on distributed computing for functions with certain structures the problem of distributed function computation is studied where functions to be computed is not necessarily symbol wise a new method to derive a converse bound for distributed computing is proposed from the structure of functions to be computed information that is inevitably conveyed to the decoder is identified and the bound is derived in terms of the optimal rate needed to send that information the class of informative functions is introduced and for the class of smooth sources the optimal rate for computing those functions is characterized furthermore for i i d sources with joint distribution that may not be full support functions that are composition of symbol wise function and the type of a sequence are considered and the optimal rate for computing those functions is characterized in terms of the hypergraph entropy as a byproduct our method also provides a conceptually simple proof of the known fact that computing a boolean function may require as large rate as reproducing the entire source

preconditioned minimal residual methods for chebyshev spectral calculations abstract the problem of preconditioning the pseudospectral chebyshev approximation of an elliptic operator is considered the numerical sensitivity to variations of the coefficients of the operator are investigated for two classes of preconditioning matrices one arising from finite differences the other from finite elements the preconditioned system is solved by a conjugate gradient type method and by a dufort frankel method with dynamical parameters the methods are compared on some test problems with the richardson method 13 and with the minimal residual richardson method 21

finite element preconditioning for spectral multigrid methods abstract the good performance of finite element preconditioning for the algebraic spectral systems is demonstrated different types of multigrid strategies are compared the most efficient strategy consists of a pure richardson relaxation preconditioned by one v cycle of a finite element multigrid method the good smoothing properties of line relaxation become obvious the computational work for the various methods is estimated

multigrid methods for combined finite difference and fourier problems abstract multigrid methods for combined finite difference and fourier pseudospectral problems are considered applications to two and three dimensional elliptic systems are given suitable relaxation schemes for isotropic and anisotropic problems are presented numerical results demonstrate the efficiency of these methods

a posteriori steered p robust multigrid with optimal step sizes and adaptive number of smoothing steps we develop a multigrid solver steered by an a posteriori estimator of the algebraic error we adopt the context of a second order elliptic diffusion problem discretized by conforming finite elements of arbitrary polynomial degree p 1 our solver employs zero pre and one post smoothing by the overlapping schwarz block jacobi method and features an optimal choice of the step sizes in the smoothing correction on each level by line search this leads to a simple pythagorean formula of the algebraic error in the next step in terms of the current error and level wise and patch wise error reductions we show the two following results and their equivalence the solver contracts the algebraic error independently of the polynomial degree p and the estimator represents a two sided p robust bound on the algebraic error the p robustness results are obtained by carefully applying the results of schoberl et al ima j numer anal 28 2008 pp 1 24 for one mesh combined with a multilevel stable decomposition for piecewise affine polynomials of xu et al multiscale nonlinear and adaptive approximation springer berlin 2009 pp 599 659 we consider quasi uniform or graded bisection simplicial meshes and prove mild dependence on the number of mesh levels for minimal h 1 regularity and complete independence for h 2 regularity we also present a simple and effective way for the solver to adaptively choose the number of post smoothing steps necessary at each individual level yielding a yet improved error reduction numerical tests confirm p robustness and show the benefits of the adaptive number of smoothing steps

improved condition number for spectral methods introduction d une methode spectrale avec un nombre de conditionnement o n 2 n etant le degre maximum des polynomes

on the redundancy of slepian wolf coding in this paper the redundancy of both variable and fixed rate slepian wolf coding is considered given any jointly memoryless source side information pair xi yi i 1 infin with finite alphabet the redundancy rn isinn of variable rate slepian wolf coding of x1 n with decoder only side information y1 n depends on both the block length n and the decoding block error probability isinn and is defined as the difference between the minimum average compression rate of order n variable rate slepian wolf codes having the decoding block error probability less than or equal to isinn and the conditional entropy h x y where h x y is the conditional entropy rate of the source given the side information the redundancy of fixed rate slepian wolf coding of x1 n with decoder only side information y1 n is defined similarly and denoted by rf n isinn it is proved that under mild assumptions about isinn rn isinn dvradic log isinn n oradic log isinn n and rf n isinn dfradic log isinn n o radic log isinn n where df and dnu are two constants completely determined by the joint distribution of the source side information pair since dv is generally smaller than df our results show that variable rate slepian wolf coding is indeed more efficient than fixed rate slepian wolf coding

defect correction for the advection diffusion equation abstract a defect correction method for the advection diffusion equation is presented the discretization is performed by 2nd order finite difference schemes schemes where the 2nd order upstream scheme is combined with the standard central scheme for preconditioning the usual first order upstream scheme is employed the defect correction iteration is used for relaxation inside a multigrid procedure it is shown that the smoothing analysis yields rather pessimistic results with rates of about 0 7 0 8 in the practical computation the discretization error is reached in 2 v cycles numerical results are presented which demonstrate the high efficiency of our treatment

functional identities in upper triangular matrix rings abstract let r be a subring of a ring q both having the same unity we prove that if r is a d free subset of q then the upper triangular matrix ring t n r is a d free subset of t n q for any n n

splitting techniques for the pseudospectral approximation of the unsteady stokes equations a pseudospectral or collocation approximation of the unsteady stokes equations is considered using the uzawa algorithm the spectral system is decoupled into a helmholtz equation and an equation with a pseudo laplace operator it is proven that the eigenvalues of the pseudo laplacian are real and negative except one zero eigenvalue belonging to the constant mode the pseudospectral method avoids spurious modes since the pressure is approximated with lower order two degrees lower polynomials than the velocity only one grid no staggered grids with the standard chebyshev gauss lobatto nodes is used for discretization for the iterative solution of the spectral system an effective preconditioner is necessary here effective finite difference preconditioners for the spectral pseudo laplacian are presented

commuting traces and commutativity preserving maps on triangular algebras abstract let a be a triangular algebra the problem of describing the form of a bilinear map b a a a satisfying b x x x x b x x for all x a is considered as an application commutativity preserving maps and lie isomorphisms of certain triangular algebras e g upper triangular matrix algebras and nest algebras are determined

spectral multigrid methods for domain decomposition problems using patching techniques abstract we consider spectral domain decomposition problems that use patching techniques the resulting discrete systems are solved by the very efficient spectral multigrid methods for relaxation we employ a preconditioned richardson iteration effective preconditioning is achieved by one or more steps of a line relaxation method for the corresponding finite difference scheme numerical results are presented for rectangular and l shaped domains furthermore a cross point problem is considered

wyner ziv theory for a general function of the correlated sources corresp a source coding problem is considered for the wyner ziv type system where the decoder is required to estimate the value of some function of the encoder input and the side information the rate distortion function is established for this system and for some binary cases parametric expressions are obtained to enable numerical calculations

a spectral multigrid method for the stokes problem in streamfunction formulation abstract suitable spectral multigrid components for the stokes problem in streamfunction formulation are presented we split the stokes problem into a system of two equations with the laplace operator we discretize by means of a collocation or pseudo spectral method preconditioning with finite differences is employed numerical results for the spectral multigrid method are presented

a dichotomy of functions f x y of correlated sources x y consider separate encoding of correlated sources x n x l cdots x n y n y l cdots y n for the decoder to reliably reproduce a function f x i y i n i 1 we establish the necessary and sufficient condition for the set of all achievable rates to coincide with the slepian wolf region whenever the probability density p x y is positive for all x y

moderate deviations in channel coding we consider block codes whose rate converges to the channel capacity with increasing blocklength at a certain speed and examine the best possible decay of the probability of error for discrete memoryless channels we prove that a moderate deviation principle holds for all convergence rates between the large deviation and the central limit theorem regimes

functional identities of degree 2 in triangular rings abstract let r be a triangular ring the problem of describing the form of additive maps f 1 f 2 g 1 g 2 r r satisfying functional identity f 1 x y f 2 y x xg 2 y yg 1 x 0 for all x y r is considered as an application generalized inner biderivations and commuting additive maps of certain triangular rings are determined

an information spectrum approach to weak variable length source coding with side information this paper studies variable length vl source coding of general sources with side information novel one shot coding bounds for slepian wolf sw coding which give nonasymptotic tradeoff between the error probability and the codeword length of vl sw coding are established one shot results are applied to asymptotic analysis and a general formula for the optimal coding rate achievable by weakly lossless vl sw coding i e vl sw coding with vanishing error probability is derived our general formula reveals how the encoder side information and or vl coding improve the optimal coding rate in the general setting in addition it is shown that if the encoder side information is useless in weakly lossless vl coding then it is also useless even in the case where the error probability may be positive asymptotically

spectral collocation on triangular elements we consider the poisson problem on a segment of the unit disc and on triangles on the segment we transform the poisson problem by means of polar coordinates in these new coordinates we have a problem in a rectangle which can easily be mapped onto the square here standard chebyshev collocation techniques can be applied then the segment is mapped onto a triangle where the same spectral scheme may be used by numerical tests we observed the expected high spectral accuracy due to the corner singularity a singular behaviour of the solution can be expected here we improved the accuracy by auxiliary mapping techniques further it is shown that finite difference preconditioning can be successfully applied in order to construct an efficient iterative solver finally a domain decomposition technique is applied to the patching of a rectangular and a triangular element

a simple proof of the ahlswede csiszar one bit theorem corresp it is proved that if x y are two finite alphabet correlated sources with p x y 0 for all x y in cal x times cal y and if a function f x y is alpha sensitive then the rate r of transmission from x to y necessary to compute f x y reliably must be greater than h x y the same result holds if the function is highly sensitive and for every x 1 neq x 2 in cal x then the number of elements y in cal y with p x l y cdot p x 2 y 0 is different from one

improved spectral multigrid methods for periodic elliptic problems abstract the spectral multigrid method for periodic elliptic problems is examined several modifications are introduced including a midpoint pseudospectral discretization which eliminates the need for filtering the highest fourier mode and new relaxation schemes for isotropic and anisotropic problems numerical results are presented demonstrating substantial increases in efficiency and accuracy over previous methods

spectral collocation methods and polar coordinate singularities this paper considers the numerical solution of elliptic differential equations on the unit disk using polar coordinates the disk is mapped onto a rectangle the resulting transformed problem is solved by a method related to collocation since the origin is a coordinate singularity some natural trial functions are singular there and a special technique is applied to use zero as a collocation point for poisson and helmholtz equations a fast algorithm with an operation count of o n2 log n is presented numerical results show the different stability and convergence properties of the algorithms

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

a dichotomy of functions in distributed coding an information spectral approach the problem of distributed data compression for function computation is considered where 1 the function to be computed is not necessarily symbolwise function and 2 the information source has memory and may not be stationary nor ergodic we introduce the class of smooth sources and give a sufficient condition on functions so that the achievable rate region for computing coincides with the slepian wolf region i e the rate region for reproducing the entire source for any smooth sources moreover for symbolwise functions the necessary and sufficient condition for the coincidence is established our result for the full side information case is a generalization of the result by ahlswede and csiszar to sources with memory our dichotomy theorem is different from han and kobayashi s dichotomy theorem which reveals an effect of memory in distributed function computation all results are given not only for fixed length coding but also for variable length coding in a unified manner furthermore for the full side information case the error probability in the moderate deviation regime is also investigated

chebyshev 3 d spectral and 2 d pseudospectral solvers for the helmholtz equation two chebyshev solvers are presented for the linear helmholtz equation the first algorithm is a 3 d direct spectral solver based on a diagonalization technique whilst the second performs an iterative pseudospectral 2 d calculation with finite difference preconditioning both techniques handle general nonhomogeneous boundary conditions computing times and accuracies of the two methods are compared

functional identities on tensor products of algebras abstract let r and s be unital algebras we show that if x is a d free subset of r and s is finite dimensional then the set x x s x x s s is a d free subset of the algebra r s the assumption that s is finite dimensional turns out to be necessary in general however we show that some important functional identities have only standard solutions on x even when s is infinite dimensional

spectral multigrid techniques for the navier stokes equations abstract we present a spectral multigrid method for the effective solution of the navier stokes equations for incompressible fluids here we consider the vorticity streamfunction formulation discretization in time is performed by a second order semi implicit scheme where the diffusive term is treated in an implicit manner while the convective term is evaluated explicitly for the discretization of the resulting stokes type problem we introduce a new spectral collocation method this scheme is highly accurate and allows efficient solution by spectral multigrid methods we propose an efficient finite difference preconditioner numerical results are presented which substantiate the usefulness of our concept

splitting techniques for the unsteady stokes equations a pseudospectral or collocation approximation of the unsteady stokes equations is presented using the uzawa algorithm the spectral system is decoupled into helmholtz equations for the velocity components and an equation with the pseudo laplacian for the pressure both the velocity components and the pressure are approximated by polynomials of the same degree n compatible velocity and pressure approximations are constructed to ensure that the discrete problem is well posed the resulting discrete velocity field is identically divergence free only one grid no staggered grids with the standard chebyshev gauss lobatto nodes is used here we further compare our treatment with an inhomogeneous neumann condition for the pressure this scheme was proposed in g karniadakis m israeli and s a orszag j comp phys 97 1991 pp 414 443 the comparison shows that the neumann condition yields somewhat better approximations for the pressure in the time discretization a high order backward differentiation scheme for the intermediate velocity is combined with a high order extrapolant for the pressure it is numerically shown that a stable third order method in time can be achieved in the case of periodic boundary conditions this result is confirmed by fourier analysis

biderivations of triangular algebras abstract let a be a triangular algebra a bilinear map a a a is called a biderivation if it is a derivation with respect to both arguments in this paper we define the concept of an extremal biderivation and prove that under certain conditions a biderivation of a triangular algebra a is a sum of an extremal and an inner biderivation the main result is then applied to block upper triangular matrix algebras and nest algebras we also consider the question when a derivation of a triangular algebra is an inner derivation

line relaxation for spectral multigrid methods abstract spectral multigrid methods for elliptic problems with dirichlet and periodic boundary conditions are examined the multigrid convergence is improved by certain line relaxation techniques which are suggested by a smoothing analysis furthermore several choices of relaxation parameters are compared numerical results are presented showing the gain in efficiency and accuracy over previous methods

spectral methods with sparse matrices spectral methods employ global polynomials for approximation hence they give very accurate approximations for smooth solutions unfortunately for dirichlet problems the matrices involved are dense and have condition numbers growing aso n 4 for polynomials of degree n in each variable we propose a new spectral method for the helmholtz equation with a symmetric and sparse matrix whose condition number grows only aso n 2 certain algebraic spectral multigrid methods can be efficiently used for solving the resulting system numerical results are presented which show that we have probably found the most effective solver for spectral systems

generalized biderivations of nest algebras let n be a nest on a complex separable hilbert space h and n be the associated nest algebra in this paper we prove that every biderivation of n is an inner biderivation if and only if dim 0 1 or dimh 1 and that every generalized biderivation of n is an inner generalized biderivation if dim 0 1 and dimh 1

on biderivations of upper triangular matrix rings abstract let r and s be rings with identity m be a unitary r s bimodule and t r m 0 s be the upper triangular matrix ring determined by r s and m let e ij be the standard matrix unit in this paper we show that every biderivation of t is decomposed into the sum of three biderivations d and where d e 11 e 11 0 is an extremal biderivation and is a special kind of biderivation using this characterization we determine the structure of biderivations of the ring t n r n 2 of all n n upper triangular matrices over r and show that in the special case when r is a noncommutative prime ring every biderivation of t n r is inner this extends some results of benkovic 2009 1

spectral multigrid methods for elliptic equations ii a detailed description of spectral multigrid methods is provided this includes the interpolation and coarse grid operators for both periodic and dirichlet problems the spectral methods for periodic problems use fourier series and those for dirichlet problems are based upon chebyshev polynomials an improved preconditioning for dirichlet problems is given numerical examples and practical advice are included

defect correction for convection dominated flow a defect correction scheme with the first order upwind preconditioner is considered by fourier analysis the preconditioning properties for the second order upwind scheme the central scheme and spectral methods are examined since the eigenvalues of the preconditioned operator are complex the generalized minimal residual iteration is used for the iterative solution this procedure is applied to the boussinesq flow problem in vorticity streamfunction formulation numerical results are presented for increasing rayleigh numbers

functional identities of degree 2 in arbitrary triangular rings abstract the aim of the paper is to give the solutions of functional identities of degree 2 in arbitrary triangular rings which generalizes a recent result due to eremita as applications commuting additive maps and generalized inner biderivations of arbitrary triangular rings are determined

a 3d spectral multigrid method we consider a 3d spectral multigrid method for 3d elliptic problems with arbitrary anisotropies the smoothing method is a preconditioned richardson iteration where a pseudospectral discretization is used on the right hand side and a finite difference discretization on the left hand side in each step of this iteration a finite difference problem is solved approximately by plane relaxation sweeps the plane relaxation requires the solution of 2d finite difference problems these problems are solved approximately using 2d multigrid cycles

commuting centralizing traces and lie triple isomorphisms on triangular algebras revisited abstract the aim of the paper is to give a description of commuting centralizing traces of bilinear maps and lie triple isomorphisms on certain triangular algebras as consequences commuting centralizing traces of bilinear maps and lie triple isomorphisms on upper triangular matrix algebras over a 2 torsion free commutative ring are completely determined

to get a bit of information may be as hard as to get full information the following coding problem for correlated discrete memoryless sources is considered the two sources can be separately block encoded and the values of the encoding functions are available to a decoder who wants to answer a certain question concerning the source outputs typically this question has only a few possible answers even as few as two the rates of the encoding functions must be found that enable the decoder to answer this question correctly with high probability it is proven that these rates are often as large as those needed for a full reproduction of the outputs of both sources furthermore if one source is completely known at the decoder this phenomenon already occurs when what is asked for is the joint type joint composition of the two source output blocks or some function thereof such as the hamming distance of the two blocks or for alphabet size at least three just the parity of this hamming distance

lie gradings on associative algebras abstract in this paper we apply the method of functional identities to the study of group gradings by an abelian group g on simple lie algebras under very mild restrictions on the grading group or the base field of coefficients