a novel multigrid based preconditioner for heterogeneous helmholtz problems an iterative solution method in the form of a preconditioner for a krylov subspace method is presented for the helmholtz equation the preconditioner is based on a helmholtz type differential operator with a complex term a multigrid iteration is used for approximately inverting the preconditioner the choice of multigrid components for the corresponding preconditioning matrix with a complex diagonal is validated with fourier analysis multigrid analysis results are verified by numerical experiments high wavenumber helmholtz problems in heterogeneous media are solved indicating the performance of the preconditioner

dynamical behavior of susceptible infected recovered susceptible epidemic model on weighted networks abstract we study susceptible infected recovered susceptible epidemic model in weighted regular and random complex networks we institute a pairwise type mathematical model with a general transmission rate to evaluate the influence of the link weight distribution on the spreading process furthermore we develop a dimensionality reduction approach to derive the condition for the contagion outbreak finally we analyze the influence of the heterogeneity of weight distribution on the outbreak condition for the scenario with a linear transmission rate our theoretical analysis is in agreement with stochastic simulations showing that the heterogeneity of link weight distribution can have a significant effect on the epidemic dynamics

efficient iterative solution of the three dimensional helmholtz equation we examine preconditioners for the discrete indefinite helmholtz equation on a three dimensional box shaped domain with sommerfeld like boundary conditions the preconditioners are of two types the first is derived by discretization of a related continuous operator that differs from the original only in its boundary conditions the second is derived by a block toeplitz approximation to the descretized problem the resulting preconditioning matrices allow the use of fast transform methods and differ from the discrete helmholtz operator by an operator of low rank we present experimental results demonstrating that when these methods are combined with krylov subspace iteration convergence rates depend only mildly on both the wave number and discretization mesh size in addition the methods display high efficiencies in an implementation on an ibm sp 2 parallel computer

an iterative method for the helmholtz equation an iterative algorithm for the solution of the helmholtz equation is developed the algorithm is based on a preconditioned conjugate gradient iteration for the normal equations the preconditioning is based on an ssor sweep for the discrete laplacian numerical results are presented for a wide variety of problems of physical interest and demonstrate the effectiveness of the algorithm

sweeping preconditioner for the helmholtz equation hierarchical matrix representation the paper introduces the sweeping preconditioner which is highly efficient for iterative solutions of the variable coefficient helmholtz equation including very high frequency problems the first central idea of this novel approach is to construct an approximate factorization of the discretized helmholtz equation by sweeping the domain layer by layer starting from an absorbing layer or boundary condition given this specific order of factorization the second central idea is to represent the intermediate matrices in the hierarchical matrix framework in two dimensions both the construction and the application of the preconditioners are of linear complexity the generalized minimal residual method gmres solver with the resulting preconditioner converges in an amazingly small number of iterations which is essentially independent of the number of unknowns this approach is also extended to the three dimensional case with some success numerical results are provided in both two and three dimensions to demonstrate the efficiency of this new approach

spectral analysis of the discrete helmholtz operator preconditioned with a shifted laplacian shifted laplace preconditioners have attracted considerable attention as a technique to speed up convergence of iterative solution methods for the helmholtz equation in this paper we present a comprehensive spectral analysis of the helmholtz operator preconditioned with a shifted laplacian our analysis is valid under general conditions the propagating medium can be heterogeneous and the analysis also holds for different types of damping including a radiation condition for the boundary of the computational domain by combining the results of the spectral analysis of the preconditioned helmholtz operator with an upper bound on the gmres residual norm we are able to provide an optimal value for the shift and to explain the mesh dependency of the convergence of gmres preconditioned with a shifted laplacian we illustrate our results with a seismic test problem

eigenanalysis of some preconditioned helmholtz problems in this work we calculate the eigenvalues obtained by preconditioning the discrete helmholtz operator with sommerfeld like boundary conditions on a rectilinear domain by a related operator with boundary conditions that permit the use of fast solvers the main innovation is that the eigenvalues for two and three dimensional domains can be calculated exactly by solving a set of one dimensional eigenvalue problems this permits analysis of quite large problems for grids fine enough to resolve the solution for a given wave number preconditioning using neumann boundary conditions yields eigenvalues that are uniformly bounded located in the first quadrant and outside the unit circle in contrast dirichlet boundary conditions yield eigenvalues that approach zero as the product of wave number with the mesh size is decreased these eigenvalue properties yield the first insight into the behavior of iterative methods such as gmres applied to these preconditioned problems

sampling methods for the quasistationary regime of epidemic processes on regular and complex networks a major hurdle in the simulation of the steady state of epidemic processes is that the system will unavoidably visit an absorbing disease free state at sufficiently long times due to the finite size of the networks where epidemics evolves in the present work we compare different quasistationary qs simulation methods where the absorbing states are suitably handled and the thermodynamical limit of the original dynamics can be achieved we analyze the standard qs sqs method where the sampling is constrained to active configurations the reflecting boundary condition rbc where the dynamics returns to the pre absorbing configuration and hub reactivation hr where the most connected vertex of the network is reactivated after a visit to an absorbing state we apply the methods to the contact process cp and susceptible infected susceptible sis models on regular and scale free networks the investigated methods yield the same epidemic threshold for both models for cp that undergoes a standard collective phase transition the methods are equivalent for sis whose phase transition is ruled by the hub mutual reactivation the sqs and hr methods are able to capture localized epidemic phases while rbc is not we also apply the autocorrelation time as a tool to characterize the phase transition and observe that this analysis provides the same finite size scaling exponents for the critical relaxation time for the investigated methods finally we verify the equivalence between rbc method and a weak external field for epidemics on networks

two transparent boundary conditions for the electromagnetic scattering from two dimensional overfilled cavities we consider electromagnetic scattering from two dimensional 2d overfilled cavities embedded in an infinite ground plane the unbounded computational domain is truncated to a bounded one by using a transparent boundary condition tbc proposed on a semi ellipse for overfilled rectangular cavities with homogeneous media another tbc is introduced on the cavity apertures which produces a smaller computational domain the existence and uniqueness of the solutions of the variational formulations for the transverse magnetic and transverse electric polarizations are established in the exterior domain the 2d scattering problem is solved in the elliptic coordinate system using the mathieu functions in the interior domain the problem is solved by a finite element method numerical experiments show the efficiency and accuracy of the new boundary conditions

fast directional multilevel algorithms for oscillatory kernels this paper introduces a new directional multilevel algorithm for solving n body or n point problems with highly oscillatory kernels these systems often result from the boundary integral formulations of scattering problems and are difficult due to the oscillatory nature of the kernel and the non uniformity of the particle distribution we address the problem by first proving that the interaction between a ball of radius r and a well separated region has an approximate low rank representation as long as the well separated region belongs to a cone with a spanning angle of o 1 r and is at a distance which is at least o r 2 away from from the ball we then propose an efficient and accurate procedure which utilizes random sampling to generate such a separated low rank representation based on the resulting representations our new algorithm organizes the high frequency far field computation by a multidirectional and multiscale strategy to achieve maximum efficiency the algorithm performs well on a large group of highly oscillatory kernels our algorithm is proved to have o n log n computational complexity for any given accuracy when the points are sampled from a two dimensional surface we also provide numerical results to demonstrate these properties

free field representation for massive integrable models a new approach to massive integrable models is considered it allows one to find symmetry algebras which define the spaces of local operators and to get general integral representations for form factors in thesu 2 thirring and sine gordon models

high prevalence regimes in the pair quenched mean field theory for the susceptible infected susceptible model on networks reckoning of pairwise dynamical correlations significantly improves the accuracy of mean field theories and plays an important role in the investigation of dynamical processes in complex networks in this work we perform a nonperturbative numerical analysis of the quenched mean field theory qmf and the inclusion of dynamical correlations by means of the pair quenched mean field pqmf theory for the susceptible infected susceptible model on synthetic and real networks we show that the pqmf considerably outperforms the standard qmf theory on synthetic networks of distinct levels of heterogeneity and degree correlations providing extremely accurate predictions when the system is not too close to the epidemic threshold while the qmf theory deviates substantially from simulations for networks with a degree exponent 2 5 the scenario for real networks is more complicated still with pqmf significantly outperforming the qmf theory however despite its high accuracy for most investigated networks in a few cases pqmf deviations from simulations are not negligible we found correlations between accuracy and average shortest path while other basic network metrics seem to be uncorrelated with the theory accuracy our results show the viability of the pqmf theory to investigate the high prevalence regimes of recurrent state epidemic processes in networks a regime of high applicability

a domain decomposition solver for acoustic scattering by elastic objects in layered media a finite element solution procedure is presented for accurately computing time harmonic acoustic scattering by elastic targets buried in sediment an improved finite element discretization based on trilinear basis functions leading to fourth order phase accuracy is described for sufficiently accurate discretizations 100 million to 1 billion unknowns are required the resulting systems of linear equations are solved iteratively using the gmres method with a domain decomposition preconditioner employing a fast direct solver due to the construction of the discretization and preconditioner iterations can be reduced onto a sparse subspace associated with the interfaces numerical experiments demonstrate capability to evaluate the scattered field with hundreds of wavelengths

dynamical correlations and pairwise theory for the symbiotic contact process on networks the two species symbiotic contact process 2scp is a stochastic process in which each vertex of a graph may be vacant or host at most one individual of each species vertices with both species have a reduced death rate representing a symbiotic interaction while the dynamics evolves according to the standard single species contact process rules otherwise we investigate the role of dynamical correlations on the 2scp on homogeneous and heterogeneous networks using pairwise mean field theory this approach is compared with the ordinary one site theory and stochastic simulations we show that our approach significantly outperforms the one site theory in particular the stationary state of the 2scp model on random regular networks is very accurately reproduced by the pairwise mean field even for relatively small values of vertex degree where expressive deviations of the standard mean field are observed the pairwise approach is also able to capture the transition points accurately for heterogeneous networks and provides rich phase diagrams with transitions not predicted by the one site method our theoretical results are corroborated by extensive numerical simulations

a finite element capacitance matrix method for exterior helmholtz problems we introduce an algorithm for the efficient numerical solution of exterior boundary value problems for the helmholtz equation the problem is reformulated as an equivalent one on a bounded domain using an exact non local boundary condition on a circular artificial boundary an fft based fast helmholtz solver is then derived for a finite element discretization on an annular domain the exterior problem for domains of general shape are treated using an imbedding or capacitance matrix method the imbedding is achieved in such a way that the resulting capacitance matrix has a favorable spectral distribution leading to mesh independent convergence rates when krylov subspace methods are used to solve the capacitance matrix equation

a fast method for the solution of the helmholtz equation in this paper we consider the numerical solution of the helmholtz equation arising from the study of the wave equation in the frequency domain the approach proposed here differs from those recently considered in the literature in that it is based on a decomposition that is exact when considered analytically so the only degradation in computational performance is due to discretization and roundoff errors in particular we make use of a multiplicative decomposition of the solution of the helmholtz equation into an analytical plane wave and a multiplier which is the solution of a complex valued advection diffusion reaction equation the use of fast multigrid methods for the solution of this equation is investigated numerical results show that this is an efficient solution algorithm for a reasonable range of frequencies

a parallel multigrid based preconditioner for the 3d heterogeneous high frequency helmholtz equation we investigate the parallel performance of an iterative solver for 3d heterogeneous helmholtz problems related to applications in seismic wave propagation for large 3d problems the computation is no longer feasible on a single processor and the memory requirements increase rapidly therefore parallelization of the solver is needed we employ a complex shifted laplace preconditioner combined with the bi cgstab iterative method and use a multigrid method to approximate the inverse of the resulting preconditioning operator a 3d multigrid method with 2d semi coarsening is employed we show numerical results for large problems arising in geophysical applications

a fast algorithm for the electromagnetic scattering from a large cavity a fast algorithm is presented for solving electromagnetic scattering from a rectangular open cavity embedded in an infinite ground plane the medium inside the cavity is assumed to be vertically layered by introducing a transparent artificial boundary condition the problem in the open cavity is reduced to a bounded domain problem a simple finite difference method is then applied to solve the model helmholtz equation the fast algorithm is designed for solving the resulting discrete system in terms of the discrete fourier transform in the horizontal direction a gaussian elimination in the vertical direction and a preconditioning conjugate gradient method with a complex diagonal preconditioner for the indefinite interface system the existence and uniqueness of the finite difference solution are established for arbitrary wave numbers our numerical experiments for large numbers of mesh points up to 16 million unknowns and for large wave numbers e g between 100 and 200 wavelengths show that the algorithm is extremely efficient the cost for calculating the radar cross section which is of significant interest in practice is o m2 for an m times m mesh the proposed algorithm may be extended easily to solve discrete systems from other discretization methods of the model problem

robustness and fragility of the susceptible infected susceptible epidemic models on complex networks we analyze two alterations of the standard susceptible infected susceptible sis dynamics that preserve the central properties of spontaneous healing and infection capacity of a vertex increasing unlimitedly with its degree all models have the same epidemic thresholds in mean field theories but depending on the network properties simulations yield a dual scenario in which the epidemic thresholds of the modified sis models can be either dramatically altered or remain unchanged in comparison with the standard dynamics for uncorrelated synthetic networks having a power law degree distribution with exponent 5 2 differences are more remarkable for 3 where a finite threshold is found in the modified models in contrast with the vanishing threshold of the original one this duality is elucidated in terms of epidemic lifespan on star graphs we verify that the activation of the modified sis models is triggered in the innermost component of the network given by a k core decomposition for 3 the activation in the modified dynamics is collective involving essentially the whole network while it is triggered by hubs in the standard sis the duality also appears in the finite size scaling of the critical quantities where mean field behaviors are observed for the modified but not for the original dynamics our results feed the discussions about the most proper conceptions of epidemic models to describe real systems and the choices of the most suitable theoretical approaches to deal with these models

a composite preconditioner for the electromagnetic scattering from a large cavity in this paper we propose a composite preconditioner for the electromagnetic scattering from a large cavity the electromagnetic cavity problem is described by the helmholtz equation with a nonlocal boundary condition on the aperture of the cavity and dirichlet or neumann boundary conditions on the walls of the cavity the preconditioner proposed here combines the optimal sine transform based approximation with a layered medium model using fast fourier transforms the computational cost of every iteration is o n 2logn on an nxn uniform partition of the unit square numerical results for a model problem show that the new preconditioner is more efficient than those recently considered in the literature for the cavity with a small portion of non layered media we propose a sparse preconditioned conjugate orthogonal conjugate gradient solver combined with the new preconditioner numerical results for a model problem are reported to demonstrate the efficiency of the sparse solver

a domain decomposition method for the helmholtz equation in a multilayer domain the two dimensional helmholtz equation for problems where the physical domain consists of layers with different material properties is studied an efficient preconditioner for iterative solution of the problem is constructed the problem is discretized with fourth order accurate finite difference operators for the construction of the radiation boundary conditions a fourth order finite element method also is used the large sparse complex indefinite and ill conditioned system of equations that arises is solved with preconditioned restarted gmres a domain decomposition method is used in which the preconditioning is based on the schur complement algorithm with fast poisson type preconditioners for the subdomains the memory requirements for the preconditioner are nearly linear in the number of unknowns the arithmetic complexity for each iteration is low whereas the construction of the preconditioner is a bit more expensive electromagnetic wave propagation in a three layered waveguide is used as a model problem numerical experiments show that convergence is achieved in a few iterations compared with banded gaussian elimination which is a standard solution method the iterative method shows significant gain in both memory requirements and arithmetic complexity furthermore the relative gain grows when the problem size increases

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

metastable localization of diseases in complex networks we describe the phenomenon of localization in the epidemic susceptible infective susceptible model on highly heterogeneous networks in which strongly connected nodes hubs play the role of centers of localization we find that in this model the localized states below the epidemic threshold are metastable the longevity and scale of the metastable outbreaks do not show a sharp localization transition instead there is a smooth crossover from localized to delocalized states as we approach the epidemic threshold from below analyzing these long lasting local outbreaks for a random regular graph with a hub we show how this localization can be detected from the shape of the distribution of the number of infective nodes

compact optimal quadratic spline collocation methods for the helmholtz equation quadratic spline collocation methods are formulated for the numerical solution of the helmholtz equation in the unit square subject to non homogeneous dirichlet neumann and mixed boundary conditions and also periodic boundary conditions the methods are constructed so that they are a of optimal accuracy and b compact that is the collocation equations can be solved using a matrix decomposition algorithm involving only tridiagonal linear systems using fast fourier transforms the computational cost of such an algorithm is o n 2logn on an nxn uniform partition of the unit square the results of numerical experiments demonstrate the optimal global accuracy of the methods as well as superconvergence phenomena in particular it is shown that the methods are fourth order accurate at the nodes of the partition

phase transition of the susceptible infected susceptible dynamics on time varying configuration model networks we present a degree based theoretical framework to study the susceptible infected susceptible sis dynamics on time varying rewired configuration model networks using this framework on a given degree distribution we provide a detailed analysis of the stationary state using the rewiring rate to explore the whole range of the time variation of the structure relative to that of the sis process this analysis is suitable for the characterization of the phase transition and leads to three main contributions 1 we obtain a self consistent expression for the absorbing state threshold able to capture both collective and hub activation 2 we recover the predictions of a number of existing approaches as limiting cases of our analysis providing thereby a unifying point of view for the sis dynamics on random networks 3 we obtain bounds for the critical exponents of a number of quantities in the stationary state this allows us to reinterpret the concept of hub dominated phase transition within our framework it appears as a heterogeneous critical phenomenon observables for different degree classes have a different scaling with the infection rate this phenomenon is followed by the successive activation of the degree classes beyond the epidemic threshold

a multigrid method enhanced by krylov subspace iteration for discrete helmholtz equations standard multigrid algorithms have proven ineffective for the solution of discretizations of helmholtz equations in this work we modify the standard algorithm by adding gmres iterations at coarse levels and as an outer iteration we demonstrate the algorithm s effectiveness through theoretical analysis of a model problem and experimental results in particular we show that the combined use of gmres as a smoother and outer iteration produces an algorithm whose performance depends relatively mildly on wave number and is robust for normalized wave numbers as large as 200 for fixed wave numbers it displays grid independent convergence rates and has costs proportional to the number of unknowns

collective versus hub activation of epidemic phases on networks we consider a general criterion to discern the nature of the threshold in epidemic models on scale free sf networks comparing the epidemic lifespan of the nodes with largest degrees with the infection time between them we propose a general dual scenario in which the epidemic transition is either ruled by a hub activation process leading to a null threshold in the thermodynamic limit or given by a collective activation process corresponding to a standard phase transition with a finite threshold we validate the proposed criterion applying it to different epidemic models with waning immunity or heterogeneous infection rates in both synthetic and real sf networks in particular a waning immunity irrespective of its strength leads to collective activation with finite threshold in scale free networks with large degree exponent at odds with canonical theoretical approaches

sine transform based preconditioners for symmetric toeplitz systems abstract the optimal circulant preconditioner for a given matrix a is defined to be the minimizer of c a f over the set of all circulant matrices c here f is the frobenius norm optimal circulant preconditioners have been proved to be good preconditioners in solving toeplitz systems with the preconditioned conjugate gradient method in this paper we construct an optimal sine transform based preconditioner which is defined to be the minimizer of b a f over the set of matrices b that can be diagonalized by sine transforms we will prove that for general n by n matrices a these optimal preconditioners can be constructed in o n 2 real operations and in o n real operations if a is toeplitz we will also show that the convergence properties of these optimal sine transform preconditioners are the same as that of the optimal circulant ones when they are employed to solve toeplitz systems numerical examples are given to support our convergence analysis

numerical solution of electromagnetic scattering from a large partly covered cavity the paper focuses on the numerical study of electromagnetic scattering from two dimensional 2d large partly covered cavities which is described by the helmholtz equation with a nonlocal boundary condition on the aperture the classical five point finite difference method is applied for the discretization of the helmholtz equation and a linear approximation is used for the nonlocal boundary condition we prove the existence and uniqueness of the numerical solution when the medium in the cavity is y direction layered or the number of the mesh points on the aperture is large enough the fast algorithm proposed in bao and sun 2005 2 for open cavity models is extended to solving the partly covered cavity problem with vertically layered media a preconditioned krylov subspace method is proposed to solve the partly covered cavity problem with a general medium in which a layered medium model is used as a preconditioner of the general model numerical results for several types of partly covered cavities with different wave numbers are reported and compared with those by ilu type preconditioning algorithms our numerical experiments show that the proposed preconditioning algorithm is more efficient for partly covered cavity problems particularly with large wave numbers