a semi implicit numerical scheme for reacting flow an additive semi implicit projection scheme for the simulation of unsteady combustion in two dimensions is constructed the scheme relies on a zero mach number formulation of the compressible conservation equations with detailed chemistry the governing equations are discretized in space using second order differences and integrated in time using a semi implicit approach time integration of the evolution equations for species mass fraction thermodynamic pressure and density is performed using a semi implicit nonsplit scheme that combines a second order predictor corrector treatment of convection and diffusion terms and a stiff integrator for the reaction source terms meanwhile the momentum equations are integrated using a second order projection scheme the projection scheme is based on a predictor corrector approach that couples the evolution of the velocity and density fields in order to stabilize computations of reacting flows with large density variations a pressure poisson equation is inverted following both the predictor and corrector steps using a fast solver the advantages of the stiff integration of reaction source terms are analyzed by comparing the performance of the scheme to that of a predictor corrector scheme in which reaction and diffusion are integrated in a similar nonstiff fashion the comparison in based on both one dimensional 1d unsteady tests of a premixed methane air flame and unsteady two dimensional tests of the same flame interacting with a counterrotating vortex pair in both cases the grimech1 2 reaction mechanism with 32 species and 177 elementary reactions is used computed results show that the stiff reaction scheme enables selection of larger time steps and thus leads to substantial improvement in the performance of the computations for the present reaction mechanism and flame conditions speedup factors of about 10 are achieved in the 1d tests and about five in two dimensions possible extensions of the present scheme to further improve efficiency are also discussed

an integral equation method for epitaxial step flow growth simulations in this paper we describe an integral equation approach for simulating diffusion problems with moving interfaces the solutions are represented as moving layer potentials where the unknowns are only defined on the interfaces the resulting integro differential equation ide system is solved using spectral deferred correction sdc techniques developed for general differential algebraic equations daes and the time dependent potentials are evaluated efficiently using fast convolution algorithms the numerical solver is applied to the bcf model for the epitaxial step flow growth of crystals for which the solutions are calculated accurately instead of using quasi static approximations numerical results in 1 1 dimensions are compared with available results in the literature

space time block codes from orthogonal designs we introduce space time block coding a new paradigm for communication over rayleigh fading channels using multiple transmit antennas data is encoded using a space time block code and the encoded data is split into n streams which are simultaneously transmitted using n transmit antennas the received signal at each receive antenna is a linear superposition of the n transmitted signals perturbed by noise maximum likelihood decoding is achieved in a simple way through decoupling of the signals transmitted from different antennas rather than joint detection this uses the orthogonal structure of the space time block code and gives a maximum likelihood decoding algorithm which is based only on linear processing at the receiver space time block codes are designed to achieve the maximum diversity order for a given number of transmit and receive antennas subject to the constraint of having a simple decoding algorithm the classical mathematical framework of orthogonal designs is applied to construct space time block codes it is shown that space time block codes constructed in this way only exist for few sporadic values of n subsequently a generalization of orthogonal designs is shown to provide space time block codes for both real and complex constellations for any number of transmit antennas these codes achieve the maximum possible transmission rate for any number of transmit antennas using any arbitrary real constellation such as pam for an arbitrary complex constellation such as psk and qam space time block codes are designed that achieve 1 2 of the maximum possible transmission rate for any number of transmit antennas for the specific cases of two three and four transmit antennas space time block codes are designed that achieve respectively all 3 4 and 3 4 of maximum possible transmission rate using arbitrary complex constellations the best tradeoff between the decoding delay and the number of transmit antennas is also computed and it is shown that many of the codes presented here are optimal in this sense as well

optimal designs for space time linear precoders and decoders we introduce a new paradigm for the design of transmitter space time coding that we refer to as linear precoding it leads to simple closed form solutions for transmission over frequency selective multiple input multiple output mimo channels which are scalable with respect to the number of antennas size of the coding block and transmit average peak power the scheme operates as a block transmission system in which vectors of symbols are encoded and modulated through a linear mapping operating jointly in the space and time dimension the specific designs target minimization of the symbol mean square error and the approximate maximization of the minimum distance between symbol hypotheses under average and peak power constraints the solutions are shown to convert the mimo channel with memory into a set of parallel flat fading subchannels regardless of the design criterion while appropriate power bits loading on the subchannels is the specific signature of the different designs the proposed designs are compared in terms of various performance measures such as information rate ber and symbol mean square error

numerical methods for solving moment equations in kinetic theory of neuronal network dynamics recently developed kinetic theory and related closures for neuronal network dynamics have been demonstrated to be a powerful theoretical framework for investigating coarse grained dynamical properties of neuronal networks the moment equations arising from the kinetic theory are a system of 1 1 dimensional nonlinear partial differential equations pde on a bounded domain with nonlinear boundary conditions the pdes themselves are self consistently specified by parameters which are functions of the boundary values of the solution the moment equations can be stiff in space and time numerical methods are presented here for efficiently and accurately solving these moment equations the essential ingredients in our numerical methods include i the system is discretized in time with an implicit euler method within a spectral deferred correction framework therefore the pdes of the kinetic theory are reduced to a sequence in time of boundary value problems bvps with nonlinear boundary conditions ii a set of auxiliary parameters is introduced to recast the original bvp with nonlinear boundary conditions as bvps with linear boundary conditions with additional algebraic constraints on the auxiliary parameters iii a careful combination of two newton s iterates for the nonlinear bvp with linear boundary condition interlaced with a newton s iterate for solving the associated algebraic constraints is constructed to achieve quadratic convergence for obtaining the solutions with self consistent parameters it is shown that a simple fixed point iteration can only achieve a linear convergence for the self consistent parameters the practicability and efficiency of our numerical methods for solving the moment equations of the kinetic theory are illustrated with numerical examples it is further demonstrated that the moment equations derived from the kinetic theory of neuronal network dynamics can very well capture the coarse grained dynamical properties of integrate and fire neuronal networks

space time codes for high data rate wireless communication performance criteria in the presence of channel estimation errors mobility and multiple paths space time coding is a bandwidth and power efficient method of communication over fading channels that realizes the benefits of multiple transmit antennas specific codes have been constructed using design criteria derived for quasi static flat rayleigh or rician fading where channel state information is available at the receiver it is evident that the practicality of space time codes will be greatly enhanced if the derived design criteria remain valid in the absence of perfect channel state information it is even more desirable that the design criteria not be unduly sensitive to frequency selectivity and to the doppler spread this paper presents a theoretical study of these issues beginning with the effect of channel estimation error here it is assumed that a channel estimator extracts fade coefficients at the receiver and for constellations with constant energy it is proved that in the absence of ideal channel state information the design criteria for space time codes is still valid the analysis also demonstrates that standard channel estimation techniques can be used in conjunction with space time codes provided that the number of transmit antennas is small we also derive the maximum likelihood detection metric in the presence of channel estimation errors next the effect of multiple paths on the performance of space time codes is studied for a slowly changing rayleigh channel it is proved that the presence of multiple paths does not decrease the diversity order guaranteed by the design criteria used to construct the space time codes similar results hold for rapid fading channels with or without multiple paths the conclusion is that the diversity order promised by space time coding is achieved under a variety of mobility conditions and environmental effects

solvability of concatenated runge kutta equations for second order nonlinear pdes in this paper we consider the solvability of concatenated runge kutta crk equations which are the nonlinear algebraic equations arising from crk methods applied to some kind of second order nonlinear partial differential equations to guarantee the existence and uniqueness of the solution of crk equations temporal and spatial step sizes of crk methods should be chosen appropriately under different assumptions with the homotopy continuation method we choose different step sizes to ensure the solvability of crk equations

linear precoding for finite alphabet inputs over mimo fading channels with statistical csi this paper investigates the linear precoder design that maximizes the average mutual information of multiple input multiple output fading channels with statistical channel state information known at the transmitter it formulates the design from the standpoint of finite alphabet inputs which leads to a problem that is very important in practice but extremely difficult in theory first the average mutual information lacks closed form expression and involves prohibitive computational burden second the optimization over the precoder is nonconcave and thus easily gets stuck in local maxima to address these issues this study first derives lower and upper bounds for the average mutual information in which the computational complexity is reduced by several orders of magnitude compared to calculating the average mutual information directly it proves that maximizing the bounds is asymptotically optimal and shows that with a constant shift the lower bound actually offers a very accurate approximation to the average mutual information for various fading channels this paper further proposes utilizing the lower bound as a low complexity and accurate alternative for developing a two step algorithm to find a near global optimal precoder numerical examples demonstrate the convergence and efficacy of the proposed algorithm compared to its conventional counterparts the proposed linear precoding method provides significant performance gain over existing precoding algorithms the gain becomes more substantial when the spatial correlation of mimo channels increases

defect deferred correction method for the two domain convection dominated convection diffusion problem abstract we present a method for solving a fluid fluid interaction problem two convection dominated convection diffusion problems adjoined by an interface which is a simplified version of the atmosphere ocean coupling problem the method resolves some of the issues that can be crucial to the fluid fluid interaction problems it is a partitioned time stepping method yet it is of high order accuracy in both space and time the two step algorithm considered in this report provides second order accuracy it allows for the usage of the legacy codes which is a common requirement when resolving flows in complex geometries yet it can be applied to the problems with very small viscosity diffusion coefficients this is achieved by combining the defect correction technique for increased spatial accuracy and for resolving the issue of high convection to diffusion ratio with the deferred correction in time which allows for the usage of the computationally attractive partitioned scheme yet the time accuracy is increased beyond the usual result of partitioned methods being only first order accurate into the defect deferred correction method ddc the results are readily extendable to the higher order accuracy cases by adding more correction steps both the theoretical results and the numerical tests provided demonstrate that the computed solution is unconditionally stable and the accuracy in both space and time is improved after the correction step

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

multidimensional upwind methods for hyperbolic conservation laws abstract we present a class of second order conservative finite difference algorithms for solving numerically time dependent problems for hyperbolic conservation laws in several space variables these methods are upwind and multidimensional in that the numerical fluxes are obtained by solving the characteristic form of the full multidimensional equations at the zone edge and that all fluxes are evaluated and differenced at the same time in particular operator splitting is not used correct behavior at discontinuities is obtained by the use of solutions to the riemann problem and by limiting some of the second order terms numerical results are presented which show that the methods described here yield the same high resolution as the corresponding operator split methods

generalized linear precoder and decoder design for mimo channels using the weighted mmse criterion we address the problem of designing jointly optimum linear precoder and decoder for a mimo channel possibly with delay spread using a weighted minimum mean squared error mmse criterion subject to a transmit power constraint we show that the optimum linear precoder and decoder diagonalize the mimo channel into eigen subchannels for any set of error weights furthermore we derive the optimum linear precoder and decoder as functions of the error weights and consider specialized designs based on specific choices of error weights we show how to obtain 1 the maximum information rate design 2 qos based design we show how to achieve any set of relative snrs across the subchannels and 3 the unweighted mmse and equal error design for fixed rate systems

false neighbors and false strands a reliable minimum embedding dimension algorithm the time delay reconstruction of the state space of a system from observed scalar data requires a time lag and an integer embedding dimension we demonstrate a reliable method to estimate the minimum necessary embedding dimension that improves upon previous methods by correcting for systematic effects due to temporal oversampling autocorrelation and changing time lag the method gives a sharp and reliable indication of the proper dimension with little computational cost the method also distinguish easily between infinite dimensional colored noise including noisy periodicity and low dimensional dynamics

arbitrary order krylov deferred correction methods for differential algebraic equations in this paper a new framework for the construction of accurate and efficient numerical methods for differential algebraic equation dae initial value problems is presented the methods are based on applying spectral deferred correction techniques as preconditioners to a picard integral collocation formulation for the solution the resulting preconditioned nonlinear system is solved using newton krylov schemes such as the newton gmres method least squares based orthogonal polynomial approximations are computed using gaussian type quadratures and spectral integration is used to avoid the numerically unstable differentiation operator the resulting krylov deferred correction kdc methods are of arbitrary order of accuracy and very stable preliminary results show that these new methods are very competitive with existing dae solvers particularly when high precision is desired

a high order time splitting method based on integral deferred correction for semi lagrangian vlasov simulations semi lagrangian schemes with various splitting methods and with different reconstruction interpolation strategies have been applied to kinetic simulations for example the order of spatial accuracy of the algorithms proposed in qiu and christlieb 2010 29 is very high as high as ninth order however the temporal error is dominated by the operator splitting error which is second order for strang splitting it is therefore important to overcome such low order splitting error in order to have numerical algorithms that achieve higher orders of accuracy in both space and time in this paper we propose to use the integral deferred correction idc method to reduce the splitting error specifically the temporal order accuracy is increased by r with each correction loop in the idc framework where r 1 2r 1 2 for coupling the first order splitting and the strang splitting respectively the proposed algorithm is applied to the vlasov poisson system the guiding center model and two dimensional incompressible flow simulations in the vorticity stream function formulation we show numerically that the idc procedure can automatically increase the order of accuracy in time we also investigate numerical stability of the proposed algorithm via performing fourier analysis to a linear model problem

high order multi implicit spectral deferred correction methods for problems of reactive flow models for reacting flow are typically based on advection diffusion reaction a d r partial differential equations many practical cases correspond to situations where the relevant time scales associated with each of the three sub processes can be widely different leading to disparate time step requirements for robust and accurate time integration in particular interesting regimes in combustion correspond to systems in which diffusion and reaction are much faster processes than advection the numerical strategy introduced in this paper is a general procedure to account for this time scale disparity the proposed methods are high order multi implicit generalizations of spectral deferred correction methods misdc methods constructed for the temporal integration of a d r equations spectral deferred correction methods compute a high order approximation to the solution of a differential equation by using a simple low order numerical method to solve a series of correction equations each of which increases the order of accuracy of the approximation the key feature of misdc methods is their flexibility in handling several sub processes implicitly but independently while avoiding the splitting errors present in traditional operator splitting methods and also allowing for different time steps for each process the stability accuracy and efficiency of misdc methods are first analyzed using a linear model problem and the results are compared to semi implicit spectral deferred correction methods furthermore numerical tests on simplified reacting flows demonstrate the expected convergence rates for misdc methods of orders three four and five the gain in efficiency by independently controlling the sub process time steps is illustrated for nonlinear problems where reaction and diffusion are much stiffer than advection although the paper focuses on this specific time scales ordering the generalization to any ordering combination is straightforward

high order operator splitting methods based on an integral deferred correction framework integral deferred correction idc methods have been shown to be an efficient way to achieve arbitrary high order accuracy and possess good stability properties in this paper we construct high order operator splitting schemes using the idc procedure to solve initial value problems ivps we present analysis to show that the idc methods can correct for both the splitting and numerical errors lifting the order of accuracy by r with each correction where r is the order of accuracy of the method used to solve the correction equation we further apply this framework to solve partial differential equations pdes numerical examples in two dimensions of linear and nonlinear initial boundary value problems are presented to demonstrate the performance of the proposed idc approach

minimum euclidean distance based precoders for mimo systems using rectangular qam modulations from the feedback of the channel state information csi precoding techniques improve the performance of multiple input multiple output mimo systems by optimizing various criteria in this correspondence an efficient precoder that maximizes the minimum distance dmin of two received vectors is studied this criterion leads to a nondiagonal precoding scheme and allows achieving a full diversity order however the optimized solution for mimo systems using a high order qam modulation is rather complex and changes for different constellations therefore we propose herein a general form of minimum euclidean distance based precoders for all rectangular qam modulations it is shown that the new solution optimizes the distance dmin for small and large dispersive channels

error analysis and applications of the fourier galerkin runge kutta schemes for high order stiff pdes an integrating factor mixed with runge kutta technique is a time integration method that can be efficiently combined with spatial spectral approximations to provide a very high resolution to the smooth solutions of some linear and nonlinear partial differential equations in this paper the novel hybrid fourier galerkin runge kutta scheme with the aid of an integrating factor is proposed to solve nonlinear high order stiff pdes error analysis and properties of the scheme are provided application to the approximate solution of the nonlinear stiff korteweg de vries the 3rd order pde dispersive equation kuramoto sivashinsky the 4th order pde dissipative equation and kawahara the 5th order pde equations are presented comparisons are made between this proposed scheme and the competing method given by kassam and trefethen it is found that for kdv ks and kawahara equations the proposed method is the best

semi implicit krylov deferred correction methods for differential algebraic equations in the recently developed krylov deferred correction kdc methods for differential algebraic equation initial value problems 33 a picardtype collocation formulation is preconditioned using low order time integration schemes based on spectral deferred correction sdc and the resulting system is solved efficiently using newton krylov methods kdc methods have the advantage that methods with arbitrarily high order of accuracy can be easily constructed which have similar computational complexity as lower order methods in this paper we investigate semi implicit kdc si kdc methods in which the stiff component of the preconditioner is treated implicitly and the non stiff parts explicitly for certain types of problems such a semi implicit treatment can significantly reduce the computational cost of the preconditioner compared to fully implicit kdc fi kdc methods preliminary analysis and numerical experiments show that the convergence of newton krylov iterations in the si kdc methods is similar to that in fi kdc and hence the si kdc methods offer a reduction in overall computational cost for such problems

adaptive time stepping for vesicle suspensions we present an adaptive high order accurate time stepping numerical scheme for the flow of vesicles suspended in stokesian fluids our scheme can be summarized as an approximate implicit spectral deferred correction sdc method applying a fully implicit sdc scheme to vesicle flows is prohibitively expensive for this reason we introduce several approximations our scheme is based on a semi implicit linearized low order time stepping method our discretization is spectrally accurate in space we expect that the accuracy can be arbitrary order but our examples suffer from order reduction which limits the observed accuracy to third order we also use invariant properties of vesicle flows constant area and boundary length in two dimensions to reduce the computational cost of error estimation for adaptive time stepping we present results in two dimensions for single vesicle flows constricted geometry flows converging flows and flows in a couette apparatus we experimentally demonstrate that the proposed scheme enables automatic selection of the time step size and high order accuracy

explicit time scale splitting algorithm for stiff problems auto ignition of gaseous mixtures behind a steady shock a new explicit algorithm based on the computational singular perturbation csp method is presented this algorithm is specifically designed to solve stiff problems and its performance increases with stiffness the key concept in its structure is the splitting of the fast from the slow time scales in the problem realized by embedding csp concepts into an explicit scheme in simple terms the algorithm marches in time with only the terms producing the slow time scales while the contribution of the terms producing the fast time scales is taken into account at the end of each integration step as a correction the new algorithm is designed for the integration of stiff systems of pdes by means of explicit schemes for simplicity in the presentation and discussion of the different features of the new algorithm a simple test case is considered involving the auto ignition of a methane air mixture behind a normal shock wave which is described by a system of odes the performance of the new algorithm accuracy and computational efficiency is then compared with the well known lsode package its merits when used for the solution of systems of pdes are discussed although when dealing with a stiff system of odes the new algorithm is shown to provide equal accuracy with that delivered by lsode at the cost of higher execution time the results indicate that its performance could be superior when facing a stiff system of pdes

a unified approach for calculating error rates of linearly modulated signals over generalized fading channels we present a unified analytical framework to determine the exact average symbol error rate ser of linearly modulated signals over generalized fading channels the results are applicable to systems employing coherent demodulation with maximal ratio combining multichannel reception the analyses assume independent fading paths which are not necessarily identically distributed in all cases the proposed approach leads to an expression of the average ser involving a single finite range integral which can be easily computed numerically in addition as special cases ser expressions for single channel reception are obtained these expressions reduce to well known solutions give alternative often simpler expressions for previous results or provide new formulas that are either closed form expressions or simple to compute numerically

gaussian multiaccess channels with isi capacity region and multiuser water filling the capacity region of a two user gaussian multiaccess channel with intersymbol interference isi in which the inputs pass through respective linear systems and are superimposed before being corrupted by an additive gaussian noise process is discussed a geometrical method for obtaining the optimal input power spectral densities and the capacity region is presented this method can be viewed as a nontrivial generalization of the single user water filling argument it is shown that as in the traditional memoryless multiaccess channel frequency division multiaccess fdma with optimally selected frequency bands for each user achieves the total capacity of the multiuser gaussian multiaccess channel with isi however the capacity region of the two user channel with memory is in general not a pentagon unless the channel transfer functions for both users are identical

a high order boris integrator this work introduces the high order boris sdc method for integrating the equations of motion for electrically charged particles in electric and magnetic fields boris sdc relies on a combination of the boris integrator with spectral deferred corrections sdc sdc can be considered as preconditioned picard iteration to compute the stages of a collocation method in this interpretation inverting the preconditioner corresponds to a sweep with a low order method in boris sdc the boris method a second order lorentz force integrator based on velocity verlet is used as a sweeper preconditioner the presented method provides a generic way to extend the classical boris integrator which is widely used in essentially all particle based plasma physics simulations involving magnetic fields to a high order method stability convergence order and conservation properties of the method are demonstrated for different simulation setups boris sdc reproduces the expected high order of convergence for a single particle and for the center of mass of a particle cloud in a penning trap and shows good long term energy stability introduces boris sdc an arbitrary order extension of the boris integrator boris sdc is based on merging the boris method with spectral deferred corrections sdc for a 2nd order problem using interpretation as preconditioned iteration analyzes properties of high order boris sdc for particles in a penning trap

locally embedded presages of global network bursts spontaneous synchronous bursting of neural population is a widely observed phenomenon in nervous networks which is considered important for functions and dysfunctions of the brain however how the global synchrony across a large number of neurons emerges from an initially nonbursting network state is not fully understood in this study we develop a state space reconstruction method combined with high resolution recordings of cultured neurons this method extracts deterministic signatures of upcoming global bursts in local dynamics of individual neurons during nonbursting periods we find that local information within a single cell time series can compare with or even outperform the global mean field activity for predicting future global bursts moreover the intercell variability in the burst predictability is found to reflect the network structure realized in the nonbursting periods these findings suggest that deterministic local dynamics can predict seemingly stochastic global events in self organized networks implying the potential applications of the present methodology to detecting locally concentrated early warnings of spontaneous seizure occurrence in the brain

the optimum combination of block codes and receivers for arbitrary channels a determination is made of the combinations of transmitted block codewords and matching receiver filters that maximize the signal to noise ratio at the detector with arbitrary channel impulse response and arbitrary noise spectrum this permits representing the channel as a set of parallel uncoupled subchannels with uncorrelated noise the representation is used to calculate the capacity of an arbitrary block coded time discrete channel it is shown that substantial coding gains compared to symbol by symbol processing can be achieved the block code used can be concatenated with trellis coded modulation so that the two coding gains are additive

spatio temporal coding for wireless communication multipath signal propagation has long been viewed as an impairment to reliable communication in wireless channels this paper shows that the presence of multipath greatly improves achievable data rate if the appropriate communication structure is employed a compact model is developed for the multiple input multiple output mimo dispersive spatially selective wireless communication channel the multivariate information capacity is analyzed for high signal to noise ratio snr conditions the mimo channel can exhibit a capacity slope in bits per decibel of power increase that is proportional to the minimum of the number multipath components the number of input antennas or the number of output antennas this desirable result is contrasted with the lower capacity slope of the well studied case with multiple antennas at only one side of the radio link a spatio temporal vector coding stvc communication structure is suggested as a means for achieving mimo channel capacity the complexity of stvc motivates a more practical reduced complexity discrete matrix multitone dmmt space frequency coding approach both of these structures are shown to be asymptotically optimum an adaptive lattice trellis coding technique is suggested as a method for coding across the space and frequency dimensions that exist in the dmmt channel experimental examples that support the theoretical results are presented

krylov implicit integration factor methods for spatial discretization on high dimensional unstructured meshes application to discontinuous galerkin methods abstract integration factor methods are a class of exactly linear part time discretization methods in q nie y t zhang r zhao efficient semi implicit schemes for stiff systems journal of computational physics 214 2006 521 537 a class of efficient implicit integration factor iif methods were developed for solving systems with both stiff linear and nonlinear terms arising from spatial discretization of time dependent partial differential equations pdes with linear high order terms and stiff lower order nonlinear terms the tremendous challenge in applying iif temporal discretization for pdes on high spatial dimensions is how to evaluate the matrix exponential operator efficiently for spatial discretization on unstructured meshes to solve pdes on complex geometrical domains how to efficiently apply the iif temporal discretization was open in this paper we solve this problem by applying the krylov subspace approximations to the matrix exponential operator then we apply this novel time discretization technique to discontinuous galerkin dg methods on unstructured meshes for solving reaction diffusion equations numerical examples are shown to demonstrate the accuracy efficiency and robustness of the method in resolving the stiffness of the dg spatial operator for reaction diffusion pdes application of the method to a mathematical model in pattern formation during zebrafish embryo development shall be shown

state space reconstruction in the presence of noise takens theorem demonstrates that in the absence of noise a multidimensional state space can be reconstructed from a scalar time series this theorem gives little guidance however about practical considerations for reconstructing a good state space we extend takens treatment applying statistical methods to incorporate the effects of observational noise and estimation error we define the distortion matrix which is proportional to the conditional covariance of a state given a series of noisy measurements and the noise amplification which is proportional to root mean square time series prediction errors with an ideal model we derive explicit formulae for these quantities and we prove that in the low noise limit minimizing the distortion is equivalent to minimizing the noise amplification we identify several different scaling regimes for distortion and noise amplification and derive asymptotic scaling laws when the dimension and lyapunov exponents are sufficiently large these scaling laws show that no matter how the state space is reconstructed there is an explosion in the noise amplification from a practical point of view determinism is lost and the time series is effectively a random process in the low noise large data limit we show that the technique of local singular value decomposition is an optimal coordinate transformation in the sense that it achieves the minimum distortion in a state space of the lowest possible dimension however in numerical experiments we find that estimation error complicates this issue for local approximation methods we analyze the effect of reconstruction on estimation error derive a scaling law and suggest an algorithm for reducing estimation errors

accelerating the convergence of spectral deferred correction methods in the recent paper by dutt greengard and rokhlin a variant of deferred or defect correction methods is presented which couples gaussian quadrature with the picard integral equation formulation of the initial value ordinary differential equation the resulting spectral deferred correction sdc methods have been shown to possess favorable accuracy and stability properties even for versions with very high order of accuracy in this paper we show that for linear problems the iterations in the sdc algorithm are equivalent to constructing a preconditioned neumann series expansion for the solution of the standard collocation discretization of the ode this observation is used to accelerate the convergence of sdc using the gmres krylov subspace method for nonlinear problems the gmres acceleration is coupled with a linear implicit approach stability and accuracy analyses show the accelerated scheme provides an improvement in the accuracy efficiency and stability of the original sdc approach furthermore preliminary numerical experiments show that accelerating the convergence of sdc methods can effectively eliminate the order reduction previously observed for stiff ode systems

an error embedded method based on generalized chebyshev polynomials in this paper we develop an error embedded method based on generalized chebyshev polynomials for solving stiff initial value problems the solution and the error at each integration step are calculated by generalized chebyshev polynomials of two consecutive degrees having overlapping zeros which enables us to minimize overall computational costs further the errors at each integration step are embedded in the algorithm itself in terms of concrete convergence and stability analysis the constructed algorithm turns out to have the 6th order convergence and an almost l stability we assess the proposed method with several numerical results showing that it uses larger time step sizes and is numerically more efficient

transmit equal gain precoding in rayleigh fading channels precoding with limited feedback information can achieve satisfactory performance while the amount of feedback information is kept small in this paper we analyze the theoretical performance of equal gain precoder and find that its performance is at most 1 049 db worse than the optimal precoder no matter how the number of transmit antennas increases moreover we analyze the performance degradation of the equal gain precoder due to scalar quantization theoretically the result shows that 2 3 bits per transmit antenna excluding the first antenna can achieve 0 5 0 25 db performance gap compared with the same scheme without quantization furthermore we found that the equal gain precoder in general can achieve comparable performance with the grassmannian precoder in the same moderate feedback bits simulation results are provided to corroborate the theoretical results

optimum space time processors with symbol rate sampling are bandlimited this correspondence considers the problem of joint transmitter and receiver design for multiple input multiple output mimo channels under the mean square error mse criterion when the system is not necessarily bandlimited it is proven that the optimum space time filters are strictly bandlimited to a frequency set the aggregated bandwidth of this set is at most the number of the substreams times symbol rate the optimal frequency sets corresponding to different substreams are disjoint if two maximum channel eigenmodes are not equal for the same nyquist translate

performance analysis of mmse based amplify and forward spatial multiplexing mimo relaying systems in this paper we propose a general framework to quantify the average error probability of the minimum mean squared error based precoding schemes in amplify and forward relay networks where all nodes are equipped with multiple antennas especially we investigate spatial multiplexing schemes which transmit multiple data streams simultaneously due to difficulty in finding an exact expression of the average error rate we exploit the high signal to noise ratio snr based approach which allows a simple and accurate characterization of the performance then we derive new closed form expressions for bit error rate performance of both the optimal source relay joint precoding schemes and the optimal relay only precoding schemes in terms of a coding gain as well as a diversity gain taking a different pathloss in each hop into consideration we evaluate the performance in a generalized environment through our analysis we discuss several interesting observations and provide a helpful guideline for designing mmse based relaying systems monte carlo simulations show that our analytical work accurately predicts numerical results

numerical analysis of a second order algorithm for a non stationary navier stokes darcy model abstract in this paper we construct a second order algorithm based on the semi implicit spectral deferred correction method for the non stationary coupled navier stokes darcy equations with the beavers joseph saffman s interface condition we present a complete theoretical analysis to prove that this algorithm is stable and possesses second order accuracy in time numerical examples confirm the convergence rate and the effectiveness of our algorithm

reduced feedback mimo ofdm precoding and antenna selection transmitter precoding for multiple input multiple output orthogonal frequency division multiplexing mimo ofdm is an effective way of leveraging the diversity gains afforded by a multiple transmit multiple receive antenna system in a frequency selective environment in the limited feedback scenario optimal precoder representation for narrowband mimo systems using moderately sized codebooks designed on the grassmann manifold has been shown to perform remarkably well in mimo ofdm systems precoder matrices have to be designed for all subcarriers and the amount of feedback can get prohibitively large this is especially true for next generation wireless local area networks and wireless metropolitan area networks which have a large number of subcarriers in this paper we present techniques to reduce this feedback requirement and the performance of these algorithms is numerically shown to provide improvement over existing schemes

an analysis of operator splitting techniques in the stiff case operator splitting methods are commonly used in many applications we focus here on the case where the evolution equations to be simulated are stiff we will more particularly consider the case of two operators a stiff one and a nonstiff one this occurs in numerous application fields e g combustion air pollution and reactive flows the classical analysis of the splitting error may then fail since the chosen splitting timestep t is in practice much larger than the fastest time scales the asymptotic expansion t 0 is therefore no longer valid we show here that singular perturbation theory provides an interesting framework for the study of splitting error some new results concerning the order of local errors are derived the main result deals with the choice of the sequential order for the operators the stiff operator must always be last in the splitting scheme

analysis of operator splitting for advection diffusion reaction problems from air pollution modelling abstract operator or time splitting is often used in the numerical solution of initial boundary value problems for differential equations it is for example standard practice in computational air pollution modelling where we encounter systems of three dimensional time dependent partial differential equations of the advection diffusion reaction type for such systems little attention has been devoted to the analysis of splitting and to the question why splitting can work so well from the theoretical point of view the success of splitting is primarily determined by the splitting error this paper presents an analysis of operator splitting aimed at providing insight into the splitting error using the lie operator formalism a general expression is derived for a three term strang splitting in the pure initial value case for a class of advection diffusion reaction problems the splitting error is analyzed in greater detail a special case is discussed in which the splitting error can be reduced also some attention is paid to the use of operator splitting in initial boundary value problems

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

l2 stability analysis of novel etd scheme for kuramoto sivashinsky equations the aim of this paper is to study the stability analysis of novel etd scheme proposed by the authors based on spectral methods the exponential time differencing and taylor expansion stability issue of the proposed numerical scheme is related to an analysis of the stability of the corresponding ode system for time marching approach it is proved that the novel scheme is l 2 stable in solving the kuramoto sivashinsky model problems the truncation error and the stability region for the novel scheme are provided comparisons with available literature are made

modeling water transport across elastic boundaries using an explicit jump method a mathematical model is presented to simulate water and solute transport in a highly viscous fluid with a water permeable elastic immersed membrane in this model fluid motion is described by stokes flow whereas water fluxes across the membrane are driven by transmural pressure and solute concentration differences the elastic forces arising from the membrane being distorted from its relaxed configuration and the transmembrane water fluxes introduce into model solutions discontinuities across the membrane such discontinuities are faithfully captured using a second order explicit jump method a mayo siam j numer anal 21 1984 pp 285 299 in which jumps in the solution and its derivatives are incorporated into a finite difference scheme numerical results suggest that the method exhibits desirable volume accuracy and mass conservation

accuracy of operator splitting for advection dispersion reaction problems an operator splitting approach is often used for the numerical solution of advection dispersion reaction problems operationally this approach advances the solution over a single time step in two stages one involving the solution of the nonreactive advection dispersion equation and the other the solution of the reaction equations the first stage is usually solved with a finite difference finite element or related technique while the second stage is normally solved with an ordinary differential equation integrator the only generally published guidelines on numerical accuracy suggest that the discretization errors associated with each stage must be small in order to achieve high accuracy of the overall solution however in this note we demonstrate that there is an inherent mass balance error present in the operator splitting algorithm for problems involving continuous mass influx boundary conditions the mass balance error does not exist for instantaneous mass input problems these conclusions are based upon analysis of a simple first order decay problem for which each stage of the calculation can be performed analytically i e without discretization error for this linear decay problem we find that the product of the first order decay coefficient k times t must be less than approximately 0 1 in order for the mass balance error to be less than 5 we also present a variant of the normal operator splitting algorithm in which the order of solving the advection dispersion and reaction operators is reversed at each time step this modification reduces the mass balance error by more than a factor of 10 for a wide range of k t values

vector coding for partial response channels a linear technique for combining equalization and coset codes on partial response channels with additive white gaussian noise is developed the technique vector coding uses a set of transmit filters or vectors to partition the channel into an independent set of parallel intersymbol interference isi free channels for any given finite or infinite block length the optimal transmit vectors for such channel partitioning are shown to be the eigenvectors of the channel covariance matrix for the specified block length and the gains of the individual channels are the eigenvalues an optimal bit allocation and energy distribution are derived for the set of parallel channels under an accurate extension of the continuous approximation for power in optimal multidimensional signal sets for constellations with unequal signal spacing in different dimensions examples are presented that demonstrate performance advantages with respect to zero forcing decision feedback methods that use the same coset code on the same partial response channel only resampling the channel at an optimal rate and assuming no errors in the feedback path will bring the performance of the decision feedback methods up to the level of the vector coded system

on the removal of boundary errors caused by runge kutta integration of non linear partial differential equations the temporal integration of hyperbolic partial differential equations pdes has been shown to lead sometimes to the deterioration of accuracy of the solution because of boundary conditions a procedure for removal of this error in the linear case has been established previously in this paper we consider hyperbolic pdes linear and nonlinear whose boundary treatment is accomplished via the simultaneous approximation term sat procedure a methodology is presented for recovery of the full order of accuracy and has been applied to the case of a fourth order explicit finite difference scheme

the theoretical accuracy of runge kutta time discretizations for the initial boundary value problem a study of the boundary error the conventional method of imposing time dependent boundary conditions for runge kutta rk time advancement reduces the formal accuracy of the space time method to first order locally and second order globally independently of the spatial operator this counter intuitive result is analyzed in this paper two methods of eliminating this problem are proposed for the linear constant coefficient case 1 impose the exact boundary condition only at the end of the complete rk cycle 2 impose consistent intermediate boundary conditions derived from the physical boundary condition and its derivatives the first method while retaining the rk accuracy in all cases results in a scheme with much reduced cfl condition rendering the rk scheme less attractive the second method retains the same allowable time step as the periodic problem however it is a general remedy only for the linear case for non linear hyperbolic equations the second method is effective only for for rk schemes of third order accuracy or less numerical studies are presented to verify the efficacy of each approach

filterbank transceivers optimizing information rate in block transmissions over dispersive channels optimal finite impulse response fir transmit and receive filterbanks are derived for block based data transmissions over frequency selective additive gaussian noise agn channels by maximizing mutual information subject to a fixed transmit power constraint both fir and pole zero channels are considered the inherent flexibility of the proposed transceivers is exploited to derive as special cases zero forcing zf and minimum mean square error receive filterbanks the transmit filterbank converts transmission over a frequency selective fading channel affected by additive colored noise into a set of independent flat fading subchannels with uncorrelated noise samples two loading algorithms are also developed to distribute transmit power and number of bits across the usable subchannels while adhering to an upper bound on the bit error rate ber reduction of the signal to noise ratio snr margin required to satisfy the prescribed ber is achieved by coding each subchannel s bit stream the potential of the proposed transceivers is illustrated and compared to discrete multitone dmt with simulated examples

integral deferred correction methods constructed with high order runge kutta integrators spectral deferred correction sdc methods for solving ordinary differential equations odes were introduced by dutt greengard and rokhlin 2000 it was shown in that paper that sdc methods can achieve arbitrary high order accuracy and possess nice stability properties their sdc methods are constructed with low order integrators such as forward euler or backward euler and are able to handle stiff and non stiff terms in the odes in this paper we use high order runge kutta rk integrators to construct a family of related methods which we refer to as integral deferred correction idc methods the distribution of quadrature nodes is assumed to be uniform and the corresponding local error analysis is given the smoothness of the error vector associated with an idc method measured by the discrete sobolev norm is a crucial tool in our analysis the expected order of accuracy is demonstrated through several numerical examples superior numerical stability and accuracy regions are observed when high order rk integrators are used to construct idc methods

on the use of higher order projection methods for incompressible turbulent flow an important issue in the development of higher order methods for incompressible flow is how they perform when the flow is turbulent a useful diagnostic of a method for turbulent flow is the minimum resolution that is required to adequately resolve the turbulent energy cascade at a given reynolds number in this paper we present careful numerical experiments to assess the utility of higher order numerical methods based on this metric we first introduce a numerical method for the incompressible navier stokes equations based on fourth order discretizations in both space and time the method is based on an auxiliary variable formulation and combines fourth order finite volume differencing with a semi implicit spectral deferred correction temporal integration scheme we also introduce for comparison purposes versions based on second order spatial and or temporal discretizations we demonstrate that for smooth problems each of the methods exhibits the expected order of convergence in time and space we

a multirate time integrator for regularized stokeslets the method of regularized stokeslets is a numerical approach to approximating solutions of fluid structure interaction problems in the stokes regime regularized stokeslets are fundamental solutions to the stokes equations with a regularized point force term that are used to represent forces generated by a rigid or elastic object interacting with the fluid due to the linearity of the stokes equations the velocity at any point in the fluid can be computed by summing the contributions of regularized stokeslets and the time evolution of positions can be computed using standard methods for ordinary differential equations rigid or elastic objects in the flow are usually treated as immersed boundaries represented by a collection of regularized stokeslets coupled together by virtual springs which determine the forces exerted by the boundary in the fluid for problems with boundaries modeled by springs with large spring constants the resulting ordinary differential equations become stiff and hence the time step for explicit time integration methods is severely constrained unfortunately the use of standard implicit time integration methods for the method of regularized stokeslets requires the solution of dense nonlinear systems of equations for many relevant problems here an alternate strategy using an explicit multirate time integration scheme based on spectral deferred corrections is incorporated that in many cases can significantly decrease the computational cost of the method the multirate methods are higher order methods that treat different portions of the ode explicitly with different time steps depending on the stiffness of each component numerical examples on two nontrivial three dimensional problems demonstrate the increased efficiency of the multi explicit approach with no significant increase in numerical error

conservative multi implicit spectral deferred correction methods for reacting gas dynamics in most models of reacting gas dynamics the characteristic time scales of chemical reactions are much shorter than the hydrodynamic and diffusive time scales rendering the reaction part of the model equations stiff moreover non linear forcings may introduce into the solutions sharp gradients or shocks the robust behavior and correct propagation of which require the use of specialized spatial discretization procedures this study presents high order conservative methods for the temporal integration of model equations of reacting flows by means of a method of lines discretization on the flux difference form of the equations these methods compute approximations to the cell averaged or finite volume solution the temporal discretization is based on a multi implicit generalization of spectral deferred correction methods the advection term is integrated explicitly and the diffusion and reaction terms are treated implicitly but independently with the splitting errors reduced via the spectral deferred correction procedure to reduce computational cost different time steps may be used to integrate processes with widely differing time scales numerical results show that the conservative nature of the methods allows a robust representation of discontinuities and sharp gradients the results also demonstrate the expected convergence rates for the methods of orders three four and five for smooth problems