hllc type and path conservative schemes for a single velocity six equation two phase flow model a comparative study abstract the present article deals with the numerical integration of a six equation single velocity two phase flow model with stiff mechanical relaxation this model can be employed to approximate efficiently the well known single velocity single pressure five equation model of kapila et al 2001 work in the literature has shown the efficiency of the six equation model in simulating complex two phase flows involving cavitation and evaporation processes the aim of this work is to present and discuss various numerical schemes for this two phase model focusing on the integration of the nonconservative terms appearing in the phasic energy equations in fact previous work has suggested that the choice of the discretization method for the nonconservative terms often does not play a significant role two new methods are proposed a path conservative hllc type scheme that is based on the dal maso lefloch murat theory and a generalized hllc type scheme that is based on a suliciu s riemann solver the latter scheme has the important property of preserving the positivity of the intermediate states of the conserved quantities moreover we also approximate solutions of the six equation model by applying two path conservative schemes recently proposed in the literature which have been derived from the osher and hllem riemann solvers we show comparisons of the different numerical schemes for several test cases including cavitation problems and shock tubes an efficiency study for first and second order schemes is also presented numerical results show that different methods corresponding to different numerical treatments of the nonconservative terms give analogous results and they are all able to produce accurate approximations of solutions of the kapila s five equation model except as expected for shocks in two phase mixtures with very high pressure ratios

vanishing viscosity solutions of nonlinear hyperbolic systems we consider the cauchy problem for a strictly hyperbolic n n system in one space dimension ut a u ux 0 assuming that the initial data have small total variation we show that the solutions of the viscous approximations ut a u ux euxx are defined globally in time and satisfy uniform bv estimates independent of e moreover they depend continuously on the initial data in the l 1 distance with a lipschitz constant independent of t e letting e 0 these viscous solutions converge to a unique limit depending lipschitz continuously on the initial data in the conservative case where a df is the jacobian

hyperbolic conservation laws with relaxation the effect of relaxation is important in many physical situations it is present in the kinetic theory of gases elasticity with memory gas flow with thermo non equilibrium water waves etc the governing equations often take the form of hyperbolic conservation laws with lower order terms in this article we present and analyze a simple model of hyperbolic conservation laws with relaxation effects dynamic subcharacteristics governing the propagation of disturbances over strong wave forms are identified stability criteria for diffusion waves expansion waves and traveling waves are found and justified nonlinearly time asymptotic expansion and the energy method are used in the analysis for dissipative waves the expansion is similar in spirit to the chapman enskog expansion in the kinetic theory for shock waves however a different approach is needed

consistency of spectral clustering consistency is a key property of all statistical procedures analyzing randomly sampled data surprisingly despite decades of work little is known about consistency of most clustering algorithms in this paper we investigate consistency of the popular family of spectral clustering algorithms which clusters the data with the help of eigenvectors of graph laplacian matrices we develop new methods to establish that for increasing sample size those eigenvectors converge to the eigenvectors of certain limit operators as a result we can prove that one of the two major classes of spectral clustering normalized clustering converges under very general conditions while the other unnormalized clustering is only consistent under strong additional assumptions which are not always satisfied in real data we conclude that our analysis provides strong evidence for the superiority of normalized spectral clustering

a weak formulation of roe s approximate riemann solver abstract this paper presents a weak formulation of roe s approximate riemann solver based on a notion of weak solution to nonlinear hyperbolic systems in nonconservative form this weak formulation is applied to the euler equations for real gases and the homogeneous equilibrium two phase flow model numerical results for a two phase flow heat addition problem are given

network cross validation for determining the number of communities in network data abstractthe stochastic block model sbm and its variants have been a popular tool for analyzing large network data with community structures in this article we develop an efficient network cross validation ncv approach to determine the number of communities as well as to choose between the regular stochastic block model and the degree corrected block model dcbm the proposed ncv method is based on a block wise node pair splitting technique combined with an integrated step of community recovery using sub blocks of the adjacency matrix we prove that the probability of under selection vanishes as the number of nodes increases under mild conditions satisfied by a wide range of popular community recovery algorithms the solid performance of our method is also demonstrated in extensive simulations and two data examples supplementary materials for this article are available online

why nonconservative schemes converge to wrong solutions error analysis this paper attempts to give a qualitative and quailtitative description of the numerical error introduced by using finite difference schemes in nonconservative form for scalar conservation laws we show that these schemes converge strongly in li norm to the solution of an inhomogeneous conservation law containing a borel measure source term moreover we analyze the properties of this borel measure and derive a sharp estimate for the l1 error between the limit function given by the scheme and the correct solution in general the measure source term is of the order of the entropy dissipation measure associated with the scheme in certain cases the error can be small for short times which makes it difficult to detect numerically but generically such an error will grow in time and this would lead to a large error for large time calculations finally we show that a local correction of any high order accurate scheme in nonconservative form is sufficient to ensure its convergence to the correct solution

pseudo likelihood methods for community detection in large sparse networks many algorithms have been proposed for fitting network models with communities but most of them do not scale well to large networks and often fail on sparse networks here we propose a new fast pseudo likelihood method for fitting the stochastic block model for networks as well as a variant that allows for an arbitrary degree distribution by conditioning on degrees we show that the algorithms perform well under a range of settings including on very sparse networks and illustrate on the example of a network of political blogs we also propose spectral clustering with perturbations a method of independent interest which works well on sparse networks where regular spectral clustering fails and use it to provide an initial value for pseudo likelihood we prove that pseudo likelihood provides consistent estimates of the communities under a mild condition on the starting value for the case of a block model with two communities

simplified algorithms for optimizing multiuser multi hop mimo relay systems in this paper we address the issue of multiaccess communication through multi hop linear non regenerative relays where all users all relay nodes and the destination node may have multiple antennas using a linear minimal mean squared error mmse receiver at the destination node we demonstrate that the optimal amplifying matrix at each relay node can be viewed as a linear mmse filter concatenated with another linear filter as a consequence the mse matrix of the signal waveform estimation at the destination node is decomposed into the sum of the mse matrices at all relay nodes we show that at a high signal to noise ratio snr environment this mse matrix decomposition significantly simplifies the solution to the problem of optimizing the source precoding matrices and relay amplifying matrices simulation results show that even at the low to medium snr range the simplified optimization algorithms have only a marginal performance degradation but a greatly reduced computational complexity and signalling overhead compared with the existing optimal iterative algorithm and thus are of great interest for practical relay systems

the capacity region of the gaussian multiple input multiple output broadcast channel the gaussian multiple input multiple output mimo broadcast channel bc is considered the dirty paper coding dpc rate region is shown to coincide with the capacity region to that end a new notion of an enhanced broadcast channel is introduced and is used jointly with the entropy power inequality to show that a superposition of gaussian codes is optimal for the degraded vector broadcast channel and that dpc is optimal for the nondegraded case furthermore the capacity region is characterized under a wide range of input constraints accounting as special cases for the total power and the per antenna power constraints

power allocation for a mimo relay system with multiple antenna users a power allocation or scheduling problem is studied for a multiuser multiple input multiple output mimo wireless relay system where there is a non regenerative relay between one access point and multiple users each node in the system is equipped with multiple antennas the purpose of this study is to develop fast algorithms to compute the source covariance matrix or matrices and the relay transformation matrix to optimize a system performance we consider the minimization of power consumption subject to rate constraint and also the maximization of system throughput subject to power constraint these problems are nonconvex and apparently have no simple solutions in this paper a number of computational strategies are presented and their performances are investigated both uplink and downlink cases are considered the use of multiple carriers is also discussed moreover a generalized water filling gwf algorithm is developed to solve a special class of convex optimization problems the gwf algorithm is used for two of the strategies shown in this paper

a relaxation projection method for compressible flows part ii artificial heat exchanges for multiphase shocks the relaxation projection method developed in saurel et al r saurel e franquet e daniel o le metayer a relaxation projection method for compressible flows part i the numerical equation of state for the euler equations j comput phys 2007 822 845 is extended to the non conservative hyperbolic multiphase flow model of kapila et al a k kapila menikoff j b bdzil s f son d s stewart two phase modeling of deflagration to detonation transition in granular materials reduced equations physics of fluids 13 10 2001 3002 3024 this model has the ability to treat multi temperatures mixtures evolving with a single pressure and velocity and is particularly interesting for the computation of interface problems with compressible materials as well as wave propagation in heterogeneous mixtures the non conservative character of this model poses however computational challenges in the presence of shocks the first issue is related to the riemann problem resolution that necessitates shock jump conditions thanks to the rankine hugoniot relations proposed and validated in saurel et al r saurel o le metayer j massoni s gavrilyuk shock jump conditions for multiphase mixtures with stiff mechanical relaxation shock waves 16 3 2007 209 232 exact and approximate 2 shocks riemann solvers are derived however the riemann solver is only a part of a numerical scheme and non conservative variables pose extra difficulties for the projection or cell average of the solution it is shown that conventional godunov schemes are unable to converge to the exact solution for strong multiphase shocks this is due to the incorrect partition of the energies or entropies in the cell averaged mixture to circumvent this difficulty a specific lagrangian scheme is developed the correct partition of the energies is achieved by using an artificial heat exchange in the shock layer with the help of an asymptotic analysis this heat exchange takes a similar form as the pseudoviscosity introduced by von neumann and richtmyer j von neumann r d richtmyer a method for the numerical calculation of hydrodynamic shocks j appl phys 21 1950 232 237 the present lagrangian numerical scheme thus combines riemann solvers and artificial heat exchanges an eulerian variant is then obtained by using the relaxation projection method developed earlier by the authors for the euler equations the method is validated against exact solutions based on the multiphase shock relations as well as exact solutions of the euler equations in the context of interface problems the method is able to solve interfaces separating pure fluids or heterogeneous mixtures with very large density ratio and with very strong shocks

finite volume approximation of two phase fluid flows based on an approximate roe type riemann solver abstract we introduce an approximate roe type riemann solver for the numerical simulation of two phase fluid flows composed of liquid droplets suspended in gas we compute a roe linearization of some well conditioned approximate rankine hugoniot relations in non conservation form the computed solutions are found to be in good agreement with the exact solution in one dimension slab geometry we extend this solver to two dimensional geometries using a finite volume formulation

latent space approaches to social network analysis network models are widely used to represent relational information among interacting units in studies of social networks recent emphasis has been placed on random graph models where the nodes usually represent individual social actors and the edges represent the presence of a specified relation between actors we develop a class of models where the probability of a relation between actors depends on the positions of individuals in an unobserved social space we make inference for the social space within maximum likelihood and bayesian frameworks and propose markov chain monte carlo procedures for making inference on latent positions and the effects of observed covariates we present analyses of three standard datasets from the social networks literature and compare the method to an alternative stochastic blockmodeling approach in addition to improving on model fit for these datasets our method provides a visual and interpretable model based spatial representation of social relationships and improv

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

strong consistency of spectral clustering for stochastic block models in this paper we prove the strong consistency of several methods based on the spectral clustering techniques that are widely used to study the community detection problem in stochastic block models sbms we show that under some weak conditions on the minimal degree the number of communities and the eigenvalues of the probability block matrix the k means algorithm applied to the eigenvectors of the graph laplacian associated with its first few largest eigenvalues can classify all individuals into the true community uniformly correctly almost surely extensions to both regularized spectral clustering and degree corrected sbms are also considered we illustrate the performance of different methods on simulated networks

accurate and efficient p value calculation via gaussian approximation a novel monte carlo method abstractit is of fundamental interest in statistics to test the significance of a set of covariates for example in genome wide association studies a joint null hypothesis of no genetic effect is tested for a set of multiple genetic variants the minimum p value method higher criticism and berk jones tests are particularly effective when the covariates with nonzero effects are sparse however the correlations among covariates and the nongaussian distribution of the response pose a great challenge toward the p value calculation of the three tests in practice permutation is commonly used to obtain accurate p values but it is computationally very intensive especially when we need to conduct a large amount of hypothesis testing in this paper we propose a gaussian approximation method based on a monte carlo scheme which is computationally more efficient than permutation while still achieving similar accuracy we derive nonasymptotic approximation error bounds that could vanish in the limit even if

duality and optimization for generalized multi hop mimo amplify and forward relay networks with linear constraints we consider a generalized multi hop mimo amplify and forward af relay network with multiple sources destinations and arbitrarily number of relays we establish two dualities and the corresponding dual transformations between such a network and its dual respectively under single network linear constraint and per hop linear constraint a unified optimization framework is proposed to find a stationary point for an important class of nonconvex optimization problems of af relay networks based on a local lagrange dual method where the primal algorithm only finds a stationary point for the inner loop problem of maximizing the lagrangian w r t the primal variables the input covariance matrices are shown to satisfy a polite water filling structure at a stationary point of the inner loop problem the duality and polite water filling are exploited to design fast primal algorithms compared with the existing algorithms the proposed optimization framework with duality based primal algorithms can be used to solve more general problems with lower computation cost

duality achievable rates and sum rate capacity of gaussian mimo broadcast channels we consider a multiuser multiple input multiple output mimo gaussian broadcast channel bc where the transmitter and receivers have multiple antennas since the mimo bc is in general a nondegraded bc its capacity region remains an unsolved problem we establish a duality between what is termed the dirty paper achievable region the caire shamai see proc ieee int symp information theory washington dc june 2001 p 322 achievable region for the mimo bc and the capacity region of the mimo multiple access channel mac which is easy to compute using this duality we greatly reduce the computational complexity required for obtaining the dirty paper achievable region for the mimo bc we also show that the dirty paper achievable region achieves the sum rate capacity of the mimo bc by establishing that the maximum sum rate of this region equals an upper bound on the sum rate of the mimo bc

attribute fusion in a latent process model for time series of graphs hypothesis testing on time series of attributed graphs has applications in diverse areas e g social network analysis wherein vertices represent individual actors or organizations connectome inference wherein vertices are neurons or brain regions and text processing wherein vertices represent authors or documents we consider the problem of anomaly change point detection given the latent process model for time series of graphs with categorical attributes on the edges presented in n h lee and c e priebe a latent process model for time series of attributed random graphs statist inference stoch process vol 14 pp 231 253 2011 various attributed graph invariants are considered and their power for detection as a function of a linear fusion parameter is presented our main result is that inferential performance in mathematically tractable first order and second order approximation models does provide guidance for methodological choices applicable to the exact realistic but intractable model furthermore to the extent that the exact model is realistic we may tentatively conclude that approximation model investigations have some bearing on real data applications

stochastic blockmodels for directed graphs abstract holland and leinhardt 1981 proposed the p 1 model for the analysis of binary directed graph data in network studies such a model provides information about the attractiveness and expansiveness of the individual nodes in the network as well as the tendency of a pair of nodes to reciprocate relational ties when the nodes are a priori partitioned into subgroups based on attributes such as race and sex the density of ties from one subgroup to another can differ considerably from that relating another pair of subgroups thus creating a situation called blocking in social networks the p 1 model completely ignores this extra piece of information and is therefore unable to explain the block structure blockmodels that are simple extensions of the p 1 model are proposed specifically for such data an iterative scaling algorithm is presented for fitting the model parameters by maximum likelihood the methodology is illustrated in detail on two empirical examples

the two to infinity norm and singular subspace geometry with applications to high dimensional statistics the singular value matrix decomposition plays a ubiquitous role throughout statistics and related fields myriad applications including clustering classification and dimensionality reduction involve studying and exploiting the geometric structure of singular values and singular vectors this paper provides a novel collection of technical and theoretical tools for studying the geometry of singular subspaces using the two to infinity norm motivated by preliminary deterministic procrustes analysis we consider a general matrix perturbation setting in which we derive a new procrustean matrix decomposition together with flexible machinery developed for the two to infinity norm this allows us to conduct a refined analysis of the induced perturbation geometry with respect to the underlying singular vectors even in the presence of singular value multiplicity our analysis yields singular vector entrywise perturbation bounds for a range of popular matrix noise models each of which has a meaningful associated statistical inference task in addition we demonstrate how the two to infinity norm is the preferred norm in certain statistical settings specific applications discussed in this paper include covariance estimation singular subspace recovery and multiple graph inference both our procrustean matrix decomposition and the technical machinery developed for the two to infinity norm may be of independent interest

the capacity and optimal signaling for gaussian mimo channels under interference constraints gaussian mimo channel under the joint transmit tx and interference power constraints tpc and ipc is studied a closed form solution for the optimal tx covariance matrix in the general case is obtained using the kkt based approach up to dual variables a number of more explicit closed form solutions are given with optimal dual variables including full rank and rank 1 beamforming cases as well as the case of identical eigenvectors typical for massive mimo settings which differer significantly from the standard water filling solution a whitening filter is shown to be an important part of optimal precoding under interference constraints sufficient and necessary conditions for each constraint to be redundant are given capacity scaling with the snr is shown to be determined by a natural linear algebraic structure of sub spaces induced by channel matrices of multiple users a number of unusual properties of optimal tx covariance matrix under the joint tpc and ipc are pointed out and a bound on its rank is established an interplay between the tpc and ipc is studied including the transition from power limited to interference limited regimes as the tx power increases while closed from solutions for optimal dual variables are given in some special cases an iterative bisection algorithm iba is proposed to find optimal dual variables in the general case and its convergence is proved for some special cases numerical experiments illustrate its efficient performance bounds for the optimal dual variables are given

signal plus noise matrix models eigenvector deviations and fluctuations summary estimating eigenvectors and low dimensional subspaces is of central importance for numerous problems in statistics computer science and applied mathematics in this paper we characterize the behaviour of perturbed eigenvectors for a range of signal plus noise matrix models encountered instatistical and random matrix theoretic settings we establish both first order approximation results i e sharp deviations and second order distributional limit theory i e fluctuations the concise methodology presented in this paper synthesizes tools rooted in two core concepts namely deterministic decompositions of matrix perturbations and probabilistic matrix concentration phenomena we illustrate our theoretical results with simulation examples involving stochastic block model random graphs

estimation and prediction for stochastic blockstructures a statistical approach to a posteriori blockmodeling for digraphs and valued digraphs is proposed the probability model assumes that the vertices of the digraph are partitioned into several unobserved latent classes and that the probability distribution of the relation between two vertices depends only on the classes to which they belong a bayesian estimator based on gibbs sampling is proposed the basic model is not identified because class labels are arbitrary the resulting identifiability problems are solved by restricting inference to the posterior distributions of invariant functions of the parameters and the vertex class membership in addition models are considered where class labels are identified by prior distributions for the class membership of some of the vertices the model is illustrated by an example from the social networks literature kapferer s tailor shop

propagating phase boundaries formulation of the problem and existence via the glimm method this paper treats the hyperbolic elliptic system of two conservation laws which describes the dynamics of an elastic material having a non monotone strain stress function followingabeyaratne knowles we propose a notion of admissible weak solution for this system in the class of functions of bounded variation the formulation includes an entropy inequality a kinetic relation imposed along any subsonic phase boundary and an initiation criterion for the appearance of new phase boundaries we prove thel1 continuous dependence of the solution to the riemann problem our main result yields the existence and the stability of propagating phase boundaries the proofs are based onglimm s scheme and in particular on the techniques ofglimm andlax in order to deal with the kinetic relation we prove a result of pointwise convergence of the phase boundary

co clustering of nonsmooth graphons performance bounds are given for exploratory co clustering blockmodeling of bipartite graph data where we assume the rows and columns of the data matrix are samples from an arbitrary population this is equivalent to assuming that the data is generated from a nonsmooth graphon it is shown that co clusters found by any method can be extended to the row and column populations or equivalently that the estimated blockmodel approximates a blocked version of the generative graphon with estimation error bounded by o p n 1 2 analogous performance bounds are also given for degree corrected blockmodels and random dot product graphs with error rates depending on the dimensionality of the latent variable space

optimization via low rank approximation for community detection in networks author s le can m levina elizaveta vershynin roman abstract community detection is one of the fundamental problems of network analysis for which a number of methods have been proposed most model based or criteria based methods have to solve an optimization problem over a discrete set of labels to find communities which is computationally infeasible some fast spectral algorithms have been proposed for specific methods or models but only on a case by case basis here we propose a general approach for maximizing a function of a network adjacency matrix over discrete labels by projecting the set of labels onto a subspace approximating the leading eigenvectors of the expected adjacency matrix this projection onto a low dimensional space makes the feasible set of labels much smaller and the optimization problem much easier we prove a general result about this method and show how to apply it to several previously proposed community detection criteria establishing its consistency for label estimation in each case and demonstrating the fundamental connection between spectral properties of the network and various model based approaches to community detection simulations and applications to real world data are included to demonstrate our method performs well for multiple problems over a wide range of parameters

stochastic blockmodels with a growing number of classes we present asymptotic and finite sample results on the use of stochastic blockmodels for the analysis of network data we show that the fraction of misclassified network nodes converges in probability to zero under maximum likelihood fitting when the number of classes is allowed to grow as the root of the network size and the average network degree grows at least poly logarithmically in this size we also establish finite sample confidence bounds on maximum likelihood blockmodel parameter estimates from data comprising independent bernoulli random variates these results hold uniformly over class assignment we provide simulations verifying the conditions sufficient for our results and conclude by fitting a logit parameterization of a stochastic blockmodel with covariates to a network data example comprising self reported school friendships resulting in block estimates that reveal residual structure

network cross validation by edge sampling summary while many statistical models and methods are now available for network analysis resampling of network data remains a challenging problem cross validation is a useful general tool for model selection and parameter tuning but it is not directly applicable to networks since splitting network nodes into groups requires deleting edges and destroys some of the network structure in this paper we propose a new network resampling strategy based on splitting node pairs rather than nodes that is applicable to cross validation for a wide range of network model selection tasks we provide theoretical justification for our method in a general setting and examples of how the method can be used in specific network model selection and parameter tuning tasks numerical results on simulated networks and on a statisticians citation network show that the proposed cross validation approach works well for model selection

a consistent adjacency spectral embedding for stochastic blockmodel graphs we present a method to estimate block membership of nodes in a random graph generated by a stochastic blockmodel we use an embedding procedure motivated by the random dot product graph model a particular example of the latent position model the embedding associates each node with a vector these vectors are clustered via minimization of a square error criterion we prove that this method is consistent for assigning nodes to blocks as only a negligible number of nodes will be misassigned we prove consistency of the method for directed and undirected graphs the consistent block assignment makes possible consistent parameter estimation for a stochastic blockmodel we extend the result in the setting where the number of blocks grows slowly with the number of nodes our method is also computationally feasible even for very large graphs we compare our method with laplacian spectral clustering through analysis of simulated data and a graph derived from wikipedia documents

duality and rate optimization for multiple access and broadcast channels with amplify and forward relays in this paper we consider multihop multiple access mac and broadcast channels bc where communication takes place with the assistance of relays that amplify and forward af their received signals for a two hop parallel af relay mac assuming a sum power constraint across all relays we characterize optimal relay amplification factors and the resulting optimal rate regions we find the maximum sum rate and the maximum rate for each user in closed form and express the optimal rate pair r1 r2 that maximizes mu1r1 mu2r2 as the solution of a pair of simultaneous equations we find that the parallel af relay mac with total transmit power of the two users p1 p2 p and total relay power pr is the dual of the parallel af relay bc where the mac source nodes become the bc destination nodes the mac destination node becomes the bc source node the dual bc source transmit power is pr and the total transmit power of the af relays is p the duality means that the rate region of the af relay mac with a sum power constraint p on the transmitters is the same as that of the dual bc the duality relationship is found to be useful in characterizing the rate region of the af relay bc as the union of mac rate regions the duality is established for distributed multiple antenna af relay nodes and multiple more than 2 hops as well

a five equation reduced model for compressible two phase flow problems this paper studies an eulerian diffuse interface model for the simulation of compressible multifluid and two phase flow problems we first show how to derive this model from a seven equation two pressure two velocity model of baer nunziato type using an asymptotic analysis in the limit of zero relaxation time we then study the mathematical properties of the system the structure of the waves the expression of the riemann s invariants and the existence of a mathematical entropy we also describe two different numerical approximation schemes for this system the first one relies on a linearized riemann solver while the second uses more heavily the mathematical structure of the system and relies on a linearization of the characteristic relations finally we present some numerical experiments and comparisons with the results obtained by the two pressure two velocity model as well as some test cases and comparisons with another five equation model recently proposed for interface computations between compressible fluids

a multiphase godunov method for compressible multifluid and multiphase flows we propose a new model and a solution method for two phase compressible flows the model involves six equations obtained from conservation principles applied to each phase completed by a seventh equation for the evolution of the volume fraction this equation is necessary to close the overall system the model is valid for fluid mixtures as well as for pure fluids the system of partial differential equations is hyperbolic hyperbolicity is obtained because each phase is considered to be compressible two difficulties arise for the solution one of the equations is written in non conservative form non conservative terms exist in the momentum and energy equations we propose robust and accurate discretisation of these terms the method solves the same system at each mesh point with the same algorithm it allows the simulation of interface problems between pure fluids as well as multiphase mixtures several test cases where fluids have compressible behavior are shown as well as some other test problems where one of the phases is incompressible the method provides reliable results is able to compute strong shock waves and deals with complex equations of state

why many theories of shock waves are necessary convergence error in formally path consistent schemes we are interested in nonlinear hyperbolic systems in nonconservative form arising in fluid dynamics and for solutions containing shock waves we investigate the convergence of finite difference schemes applied to such systems according to dal maso lefloch and murat s theory a shock wave theory for a given nonconservative system requires prescribing a priori a family of paths in the phase space in the present paper we consider schemes that are formally consistent with a given family of paths and we investigate their limiting behavior as the mesh is refined we first generalize to systems a property established earlier by hou and lefloch for scalar conservation laws and we prove that nonconservative schemes generate at the level of the limiting hyperbolic system an convergence error source term which provided the total variation of the approximations remains uniformly bounded is a locally bounded measure this convergence error measure is supported on the shock trajectories and as we demonstrate here is usually small in the special case that the scheme converges in the sense of graphs a rather strong convergence property often violated in practice then this measure source term vanishes we also discuss the role of the equivalent equation associated with a difference scheme here the distinction between scalar equations and systems appears most clearly since for systems the equivalent equation of a scheme that is formally path consistent depends upon the prescribed family of paths the core of this paper is devoted to investigate numerically the approximation of several simplified or full hyperbolic models arising in fluid dynamics this leads us to the conclusion that for systems having nonconservative products associated with linearly degenerate characteristic fields the convergence error vanishes for more general models this measure is evaluated very accurately especially by plotting the shock curves associated with each scheme under consideration as we demonstrate plotting the shock curves provide a convenient approach for evaluating the range of validity of a given scheme

spectral clustering and the high dimensional stochastic blockmodel networks or graphs can easily represent a diverse set of data sources that are characterized by interacting units or actors social ne tworks representing people who communicate with each other are one example communities or clusters of highly connected actors form an essential feature in the structure of several empirical networks spectral clustering is a popular and computationally feasi ble method to discover these communities the stochastic block model holland et al 1983 is a social network model with well defined communities each node is a member of one community for a network generated from the stochastic block model we bound the number of nodes misclus tered by spectral clustering the asymptotic results in th is paper are the first clustering results that allow the number of clusters in the model to grow with the number of nodes hence the name high dimensional in order to study spectral clustering under the stochastic block model we first show that under the more general latent space model the eigenvectors of the normalized graph laplacian asymptotically converge to the eigenvectors of a population normal ized graph laplacian aside from the implication for spectral clustering this provides insight into a graph visualization technique our method of studying the eigenvectors of random matrices is original ams 2000 subject classifications primary 62h30 62h25 secondary 60b20

a class of computationally fast first order finite volume solvers pvm methods in this work we present a class of fast first order finite volume solvers called pvm polynomial viscosity matrix for balance laws or more generally for nonconservative hyperbolic systems they are defined in terms of viscosity matrices computed by a suitable polynomial evaluation of a roe matrix these methods have the advantage that they only need some information about the eigenvalues of the system to be defined and no spectral decomposition of a roe matrix is needed as a consequence they are faster than the roe method these methods can be seen as a generalization of the schemes introduced by degond et al in c r acad sci paris ser l math 328 1999 pp 479 483 for balance laws and nonconservative systems the first order path conservative methods to be designed here are intended to be used as the basis for higher order methods for multidimensional problems in this work some well known solvers such as rusanov lax friedrichs force see e f toro and s j billett ima j num

a closed form power allocation and signal alignment for a diagonalized mimo two way relay channel with linear receivers a novel channel diagonalization scheme for an amplify and forward multiple input multiple output mimo two way relay channel twrc is proposed using generalized singular value decomposition gsvd diagonalization of the mimo twrc is a sub optimal approach that achieves two main purposes reducing the computational complexity for optimizing the linear precoders at each node and reducing the detection complexity at the source nodes by separating the multiple data streams for the given diagonalized structure we first align the entries of the diagonalized channels using a permutation to maximize a lower bound of average achievable sum rate asr and a joint source relay power allocation is then performed to maximize the asr of the aligned twrc the overall problem is divided into two convex subproblems the solutions to which are provided in closed form our analysis for the proposed scheme underlines the benefits of acquiring channel state information simulation results demonstrate that the proposed gsvd based relaying scheme with the signal alignment and closed form power allocation significantly improves the asr while retaining the diagonalized channel structure in addition the proposed scheme achieves the same level of asr with much less computational complexity as compared to the iterative schemes

eigenvalue inequalities for matrix product we present a family of eigenvalue inequalities for the product of a hermitian matrix and a positive semidefinite matrix our theorem contains or extends some existing results on trace and eigenvalues

short note a comment on the computation of non conservative products we are interested in the solution of non conservative hyperbolic systems and consider in particular the so called path conservative schemes see e g 2 3 which rely on the theoretical work in 1 the example of the standard euler equations for a perfect gas is used to illuminate some computational issues and shortcomings of this approach

impact of regularization on spectral clustering the performance of spectral clustering can be considerably improved via regularization as demonstrated empirically in amini et al ann statist 41 2013 2097 2122 here we provide an attempt at quantifying this improvement through theoretical analysis under the stochastic block model sbm and its extensions previous results on spectral clustering relied on the minimum degree of the graph being sufficiently large for its good performance by examining the scenario where the regularization parameter tau is large we show that the minimum degree assumption can potentially be removed as a special case for an sbm with two blocks the results require the maximum degree to be large grow faster than log n as opposed to the minimum degree more importantly we show the usefulness of regularization in situations where not all nodes belong to well defined clusters our results rely on a bias variance like trade off that arises from understanding the concentration of the sample laplacian and the eigengap as a function of the regularization parameter as a byproduct of our bounds we propose a data driven technique dkest standing for estimated davis kahan bounds for choosing the regularization parameter this technique is shown to work well through simulations and on a real data set

a hllc scheme for ripa model we consider the one dimensional system of shallow water equations with horizontal temperature gradients the ripa system we derive a hllc scheme for ripa system which falls into the theory of path conservative approximate riemann solvers the resulting scheme is robust easy to implement well balanced positivity preserving and entropy dissipative for the case of flat or continuous bottom

discontinuous galerkin finite element method for shallow two phase flows we present a discontinuous galerkin finite element method for two depth averaged two phase flow models one of these models contains nonconservative products for which we developed a discontinuous galerkin finite element formulation in rhebergen et al 2008 j comput phys 227 1887 1922 the other model is a new depth averaged two phase flow model we introduce for shallow two phase flows that does not contain nonconservative products we will compare numerical results of both models and qualitatively validate the models against a laboratory experiment furthermore because of spurious oscillations that may occur near discontinuities a weno slope limiter is applied in conjunction with a discontinuity detector to detect regions where spurious oscillations appear

universally consistent vertex classification for latent positions graphs in this work we show that using the eigen decomposition of the adjacency matrix we can consistently estimate feature maps for latent position graphs with positive definite link function provided that the latent positions are i i d from some distribution f we then consider the exploitation task of vertex classification where the link

consistency of spectral clustering in stochastic block models we analyze the performance of spectral clustering for community extraction in stochastic block models we show that under mild conditions spectral clustering applied to the adjacency matrix of the network can consistently recover hidden communities even when the order of the maximum expected degree is as small as log n with n the number of nodes this result applies to some popular polynomial time spectral clustering algorithms and is further extended to degree corrected stochastic block models using a spherical k median spectral clustering method a key component of our analysis is a combinatorial bound on the spectrum of binary random matrices which is sharper than the conventional matrix bernstein inequality and may be of independent interest

using maximum entry wise deviation to test the goodness of fit for stochastic block models the stochastic block model is widely used for detecting community structures in network data how to test the goodness of fit of the model is one of the fundamental problems and has gained growing