for time dependent elastic acoustic coupled problems high order numerical methods for solving time dependent acoustic elastic coupled problems are introduced these methods based on finite element techniques allow for a flexible coupling between the fluid and the solid domain by using non conforming meshes and curved elements since characteristic waves travel at different speeds through different media specific levels of granularity for the mesh discretization are required on each domain making impractical a possible conforming coupling in between advantageously physical domains may be independently discretized in our framework due to the non conforming feature consequently an important increase in computational efficiency may be achieved compared to other implementations based on conforming techniques namely by reducing the total number of degrees of freedom differently from other non conforming approaches proposed so far our technique is relatively simpler and requires only a geometrical adjustment at the coupling interface at a preprocessing stage so that no extra computations are necessary during the time evolution of the simulation on the other hand as an advantage of using curvilinear elements the geometry of the coupling interface between the two media of interest is faithfully represented up to the order of the scheme used in other words higher order schemes are in consonance with higher order approximations of the geometry concerning the time discretization we analyze both explicit and implicit schemes these schemes are energy conserving and for the explicit case the stability is guaranteed by a cfl condition in order to illustrate the accuracy and convergence of these methods a set of representative numerical tests are presented

higher order godunov methods for general systems of hyperbolic conservation laws abstract we describe an extension of higher order godunov methods to general systems of hyperbolic conservation laws this extension allow the method to be applied to problems that are not strictly hyperbolic and exhibit local linear degeneracies in the wave fields the method constructs an approximation of the riemann problem from local wave information a generalization of the engquist osher flux for systems is then used to compute a numerical flux based on this approximation this numerical flux replaces the godunov numerical flux in the algorithm thereby eliminating the need for a global riemann problem solution the additional modifications to the godunov methodology that are needed to treat loss of strict hyperbolicity are described in detail the method is applied to some simple model problems for which the global analytic structure is known the method is also applied to the black oil model for multiphase flow in petroleum reservoirs

enriched multi point flux approximation for general grids it is well known that the two point flux approximation a numerical scheme used in most commercial reservoir simulators has o 1 error when grids are not k orthogonal in the last decade the multi point flux approximations have been developed as a remedy however non physical oscillations can appear when the anisotropy is really strong we found out the oscillations are closely related to the poor approximation of pressure gradient in the flux computation in this paper we propose the control volume enriched multi point flux approximation empfa for general diffusion problems on polygonal and polyhedral meshes non physical oscillations are not observed for realistic and strongly anisotropic heterogeneous material properties described by a full tensor exact linear solutions are recovered for grids with non planar interfaces and a first and second order convergence are achieved for the flux and scalar unknowns respectively

a nine point scheme for the approximation of diffusion operators on distorted quadrilateral meshes a nine point scheme is presented for discretizing diffusion operators on distorted quadrilateral meshes the advantage of this method is that highly distorted meshes can be used without the numerical results being altered remarkably and it treats material discontinuities rigorously and offers an explicit expression for the face centered flux moreover it has only the cell centered unknowns we prove that the method is stable and has first order convergence on distorted meshes numerical experiments show that the method has second order or nearly second order accuracy on distorted meshes

periodicity of lively quantum walks on cycles with generalized grover coin abstract in this paper we extend the study of three state lively quantum walks on cycles by considering the coin operator as linear sum of permutation matrices which is a generalization of the grover matrix first we provide a complete characterization of orthogonal matrices of order 3 3 which are linear sum of permutation matrices consequently we determine several groups of complex real and rational orthogonal matrices we establish that an orthogonal matrix of order 3 3 is a linear sum of permutation matrices if and only if it is permutative finally we determine period of lively quantum walk on cycles when the coin operator belongs to the orthogonal group of real linear sum of permutation matrices

a cell functional minimization scheme for domain decomposition method on non orthogonal and non matching meshes we are interested in a robust and accurate domain decomposition algorithm with interface conditions of robin type on non matching multiblock grids using a cell functional minimization scheme which has a good performance on non orthogonal meshes in order to treat the non matching grids at the interface we introduce the l 2 l 2 projection operator to ensure weak continuity of the primary unknown and of the normal flux across the non matching interface furthermore we prove the wellposedness of local and global problems and obtain as well an error estimate of first order in a discrete h 1 h 1 norm only using the l 2 l 2 projection operator on the non matching interface as done in the matching case numerical results are presented in confirmation of the theoretical results

spectral transition for random quantum walks on trees we define and analyze random quantum walks on homogeneous trees of degree q geq 3 such walks describe the discrete time evolution of a quantum particle with internal degree of freedom in mathbb c q hopping on the neighboring sites of the tree in the presence of static disorder the one time step random unitary evolution operator of the particle depends on a unitary matrix c in u q which monitors the strength of the disorder we prove for any q that there exist open sets of matrices in u q for which the random evolution has either pure point spectrum almost surely or purely absolutely continuous spectrum thereby showing the existence of a spectral transition driven by c in u q for q in 3 4 we establish properties of the spectral diagram which provide a description of the spectral transition

any and or formula of size n can be evaluated in time n1 2 o 1 on a quantum computer consider the problem of evaluating an and or formula on an n bit black box input we present a bounded error quantum algorithm that solves this problem in time n 1 2 o 1 in particular approximately balanced formulas can be evaluated in o n queries which is optimal the idea of the algorithm is to apply phase estimation to a discrete time quantum walk on a weighted tree whose spectrum encodes the value of the formula

search via quantum walk we propose a new method for designing quantum search algorithms for finding a marked element in the state space of a classical markov chain the algorithm is based on a quantum walk a la szegedy quantum speed up of markov chain based algorithms in proceedings of the 45th ieee symposium on foundations of computer science ieee computer society press 2004 pp 32 41 that is defined in terms of the markov chain the main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator this operator is then used in an amplitude amplification scheme as a result we considerably expand the scope of the previous approaches of ambainis quantum walk algorithm for element distinctness in proceedings of the 45th ieee symposium on foundations of computer science ieee computer society press 2004 pp 22 31 and szegedy 2004 our algorithm combines the benefits of these approaches in terms of being able to find marked elements incurring the smaller cost of the two and being applicable to a larger class of markov chains in addition it is conceptually simple and avoids some technical difficulties in the previous analyses of several algorithms based on quantum walk

mathcal h matrices compressed multiplicative schwarz preconditioner for nonconformal fem bem ddm an mathcal h matrices compressed multiplicative schwarz ms preconditioner for the nonconformal finite element method fem boundary element method bem domain decomposition method ddm is proposed to analyze the electromagnetic radiation and scattering from periodic structures such as frequency selective surfaces fsss or antenna arrays to improve the overall convergence an ms preconditioner is proposed for the ddm matrix the hierarchical matrix mathcal h matrices technique is introduced into the preconditioned system to compress the dense matrices to further improve the efficiency moreover a finite element tearing and interconnecting like algorithm is adopted to eliminate the interior volumetric fem unknowns and the multilevel fast multipole algorithm is integrated to reduce the computational complexity of bem numerical examples are presented to validate the proposed method it is observed that this method exhibits superior performance and consumes far less memory and cpu time than the traditional implementation

occurrence repetition and matching of patterns in the low temperature ising model we continue our study of the exponential law for occurrences and returns of patterns in the context of gibbsian random fields for the low temperature plus phase of the ising model we prove exponential laws with error bounds for occurrence return waiting and matching times moreover we obtain a poisson law for the number of occurrences of large cylindrical events and a gumbel law for the maximal overlap between two independent copies as a by product we derive precise fluctuation results for the logarithm of waiting and return times the main technical tool we use in order to control mixing is disagreement percolation

an iterative procedure with interface relaxation for domain decomposition methods a domain decomposition method for second order elliptic problems is considered an iterative procedure that reduces the problem to a sequence of mixed boundary value problems on each subdomain is proposed at each iteration a relaxation is accomplished at the subdomain interfaces in several circumstances a value of the relaxation parameter that yields exact convergence in a finite number of iterations is explicitly found moreover when such a value is not available an appropriate strategy for the automatic selection of the relaxation parameter at each iteration is indicated and analyzed this iterative method is then applied to the spectral collocation approximation of the differential problem the same kind of convergence results are proven many numerical experiments show the effectiveness of the method proposed here

localization for a one dimensional split step quantum walk with bound states robust against perturbations for given two unitary and self adjoint operators on a hilbert space a spectral mapping theorem was proved in the work of higuchi et al e print arxiv 1506 06457 see also e segawa and a suzuki quantum stud math found 3 11 2016 in this paper as an application of the spectral mapping theorem we investigate the spectrum of a one dimensional split step quantum walk we give a criterion for when there are no eigenvalues around 1 in terms of a discriminant operator we also provide a criterion for when eigenvalues 1 exist in terms of birth eigenspaces moreover we prove that eigenvectors from the birth eigenspaces decay exponentially at spatial infinity and that the birth eigenspaces are robust against perturbations

optimal schwarz waveform relaxation for the one dimensional wave equation we introduce a nonoverlapping variant of the schwarz waveform relaxation algorithm for wave propagation problems with variable coefficients in one spatial dimension we derive transmission conditions which lead to convergence of the algorithm in a number of iterations equal to the number of subdomains independently of the length of the time interval these optimal transmission conditions are in general nonlocal but we show that the nonlocality depends on the time interval under consideration and we introduce time windows to obtain optimal performance of the algorithm with local transmission conditions in the case of piecewise constant wave speed we show that convergence in two iterations can be achieved independently of the number of subdomains in that case the algorithm thus scales optimally with the number of subdomains provided the time windows are chosen appropriately for continuously varying coefficients we prove convergence of the algorithm with local transmission conditions using energy estimates we then introduce a finite volume discretization which permits computations on nonmatching grids and we prove convergence of the fully discrete schwarz waveform relaxation algorithm we finally illustrate our analysis with numerical experiments

poisson approximation for large contours in low temperature ising models we consider the contour representation of the infinite volume ising model at any fixed inverse temperature the solution of 0e 1 let be the infinite volume measure fix v zd 0 and a large n such that calling gn v the set of contours of length at least n intersecting v there are in average contours in gn v under we show that the total variation distance between the law of gn v under and a poisson process is bounded by a constant depending on and times e n the proof builds on the chen stein method as presented by arratia goldstein and gordon the control of the correlations is obtained through the loss network space time representation of contours due to fernandez ferrari and garcia

optimized schwarz waveform relaxation and discontinuous galerkin time stepping for heterogeneous problems we design and analyze a schwarz waveform relaxation algorithm for domain decomposition of advection diffusion reaction problems with strong heterogeneities the interfaces are curved and we use optimized ventcell transmission conditions we analyze the semidiscretization in time with discontinuous galerkin as well we also show two dimensional numerical results using generalized mortar finite elements in space

convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes the stability and convergence properties of the mimetic finite difference method for diffusion type problems on polyhedral meshes are analyzed the optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved

a finite volume ventcell schwarz algorithm for advection diffusion equations this paper provides a new fully discrete domain decomposition algorithm for the advection diffusion reaction equation it relies on the optimized ventcell schwarz algorithm with a finite volume discretization of the subdomain problems the scheme includes a wide range of advection fluxes with a special treatment on the boundary a complete analysis of the scheme is presented and the convergence of the algorithm to a discrete approximation of the equation using a modified convective flux is proved numerical illustrations of the efficiency of the discrete ventcell schwarz algorithm are given

disagreement percolation in the study of markov fields recently one of the authors van den berg has obtained a uniqueness condition for gibbs measures in terms of disagreement percolation involving two independent realizations in the present paper we study the dependence of markov fields on boundary conditions by taking a more suitable coupling this coupling leads to a new uniqueness condition which improves the one mentioned above we also compare it with the dobrushin uniqueness condition in the case of the ising model our coupling shares certain properties with the fortuin kasteleyn representation it gives an explicit expression of the boundary effect on a certain vertex in terms of percolation like probabilities

localization of the grover walks on spidernets and free meixner laws a spidernet is a graph obtained by adding large cycles to an almost regular tree and considered as an example having intermediate properties of lattices and trees in the study of discrete time quantum walks on graphs we introduce the grover walk on a spidernet and its one dimensional reduction we derive an integral representation of the n step transition amplitude in terms of the free meixner law which appears as the spectral distribution as an application we determine the class of spidernets which exhibit localization our method is based on quantum probabilistic spectral analysis of graphs

exponential distribution for the occurrence of rare patterns in gibbsian random fields we study the distribution of the occurrence of rare patterns in sufficiently mixing gibbs random fields on the lattice d d 2 a typical example is the high temperature ising model this distribution is shown to converge to an exponential law as the size of the pattern diverges our analysis not only provides this convergence but also establishes a precise estimate of the distance between the exponential law and the distribution of the occurrence of finite patterns a similar result holds for the repetition of a rare pattern we apply these results to the fluctuation properties of occurrence and repetition of patterns we prove a central limit theorem and a large deviation principle

elimination of adaptive grid interface errors in the discrete cell centered pressure equation classical cell centered discretization of the reservoir simulation pressure equation on anh adaptive grid results in ano 1 h leading truncation error at the grid interface a new flux continuous finite volume correction is presented together with an improved version of a previously proposed correction while both corrections eliminate the leading error the new correction exhibits the best convergence rates and has the following properties the resulting matrix is in general symmetric positive definite diagonally dominant for locally isotropic spatially varying coefficients convergence too h is demonstrated and support of the standard approximation is retained ensuring a fully implicit implementation results computed by the uncorrected classical scheme and both correction schemes are compared for two phase flow simulations with multilevel dynamic local grid refinement

a preconditioned dual primal finite element tearing and interconnecting method for solving three dimensional time harmonic maxwell s equations a new preconditioned dual primal nonoverlapping domain decomposition method is proposed for the finite element solution of three dimensional large scale electromagnetic problems with the aid of two lagrange multipliers the new method converts the original volumetric problem to a surface problem by using a higher order transmission condition at the subdomain interfaces to significantly improve the convergence of the iterative solution of the global interface equation similar to the previous version a global coarse problem related to the degrees of freedom at the subdomain corner edges is formulated to propagate the residual error to the whole computational domain at each iteration which further increases the rate of convergence in addition a fully algebraic preconditioner based on matrix splitting is constructed to make the proposed domain decomposition method even more robust and scalable perfectly matched layers pmls are considered for the boundary truncation when solving open region problems the influence of the pml truncation on the convergence performance is investigated by examining the convergence of the transmission condition for an interface inside the pml numerical examples including wave propagation and antenna radiation problems truncated with pmls are presented to demonstrate the validity and the capability of this method

completely analytical interactions constructive description an interactionu is called a completely analytical ca interaction if it satisfies one of 12 given conditions formulated in terms of analyticity properties of the partition functions zv u or correlation decay or truncated correlation bounds or asymptotic behavior of ln zv u v the 12 conditions are presented together with part of the proof of their equivalence the main result of the paper is that each condition is constructive in the following sense instead of checking it in all finite volumesv v it is enough to consider only a finite amount of volumes with restricted size in particular the partition functions z v u for the complex perturbationsu ofu do not vanish for all v v and all with provided this is true only forv with diam v c and but with

unveiling and exemplifying the unitary equivalence of discrete time quantum walk models the two major discrete time formulations for quantum walks coined and scattering are unitarily equivalent for arbitrary position dependent transition amplitudes and any topology andrade et al 2009 phys rev a 80 052301 although the proof explicitly describes the mapping obtention its high technicality may hinder relevant physical aspects involved in the equivalence discussing concrete examples the most general constructions for the line square and honeycomb lattices here we unveil the similarities and differences of these two versions of quantum walks we moreover show how to derive the dynamics of one from the other by means of proper projections we perform calculations for different probability amplitudes such as hadamard grover discrete fourier transform and the uncommon in the area but interesting discrete hartley transform comparing the evolutions our study illustrates the models interplay an important issue for implementations and applications of such systems

finite volume method for 2d linear and nonlinear elliptic problems with discontinuities in this paper we study the approximation of solutions to linear and nonlinear elliptic problems with discontinuous coefficients in the discrete duality finite volume ddfv framework this family of schemes allows very general meshes and inherits the main properties of the continuous problem in order to take into account the discontinuities and to prevent consistency defect in the scheme we propose to modify the definition of the numerical fluxes on the edges of the mesh where the discontinuity occurs we first illustrate our approach by the study of the 1d situation then we show how to design our new scheme called m ddfv and we propose its analysis we also describe an iterative solver whose convergence is proved which can be used to solve the nonlinear discrete equations defining the finite volume scheme finally we provide numerical results which confirm that the m ddfv scheme significantly improves the convergence rate of the usual ddfv method for both linear and nonlinear problems

sharp error terms and neccessary conditions for exponential hitting times in mixing processes we prove an upper bound for the error in the exponential approximation of the hitting time law of a rare event in mixing processes with exponential decay mixing processes with a summable function and for general mixing processes with a finite alphabet in the first case the bound is uniform as a function of the measure of the event in the last two cases the bound depends also on the time scale t this allows us to get further statistical properties as the ratio convergence of the expected hitting time and the expected return time a uniform bound is a consequence we present an example that shows that this bound is sharp we also prove that second moments are not necessary for having the exponential law moreover we prove a necessary condition for having the exponential limit law

a new discretization methodology for diffusion problems on generalized polyhedral meshes we develop a family of inexpensive discretization schemes for diffusion problems on generalized polyhedral meshes with elements having non planar faces the material properties are described by a full tensor we also prove superconvergence for the scalar pressure variable under very general assumptions the theoretical results are confirmed with numerical experiments in the practically important case of logically cubic meshes with randomly perturbed nodes the mixed finite element with the lowest order raviart thomas elements does not converge while the proposed mimetic method has the optimal convergence rate 2007 elsevier b v all rights reserved

local flux mimetic finite difference methods we develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral meshes to approximate the velocity vector variable the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per n gonal mesh face in three dimensions to approximate the pressure scalar variable the method uses one degree of freedom per element a specially chosen quadrature rule for the l 2 product of vector functions allows for a local flux elimination and reduction of the method to a cell centered finite difference scheme for the pressure unknowns under certain assumptions first order convergence is proved for both variables and second order convergence is proved for the pressure the assumptions are verified on simplicial meshes for a particular quadrature rule that leads to a symmetric method for general polyhedral meshes non symmetric methods are constructed based on quadrature rules that are shown to satisfy some of the assumptions numerical results confirm the theory

a monotone finite volume scheme for diffusion equations on general non conforming meshes a nonlinear monotone finite volume scheme on general non conforming meshes for diffusion equations is introduced which deals with discontinuous tensor coefficients rigorously since the expression of normal flux depends on auxiliary unknowns defined at cell vertex including hanging nodes we propose a new method to eliminate vertex unknown by using primary unknowns at the centers of the cells sharing the vertex especially the unknowns defined on hanging nodes are eliminated by flux continuous conditions the resulting scheme is monotone and preserves positivity of analytical solutions for strongly anisotropic and heterogeneous full tensor coefficient problems numerical results show that the convergent order of the monotone scheme by different methods of eliminating vertex unknowns will vary remarkably and our new method can assure that it has almost second order accuracy and more accurate than some existing methods

coupling spectral and finite elements for second order elliptic three dimensional equations this paper deals with the approximation of elliptic second order problems with a nonconforming domain decomposition method that allows for coupling spectral and finite element discretizations in three dimensions we present two kinds of matching conditions at the interfaces between the subdomains the detailed numerical analysis of the error committed upon the exact solution turns out to be optimal we describe also the implementation and give some numerical results confirming the theoretical predictions

efficient quantum circuits for continuous time quantum walks on composite graphs in this paper we investigate the simulation of continuous time quantum walks on specific classes of graphs for which it is possible to fast forward the time evolution operator to achieve constant time simulation complexity and to perform the simulation exactly i e while maintaining efficiency in particular we discuss two classes of composite graphs commuting graphs and cartesian product of graphs that contain classes of graphs which can be simulated in this fashion this allows us to identify new families of graphs that we can efficiently simulate in a quantum circuit framework providing practical and explicit means to explore quantum walk based algorithms in laboratories

the mortar finite element method with lagrange multipliers the present paper deals with a variant of a non conforming domain decomposition technique the mortar finite element method in the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices and the edges in 3d of subdomains it is shown that written under primal hybrid formulation the approximation problem issued from a discretization of a second order elliptic equation in 2d is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms finally the parallelization advantages consequence of this variant are also addressed

a fast domain decomposition method for the solution of electromagnetic scattering by large objects a nonoverlapping domain decomposition method is proposed for the finite element solution of the scattering problem by electrically large inhomogeneous infinite cylinders of arbitrary cross section to minimize the size of the total computational domain a second order absorbing boundary condition abc is applied upon an outer boundary of arbitrary shape which may be conformal to the surface of the scatterer this domain is then partitioned into concentric subdomains circumscribing the object a second order transmission condition derived from the abc is prescribed upon the interfaces between two adjacent subdomains this particular configuration is responsible for the fast convergence of the domain decomposition iterative algorithm which is parallelizable numerical results obtained with a nonparallelized computer code are presented which emphasize the superiority of this technique in terms of memory storage requirements and computing times over the standard finite element approach as well as over the rigorous hybrid finite element integral equation formulation

cross flow tensors and finite volume approximation with by deferred correction abstract a general framework for full tensor finite volume schemes is presented along with properties of the schemes that in turn provide motivation for a deferred correction scheme the approach is illustrated by writing a general nine node discretization as a five node scheme plus a residual term involving the extra nodes a possible solution procedure involves solving the standard five node matrix problem to yield an initial solution then re solving the problem with the residual determined by the first iterate of the pressure field relaxation using the pressure from the previous impes time step is also examined stability of the inner iteration is considered and strategies are discussed this method is coupled with a higher order godunov scheme for integrating the convective components of the flow equations results presented include two phase flow examples with full tensor fields and serve to demonstrate the potential of the method

exponentially fitted methods for solving time fractional nonlinear reaction diffusion equation abstract in this article we will develop a new numerical scheme with the second order in time and a class of fourth order or sixth order in space based on the exponential fitting techniques to approximate the nonlinear time fractional reaction diffusion equation with fixed order and distributed order derivatives these techniques depend on a parameter which will be used to annihilate the error and increase the order of accuracy the proposed methods are proved to be unconditionally stable and convergent by fourier analysis also the theoretical results and the effectiveness of the numerical scheme are confirmed by numerical test problems and a comparison with other methods is presented

space time domain decomposition methods for diffusion problems in mixed formulations this paper is concerned with global in time nonoverlapping domain decomposition methods for the mixed formulation of the diffusion problem two approaches are considered one uses the time dependent steklov poincare operator and the other uses optimized schwarz waveform relaxation oswr based on robin transmission conditions for each method a mixed formulation of an interface problem on the space time interfaces between subdomains is derived and different time grids are employed to adapt to different time scales in the subdomains demonstrations of the well posedness of the robin subdomain problems involved in the oswr method and a convergence proof of the oswr algorithm are given for the mixed formulation numerical results for two dimensional problems with strong heterogeneities are presented to illustrate the performance of the two methods

quantum algorithms for finding constant sized sub hypergraphs we develop a general framework to construct quantum algorithms that detect if a 3 uniform hypergraph given as input contains a sub hypergraph isomorphic to a prespecified constant sized hypergraph this framework is based on the concept of nested quantum walks recently proposed by jeffery kothari and magniez 2013 12 and extends the methodology designed by lee magniez and santha 2013 18 for similar problems over graphs as applications we obtain a quantum algorithm for finding a 4 clique in a 3 uniform hypergraph on n vertices with query complexity o n1 883 and a quantum algorithm for determining if a ternary operator over a set of size n is associative with query complexity o n2 113

on concentration inequalities and their applications for gibbs measures in lattice systems we consider gibbs measures on the configuration space s mathbb z d where mostly d ge 2 and s is a finite set we start by a short review on concentration inequalities for gibbs measures in the dobrushin uniqueness regime we have a gaussian concentration bound whereas in the ising model and related models at sufficiently low temperature we control all moments and have a stretched exponential concentration bound we then give several applications of these inequalities whereby we obtain various new results amongst these applications we get bounds on the speed of convergence of the empirical measure in the sense of kantorovich distance fluctuation bounds in the shannon mcmillan breiman theorem fluctuation bounds for the first occurrence of a pattern as well as almost sure central limit theorems

a uniqueness condition for gibbs measures with application to the 2 dimensional ising antiferromagnet a uniqueness condition for gibbs measures is given this condition is stated in terms of absence of a certain type of percolation involving two independent realisations this result can be applied in certain concrete situations by comparison with ordinary percolation in this way we prove that the ising antiferromagnet on a square lattice has a unique gibbs measure if 4 h 1 2ln pc 1 pc whereh denotes the external magnetic field the inverse temperature andpc the critical probability for site percolation on that lattice sincepc is larger than 1 2 this extends a result by dobrushin kolafa and shlosman whose proof was computer assisted

a schwarz alternating procedure for singular perturbation problems we show that the schwarz alternating procedure offers a good algorithm for the numerical solution of singular perturbation problems provided the domain decomposition is properly designed to resolve the boundary and transition layers we give sharp estimates for the optimal position of the domain boundaries and study the convergence rates of the algorithm for various linear second order singular perturbation problems we report on implementation results for a turning point problem and a combustion problem

a domain decomposition method for the helmholtz equation and related optimal control problems we present an iterative domain decomposition method to solve the helmholtz equation and related optimal control problems the proof of convergence of this method relies on energy techniques this method leads to efficient algorithms for the numerical resolution of harmonic wave propagation problems in homogeneous and heterogeneous media

an octree multigrid method for quasi static maxwell s equations with highly discontinuous coefficients in this paper we develop an octree discretization for maxwell s equations in the quasi static regime we then use this discretization in order to develop a multigrid method for maxwell s equations with highly discontinuous coefficients we test our algorithms and compare it to other multilevel algorithms

double families of quasi positive darcy flux approximations with highly anisotropic tensors on structured and unstructured grids this paper focuses on flux continuous pressure equation approximation for strongly anisotropic media previous work on families of flux continuous schemes for solving the general geometry permeability tensor pressure equation has focused on single parameter families these schemes have been shown to remove the o 1 errors introduced by standard two point flux reservoir simulation schemes when applied to full tensor flow approximation improved convergence of the schemes has also been established for specific quadrature points however these schemes have conditional m matrices depending on the strength of the off diagonal tensor coefficients when applied to cases involving full tensors arising from strongly anisotropic media the point wise continuous schemes can fail to satisfy the maximum principle and induce severe spurious oscillations in the numerical pressure solution new double family flux continuous locally conservative schemes are presented for the general geometry permeability tensor pressure equation the new double family formulation is shown to expand on the current single parameter range of existing conditional m matrix schemes the conditional m matrix bounds on a double family formulation are identified for both quadrilateral and triangle cell grids a quasi positive qm matrix analysis is presented that classifies the behaviour of the new schemes with respect to double family quadrature in regions beyond the m matrix bounds the extension to double family quadrature is shown to be beneficial resulting in novel optimal anisotropic quadrature schemes the new methods are applied to strongly anisotropic full tensor field problems and yield results with sharp resolution with only minor or practically zero spurious oscillations

a mortar mimetic finite difference method on non matching grids we consider mimetic finite difference approximations to second order elliptic problems on non matching multiblock grids mortar finite elements are employed on the non matching interfaces to impose weak flux continuity optimal convergence and in certain cases superconvergence is established for both the scalar variable and its flux the theory is confirmed by computational results

maximal clusters in non critical percolation and related models we investigate the maximal non critical cluster in a big box in various percolation type models we investigate its typical size and the fluctuations around this typical size the limit law of these fluctuations is related to maxima of independent random variables with law described by a single cluster

a non mortar mixed finite element method for elliptic problems on non matching multiblock grids abstract we consider the approximation of second order elliptic equations on domains that can be described as a union of sub domains or blocks we assume that a grid is defined on each block independently so that the resulting grid over the entire domain need not be conforming i e match across the block boundaries several techniques have been developed to approximate elliptic equations on multiblock grids that utilize a mortar finite element space defined on the block boundary interface itself we define a mixed finite element method that does not use such a mortar space the method has an advantage in the case where adaptive local refinement techniques will be used in that there is no mortar grid to refine as is typical of mixed methods our method is locally conservative element by element it is also globally conservative across the block boundaries theoretical results show that the approximate solution converges at the optimal rate to the true solution we present computational results to illustrate and confirm the theory

grover search with lackadaisical quantum walks the lazy random walk where the walker has some probability of staying put is a useful tool in classical algorithms we propose a quantum analogue the lackadaisical quantum walk where each vertex is given l self loops and we investigate its effects on grover s algorithm when formulated as search for a marked vertex on the complete graph of n vertices for the discrete time quantum walk using the phase flip coin adding a self loop to each vertex boosts the success probability from 1 2 to 1 additional self loops however decrease the success probability using instead the shenvi kempe and whaley 2003 coin adding self loops simply slows down the search these coins also differ in that the first is faster than classical when l scales less than n while the second requires that l scale less than n 2 finally continuous time quantum walks differ from both of these discrete time examples the self loops make no difference at all these behaviors generalize to multiple marked vertices

bulk edge correspondence of one dimensional quantum walks we outline a theory of symmetry protected topological phases of one dimensional quantum walks we assume spectral gaps around the symmetry distinguished points 1 and 1 in which only discrete eigenvalues are allowed the phase classification by integer or binary indices extends the classification known for translation invariant systems in terms of their band structure however our theory requires no translation invariance whatsoever and the indices we define in this general setting are invariant under arbitrary symmetric local perturbations even those that cannot be continuously contracted to the identity more precisely we define two indices for every walk characterizing the behavior far to the right and far to the left respectively their sum is a lower bound on the number of eigenstates at 1 and 1 for a translation invariant system the indices add up to zero so one of them already characterizes the phase by joining two bulk phases with different indices we get a walk in which the right and left indices no longer cancel so the theory predicts bound states at 1 or 1 this is a rigorous statement of bulk edge correspondence the results also apply to the hamiltonian case with a single gap at zero

recurrence and polya number of quantum walks we analyze the recurrence probability polya number for d dimensional unbiased quantum walks a sufficient condition for a quantum walk to be recurrent is derived as a by product we find a simple criterion for localization of quantum walks in contrast with classical walks where the polya number is characteristic for the given dimension the recurrence probability of a quantum walk depends in general on the topology of the walk choice of the coin and the initial state this allows us to change the character of the quantum walk from recurrent to transient by altering the initial state

a local regularization operator for triangular and quadrilateral finite elements this paper develops a local regularization operator on triangular or quadrilateral finite elements built on structured or unstructured meshes this operator is a variant of the regularization operator of clement however ours is constructed via a local projection in a reference domain we prove in this paper that it has the same optimal approximation properties as the standard interpolation operator and we present some applications

a two level domain decomposition method for the iterative solution of high frequency exterior helmholtz problems summary we present a lagrange multiplier based two level domain decomposition method for solving iteratively large scale systems of equations arising from the finite element discretization of high frequency exterior helmholtz problems the proposed method is essentially an extension of the regularized feti finite element tearing and interconnecting method to indefinite problems its two key ingredients are the regularization of each subdomain matrix by a complex interface lumped mass matrix and the preconditioning of the interface problem by an auxiliary coarse problem constructed to enforce at each iteration the orthogonality of the residual to a set of carefully chosen planar waves we show numerically that the proposed method is scalable with respect to the mesh size the subdomain size and the wavenumber we report performance results for a submarine application that highlight the efficiency of the proposed method for the solution of high frequency acoustic scattering problems discretized by finite elements

iterative substructuring preconditioners for mortar element methods in two dimensions the mortar methods are based on domain decomposition and they allow for the coupling of different variational approximations in different subdomains the resulting methods are nonconforming but still yield optimal approximations in this paper we will discuss iterative substructuring algorithms for the algebraic systems arising from the discretization of symmetric second order elliptic equations in two dimensions both spectral and finite element methods for geometrically conforming as well as nonconforming domain decompositions are studied in each case we obtain a polylogarithmic bound on the condition number of the preconditioned matrix

a cell functional minimization scheme for parabolic problem we are interested in a robust and accurate finite volume scheme for 2 d parabolic problems derived from the cell functional minimization approach the scheme has a local stencil is locally conservative treats discontinuity rigorously and leads to a symmetric positive definite linear system since the scheme has both cell centered unknowns and cell edge unknowns the computational cost is an issue and a parallel algorithm is then suggested based on nonoverlapping domain decomposition approach the interface condition is of the dirichlet robin type and has a parameter l by choosing this parameter properly the convergence of the iteration process could be sped up numerical results for linear and nonlinear problems demonstrate the good performance of the cell functional minimization scheme and its parallel version on distorted meshes

a second order discretization of maxwell s equations in the quasi static regime on octree grids in this study we consider adaptive mesh refinement for the solution of maxwell s equations in the quasi static or diffusion regime we propose a new finite volume octree discretization for the problem and show how to construct second order stencils on yee grids extending the known first order discretization stencils we then develop an effective preconditioner to the problem we show that our preconditioner performs well for discontinuous conductivities as well as for a wide range of frequencies

a fem domain decomposition method for photonic and electromagnetic band gap structures this paper introduces an efficient domain decomposition algorithm for the solution of time harmonic electromagnetic fields arising in three dimensional finite size photonic band gap and electromagnetic band gap structures the method is based on the finite element approximation and a nonoverlapping domain decomposition method a set of cement unknowns on the inter domain interfaces has been explicitly introduced to enforce the appropriate field continuities the introduction of these extra unknowns allows for nonconforming nonmatching triangulations across domain eliminating the need for periodic mesh to ensure and improve the convergence of the outer iteration loop robin transmission condition is used to communicate information across interfaces the resulting system of equations is solved with a fast algorithm that loosely resembles the well known finite element tearing and interconnecting algorithm in this algorithm the method solves in the preprocessing step for the robin primal subdomain problems multiple times by exciting one dual unknown at a time this step generates an iteration matrix that is then used to update the dual unknowns in the outer loop iteration the present method becomes extremely efficient for problems with geometric repetitions such as photonic and electromagnetic band gap structures

finite volume methods for domain decomposition on nonmatching grids with arbitrary interface conditions we are interested in a robust and accurate domain decomposition method with arbitrary interface conditions on nonmatching multiblock grids using a finite volume discretization to take into account the nonmatching grids at the interface we introduce transmission operators dirichlet neumann interface conditions and arbitrary equivalent interface conditions for example robin interface conditions under a compatibility assumption on the transmission operators we prove the equivalence between the different types of interface conditions and the well posedness of the local and global problems then two error estimates are proven in terms of the discrete h1 norm the first in o h 1 2 with transmission operators based on piecewise constant functions and the second in o h as in the matching case with transmission operators using a piecewise linear rebuilding in conclusion numerical results are presented in confirmation of the theory

robust convergence of multi point flux approximation on rough grids this paper establishes the convergence of a multi point flux approximation control volume method on rough quadrilateral grids by rough grids we refer to a family of refined quadrilateral grids where the cells are not required to approach parallelograms in the asymptotic limit in contrast to previous convergence results for these methods we consider here a version where the flux approximation is derived directly in the physical space and not on a reference cell as a consequence less regular grids are allowed however the extra cost is that the symmetry of the method is lost

cell centered finite volume methods with flexible stencils for diffusion equations on general nonconforming meshes a cell centered finite volume method is presented for discretizing diffusion operator on general nonconforming meshes the node values are accurately approximated using a new weighted interpolation formula in which the calculation of the weight is adaptive to both geometric parameters and diffusion coefficients it follows that an explicit expression composed of cell centered unknowns only is obtained for the discretization of normal flux numerical results demonstrate that linear solutions are reproduced exactly on the nonconforming random grids and that the convergence rate is close to second order for non linear or discontinuous problems

mimetic finite difference methods for diffusion equations on non orthogonal non conformal meshes mimetic discretizations based on the support operators methodology are derived for non orthogonal locally refined quadrilateral meshes the second order convergence rate on non smooth meshes is verified with numerical examples