the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

improving the speed of parallel decimal multiplication hardware support for decimal computer arithmetic is regaining popularity one reason is the recent growth of decimal computations in commercial scientific financial and internet based computer applications newly commercialized decimal arithmetic hardware units use radix 10 sequential multipliers that are rather slow for multiplication intensive applications therefore the future relevant processors are likely to host fast parallel decimal multiplication circuits the corresponding hardware algorithms are normally composed of three steps partial product generation ppg partial product reduction ppr and final carry propagating addition the state of the art is represented by two recent full solutions with alternative designs for all the three aforementioned steps in addition ppr by itself has been the focus of other recent studies in this paper we examine both of the full solutions and the impact of a ppr only design on the appropriate one in order to improve the speed of parallel decimal multiplication we present a new ppg method fine tune the ppr method of one of the full solutions and the final addition scheme of the other thus assembling a new full solution logical effort analysis and 0 13 mum synthesis show at least 13 percent speed advantage but at a cost of at most 36 percent additional area consumption

low power divider the general objective of our work is to develop methods to reduce the energy consumption of arithmetic modules while maintaining the delay unchanged and keeping the increase in the area to a minimum here we illustrate some techniques for dividers realized in cmos technology the energy dissipation reduction is carried out at different levels of abstraction from the algorithm level down to the implementation or gate level we describe the use of techniques such as switching off not active blocks retiming dual voltage and equalizing the paths to reduce glitches also we describe modifications in the on the fly conversion and rounding algorithm and in the redundant representation of the residual in order to reduce the energy dissipation the techniques and modifications mentioned above are applied to a radix 4 divider realized with static cmos standard cells for which a reduction of 40 percent is obtained with respect to the standard implementation this reduction is expected to be about 60 percent if low voltage gates for dual voltage implementation are available the techniques used here should be applicable to a variety of arithmetic modules which have similar characteristics

domain decomposition methods for large linearly elliptic three dimensional problems abstract the idea of solving large problems using domain decomposition technique appears particularly attractive on present day large scale parallel computers but the performance of such techniques used on a parallel computer depends on both the numerical efficiency of the proposed algorithm and the efficiency of its parallel implementation the approach proposed herein splits the computational domain in unstructured subdomains of arbitrary shape and solves for unknown on the interface using the associated trace operator the steklov poincare operator on the continuous level or the schur complement matrix after a finite element discretization and a preconditioned conjugate gradient method this algorithm involves the solution of dirichlet and of neumann problems defined on each subdomain and which can be solved in parallel this method has been implemented on a cray 2 computer using multitasking and on an intel hypercube it was tested on a large scale industrial ill conditioned three dimensional linear elasticity problem which gives a fair indication of its performance in a real life environment in such situations the proposed method appears operational and competitive on both machines compared to standard techniques it yields faster results with far less memory requirements

high speed multioperand decimal adders there is increasing interest in hardware support for decimal arithmetic as a result of recent growth in commercial financial and internet based applications consequently new specifications for decimal floating point arithmetic have been added to the draft revision of the ieee 754 standard for floating point arithmetic this paper introduces and analyzes three techniques for performing fast decimal addition on multiple binary coded decimal bcd operands two of the techniques speculate bcd correction values and correct intermediate results while adding the input operands the first speculates over one addition the second speculates over two additions the third technique uses a binary carry save adder tree and produces a binary sum combinational logic is then used to correct the sum and determine the carry into the next more significant digit multioperand adder designs are constructed and synthesized for four to 16 input operands analyses are performed on the synthesis results and the merits of each technique are discussed finally these techniques are compared to several previous techniques for high speed decimal addition

multilevel schwarz methods for the biharmonic dirichlet problem the author considers the solution of the algebraic system of equations that result from the finite element discretization of the biharmonic equation some multilevel algorithms are designed and analyzed using a schwarz framework both additive and multiplicative variants of the algorithms are considered and condition number estimates for the additive algorithms and the energy norm estimates for the error propagation operator of the multiplicative algorithms are given it is noted that for a proper ordering the iterative operators of the multiplicative algorithms correspond to the error propagation operators of certain v cycle multigrid methods

balancing domain decomposition for mixed finite elements the rate of convergence of the balancing domain decomposition method applied to the mixed finite element discretization of second order elliptic equations is analyzed the balancing domain decomposition method introduced by mandel in 24 is a substructuring method that involves at each iteration the solution of a local problem with dirichlet data a local problem with neumann data and a coarse grid problem to propagate information globally and to insure the consistency of the neumann problems it is shown that the condition number grows at worst like the logarithm squared of the ratio of the subdomain size to the element size in both two and three dimensions and for elements of arbitrary order the bounds are uniform with respect to coefficient jumps of arbitrary size between subdomains the key component of our analysis is the demonstration of an equivalence between the norm induced by the bilinear form on the interface and the h 1 2 norm of an interplant of the boundary data computational results from a message passing parallel implementation on an intel delta machine demonstrate the scalability properties of the method and show almost optimal linear observed speed up for up to 64 processors

a bddc algorithm with deluxe scaling for three dimensional h curl problems in our paper we present and analyze a bddc algorithm for a class of elliptic problems in the three dimensional h curl space compared with existing results our condition number estimate requires fewer assumptions and also involves two fewer powers of log h h making it consistent with optimal estimates for other elliptic problems here h his the maximum of hi hi over all subdomains where hi and hi are the diameter and the smallest element diameter for the subdomain i the analysis makes use of two recent developments the first is our new approach to averaging across the subdomain interfaces while the second is a new technical tool that allows arguments involving trace classes to be avoided furthermore numerical examples are presented to confirm the theory and demonstrate the importance of the new averaging approach in certain cases

schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions domain decomposition methods provide powerful preconditioners for the iterative solution of the large systems of algebraic equations that arise in finite element or finite difference approximations

a polylogarithmic bound for an iterative substructuring method for spectral elements in three dimensions iterative substructuring methods form an important family of domain decomposition algorithms for elliptic finite element problems a p version finite element method based on continuous piecewise q p functions is considered for second order elliptic problems in three dimensions this special method can also be viewed as a conforming spectral element method an iterative method is designed for which the condition number of the relevant operator grows only in proportion to 1 log p 2 this bound is independent of jumps in the coefficient of the elliptic problem across the interfaces between the subregions numerical results are also reported which support the theory

a parallel implementation of an iterative substructuring algorithm for problems in three dimensions numerical results from a parallel implementation of a class of iterative substructuring algorithms are reported the algorithms are for solving self adjoint elliptic partial differential equations in three dimensions results are given for several variants of the algorithm in the first variant exact interior solvers are used in the second one multigrid v cycle is used to solve the interior problems approximately the results are compared with theoretical behavior of the algorithm reported in previous work a numerical experiment involving the equations of linear elasticity is also included

a fast spectral element solver combining static condensation and multigrid techniques we propose a spectral element multigrid method for the two dimensional helmholtz equation discretized on regular grids combining p multigrid with static condensation the method achieves nearly linear complexity with an order independent convergence rate for solving the condensed equations for smoothing we consider two groups of edge based relaxation schemes the best of which attains a multigrid convergence rate of 0 014 to 0 028 numerical experiments have been carried out that demonstrate the robustness of the approach for orders up to 32 and a total of 109 degrees of freedom in comparison with a fast finite difference solver the latter is clearly outperformed already for errors of one percent or lower

some estimates for a weighted l 2 projection this paper is devoted to the error estimates for some weighted l2 projections nearly optimal estimates are obtained these estimates can be applied to the analysis of the usual multigrid method multilevel preconditioner and domain decomposition method for solving elliptic boundary problems whose coefficients have large jump discontinuities

efficient variants of the vertex space domain decomposition algorithm several variants of the vertex space algorithm of smith for two dimensional elliptic problems are described the vertex space algorithm is a domain decomposition method based on nonoverlapping subregions in which the reduced schur complement system on the interface is solved using a generalized block jacobi type preconditioner with the blocks corresponding to the vertex space edges and a coarse grid two kinds of approximations are considered for the edge and vertex space subblocks one based on fourier approximation and another based on an algebraic probing technique in which sparse approximations to these subblocks are computed our motivation is to improve the efficiency of the algorithm without sacrificing the optimal convergence rate numerical and theoretical results on the performance of these algorithms including variants of an algorithm of bramble pasciak and schatz are presented

convergence estimates for product iterative methods with applications to domain decomposition in this paper we consider iterative methods for the solution of symmetric positive definite problems on a space which are defined in terms of products of operators defined with respect to a number of subspaces the simplest algorithm of this sort has an error reducing operator which is the product of orthogonal projections onto the complement of the subspaces new normreduction estimates for these iterative techniques will be presented in an abstract setting applications are given for overlapping schwarz algorithms with many subregions for finite element approximation of second order elliptic problems

discontinuous galerkin methods for a model of population dynamics with unbounded mortality discontinuous galerkin finite element methods in age time domain are proposed to approximate a possibly nonsmooth solution to the model of population dynamics with unbounded mortality coefficient function the strong stability is established and a priori l 2 error estimates are obtained we show that the error estimates are optimal both in rate and in regularity away from the maximum age nonnegativity of the solution is also shown the scheme is explicit and the solution is computed strip by strip marching in time several numerical examples are presented

digit recurrence dividers with reduced logical depth in this paper we propose a class of division algorithms with the aim of reducing the delay of the selection of the quotient digit by introducing more concurrency and flexibility in its computation from the proposed class of algorithms we select one that moves part of the selection function out of the critical path with a corresponding reduction in the critical path compared with existing alternatives we present the algorithm and describe the architectures for radix 4 and for radix 16 for radix 16 we use the scheme of overlapping two radix 4 stages in both cases radix 4 and radix 16 we show that our algorithms allow the design of units with well balanced critical paths with consequent decreases of the cycle times moreover in the radix 16 case we include some additional speculation techniques to estimate the speedup we used a rough timing model based on logical effort for both radices we estimate a speedup of about 25 percent with respect to previous implementations in the radix 4 case this is achieved by using roughly the same area while in the radix 16 case the area is increased by about 30 percent we verified our estimations by performing a synthesis of the radix 4 units

robust preconditioners for linear elasticity fem analyses summary this paper deals with two forms of preconditioner which can be easily used with a conjugate gradient solver to replace a direct solution subroutine in a traditional engineering nite element package they are tested in such a package final over a range of 2 d and 3 d elasticity problems from geotechnical engineering quadratic basis functions are used a number of modications to the basic incomplete choleski ic 0 factorization preconditioner are considered an algorithm to reduce positive o diagonal entries is shown in numerical experiments to ensure stability but at the expense of slow convergence an alternative algorithm of jennings and malik is more successful and a relaxation parameter is introduced which can make a further signicant improvement in performance while maintaining stability a heuristic for determining a near optimal value of is proposed a second form of preconditioning symmetrically scaled element by element due to bartelt is also shown to perform robustly over a range of problems it does not require assembly of the global stiness matrix and has great potential for parallelization 1997 by john wiley sons ltd

a study of decimal left shifters for binary numbers the importance of decimal floating point dfp arithmetic has been growing in the last years and specifications for it are included in the revised ieee 754 standard for floating point arithmetic ieee 754 2008 ieee 754 2008 specifies a binary encoding for decimal significands in which the significands of dfp numbers are represented as unsigned binary integers for this representation which is commonly referred to as the binary integer decimal bid encoding fast decimal left shifting of a binary number is useful for operand alignment normalization overflow avoidance and quantize operations a decimal left shift of an unsigned binary integer i by s digit positions corresponds to multiplying i by 10 s this paper presents the theory and design of decimal left shifters for binary numbers the designs perform decimal left shifting using optimized constant multiplications by selected powers of ten we propose and analyze different combinational and sequential decimal left shifter architectures for binary numbers the designs are compared to one another in terms of area and delay using both theoretical estimates and synthesis results for 16 digit 54 bit binary inputs and 4 bit shift amounts the results indicate that an optimized radix 2 decimal shifter implementing partial shifters in carry save format has 80 less area and 68 less delay than a lookup table followed by a binary multiplier to compute ix10 s when both designs are optimized for delay when both designs are optimized for area the radix 2 decimal shifter has 89 less area and 31 less delay

the neumann dirichlet domain decomposition method with inexact solvers on the subdomains we consider the neumann dirichlet domain decomposition method for the solution of linear elliptic boundary value problems we study the following question suppose that the auxiliary problems on the subdomains are not solved exactly but only with a fixed mesh size independent accuracy does the speed of convergence remain mesh size independently bounded we show that the answer is no in general but that mesh size independent convergence can be obtained if the accuracy requirement on the subsolvers becomes increasingly severe as the mesh size tends to zero

a domain decomposition algorithm for elliptic problems in three dimensions most domain decomposition algorithms have been developed for problems in two dimensions one reason for this is the difficulty in devising a satisfactory easy to implement robust method of providing global communication of information for problems in three dimensions several methods that work well in two dimension do not perform satisfactorily in three dimensions a new iterative substructuring algorithm for three dimensions is proposed it is shown that the condition number of the resulting preconditioned problem is bounded independently of the number of subdomains and that the growth is quadratic in the logarithm of the number of degrees of freedom associated with a subdomain the condition number is also bounded independently of the jumps in the coefficients of the differential equation between subdomains the new algorithm also has more potential parallelism than the iterative substructuring methods previously proposed for problems in three dimensions

schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems part i algorithms and numerical results we describe sequential and parallel algorithms based on the schwarz alternating method for the solution of mixed finite element discretizations of elliptic problems using the raviart thomas finite element spaces these lead to symmetric indefinite linear systems and the algorithms have some similarities with the traditional block gauss seidel or block jacobi methods with overlapping blocks the indefiniteness requires special treatment the sub blocks used in the algorithm correspond to problems on a coarse grid and some overlapping subdomains and is based on a similar partition used in an algorithm of dryja and widlund for standard elliptic problems if there is sufficient overlap between the subdomains the algorithm converges with a rate independent of the mesh size the number of subdomains and discontinuities of the coefficients extensions of the above algorithms to the case of local grid refinement is also described convergence theory for these algorithms will be presented in a subsequent paper

a radix 10 digit recurrence division unit algorithm and architecture in this work we present a radix 10 division unit that is based on the digit recurrence algorithm the previous decimal division designs do not include recent developments in the theory and practice of this type of algorithm which were developed for radix 2k dividers in addition to the adaptation of these features the radix 10 quotient digit is decomposed into a radix 2 digit and a radix 5 digit in such a way that only five and two times the divisor are required in the recurrence moreover the most significant slice of the recurrence which includes the selection function is implemented in radix 2 avoiding the additional delay introduced by the radix 10 carry save additions and allowing the balancing of the paths to reduce the cycle delay the results of the implementation of the proposed radix 10 division unit show that its latency is close to that of radix 16 division units comparable dynamic range of significant and it has a shorter latency than a radix 10 unit based on the newton raphson approximation

a gradient method approach to optimization based multidisciplinary simulations and nonoverlapping domain decomposition algorithms it has been shown recently that optimization based nonoverlapping domain decomposition algorithms are connected to many well known algorithms using a gradient type iterative strategy for the optimization problem we present further discussion on how to develop various algorithms that can integrate subdomain solvers into a solver for the problem in the whole domain in particular the algorithms we discuss can be used to develop efficient solvers of multidisciplinary problems which are constructed using existing subdomain solvers without the need for making changes in the latter

multilevel schwarz methods for elliptic problems with discontinuous coefficients in three dimensions multilevel schwarz methods are developed for a conforming finite element approximation of second order elliptic problems we focus on problems in three dimensions with possibly large jumps in the coefficients across the interface separating the subregions we establish a condition number estimate for the iterative operator which is independent of the coefficients and grows at most as the square of the number of levels we also characterize a class of distributions of the coefficients called quasi monotone for which the weighted l 2 projection is stable and for which we can use the standard piecewise linear functions as a coarse space in this case we obtain optimal methods i e bounds which are independent of the number of levels and subregions we also design and analyze multilevel methods with new coarse spaces given by simple explicit formulas we consider nonuniform meshes and conclude by an analysis of multilevel iterative substructuring methods

schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems part ii convergence theory in this paper we discuss bounds for the convergence rates of several domain decomposition algorithms to solve symmetric indefinite linear systems arising from mixed finite element discretizations of elliptic problems the algorithms include schwarz methods and iterative refinement methods on locally refined grids the implementation of schwarz and iterative refinement algorithms have been discussed in part i a discussion on the stability of mixed discretizations on locally refined grids is included and quantiative estimates for the convergence rates of some iterative refinement algorithms are also derived