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

parameterized algorithm for 3 path vertex cover abstract in the 3 path vertex cover problem the input is an undirected graph g and an integer k the goal is to decide whether there is a set of vertices s of size at most k such that every path with 3 vertices in g contains at least one vertex of s in this paper we give a parameterized algorithm for 3 path vertex cover whose time complexity is o 1 713 k our algorithm is faster than previous algorithms for this problem

approximation algorithms for minimum weight connected 3 path vertex cover abstract a k path vertex cover vcpk is a vertex set c of graph g such that every path of g on k vertices has at least one vertex in c because of its background in keeping data integrality of a network minimum vcpk problem minvcpk has attracted a lot of researches in recent years this paper studies the minimum weight connected vcpk problem minwcvcpk in which every vertex has a weight and the vcpk found by the algorithm induces a connected subgraph and has the minimum weight it is known that minwcvcpk is set cover hard we present two polynomial time approximation algorithms for minwcvcp3 the first one is a greedy algorithm achieving approximation ratio 3ln n the difficulty lies in its analysis dealing with a non submodular potential function the second algorithm is a 2 stage one finding a vcp3 in the first stage and then adding more vertices for connection we show that its approximation ratio is at most ln max 4 ln 2 where max is the maximum degree of the graph considering the inapproximability of this problem this ratio is asymptotically tight

ordering anisotropy and factored sparse approximate inverses we consider ordering techniques to improve the performance of factored sparse approximate inverse preconditioners concentrating on the ainv technique of m benzi and m t r u ma several practical existing unweighted orderings are considered along with a new algorithm minimum inverse penalty mip that we propose we show how good orderings such as these can improve the speed of preconditioner computation dramatically and also demonstrate a fast and fairly reliable way of testing how good an ordering is in this respect our test results also show that these orderings generally improve convergence of krylov subspace solvers but may have difficulties particularly for anisotropic problems we then argue that weighted orderings which take into account the numerical values in the matrix will be necessary for such systems after developing a simple heuristic for dealing with anisotropy we propose several practical algorithms to implement it while these show promise we conclude a better heuristic is required for robustness

bounds for multihop relayed communications in nakagami m fading we present closed form lower bounds for the performance of multihop transmissions with nonregenerative relays over not necessarily identically distributed nakagami m fading channels the end to end signal to noise ratio is formulated and upper bounded by using an inequality between harmonic and geometric means of positive random variables rvs novel closed form expressions are derived for the moment generating function the probability density function and the cumulative distribution function of the product of rational powers of statistically independent gamma rvs these statistical results are then applied to studying the outage probability and the average bit error probability for phase and frequency modulated signaling numerical examples compare analytical and simulation results verifying the tightness of the proposed bounds

efficient preconditioning of sequences of nonsymmetric linear systems we present a new approach for approximate updates of factorized nonsymmetric preconditioners for solving sequences of linear algebraic systems this approach is algebraic and it is theoretically motivated it generalizes diagonal updates introduced by benzi and bertaccini bit 43 2003 pp 231 244 and bertaccini electron trans numer anal 18 2004 pp 49 64 it is shown experimentally that this approach can be very beneficial for example it is successful in significantly decreasing the number of iterations of a preconditioned iterative method for solving subsequent systems of a sequence when compared with freezing the preconditioner from the first system of the sequence in some cases the updated preconditioners offer a rate of convergence similar to or even higher than the rate obtained when preconditioning with recomputed preconditioners since the updates are typically cheap and straightforward their use is of practical interest they can replace recomputing preconditioners which is often expensive especially in parallel and matrix free environments

a nested iterative scheme for indefinite linear systems in particulate flows abstract high fidelity large scale direct numerical simulation of particulate flows is of great value in a variety of industrial applications it is computationally intensive as it combines time integration solving nonlinear algebraic equations and the associated linear systems the finite element discretization of the coupled system of pdes on an unstructured grid using an arbitrary lagrangian eulerian moving mesh technique leads to very large nonlinear systems that are linearized by a version of newton s method the linear algebraic systems jacobians are sparse nonsymmetric and indefinite for which standard linear system solvers based on krylov subspace methods generally fail to converge without appropriate preconditioners the failure of krylov methods in production codes is currently being addressed by reducing the size of the time step this however leads to a very long simulation time and therefore is not always a viable approach in this study we design a hybrid inner outer iterative scheme for solving these indefinite systems which proves to be both efficient robust and ideally suited for parallel computing platforms even with appropriate large time steps comparisons with krylov subspace methods show the superiority of our proposed class of nested iterative schemes which is also scalable with respect to mesh size and insensitive to changes in properties of the fluid particles system

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

on the error performance of the a_ n lattices we consider the root lattice a n and derive explicit recursive formulas for the moments of its voronoi cell these formulas enable accurate prediction of the error probability of lattice codes constructed from a n

np hard graph problems and boundary classes of graphs any graph problem which is np hard in general graphs becomes polynomial time solvable when restricted to graphs in special classes when does a difficult problem become easy to answer this question we study the notion of boundary classes in the present paper we define this notion in its most general form describe several approaches to identify boundary classes and apply them to various graph problems

symmetric permutations for i matrices to delay and avoid small pivots during factorization in this article we present several new permutations for i matrices making these more suitable for incomplete lu factorization preconditioners used in solving linear systems by iterative methods a general matrix can be transformed by row permutation as well as row and columns scaling into an i matrix i e a matrix having elements of modulus 1 on the diagonal and elements of modulus of no more than 1 elsewhere reordering rows and columns by the same permutation clearly preserves i matrices in this article we consider such reordering techniques which make the permuted matrix more suitable for an incomplete lu factorization preconditioner than the original i matrix we use a multilevel iluc an incomplete lu factorization preconditioner using crout s implementation of gaussian elimination without pivoting to test these reorderings the combination of i matrix preprocessing with the various algorithms presented here and the multilevel incomplete lu factorizations forms a powerful preconditioning method for unsymmetric highly indefinite problems the c code has been made available in the software package ilu

a new pivoting strategy for gaussian elimination this paper discusses a method for determining a good pivoting sequence for gaussian elimination based on an algorithm for solving assignment problems the worst case complexity is o n3 in practice o n2 25 operations are sufficient

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 computing the minimum 3 path vertex cover and dissociation number of graphs the dissociation number of a graph g is the number of vertices in a maximum size induced subgraph of g with vertex degree at most 1 a k path vertex cover of a graph g is a subset s of vertices of g such that every path of order k in g contains at least one vertex from s the minimum 3 path vertex cover is a dual problem to the dissociation number for this problem we present an exact algorithm with a running time of o 1 5171 n on a graph with n vertices we also provide a polynomial time randomized approximation algorithm with an expected approximation ratio of 2311 for the minimum 3 path vertex cover

a robust and efficient ilu that incorporates the growth of the inverse triangular factors in this paper we present a new ilu decomposition which is based on an existing sparse direct solver in contrast to many incomplete lu decompositions this ilu incorporates information about the inverse factors l 1 and u 1 which have direct influence on the dropping strategy we demonstrate in several large scale examples that this implementation constructs a robust preconditioner

robust approximate inverse preconditioning for the conjugate gradient method we present a variant of the ainv factorized sparse approximate inverse algorithm which is applicable to any symmetric positive definite matrix the new preconditioner is breakdown free and when used in conjunction with the conjugate gradient method results in a reliable solver for highly ill conditioned linear systems we also investigate an alternative approach to a stable approximate inverse algorithm based on the idea of diagonally compensated reduction of matrix entries the results of numerical tests on challenging linear systems arising from finite element modeling of elasticity and diffusion problems are presented

preconditioning sparse nonsymmetric linear systems with the sherman morrison formula let ax b be a large sparse nonsymmetric system of linear equations a new sparse approximate inverse preconditioning technique for such a class of systems is proposed we show how the matrix a0 1 a 1 where a0 is a nonsingular matrix whose inverse is known or easy to compute can be factorized in the form u omega v t using the sherman morrison formula when this factorization process is done incompletely an approximate factorization may be obtained and used as a preconditioner for krylov iterative methods for a0 sin where in is the identity matrix and s is a positive scalar the existence of the preconditioner for m matrices is proved in addition some numerical experiments obtained for a representative set of matrices are presented results show that our approach is comparable with other existing approximate inverse techniques

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

exact algorithms for the maximum dissociation set and minimum 3 path vertex cover problems a dissociation set in a graph g v e is a vertex subset d such that the subgraph g d induced on d has vertex degree at most 1 a 3 path vertex cover in a graph is a vertex subset c such that every path of three vertices contains at least one vertex from c a vertex set d is a dissociation set if and only if v d is a 3 path vertex cover there are many applications for dissociation sets and 3 path vertex covers however it is np hard to compute a dissociation set of maximum size or a 3 path vertex cover of minimum size in graphs several exact algorithms have been proposed for these two problems and they can be solved in o 1 4658 n time in n vertex graphs in this paper we reveal some interesting structural properties of the two problems which allow us to solve them in o 1 4656 n time and polynomial space or o 1 3659 n time and exponential space

improved approximation algorithms for hitting 3 vertex paths we study the problem of deleting a minimum cost set of vertices from a given vertex weighted graph in such a way that the resulting graph has no induced path on three vertices this problem is often called cluster vertex deletion in the literature and admits a straightforward 3 approximation algorithm since it is a special case of the vertex cover problem on a 3 uniform hypergraph recently you wang and cao described an efficient 5 2 approximation algorithm for the unweighted version of the problem our main result is a 9 4 approximation algorithm for arbitrary weights using the local ratio technique we further conjecture that the problem admits a 2 approximation algorithm and give some support for the conjecture this is in sharp contrast with the fact that the similar problem of deleting vertices to eliminate all triangles in a graph is known to be ugc hard to approximate to within a ratio better than 3 as proved by guruswami and lee

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

a preconditioned krylov technique for global hydrodynamic stability analysis of large scale compressible flows the combination of iterative krylov based eigenvalue algorithms and direct numerical simulations dns has proven itself an effective and robust tool in solving complex global stability problems of compressible flows a cayley transformation is required to add flexibility to our stability solver and to allow access to specific parts of the full global spectrum which would be out of reach without such a transformation in order to robustify the overall global stability solver an efficient ilu based preconditioner has been implemented with this cayley transformed dns based krylov method two flow cases were successfully investigated i a compressible mixing layer a rather simple but well known problem which served as a test case and ii a supersonic flow about a swept parabolic body a challenging large scale flow configuration

a block version of left looking ainv preconditioner with one by one or two by two block pivots abstract in this paper we present a block format of left looking ainv preconditioner for a nonsymmetric matrix this preconditioner has block 1 1 or 2 2 pivot entries it is introduced based on a block format of gaussian elimination process which has been studied in 14 we have applied the multilevel nested dissection reordering as the preprocessing and have compared this block preconditioner by the plain left looking ainv preconditioner if we mix the multilevel nested dissection by the maximum weighted matching process then the numerical experiments indicate that the number of 2 2 pivot entries in the block preconditioner will grow up in this case the block preconditioner makes gmres method convergent in a smaller number of iterations in the numerical section we have also compared the ilut and block left looking ainv preconditioners

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

approximation algorithm for minimum weight connected k subgraph cover abstract for a given graph g the minimum weight connected k subgraph cover problem minwcksc is to find a minimum weight vertex subset c of g such that each connected subgraph of g on k vertices contains at least one vertex of c previously zhang et al 37 presented a k 1 approximation algorithm for minwcksc under the assumption that the girth of g which is the length of a shortest cycle of g is at least k in this paper we improve this result by showing that k 1 approximation can be achieved when the girth requirement is relaxed from k to 2 k 3

a sparse approximate inverse preconditioner for the conjugate gradient method a method for computing a sparse incomplete factorization of the inverse of a symmetric positive definite matrix a is developed and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient calculations it is proved that in exact arithmetic the preconditioner is well defined if a is an h matrix the results of numerical experiments are presented

algebraic tools to build modulation schemes for fading channels a unified framework is presented in order to build lattice constellations matched to both the rayleigh fading channel and the gaussian channel the method encompasses the situations where the interleaving is done on the real components or on two dimensional signals in the latter case a simple construction of lattices congruent to the densest binary lattices with respect to the euclidean distance is proposed it generalizes in a sense to be clarified later the structural construction proposed by forney 1991 these constellations are next combined with coset codes the partitioning rules and the gain formula are similar to those used for the gaussian channel

a tearing based hybrid parallel banded linear system solver a new parallel algorithm for the solution of banded linear systems is proposed the scheme tears the coefficient matrix into several overlapped independent blocks in which the size of the overlap is equal to the system s bandwidth a corresponding splitting of the right hand side is also provided the resulting independent and smaller size linear systems are solved under the constraint that the solutions corresponding to the overlap regions are identical this results in a linear system whose size is proportional to the sum of the overlap regions which we refer to as the balance system we propose a solution strategy that does not require obtaining this balance system explicitly once the balance system is solved retrieving the rest of the solution can be realized with almost perfect parallelism our proposed algorithm is a hybrid scheme that combines direct and iterative methods for solving a single banded system of linear equations on parallel architectures it has broad applications in finite element analysis particularly as a parallel solver of banded preconditioners that can be used in conjunction with outer krylov iterative schemes

approximation algorithm for the minimum weight connected k subgraph cover problem a subset f of vertices is called a connected k subgraph cover vcck if every connected subgraph on k vertices contains at least one vertex from f the minimum weight connected k subgraph cover problem mwvcck has its background in the field of security and supervisory control it is a generalization of the minimum weight vertex cover problem and is related with the minimum weight k path cover problem mwvcpk which requires that every path on k vertices has at least one vertex from f a k approximation algorithm can be easily obtained by lp rounding method assuming that the girth of the graph is at least k we reduce the approximation ratio to k 1 which is tight for our algorithm

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

a primal dual approximation algorithm for the vertex cover p3 problem we introduce the vertex coverp n vcp n problem that is the problem of finding a minimum weight set email protected v such that the graph g v f has no p n where p n is a path with n vertices the problem also has its application background in this paper we first show that the vcp n problem is np hard for any integer n 2 then we restrict our attention to the vcp 3 problem and give a 2 approximation algorithm using the primal dual method

error performance of multidimensional lattice constellations part ii evaluation over fading channels this is the second part of a two part series of papers where the error performance of multidimensional lattice constellations with signal space diversity ssd is investigated in part i following a novel combinatorial geometrical approach which is based on parallelotope geometry we have presented an exact analytical expression and two closed form bounds for the symbol error probability sep in additive white gaussian noise awgn in the present part ii we extend the analysis and present a novel analytical expression for the frame error probability fep of multidimensional lattice constellations over nakagami m fading channels as the fep of infinite lattice constellations is lower bounded by the sphere lower bound slb we propose the sphere upper bound sub for block fading channels furthermore two novel bounds for the fep of multidimensional lattice constellations over block fading channels named multiple sphere lower bound mslb and multiple sphere upper bound msub are presented the expressions for the slb and sub are given in closed form while the corresponding ones for mslb and msub are given in closed form for unitary block length numerical and simulation results illustrate the tightness of the proposed bounds which can be efficiently used to set the performance limits on the fep of lattice constellations of arbitrary structure dimension and rank

fully iterative ilu preconditioning of the unsteady navier stokes equations for gpgpu abstract in this work we investigate the numerical difficulties that arise in optimizing the efficiency of newtonian fluids simulations on a massively parallel computing hardware like a gpu in particular we will concentrate on the resulting algebraic problem we will present an approximate fully iterative ilu preconditioner and we will show that this solution is very efficient on a gpu if compared with an intrinsic massively parallel preconditioner like the diagonal preconditioner which indeed goes faster than more robust techniques like ilu despite their strong decrease in the number of iterations we refer to gmres as the iterative scheme used to solve the linear system in particular we will deal with the solution of incompressible flows with variable density and we will investigate the interplay between reynolds and atwood numbers we will show that the numerical simulation at medium high reynolds numbers produces linear systems whose matrices can be reasonably preconditioned with the diagonal preconditioner while at low reynolds numbers the higher viscosity of the fluid makes the diagonal preconditioner ineffective in the solution time requested from gmres and decreasing the reynolds number unable to let gmres converge at all in this situation we will show how an adequate iterative approach to the parallel solution of the triangular systems that result from the ilu preconditioning turns out to be robust and efficient we will show numerical results for variable density fluids discretized with the scheme described in calgaro et al 2008 in classical benchmarks and in particular in the well known rayleigh taylor instability

improved upper bounds for vertex cover this paper presents an o 1 2738 k kn time polynomial space algorithm for vertex cover improving the previous o 1 286 k kn time polynomial space upper bound by chen kanj and jia most of the previous algorithms rely on exhaustive case by case branching rules and an underlying conservative worst case scenario assumption the contribution of the paper lies in the simplicity uniformity and obliviousness of the algorithm presented several new techniques as well as generalizations of previous techniques are introduced including general folding struction tuples and local amortized analysis the algorithm also improves the o 1 2745 kk 4 kn time exponential space upper bound for the problem by chandran and grandoni

new algebraic constructions of rotated z sup n lattice constellations for the rayleigh fading channel in this correspondence we present various families of full diversity rotated z sup n lattice constellations based on algebraic number theory constructions we are able to give closed form expressions of their minimum product distance using the corresponding algebraic properties

preconditioning with direct approximate factoring of the inverse to precondition a large and sparse linear system two direct methods for approximate factoring of the inverse are devised the algorithms are fully parallelizable and appear to be more robust than the iterative methods suggested for the task a method to compute one of the matrix subspaces optimally is derived possessing a considerable amount of flexibility these approaches extend the approximate inverse preconditioning techniques in several natural ways numerical experiments are given to illustrate the performance of the preconditioners on a number of challenging benchmark linear systems

good lattice constellations for both rayleigh fading and gaussian channels recent work on lattices matched to the rayleigh fading channel has shown how to construct good signal constellations with high spectral efficiency we present a new family of lattice constellations based on complex algebraic number fields which have good performance on rayleigh fading channels some of these lattices also present a reasonable packing density and thus may be used at the same time over a gaussian channel conversely we show that particular versions of the best lattice packings d sub 4 e sub 6 e sub 8 k sub 12 spl lambda sub 16 spl lambda sub 24 constructed from totally complex algebraic cyclotomic fields present better performance over the rayleigh fading channel the practical interest in such signal constellations rises from the need to transmit information at high rates over both terrestrial and satellite links some further results in algebraic number theory related to ideals and their factorization are presented and the decoding algorithm used with these lattice constellations are illustrated together with practical results

a general approximation technique for constrained forest problems we present a general approximation technique for a large class of graph problems our technique mostly applies to problems of covering at minimum cost the vertices of a graph with trees cycles or paths satisfying certain requirements in particular many basic combinatorial optimization problems fit in this framework including the shortest path minimum cost spanning tree minimum weight perfect matching traveling salesman and steiner tree problems our technique produces approximation algorithms that run in o n 2 log n time and come within a factor of 2 of optimal for most of these problems for instance we obtain a 2 approximation algorithm for the minimum weight perfect matching problem under the triangle inequality our running time of o n 2 log n time compares favorably with the best strongly polynomial exact algorithms running in o n 3 time for dense graphs a similar result is obtained for the 2 matching problem and its variants we also derive the first approximation algorithms for many np complete problems including the non fixed point to point connection problem the exact path partitioning problem and complex location design problems moreover for the prize collecting traveling salesman or steiner tree problems we obtain 2 approximation algorithms therefore improving the previously best known performance guarantees of 2 5 and 3 respectively math programming 59 1993 pp 413 420

fast computational methods for locating fold points for the power flow equations voltage collapse in a power system can occur following a progressive decline in voltage magnitude at the system buses and mathematically this phenomenon is associated with a fold bifurcation point occurring in the nonlinear algebraic equations used to model the power system in this paper we first discuss some of the methods used to speed up the process of detecting a fold bifurcation focussing on designing test function methods to predict a performance index we then discuss some iterative methods that can be used to improve the newton iterations in particular we present new and efficient preconditioners of the two level type for the jacobian matrix numerical results are given using standard ieee test bus systems

error performance of multidimensional lattice constellations part i a parallelotope geometry based approach for the awgn channel multidimensional lattice constellations which present signal space diversity ssd have been extensively studied for single antenna transmission over fading channels with focus on their optimal design for achieving high diversity gain in this two part series of papers we present a novel combinatorial geometrical approach based on parallelotope geometry for the performance evaluation of multidimensional finite lattice constellations with arbitrary structure dimension and rank in part i we present an analytical expression for the exact symbol error probability sep of multidimensional signal sets and two novel closed form bounds named multiple sphere lower bound mlsb and multiple sphere upper bound msub part ii extends the analysis to the transmission over fading channels where multidimensional signal sets are commonly used to combat fading degradation numerical and simulation results show that the proposed geometrical approach leads to bounds that can be used for the performance evaluation and the design of multidimensional lattice constellations both in additive white gaussian noise awgn and fading channels

preconditioned iterative methods for sparse linear algebra problems arising in circuit simulation the dc operating point of a circuit may be computed by tracking the zero curve of an associated artificial parameter homotopy homotopy algorithms exist that are globally convergent with probability one for the dc operating point problem these algorithms require computing the one dimensional kernel of the jacobian matrix of the homotopy mapping at each step along the zero curve and hence the solution of a linear system of equations at each step these linear systems are typically large highly sparse non symmetric and indefinite several iterative methods which are applicable to such problems including craig s method gmres k bicgstab qmr kacz and lsqr are applied to a suite of test problems derived from simulations of actual bipolar circuits preconditioning techniques considered include incomplete lu factorization ilu sparse submatrix ilu and ilu allowing restricted fill in bands or blocks timings and convergence statistics are given for each iterative method and preconditioner

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

preconditioners for nonsymmetric indefinite linear systems abstract in this paper we develop algorithms for solving nonsymmetric indefinite linear systems by considering the augmented linear systems resulting from a weighted linear least squares problem even though the augmented system is more ill conditioned than the original linear system one can construct preconditioned gmres methods for solving these augmented systems capable of obtaining reasonable approximation of the solution in fewer iterations than the classical ilu preconditioned gmres method for solving the original linear system more specifically we present two different preconditioners for these augmented systems examine the spectral properties of these preconditioned augmented systems and report numerical results to illustrate the effectiveness of these preconditioners

kernels for packing and covering problems abstract we show how the notion of combinatorial duality related to the well known notion of duality from linear programming may be used for translating kernel results obtained for packing problems into kernel results for covering problems we exemplify this approach by having a closer look at the problems of packing a graph with vertex disjoint trees or vertex disjoint stars with r edges the case r 2 has been studied in several other papers by establishing a general notion of a crown we show how linear size vertex kernels can be efficiently achieved for the mentioned problems

universal bound on the performance of lattice codes we present a lower bound on the probability of symbol error for maximum likelihood decoding of lattices and lattice codes on a gaussian channel the bound is tight for error probabilities and signal to noise ratios of practical interest as opposed to most existing bounds that become tight asymptotically for high signal to noise ratios the bound is also universal it provides a limit on the highest possible coding gain that may be achieved at specific symbol error probabilities using any lattice or lattice code in n dimensions in particular it is shown that the effective coding gains of the densest known lattices are much lower than their nominal coding gains the asymptotic as n spl rarr spl infin behavior of the new bound is shown to coincide with the shannon 1948 limit for gaussian channels

efficient algorithm for the vertex cover pk problem on cacti abstract given a graph g v e the vertex cover p k vcp k problem is to find a minimum set f v such that every path of order k in g contains at least one vertex from f for any integer k 2 the vcp k problem for general graphs is np hard the paper presents an efficient algorithm for the vcp k problem on cacti

multilevel preconditioners constructed from inverse based ilus this paper analyzes dropping strategies in a multilevel incomplete lu decomposition context and presents a few strategies for obtaining related ilus with enhanced robustness the analysis shows that the incomplete lu factorization resulting from dropping small entries in gaussian elimination produces a good preconditioner when the inverses of these factors have norms that are not too large as a consequence a few strategies are developed whose goal is to achieve this feature a number of templates for enabling implementations of these factorizations are presented numerical experiments show that the resulting ilus offer a good compromise between robustness and efficiency

multilevel ilu with reorderings for diagonal dominance this paper presents a preconditioning method based on combining two sided permutations with a multilevel approach the nonsymmetric permutation exploits a greedy strategy to put large entries of the matrix in the diagonal of the upper leading submatrix the method can be regarded as a complete pivoting version of the incomplete lu factorization this leads to an effective incomplete factorization preconditioner for general nonsymmetric irregularly structured sparse linear systems

hitting subgraphs in p4 tidy graphs abstract vertex cover number which is one of the most basic graph invariants can be viewed as the smallest number of vertices that hit or that belong to every subgraph k2 in a graph g in this paper we consider the next two smallest cases of connected graphs which are the path p3 and the cycle c3 the problem is to minimize the set of vertices that hit every subgraph h in g where h is one of the two graphs we focus on computational aspects of these two variations of the vertex cover number we show that both problems are np complete even in the class of graphs that have no induced cycles of length t where t 4 8 in fact in the p3 hitting case the problem is np complete in even smaller family of graphs where also triangles are forbidden on the positive side we prove a complementary result that in graphs with no induced p4 the so called cographs both problems become linear moreover we consider a generalization of cographs p4 tidy graphs for which we establish an efficient algorithm for both problems

algebraic multilevel preconditioner for the helmholtz equation in heterogeneous media an algebraic multilevel ml preconditioner is presented for the helmholtz equation in heterogeneous media it is based on a multilevel incomplete ldl t factorization and preserves the inherent complex symmetry of the helmholtz equation the ml preconditioner incorporates two key components for efficiency and numerical stability symmetric maximum weight matchings and an inverse based pivoting strategy the former increases the block diagonal dominance of the system whereas the latter controls l 1 for numerical stability when applied recursively their combined effect yields an algebraic coarsening strategy similar to algebraic multigrid methods even for highly indefinite matrices the ml preconditioner is combined with a krylov subspace method and applied as a black box solver to a series of challenging two and three dimensional test problems mainly from geophysical seismic imaging the numerical results demonstrate the robustness and efficiency of the ml preconditioner even at higher frequency regimes

sparse approximate inverse preconditioners using norm minimization techniques we investigate the use of sparse approximate inverse preconditioners for the iterative solution of unsymmetric linear systems of equations we consider the approximations proposed by cosgrove diaz and griewank internat j comput math 44 1992 pp 91 110 and huckle and grote a new approach to parallel preconditioning with sparse approximate inverses tech report sccm 94 03 stanford university 1994 which are based on norm minimization techniques such methods are of particular interest because of the considerable scope for parallelization we propose a number of enhancements which may improve their performance when run in a sequential environment these methods can perform unfavorably when compared with other techniques however they can be successful when other methods fail and simulations indicate that they can be competitive when considered in a parallel environment

the k path vertex cover in cartesian product graphs and complete bipartite graphs abstract for a graph g and a positive integer k a subset s of vertices of g is called a k path vertex cover if s intersects all paths of order k in g the cardinality of a minimum k path vertex cover is denoted by k g and called the k path vertex cover number of g in this paper we study some cartesian product graphs and give several estimations and the exact values of k g

a ptas for the minimum weight connected vertex cover p 3 problem on unit disk graphs let g v e be a weighted graph i e with a vertex weight function w v r we study the problem of determining a minimum weight connected subgraph of g that has at least one vertex in common with all paths of length two in g it is known that this problem is np hard for general graphs we first show that it remains np hard when restricted to unit disk graphs our main contribution is a polynomial time approximation scheme for this problem if we assume that the problem is c local and the unit disk graphs have minimum degree of at least two

algebraic lattice constellations bounds on performance in this work we give a bound on performance of any full diversity lattice constellation constructed from algebraic number fields we show that most of the already available constructions are almost optimal in the sense that any further improvement of the minimum product distance would lead to a negligible coding gain furthermore we discuss constructions minimum product distance and bounds for full diversity complex rotated z i sup n lattices for any dimension n which avoid the need of component interleaving

every property of hyperfinite graphs is testable a k disc around a vertex v of a graph g v e is the subgraph induced by all vertices of distance at most k from v we show that the structure of a planar graph on n vertices and with constant maximum degree d is determined up to the modification insertion or deletion of at most epsilon d n edges by the frequency of k discs for certain k k epsilon d that is independent of the size of the graph we can replace planar graphs by any hyperfinite class of graphs which includes for example every graph class that does not contain a set of forbidden minors a pure combinatorial consequence of this result is that two d bounded degree graphs that have similar frequency vectors that is the ell 1 difference between the frequency vectors is small are close to isomorphic where close here means that by inserting or deleting not too many edges in one of them it becomes isomorphic to the other we also obtain the following new results in the area of property testing which are es

node deletion problems on bipartite graphs a set of problems which has attracted considerable interest recently is the set of node deletion problems the general node deletion problem can be stated as follows given a graph find the minimum number of nodes whose deletion results in a subgraph satisfying property r in ly this problem was shown to be np complete for a large class of properties the class of properties that are hereditary on induced subgraphs using a small number of reduction schemes from the node cover problem since the node cover problem becomes polynomial on bipartite graphs it might be hoped that this is the case with other node deletion problems too in this paper we characterize those properties for which the bipartite restriction of the node deletion problem is polynomial and those for which it remains np complete similar results follow for analogous problems on other structures such as families of sets hypergraphs and 0 1 matrices for example in the case of matrices our result states that ifm is a class of 0 1 matrices which is closed under permutation and deletion of rows and columns then finding the largest submatrix inm of a matrix is polynomial if the matrices ofm have bounded rank and np complete otherwise 1 introduction a common approach when faced with an np complete problem c k gj is to restrict its input domain with the hope that the problem may become solvable in polynomial time under the restriction in fact it turns out that sometimes this is the case for example the node cover problem np complete on general graphs k can be solved efficiently when restricted to bipartite graphs this follows from k6nig s theorem that the node covering number of a bipartite graph equals the number of edges in a maximum matching and moreover a minimum node cover can be found efficiently from a maximum matching when the graph is bipartite see for example la in other cases however the node cover problem continues to be np complete under restriction for example the node cover problem on planar graphs gjs much work has been done recently kd ly ldl on a class of problems called the node deletion or maximum subgraph problems the general node deletion problem can be stated as follows given a graph or digraph g find a set of nodes of minimum cardinality whose deletion results in a graph or digraph satisfying a property r several of the known np complete problems such as the node cover the max clique the feedback node set k as well as some polynomial problems such as the connectivity of a graph e it can be formulated in an obvious way as node deletion problems by specifying appropriately the property r

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

a sparse approximate inverse preconditioner for nonsymmetric linear systems this paper is concerned with a new approach to preconditioning for large sparse linear systems a procedure for computing an incomplete factorization of the inverse of a nonsymmetric matrix is developed and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient type methods some theoretical properties of the preconditioner are discussed and numerical experiments on test matrices from the harwell boeing collection and from tim davis s collection are presented our results indicate that the new preconditioner is cheaper to construct than other approximate inverse preconditioners furthermore the new technique insures convergence rates of the preconditioned iteration which are comparable with those obtained with standard implicit preconditioners

on the symbol error rates for signal space diversity schemes over a rician fading channel a signal space diversity ssd scheme is one of techniques to achieve diversity gain in fading channels this method consists of two key operations constellation rotation and component wise interleaving because of these operations the decision boundaries for the ssd are no longer perpendicular and thus different coordination approaches are required for the analysis of error rates compared to conventional rectangular coordinates in this letter we derive an exact expression of the symbol error rate for the ssd scheme in rician fading channels with m qam and m psk by defining the ratio of the standard deviation of the inphase and quadrature components we introduce a new signal model for the ssd based on this signal model we can compute the exact symbol error rate using polar coordinates the computer simulation results confirm the accuracy of our analysis for fading channels

a multi start iterated greedy algorithm for the minimum weight vertex cover p3 problem abstract given a vertex weighted graph g v e and a positive integer k 2 the minimum weight vertex cover pk mwvcpk problem is to find a vertex subset f v with minimum total weight such that every path of order k in g contains at least one vertex in f for any integer k 2 the mwvcpk problem for general graphs is np hard in this paper we restrict our attention to the mwvcp3 problem and present a multi start iterated greedy algorithm to solve the mwvcp3 problem the experiments are carried out on randomly generated instances with up to 1000 vertices and 250000 edges our work is the first one to adopt heuristic algorithms to solve the mwvcp3 problem and the experimental results show that our algorithm performs reasonably well in practice

on the hardness of approximating minimum vertex cover we prove the minimum vertex cover problem to be np hard to approximate to within a factor of 1 3606 extending on previous pcp and hardness of approximation technique to that end one needs to develop a new proof framework and to borrow and extend ideas from several fields

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

infinite dimensional representations of finite dimensional algebras and amenability we present a novel quantitative approach to the representation theory of finite dimensional algebras motivated by the emerging theory of graph limits we introduce the rank spectrum of a finite dimensional algebra r over a countable field the elements of the rank spectrum are representations of the algebra into von neumann regular rank algebras and two representations are considered to be equivalent if they induce the same sylvester rank functions on r matrices based on this approach we can divide the finite dimensional algebras into three types finite amenable and non amenable representation types we prove that string algebras are of amenable representation type but the wild kronecker algebras are not here the amenability of the rank algebras associated to the limit points in the rank spectrum plays a very important part we also show that the limit points of finite dimensional representations of algebras of amenable representation type can always be viewed as representations of the algebra in the continuous ring invented by john von neumann in the 1930 s as an application in algorithm theory we introduce and study the notion of parameter testing of modules over finite dimensional algebras that is analogous to the testing of bounded degree graphs introduced by goldreich and ron we shall see that for string algebras all the reasonable stable parameters are testable

approximation algorithm for connected bounded degree deletion problem on unit disk graphs abstract in this paper we study the minimum connected k bounded degree node deletion problem min c kbdnd for a connected graph g a constant k and a weight function w v r a vertex set c v g is a kbdnd set if the maximum degree of graph g c is at most k if furthermore the subgraph of g induced by c is connected then c is a ckbdnd set the goal of minwkbdnd resp minwckbdnd is to find a kbdnd set resp ckbdnd set with the minimum weight in this paper we focus on their cardinality versions with w v 1 v v which are denoted as minkbdnd and minckbdnd this paper presents a 1 e and a 3 76 approximation algorithm for minkbdnd and minckbdnd on unit disk graphs respectively where 0 e 1 is an arbitrary constant

a two phase preconditioning strategy of sparse approximate inverse for indefinite matrices a two phase preconditioning strategy based on a factored sparse approximate inverse is proposed for solving sparse indefinite matrices in each phase the strategy first makes the original matrix diagonally dominant to enhance the stability by a shifting method and constructs an inverse approximation of the shifted matrix by utilizing a factored sparse approximate inverse preconditioner the two inverse approximation matrices produced from each phase are then combined to be used as a preconditioner experimental results show that the presented strategy improves the accuracy and the stability of the preconditioner on solving indefinite sparse matrices furthermore the strategy ensures that convergence rate of the preconditioned iterations of the two phase preconditioning strategy is much better than that of the standard sparse approximate inverse ones for solving indefinite 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

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

a new preconditioner for indefinite and asymmetric matrices we present a novel preconditioner for numerical solutions of large sparse linear systems with indefinite and asymmetric matrices this new preconditioner named as product preconditioner ps is constructed by two fairly simple preconditioners the distribution of eigenvalues and the form of the eigenvectors of the preconditioned matrix are analyzed moreover an upper bound on the degree of the minimal polynomial is also studied numerical experiments with several examples show that the proposed ps performs better than block diagonal preconditioner bd and block triangular preconditioner bt as well as the constraint preconditioner sc in terms of the number of iteration and computational time

parallel preconditioning with sparse approximate inverses a parallel preconditioner is presented for the solution of general sparse linear systems of equations a sparse approximate inverse is computed explicitly and then applied as a preconditioner to an iterative method the computation of the preconditioner is inherently parallel and its application only requires a matrix vector product the sparsity pattern of the approximate inverse is not imposed a priori but captured automatically this keeps the amount of work and the number of nonzero entries in the preconditioner to a minimum rigorous bounds on the clustering of the eigenvalues and the singular values are derived for the preconditioned system and the proximity of the approximate to the true inverse is estimated an extensive set of test problems from scientific and industrial applications provides convincing evidence of the effectiveness of this approach

independent packings in structured graphs packing a maximum number of disjoint triangles into a given graph g is np hard even for most classes of structured graphs in contrast we show that packing a maximum number of independent that is disjoint and nonadjacent triangles is polynomial time solvable for many classes of structured graphs including weakly chordal graphs asteroidal triple free graphs polygon circle graphs and interval filament graphs these classes contain other well known classes such as chordal graphs cocomparability graphs circle graphs circular arc graphs and outerplanar graphs our results apply more generally to independent packings by members of any family of connected graphs

joint multiuser detection of multidimensional constellations over fading channels we investigate the error performance of multidimensional constellations in the multiple access and broadcast channels more specifically we provide closed form expressions for the pairwise error probability pep of the joint maximum likelihood detection for multiuser signaling in the presence of additive white gaussian noise and rayleigh fading arbitrary numbers of users and multidimensional signal sets are assumed while the provided formula for the pep is a function of the dimension wise distances of the multidimensional constellation furthermore a useful upper bound on the average symbol error probability is also obtained through the union bound the analysis is applied to the sparse code multiple access systems the analytical results are validated successfully through simulations and show their importance in the multidimensional constellation design

a modified product preconditioner for indefinite and asymmetric generalized saddle point matrices in this paper we present a new modified product preconditioner mp for a class of large sparse linear systems with indefinite and asymmetric matrices the eigenvalue distribution and form of the eigenvectors of the presented new preconditioned matrix and its minimal polynomial are investigated some numerical experiments illustrate that the proposed new preconditioner performs better than block diagonal preconditioner block triangular preconditioner constraint preconditioner and product preconditioner