orderings for conjugate gradient preconditionings many preconditioners e g ssor ilu for the conjugate gradient method require the solution of sparse triangular systems of equations for elliptic boundary value problems one approach to obtaining additional parallelism in the solution of these systems is the use of red black or multicolor orderings there has been increasing evidence however that these orderings degrade the rate of convergence compared with the natural ordering an alternative is the diagonal ordering which maintains the rate of convergence of the natural ordering but has less parallelism than multicolor orderings this paper reviews these as well as other orderings and then gives some results that help to explain why the red black ordering gives an inferior rate of convergence

experimental study of ilu preconditioners for indefinite matrices incomplete lu factorization preconditioners have been surprisingly successful for many cases of general nonsymmetric and indefinite matrices however their failure rate is still too high for them to be useful as black box library software for general matrices besides fatal breakdowns due to zero pivots the major causes of failure are inaccuracy and instability of the triangular solves when there are small pivots both these problems can occur but these problems can also occur without small pivots through examples from actual problems this paper shows how these problems evince themselves how these problems can be detected and how these problems can sometimes be circumvented through pivoting reordering scaling perturbing diagonal elements and preserving symmetric structure the goal of this paper is to gain a better practical understanding of ilu preconditioners and help improve their reliability

balanced incomplete factorization in this paper we present a new incomplete factorization of a square matrix into triangular factors in which we get standard lu or ldl t factors direct factors and their inverses inverse factors at the same time algorithmically we derive this method from the approach based on the sherman morrison formula r bru j cerdan j marin and j mas siam j sci comput 25 2003 pp 701 715 in contrast to the robust incomplete decomposition rif algorithm m benzi and m t ma numer linear algebra appl 10 2003 pp 385 400 the direct and inverse factors here directly influence each other throughout the computation consequently the algorithm to compute the approximate factors may mutually balance dropping in the factors and control their conditioning in this way for the symmetric positive definite case we derive the theory and present an algorithm for computing the incomplete ldl t factorization and we discuss experimental results we call this new approximate ldl t factorization the balanced incomplete factorization bif our experimental results confirm that this factorization is very robust and may be useful in solving difficult ill conditioned problems by preconditioned iterative methods moreover the internal coupling of the computation of direct and inverse factors results in much shorter setup times times to compute approximate decomposition than rif a method of a similar and very high level of robustness we also derive and present the theory for the general nonsymmetric case but do not discuss its implementation

bilum block versions of multielimination and multilevel ilu preconditioner for general sparse linear systems we introduce block versions of the multielimination incomplete lu ilum factorization preconditioning technique for solving general sparse unstructured linear systems these preconditioners have a multilevel structure and for certain types of problems may exhibit properties that are typically enjoyed by multigrid methods several heuristic strategies for forming blocks of independent sets are introduced and their relative merits are discussed the advantages of block ilum over point ilum include increased robustness and efficiency we compare several versions of the block ilum point ilum and the dual threshold based ilut preconditioners in particular tests with some convection diffusion problems show that it may be possible to obtain convergence that is nearly independent of the reynolds number as well as of the grid size

on the stability of relaxed incomplete lu factorizations when solving large linear systems of equations arising from the discretization of elliptic boundary value problems a combination of iterative methods and preconditioners based on incomplete lu factorizations is frequently used given a model problem with variable coefficients we investigate a class of incomplete lu factorizations depending on a relaxation parameter we show that the associated preconditioner and the factorization itself both are numerically stable the theoretical results are complemented by numerical experiments

group classifications optimal systems and exact solutions to the generalized thomas equations abstract in this paper the complete group classifications are performed on the types of thomas equations tes which arise in the study of chemical exchange progress etc all of the vector fields of the equations are presented then the optimal system of the general thomas equation is given and all of the symmetry reductions and exact solutions generated from the optimal system are investigated furthermore the exact analytic solutions to the thomas equations are obtained by the generalized power series method

symmetry group methods for fundamental solutions abstract this paper uses lie symmetry group methods to study pdes of the form u t xu xx f x u x we show that when the drift function f is a solution of a family of ricatti equations then symmetry techniques can be used to find a fundamental solution

time variant channel estimation using discrete prolate spheroidal sequences we propose and analyze a low complexity channel estimator for a multiuser multicarrier code division multiple access mc cdma downlink in a time variant frequency selective channel mc cdma is based on orthogonal frequency division multiplexing ofdm the time variant channel is estimated individually for every flat fading subcarrier assuming small intercarrier interference the temporal variation of every subcarrier over the duration of a data block is upper bounded by the doppler bandwidth determined by the maximum velocity of the users slepian showed that time limited snapshots of bandlimited sequences span a low dimensional subspace this subspace is also spanned by discrete prolate spheroidal dps sequences we expand the time variant subcarrier coefficients in terms of orthogonal dps sequences we call slepian basis expansion this enables a time variant channel description that avoids the frequency leakage effect of the fourier basis expansion the square bias of the slepian basis expansion per subcarrier is three magnitudes smaller than the square bias of the fourier basis expansion we show simulation results for a fully loaded mc cdma downlink with classic linear minimum mean square error multiuser detection the users are moving with 19 4 m s using the slepian basis expansion channel estimator and a pilot ratio of only 2 we achieve a bit error rate performance as with perfect channel knowledge

weighted matrix ordering and parallel banded preconditioners for iterative linear system solvers the emergence of multicore architectures and highly scalable platforms motivates the development of novel algorithms and techniques that emphasize concurrency and are tolerant of deep memory hierarchies as opposed to minimizing raw flop counts while direct solvers are reliable they are often slow and memory intensive for large problems iterative solvers on the other hand are more efficient but in the absence of robust preconditioners lack reliability while preconditioners based on incomplete factorizations whenever they exist are effective for many problems their parallel scalability is generally limited in this paper we advocate the use of banded preconditioners instead and introduce a reordering strategy that enables their extraction in contrast to traditional bandwidth reduction techniques our reordering strategy takes into account the magnitude of the matrix entries bringing the heaviest elements closer to the diagonal thus enabling the use of banded preconditioners when used with effective banded solvers in our case the spike solver we show that banded preconditioners i are more robust compared to the broad class of incomplete factorization based preconditioners ii deliver higher processor performance resulting in faster time to solution and iii scale to larger parallel configurations we demonstrate these results experimentally on a large class of problems selected from diverse application domains

the exact analytic solutions of a nonlinear differential iterative equation abstract this paper is concerned with a second order nonlinear iterated differential equation of the form c 0 x z c 1 x z c 2 x z x p z b x z h z by constructing a convergent power series solution of an auxiliary equation analytic solutions of the original equation are obtained we discuss not only the general case 1 but also the critical case 1 especially when is a root of unity furthermore the exact and explicit analytic solution of the original equation is investigated for the first time such equations are important in both applications and the theory of iterations

exact periodic wave solutions for the hkdv equation abstract in this paper by using the hirota bilinear method and the jacobian theta functions for the higher order kdv equation the existence of periodic wave solutions with one and two period are obtained the asymptotic properties of the periodic wave solutions are analyzed in detail it is shown that the well known soliton solutions can be reduced from the periodic wave solutions

iterative map equalization and decoding in wireless mobile coded ofdm orthogonal frequency division multiplexing ofdm system suffers extra performance degradation in fast fading channels due to intercarrier interference ici combining frequency domain equalization and bit interleaved coded modulation bicm the iterative receiver is able to harvest both temporal and frequency diversity realizing that ici channels are intrinsically isi channels this paper proposes a soft in soft out siso maximum a posteriori map equalizer by extending ungerboeck s maximum likelihood sequence estimator mlse formulation to ici channels the siso map equalizer employs bcjr algorithm and computes the bit log likelihood ratios llr for the entire received sequence by efficiently constructing a trellis that takes into account of the ici channel structure a reduced state rs formulation of the siso map equalizer which provides good performance complexity tradeoff is also described utilizing the fact that ici energy is clustered in adjacent subcarriers frequency domain equalization is made localized this paper further proposes two computational efficient linear minimum mean square error lmmse based equalization methods recursive q tap sic lmmse equalizer and recursive sliding window sw sic lmmse equalizer respectively simulations results demonstrate that the iterative siso rs map equalizer achieves the performance of no ici with normalized doppler frequency fdts up to 20 46 in realistic mobile wimax environment

regular incomplete factorizations of real positive definite matrices abstract we study the classes of matrices which admit a regular incomplete factorization with respect to any graph set we extend the construction to the class of real positive definite matrices the convergence of the basic iteration associated with the splittings we have is discussed

tight pep lower bound for constellation rotated vector ofdm under carrier frequency offset and fast fading in this paper we study pairwise error probability pep of a system employing constellation rotated vector orthogonal frequency division multiplexing crv ofdm which is an enhancement of vector ofdm v ofdm with enhanced diversity over a frequency selective channel with both carrier frequency offset cfo and doppler frequency spread first we extend the work of rugini and banelli which is known to give a good bit error rate ber approximation for the standard ofdm system only with cfo to pep evaluation for crv ofdm and discuss its limitations then we discuss the conditionally gaussian characteristi2cs of the doppler induced interference and propose a conditionally gaussian pep approximation confirming a good match of the conditionally gaussian pep approximation and with simulation results we propose a new semianalytical pep lower bound the new lower bound is computationally feasible and is shown by simulation to be tight for small to moderate cfo finally a computationally more feasible approximation of the lower bound is also proposed our discussion reveals as a by product why the gaussian assumption for the doppler induced interference works for standard ofdm systems although the assumption is actually incorrect

iterative solution of linear systems in the 20th century this paper sketches the main research developments in the area of iterative methods for solving linear systems during the 20th century although iterative methods for solving linear systems nd their origin in the early 19th century work by gauss the eld has seen an explosion of activity spurred by demand due to extraordinary technological advances in engineering and sciences the past ve decades have been particularly rich in new developments ending with the availability of large toolbox of specialized algorithms for solving the very large problems which arise in scientic and industrial computational models as in any other scientic area research in iterative methods has been a journey characterized by a chain of contributions building on each other it is the aim of this paper not only to sketch the most signicant of these contributions during the past century but also to relate them to one another c 2000 elsevier science b v all rights reserved

preconditioning highly indefinite and nonsymmetric matrices standard preconditioners like incomplete factorizations perform well when the coefficient matrix is diagonally dominant but often fail on general sparse matrices we experiment with nonsymmetric permutations and scalings aimed at placing large entries on the diagonal in the context of preconditioning for general sparse matrices the permutations and scalings are those developed by olschowka and neumaier linear algebra appl 240 1996 pp 131 151 and by duff and koster siam j matrix anal appl 20 1999 pp 889 901 tech report ral tr 99 030 rutherford appleton laboratory chilton uk 1999 we target highly indefinite nonsymmetric problems that cause difficulties for preconditioned iterative solvers our numerical experiments indicate that the reliability and performance of preconditioned iterative solvers are greatly enhanced by such preprocessing

on the rate of convergence of the preconditioned conjugate gradient method we derive new estimates for the rate of convergence of the conjugate gradient method by utilizing isolated eigenvalues of parts of the spectrum we present a new generalized version of an incomplete factorization method and compare the derived estimates of the number of iterations with the number actually found for some elliptic difference equations and for a similar problem with a model empirical distribution function

improved scaling for quantum monte carlo on insulators quantum monte carlo qmc methods are often used to calculate properties of many body quantum systems the main cost of many qmc methods for example the variational monte carlo vmc method is in constructing a sequence of slater matrices and computing the ratios of determinants for successive slater matrices recent work has improved the scaling of constructing slater matrices for insulators so that the cost of constructing slater matrices in these systems is now linear in the number of particles whereas computing determinant ratios remains cubic in the number of particles with the long term aim of simulating much larger systems we improve the scaling of computing the determinant ratios in the vmc method for simulating insulators by using preconditioned iterative solvers the main contribution of this paper is the development of a method to efficiently compute for the slater matrices a sequence of preconditioners that make the iterative solver converge rapidly this involves cheap preconditioner updates an effective reordering strategy and a cheap method to monitor instability of incomplete lu decomposition with threshold and pivoting ilutp preconditioners using the resulting preconditioned iterative solvers to compute determinant ratios of consecutive slater matrices reduces the scaling of qmc algorithms from o n 3 per sweep to roughly o n 2 where n is the number of particles and a sweep is a sequence of n steps each attempting to move a distinct particle we demonstrate experimentally that we can achieve the improved scaling without increasing statistical errors our results show that preconditioned iterative solvers can dramatically reduce the cost of vmc for large r systems

iterative solution methods for certain sparse linear systems with a non symmetric matrix arising from pde problems abstract in this paper methods are described for the solution of certain sparse linear systems with a non symmetric matrix the power of these methods is demonstrated by extensive numerical experiments application of the methods is limited to problems where the matrix has only eigenvalues with positive real part an important class of this type of matrix arises from discretisation of second order partial differential equations with first order derivative terms

orderings for factorized sparse approximate inverse preconditioners the influence of reorderings on the performance of factorized sparse approximate inverse preconditioners is considered some theoretical results on the effect of orderings on the fill in and decay behavior of the inverse factors of a sparse matrix are presented it is shown experimentally that certain reorderings like minimum degree and nested dissection can be very beneficial the benefit consists of a reduction in the storage and time required for constructing the preconditioner and of faster convergence of the preconditioned iteration in many cases of practical interest

lie symmetry analysis for a parabolic monge ampere equation in the optimal investment theory abstract in this paper lie symmetry analysis is performed on the parabolic monge ampere equation u s u y y r y u y u y y u y 2 0 arising from the optimal investment theory lie symmetry and optimal system of this equation are derived in particular based on optimal system symmetry reductions and invariant solutions are obtained

orderings for parallel conjugate gradient preconditioners the efficient solution of the linear system of equations a mathbf x mathbf b on parallel computers sometimes requires a reordering of the unknowns we investigate the effects of various orderings on the rate of convergence and the computation rate of the preconditioned conjugate method with ssor as the preconditioner as a model problem we use a generalized poisson equation nabla k nabla u f discretized with finite differencing in two and three dimensions the natural ordering has the highest rate of convergence yet results in poor to modest parallelism whereas the classical red black rb ordering fully exploits vector parallel properties but has a deleterious effect on the rate of convergence intermediate between the rb and natural orderings are the many color orderings studied by harrar and ortega but we show that these have rather poor communication properties alternatives are domain decomposition orderings which we consider with and without separator sets we investigate the effectiveness of these orderings on parallel machines the intel ipsc 860 hypercube in particular we also examine the effect of the remainder matrix on the rate of convergence

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

lie symmetry analysis and exact explicit solutions for general burgers equation in this paper the lie symmetry analysis is performed for the general burgers equation the exact solutions and similarity reductions generated from the symmetry transformations are provided furthermore the all exact explicit solutions and similarity reductions based on the lie group method are obtained some new method and techniques are employed simultaneously such exact explicit solutions and similarity reductions are important in both applications and the theory of nonlinear science

a domain decomposing parallel sparse linear system solver the solution of large sparse linear systems is often the most time consuming part of many science and engineering applications computational fluid dynamics circuit simulation power network analysis and material science are just a few examples of the application areas in which large sparse linear systems need to be solved effectively in this paper we introduce a new parallel hybrid sparse linear system solver for distributed memory architectures that contains both direct and iterative components we show that by using our solver one can alleviate the drawbacks of direct and iterative solvers achieving better scalability than with direct solvers and more robustness than with classical preconditioned iterative solvers comparisons to well known direct and iterative solvers on a parallel architecture are provided

optimum frequency domain partial response encoding in ofdm system time variance of wireless channels destroys the orthogonality among subchannels in orthogonal frequency division multiplexing ofdm systems and causes interchannel interference ici which results in an error floor in this paper we study frequency domain partial response coding prc for reducing the effect of the ici based on the general expression of the ici power for ofdm with prc the optimum weights for prc that minimize the ici power are derived from the numerical and simulation results optimum prc for ofdm can reduce the ici effectively by 4 0 db for two tap prc and 6 2 db for three tap prc respectively

lie group classifications and exact solutions for two variable coefficient equations in this paper the lie symmetry analysis and group classifications are performed for two variable coefficient equations the hanging chain equation and the bond pricing equation the symmetries for the two equations are obtained the exact explicit solutions generated from the similarity reductions are presented moreover the exact analytic solutions are considered by the power series method

bounds on the interchannel interference of ofdm in time varying impairments in this article tight and universal bounds have been derived for the interchannel interference ici of an orthogonal frequency division multiplexing ofdm signal resulting from doppler spread the universal bound depends only on the product of the maximum doppler frequency and the ofdm symbol duration the tight bound also depends on the variance of the doppler spectrum compared with the exact ici expressions derived by other researchers these bounds are easier to evaluate and can provide useful insight

preconditioning techniques for large linear systems a survey this article surveys preconditioning techniques for the iterative solution of large linear systems with a focus on algebraic methods suitable for general sparse matrices covered topics include progress in incomplete factorization methods sparse approximate inverses reorderings parallelization issues and block and multilevel extensions some of the challenges ahead are also discussed an extensive bibliography completes the paper

low complexity equalization of ofdm in doubly selective channels orthogonal frequency division multiplexing ofdm systems may experience significant inter carrier interference ici when used in time and frequency selective or doubly selective channels in such cases the classical symbol estimation schemes e g minimum mean squared error mmse and zero forcing zf estimation require matrix inversion that is prohibitively complex for large symbol lengths an analysis of the ici generation mechanism leads us to propose a novel two stage equalizer whose complexity apart from the fft is linear in the ofdm symbol length the first stage applies optimal linear preprocessing to restrict ici support and the second stage uses iterative mmse estimation to estimate finite alphabet frequency domain symbols simulation results indicate that our equalizer has significant performance and complexity advantages over the classical linear mmse estimator in doubly selective channels

windowing for multicarrier cdma systems the windows for achieving good multiple access performance of multicarrier code division multiple access mccdma systems employing minimum mean squared error receivers are studied the mean squared error mse is used as the performance metric for the optimization in a simplified two user system the rectangular window is found to be optimum while regarded as an inferior window in orthogonal frequency division multiplexing systems because of its spectral inefficiency necessary conditions are given for improving the performances of spectrally efficient windows by exploiting the cyclic prefix and oversampling in the frequency domain

lie symmetry analysis optimal systems and exact solutions to the fifth order kdv types of equations abstract in this paper the lie symmetry analysis is performed on the fifth order kdv types of equations which arise in modeling many physical phenomena the similarity reductions and exact solutions are obtained based on the optimal system method then the exact analytic solutions are considered by using the power series method

probability of error analysis of bpsk ofdm systems with random residual frequency offset in this paper we derive closed form bit error rate ber expressions for orthogonal frequency division multiplexing ofdm systems with residual carrier frequency offset cfo most of the published work treats cfo as a nonrandom parameter but in our study we consider it as a random parameter the ber performance of binary phase shift keying bpsk ofdm system is analyzed in the cases of additive white gaussian noise awgn frequency flat and frequency selective rayleigh fading channels we further discuss how these expressions can be related to systems with practical estimators the simulation results are provided to verify the accuracy of these error rate expressions

a two level ilu preconditioner for electromagnetic applications computational electromagnetics based on the solution of the integral form of maxwell s equations with boundary element methods require the solution of large and dense linear systems for large scale problems the solution is obtained by using iterative krylov type methods provided that a fast method for performing matrix vector products is available in addition for ill conditioned problems some kind of preconditioning technique must be applied to the linear system in order to accelerate the convergence of the iterative method and improve its performance for many applications it has been reported that incomplete factorizations often suffer from numerical instability due to the indefiniteness of the coefficient matrix in this context approximate inverse preconditioners based on frobenius norm minimization have emerged as a robust and highly parallel alternative in this work we propose a two level ilu preconditioner for the preconditioned gmres method the computation and application of the preconditioner is based on graph partitioning techniques numerical experiments are presented for different problems and show that with this technique it is possible to obtain robust ilu preconditioners that perform competitively compared with frobenius norm minimization preconditioners

the effect of ordering on preconditioned gmres algorithm for solving the compressible navier stokes equations we investigate the effect of the ordering of the blocks of unknowns on the rate of convergence of a preconditioned non linear gmres algorithm for solving the navier stokes equations for compressible flows using finite element methods on unstructured grids the gmres algorithm is preconditioned by an incomplete ldu block factorization of the jacobian matrix associated with the non linear problem to solve we examine a wide range of ordering methods including minimum degree reverse cuthill mckee and snake and consider preconditionings without fill in we show empirically that there can be a significant difference in the number of iterations required by the preconditioned non linear gmres method and suggest a criterion for choosing a good ordering algorithm according to the problem to solve we also consider the effect of orderings when an incomplete factorization which allows some fill in is performed we consider the effect of automatically controlling the sparsity of the incomplete factorization through the level of fill in finally following the principal ideas of non linear gmres algorithm we suggest other inexact newton methods

analysis of new and existing methods of reducing intercarrier interference due to carrier frequency offset in ofdm orthogonal frequency division multiplexing ofdm is very sensitive to frequency errors caused by frequency differences between transmitter and receiver local oscillators this sensitivity is analyzed in terms of the complex weighting coefficients which give the contribution of each transmitter subcarrier to each demodulated subcarrier previously described windowing and self intercarrier interference ici cancellation methods are analyzed in terms of these weighting coefficients new ici cancellation schemes with very much improved performance are described a condition for orthogonality of windowing schemes is derived in terms of the discrete fourier transform dft of the windowing function

multicolor iccg methods for vector computers this paper considers the incomplete choleski conjugate gradient method for symmetric positive definite linear systems the goal is to obtain implementations of this method suitable for vector computers which require long vectors for efficiency this is achieved by a multicoloring strategy which allows efficient solution of the triangular systems of the preconditioning as well as efficient matrix vector multiplication numerical results are given for three model problems on a cyber 205

orderings for incomplete factorization preconditioning of nonsymmetric problems numerical experiments are presented whereby the effect of reorderings on the convergence of preconditioned krylov subspace methods for the solution of nonsymmetric linear systems is shown the preconditioners used in this study are different variants of incomplete factorizations it is shown that certain reorderings for direct methods such as reverse cuthill mckee can be very beneficial the benefit can be seen in the reduction of the number of iterations and also in measuring the deviation of the preconditioned operator from the identity

nested grids ilu decomposition ngilu a preconditioning technique is described which shows in many cases grid independent convergence this technique only requires an ordering of the unknowns based on the different levels of multigrid and an incomplete lu decomposition based on a drop tolerance the method is demonstrated on a variety of well known elliptic test problems including strongly varying coefficients advective terms and grid refinement

incomplete lu preconditioning with the multilevel fast multipole algorithm for electromagnetic scattering iterative solution of large scale scattering problems in computational electromagnetics with the multilevel fast multipole algorithm mlfma requires strong preconditioners especially for the electric field integral equation efie formulation incomplete lu ilu preconditioners are widely used and available in several solver packages however they lack robustness due to potential instability problems in this study we consider various ilu class preconditioners and investigate the parameters that render them safely applicable to common surface integral formulations without increasing the cal o n log n complexity of mlfma we conclude that the no fill ilu 0 preconditioner is an optimal choice for the combined field integral equation cfie for efie we establish the need to resort to methods depending on drop tolerance and apply pivoting for problems with high condition estimate we propose a strategy for the selection of the parameters so that the preconditioner can be used as a black box method robustness and efficiency of the employed preconditioners are demonstrated over several test problems

parallel finite element simulations of incompressible viscous fluid flow by domain decomposition with lagrange multipliers a parallel approach to solve three dimensional viscous incompressible fluid flow problems using discontinuous pressure finite elements and a lagrange multiplier technique is presented the strategy is based on non overlapping domain decomposition methods and lagrange multipliers are used to enforce continuity at the boundaries between subdomains the novelty of the work is the coupled approach for solving the velocity pressure lagrange multiplier algebraic system of the discrete navier stokes equations by a distributed memory parallel ilu 0 preconditioned krylov method a penalty function on the interface constraints equations is introduced to avoid the failure of the ilu factorization algorithm to ensure portability of the code a message based memory distributed model with mpi is employed the method has been tested over different benchmark cases such as the lid driven cavity and pipe flow with unstructured tetrahedral grids it is found that the partition algorithm and the order of the physical variables are central to parallelization performance a speed up in the range of 5 13 is obtained with 16 processors finally the algorithm is tested over an industrial case using up to 128 processors in considering the literature the obtained speed ups on distributed and shared memory computers are found very competitive

lie group symmetries as integral transforms of fundamental solutions abstract we obtain fundamental solutions for pdes of the form u t x u x x f x u x x r u by showing that if the symmetry group of the pde is nontrivial it contains a standard integral transform of the fundamental solution we show that in this case the problem of finding a fundamental solution can be reduced to inverting a laplace transform or some other classical transform

greedy coarsening strategies for nonsymmetric problems the solution of large scale linear systems in computational science and engineering requires efficient solvers and preconditioners often the most effective such techniques are those based on multilevel splittings of the problem in this paper we consider the problem of partitioning both symmetric and nonsymmetric matrices based solely on algebraic criteria a new algorithm is proposed that combines attractive features of two previous techniques proposed by the authors it offers rigorous guarantees of certain properties of the partitioning yet is naturally compatible with the threshold based dropping known to be effective for incomplete factorizations numerical results show that the new partitioning scheme leads to improved results for a variety of problems the effects of further matrix reordering within the fine scale block are also considered

an algebraic multilevel multigraph algorithm we describe an algebraic multilevel multigraph algorithm many of the multilevel components are generalizations of algorithms originally applied to general sparse gaussian elimination indeed general sparse gaussian elimination with minimum degree ordering is a limiting case of our algorithm our goal is to develop a procedure which has the robustness and simplicity of use of sparse direct methods yet offers the opportunity to obtain the optimal or near optimal complexity typical of classical multigrid methods

intercarrier interference reduction in ofdm systems using low complexity selective mapping selective mapping slm has been employed to reduce the peak interference to carrier ratio picr in ofdm systems here we significantly reduce the computational complexity of slm using subsets of intermediate signals within the inverse fast fourier and fast fourier transforms while achieving performance close to that previously obtained

new similarity solutions for the modified boussinesq equation in this paper the author presents some new similarity solutions of the modified boussinesq equation which is a completely integrable soliton equation these new similarity solutions include reductions to the second and fourth painleve equations which are not obtainable using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method which involves no group theoretical techniques

on the design of comb spectrum code for multiple access scheme a new methodology that applies frequency orthogonality to the spreading code design has been presented the main purpose is to reduce the receiver complexity of the cyclic prefix code division multiple access cp cdma over the downlink channel and to enable parallel transmission over the uplink channel the receiver complexity has been found for requiring full band fft and ifft calculations in the channel equalizer to handle this problem we developed the comb spectrum codes organized in frequency orthogonal groups in each of which the codes have time correlation orthogonality for the downlink channel the reduction of complexity in signal processing of a cscdma receiver is due to that the separation of groups is simple and each separated group is in the form of a narrower bandwidth cp cdma thus the full band fft and ifft can be replaced by the partial fft and partial ifft as shown in this proposed approach for the uplink channel we assign the code groups to different users to form a group division multiple access gdma for enabling an independent channel equalization of each user at base station the theoretical analysis and the simulation results agree well with each other and both confirm the effectiveness of this approach

channel estimation for ofdm system with two training symbols aided and polynomial fitting a time varying channel estimation method for orthogonal frequency division multiplexing ofdm systems is presented the presented channel estimation method employs two training symbols in combination with polynomial fitting thus obtaining accurate estimation results under a large normalized doppler frequency

a parallel sparse linear system solver based on hermitian skew hermitian splitting in this paper we describe a parallel algorithm for solving large sparse nonsingular linear systems ax f of order n using the hermitian skew hermitian splitting approach for handling the augmented linear system of order 2n that arises from the linear least problem of minimizing the 2 norm of fax we use the restarted gmres as the outer iteration with the hermitian skew hermitian splitting hss preconditioner in solving systems involving this preconditioner the most time consuming part deals with handling shifted skew symmetric systems we solve such systems using the successive overrelaxation sor theoretical analysis shows that our solver always converges to the unique solution of ax f we present several numerical experiments that demonstrate the robustness of our solver compared to other schemes and show its parallel scalability on a single multicore node

a comparison of dct based ofdm and dft based ofdm in frequency offset and fading channels a precise method for calculating the bit error probability bep of a discrete cosine transform dct based orthogonal frequency division multiplexing ofdm system on additive white gaussian noise awgn channels in the presence of frequency offset is derived these accurate results are used to examine and compare the bep performance of a dct ofdm system and the conventional discrete fourier transform dft based ofdm system in an awgn environment several signaling formats such as binary phase shift keying quaternary phase shift keying and 16 ary quadrature amplitude modulation are considered the performance of a dct ofdm with a zero padding guard interval scheme is then compared with a zero padded dft ofdm with the employment of minimum mean square error mmse detection and mmse decision feedback detection with ordering scheme over frequency selective fast rayleigh fading channels analysis and simulation results show that the dct ofdm system outperforms the dft ofdm system in the presence of frequency offset and in frequency selective fast fading environments

parallelizable block diagonal preconditioners for the compressible navier stokes equations abstract we propose a new block diagonal preconditioner for solving the navier stokes equations for compressible 2d flows using finite element methods on unstructured grids and discuss the parallelization of this method the core of the navier stokes solver is the coupled solution of the nonlinear equations at each time step by a preconditioned nonlinear gmres algorithm in this work we suggest to partition the jacobian matrix a into n n blocks n being the number of available processors in practice our goal is to obtain good results on commonly available parallel workstations with from 4 to 16 processors special attention is paid to the partitioning strategy which is easily generalizable to 3d problems we approximate a by a n taking only the non zero coefficients lying in the diagonal blocks of a we show that with an appropriate partitioning strategy and a convenient node reordering the ilu 0 factorization of a n is an effective preconditioner of the nonlinear gmres algorithm because of its diagonal block structure both the factorization and the triangular system solution steps associated with the preconditioning matrix can then be parallelized numerical results are presented for several transonic and supersonic calculations around a naca 0012 aerofoil

an implicit wavelet sparse approximate inverse preconditioner wavelet based sparse approximate inverse preconditioners are considered for the linear system ax b the preconditioners are good sparse approximations to the inverse of a computed by taking advantage of the compression obtained by working in a wavelet basis when the representation of a in a single scale basis for example a finite element basis is available the formulation presented obviates computation of the representation of a in the wavelet basis and removes the associated costs efficient application for both sparse and dense a is considered

preconditioning techniques for nonsymmetric and indefinite linear systems abstract the standard preconditioning techniques for conjugate gradient methods often fail for matrices that are indefinite and or strongly nonsymmetric the most common alterative considered for these cases are either to use expensive direct solvers or to resort to one of many techniques based on the normal equations this paper examines several such alternatives and compares them in particular an incomplete lq factorization is proposed and some of its implementation details are described a number of experiments are reported to compare these methods

group analysis of kdv equation with time dependent coefficients we study the generalized kdv equation having time dependent variable coefficients of the damping and dispersion from the lie group theoretic point of view lie group classification with respect to the time dependent coefficients is performed the optimal system of one dimensional subalgebras of the lie symmetry algebras are obtained these subalgebras are then used to construct a number of similarity reductions and exact group invariant solutions including soliton solutions for some special forms of the equations

probability of error calculation of ofdm systems with frequency offset orthogonal frequency division multiplexing ofdm is sensitive to the carrier frequency offset cfo which destroys orthogonality and causes intercarrier interference ici previously two methods were available for the analysis of the resultant degradation in performance firstly the statistical average of the ici could be used as a performance measure secondly the bit error rate ber caused by cfo could be approximated by assuming the ici to be gaussian however a more precise analysis of the performance i e ber or ser degradation is desirable in this letter we propose a precise numerical technique for calculating the effect of the cfo on the ber or symbol error in an ofdm system the subcarriers can be modulated with binary phase shift keying bpsk quaternary phase shift keying qpsk or 16 ary quadrature amplitude modulation 16 qam used in many ofdm applications the bpsk case is solved using a series due to beaulieu 1990 for the qpsk and 16 qam cases we use an infinite series expression for the error function in order to express the average probability of error in terms of the two dimensional characteristic function of the ici

on the eigenvalue distribution of a class of preconditioning methods a class of preconditioning methods depending on a relaxation parameter is presented for the solution of large linear systems of equationax b wherea is a symmetric positive definite matrix the methods are based on an incomplete factorization of the matrixa and include both pointwise and blockwise factorization we study the dependence of the rate of convergence of the preconditioned conjugate gradient method on the distribution of eigenvalues ofc 1a wherec is the preconditioning matrix we also show graphic representations of the eigenvalues and present numerical tests of the methods

comparison of krylov subspace methods on the pagerank problem pagerank algorithm plays a very important role in search engine technology and consists in the computation of the eigenvector corresponding to the eigenvalue one of a matrix whose size is now in the billions the problem incorporates a parameter a that determines the difficulty of the problem in this paper the effectiveness of stationary and nonstationary methods are compared on some portion of real web matrices for different choices of a we see that stationary methods are very reliable and more competitive when the problem is well conditioned that is for small values of a however for large values of the parameter a the problem becomes more difficult and methods such as preconditioned bicgstab or restarted preconditioned gmres become competitive with stationary methods in terms of mflops count as well as in number of iterations necessary to reach convergence

a hierarchical low rank schur complement preconditioner for indefinite linear systems nonsymmetric and highly indefinite linear systems can be quite difficult to solve by iterative methods this paper combines ideas from the multilevel schur low rank preconditioner developed by y x

blind receiver design for ofdm systems over doubly selective channels we develop blind data detectors for orthogonal frequency division multiplexing ofdm systems over doubly selective channels by exploiting both frequency domain and time domain correlations of the received signal we thus derive two blind data detectors a time domain data detector and a frequency domain data detector we also contribute a reduced complexity suboptimal version of a time domain data detector that performs robustly when the normalized doppler rate is less than 3 our frequency domain data detector and suboptimal time domain data detector both result in integer least squares ls problems we propose the use of the v blast detector and the sphere decoder the time domain data detector is not limited to the doppler rates less than 3 but cannot be posed as an integer ls problem our solution is to develop an iterative algorithm that starts from the suboptimal time domain data detector output we also propose channel estimation and prediction algorithms using a polynomial expansion model and these estimators work with data detectors decision directed mode to reduce the complexity the estimators for the channel statistics and the noise variance are derived using the likelihood function for the data our blind data detectors are fairly robust against the parameter mismatch

a structural diagnosis of some ic orderings we present a novel analysis of the potential effectiveness of a matrix ordering for the incomplete cholesky factorization ic in terms of just the sparsity structure by looking at the structure of the approximate inverse implicitly created by ic we can help to explain the success of reverse cuthill mckee orderings the problems ic 0 has under red black orderings that disappear when extra fill is included and where fill must be added to make fill reducing orderings such as minimum degree effective

a transpose free quasi minimal residual algorithm for non hermitian linear systems the biconjugate gradient method bcg for solving general non hermitian linear systems ax b and its transpose free variant the conjugate gradients squared algorithm cgs both typically exhib

lie symmetry reductions and exact solutions of a coupled kdv burgers equation in this paper based on classical lie group method with the help of maple software we study a coupled kdv burgers equation and get its infinitesimal generator commutation table of lie algebra symmetry group and similarity reductions in particular solitary wave solutions and similarity solutions of the coupled kdv burgers equation are derived from the reduction equations

guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems abstract this paper presents incomplete decompositions for various types of matrices as they occur in the implicit discretisation of practical problems a review is given of methods for the usual five point discretisation of a self adjoint elliptic second order partial differential equation in two dimensions on a square the matrices which occur in this type of problem are symmetric m matrices of very regular structure the convergence behaviour of the different decompositions for this case is demonstrated by numerical experiments the paper also gives decompositions for the following type of matrices i symmetric m matrices of a different structure ii symmetric positive definite matrices iii non symmetric matrices

a stability analysis of incomplete lu factorizations abstract the combination of iterative methods with preconditionings based on incomplete lu factorizations constitutes an effective class of methods for solving the sparse linear systems arising from the discretization of elliptic partial differential equations in this paper we show that there are some settings in which the incomplete lu preconditioners are not effective and we demonstrate that their poor performance is due to numerical instability our analysis consists of an analytic and numerical study of a sample two dimensional non self adjoint elliptic problem discretized by several finite difference schemes author

fine grained parallel incomplete lu factorization this paper presents a new fine grained parallel algorithm for computing an incomplete lu factorization all nonzeros in the incomplete factors can be computed in parallel and asynchronously using one or more sweeps that iteratively improve the accuracy of the factorization unlike existing parallel algorithms the amount of parallelism is large irrespective of the ordering of the matrix and matrix ordering can be used to enhance the accuracy of the factorization rather than to increase parallelism numerical tests show that very few sweeps are needed to construct a factorization that is an effective preconditioner

extensions of certain graph based algorithms for preconditioning the original tpablo algorithms are a collection of algorithms which compute a symmetric permutation of a linear system such that the permuted system has a relatively full block diagonal with relatively large nonzero entries this block diagonal can then be used as a preconditioner we propose and analyze three extensions of this approach we incorporate a nonsymmetric permutation to obtain a large diagonal we use a more general parametrization for tpablo and we use a block gauss seidel preconditioner which can be implemented to have the same execution time as the corresponding block jacobi preconditioner experiments are presented showing that for certain classes of matrices the block gauss seidel preconditioner used with the system permuted with the new algorithm can outperform the best ilut preconditioners in a large set of experiments

analysis of incomplete factorizations for a nine point approximation to a convection diffusion model problem we study the stability of zero fill incomplete lu factorizations of a nine point coefficient matrix arising from a high order compact discretisation of a two dimensional constant coefficient convection diffusion problem nonlinear recurrences for computing entries of the lower and upper triangular matrices are derived and we show that the sequence of diagonal entries of the lower triangular factor is unconditionally convergent a theoretical estimate of the limiting value is derived and we show that this estimate is a good predictor of the computed value the unconditional convergence of the diagonal sequence of the lower triangular factor to a positive limit implies that the incomplete factorization process never encounters a zero pivot and that the other diagonal sequences are also convergent the characteristic polynomials associated with the lower and upper triangular solves that occur during the preconditioning step are studied and conditions for the stability of the triangular solves are derived in terms of the entries of the tridiagonal matrices appearing in the lower and upper subdiagonals of the block triangular system matrix and a triplet of parameters which completely determines the solution of the nonlinear recursions results of ilu preconditioned gmres iterations and the effects of orderings on their convergence are also described

iterative solutions to the steady state density matrix for optomechanical systems we present a sparse matrix permutation from graph theory that gives stable incomplete lower upper preconditioners necessary for iterative solutions to the steady state density matrix for quantum optomechanical systems this reordering is efficient adding little overhead to the computation and results in a marked reduction in both memory and runtime requirements compared to other solution methods with performance gains increasing with system size either of these benchmarks can be tuned via the preconditioner accuracy and solution tolerance this reordering optimizes the condition number of the approximate inverse and is the only method found to be stable at large hilbert space dimensions this allows for steady state solutions to otherwise intractable quantum optomechanical systems

multifrontal incomplete factorization for indefinite and complex symmetric systems a new class of preconditioners based on the adaptive threshold incomplete multifrontal factorization for indefinite and complex symmetric systems is developed numerical experiments consisting of the 3d helmholtz equations fluid structure interaction and localization problems demonstrate the excellent performance of the preconditioner copyright 2001 john wiley sons ltd

vectorizable preconditioners for elliptic difference equations in three space dimensions abstract until now research in vectorizable preconditioners has mainly focused on problems in two space dimensions employing all degrees of freedom we here present preconditioners for 3d matrices based on point line and plane factorizations both in vectorizable and for reasons of comparisons in recursive versions by a limit analysis and accompanying numerical tests of elliptic difference equations with anisotropy we find that methods based on line block factorizations are to be preferred in general among the vectorizable methods a line method where euler expansion of the factors of the pivot blocks is used is the most robust with respect to the problem parameters furthermore this method turns out to approach the best of the recursive methods in efficiency in terms of operation counts it has the added bonus of almost maximal vectorlength

analysis of parallel incomplete point factorizations abstract incomplete point factorizations are analyzed regarding their parallelism and the norm of the error matrix and the relation between the two

performance of multicarrier ds ssma over fast rayleigh fading channels with doppler diversity the significant loss in performance of multicarrier direct sequence spread spectrum multiple access systems over fast fading channels is investigated first the channel model for individual subchannels is obtained using a canonical decomposition to a wide sense stationary wss uncorrelated scattering channel next a receiver structure which features both doppler diversity and frequency diversity is presented and analyzed it is found that large doppler spreads cause the average magnitude of the desired signals to diminish dramatically expressions for the bit error rate are derived for both uncoded and coded systems by applying the standard gaussian approximation numerical results which show that doppler diversity is preferable over frequency diversity in fast fading channels are provided for different combinations of the diversity orders numerical results which show that the intersymbol and intersubchannel interference can be effectively suppressed and the multiple access interference tends to degenerate into a wss process with the statistical periodicity destroyed by the fast fading channels are also provided

low complexity equalization for doubly selective channels modeled by a basis expansion we propose a novel equalization method for doubly selective wireless channels whose taps are represented by an arbitrary basis expansion model bem we view such a channel in the time domain as a sum of product convolution operators created from the basis functions and the bem coefficients equivalently a frequency domain channel can be represented as a sum of convolution products the product convolution representation provides a low complexity memory efficient way to apply the channel matrix to a vector we compute a regularized solution of a linear system involving the channel matrix by means of the gmres and the lsqr algorithms which utilize the product convolution structure without ever explicitly creating the channel matrix our method applies to all cyclic prefix transmissions in an ofdm transmission with k subcarriers each iteration of gmres or lsqr requires only o k k flops and o k memory additionally we further accelerate convergence of both gmres and lsqr by using the single tap equalizer as a preconditioner we validate our method with numerical simulations of a wimax like system ieee 802 16e in channels with significant delay and doppler spreads the proposed equalizer achieves bers comparable to those of mmse equalization and noticeably outperforms low complexity equalizers using an approximation by a banded matrix in the frequency domain with preconditioning the lowest bers are obtained within 3 16 iterations our approach does not use any statistical information about the wireless channel