ripple compensation for a class d amplifier this paper presents the first detailed mathematical analysis of the ripple compensation technique for reducing audio distortion in a class d amplifier with negative feedback the amplifier converts a relatively low frequency audio signal to a high frequency train of rectangular pulses whose widths are slowly modulated according to the audio signal pulse width modulation pwm distortion manifests itself through unwanted audio frequency harmonics that arise in the output due to nonlinearities inherent in the design in this paper we first develop a small signal model which describes the fate of small amplitude perturbations to a constant input and demonstrate how this traditional engineering tool may be extended to allow one to infer the most significant contributions to the full output in response to a general audio input we then compute the audio output of the amplifier through a perturbation expansion based on the ratio between audio and switching frequencies our results explicitly demonstrate ho

non commutative martingale transforms abstract we prove that non commutative martingale transforms are of weak type 1 1 more precisely there is an absolute constant c such that if m is a semi finite von neumann algebra and m n n 1 is an increasing filtration of von neumann subalgebras of m then for any non commutative martingale x x n n 1 in l 1 m adapted to m n n 1 and any sequence of signs e n n 1 e 1 x 1 n 2 n e n x n x n 1 1 c x n 1 for every n 2 this generalizes a result of burkholder from classical martingale theory to non commutative setting and answers positively a question of pisier and xu as applications we get the optimal order of the unconditional martingale differences umd constants of the schatten class s p when p similarly we prove that the umd constant of the finite dimensional schatten class s n 1 is of order log n 1 we also discuss the pisier xu non commutative burkholder gundy inequalities

smoothed aggregation multigrid solvers for high order discontinuous galerkin methods for elliptic problems abstract we develop a smoothed aggregation based algebraic multigrid solver for high order discontinuous galerkin discretizations of the poisson problem algebraic multigrid is a popular and effective method for solving the sparse linear systems that arise from discretizing partial differential equations however high order discontinuous galerkin discretizations have proved challenging for algebraic multigrid the increasing condition number of the matrix and loss of locality in the matrix stencil as p increases in addition to the effect of weakly enforced dirichlet boundary conditions all contribute to the challenging algebraic setting we propose a smoothed aggregation approach that addresses these difficulties in particular the approach effectively coarsens degrees of freedom centered at the same spatial location as well as degrees of freedom at the domain boundary moreover the character of the near null space particularly at the domain boundary is captured by interpolation one classic prolongation smoothing step of weighted jacobi is also shown to be ineffective at high order and a more robust energy minimization approach is used along with block relaxation that more directly utilizes the block diagonal structure of the discontinuous galerkin discretization finally we conclude by examining numerical results in support our proposed method

a boundedness criterion for generalized calderon zygmund operators in 1965 a p calderon showed the l2 boundedness of the so called first calderon commutator this is one of the first examples of a non convolution operator associated to a kernel which satisfies certain size and smoothness properties comparable to those of the kernel of the hilbert transform these properties together with the l2 boundedness imply the lp boundedness for all p s in 1 oo many operators in analysis such as certain classes of pseudo differential operators and the cauchy integral operator on a curve are associated with kernels having these properties for these operators one of the major questions is if they are bounded on l2 we are going to give necessary and sufficient conditions for such an operator to be bounded on l2 they are essentially that the images of the function 1 under the actions of the operator and its adjoint both lie in bmo in the case of the aforementioned first commutator this can be checked by an integration by parts in the first section we present some basic notions and state the theorem which is proved in sections 2 and 3 in section 4 we show how to recover some classical results in sections 5 and 6 we construct a functional calculus for small perturbations of a and in section 7 we show a connection between the theory of calderon zygmund operators and kato s conjecture it is a pleasure to express our thanks to r r coifman and y meyer for suggesting many elegant simplifications in our proofs and most of the applications we also wish to thank stephen semmes for several pertinent remarks

embedding of the operator space oh and the logarithmic little grothendieck inequality we use voiculescu s concept of free probability to construct a completely isomorphic embedding of the operator space oh in the predual of a von neumann algebra we analyze the properties of this embedding and determine the operator space projection constant of oh n frac 1 108 sqrt frac n 1 ln n le inf p mathcal b ell 2 to oh n p 2 p left p right cb le 288 pi sqrt frac 2n 1 ln n the lower estimate is a recent result of pisier and shlyakhtenko that improves an estimate of order 1 1 lnn of the author the additional factor 1 sqrt 1 ln n indicates that the operator space oh n behaves differently than its classical counterpart ell 2 n we give an application of this formula to positive sesquilinear forms on mathcal b ell 2 this leads to logarithmic characterization of c algebras with the weak expectation property introduced by lance

mapping properties of operator valued pseudo differential operators abstract in this paper we investigate the mapping properties of pseudo differential operators with operator valued symbols we prove the boundedness of regular symbols on sobolev spaces h 2 r d l 2 m and besov spaces b p q r d l p m for r and 1 p q as well as the boundedness of forbidden symbols on h 2 r d l 2 m and b p q r d l p m for 0 and 1 p q thanks to the smooth atomic decomposition of the operator valued triebel lizorkin spaces f 1 c r d m obtained in our previous paper we establish the f 1 c regularity of regular symbols for every r and the f 1 c regularity of forbidden symbols for 0 as applications we obtain the same results on the usual and quantum tori

pseudo localization of singular integrals and noncommutative calderon zygmund theory abstract the weak type 1 1 boundedness of singular integrals acting on matrix valued functions has remained open since the 1980s mainly because the methods provided by the vector valued theory are not strong enough in fact we can also consider the action of generalized calderon zygmund operators on functions taking values in any other von neumann algebra our main tools for its solution are two first the lack of some classical inequalities in the noncommutative setting forces to have a deeper knowledge of how fast a singular integral decreases l 2 sense outside of the support of the function on which it acts this gives rise to a pseudo localization principle which is of independent interest even in the classical theory second we construct a noncommutative form of calderon zygmund decomposition by means of the recent theory of noncommutative martingales this is a corner stone in the theory as application we obtain the sharp asymptotic behavior of the constants for the strong l p inequalities also unknown up to now our methods settle some basics for a systematic study of a noncommutative calderon zygmund theory

an operator valued tb theorem we establish conditions similar to the tb theorem of david journe and semmes which guarantee the boundedness of an integral transformation t with l x valued kernel on lxp rn where 1 p and x is a banach space with the unconditionality property of martingale differences umd

interpolation between non commutative bmo and non commutative lp spaces we prove that for 1 p q the analogue of the classical result bmo lp pq lq holds in the setting of a finite von neumann algebra m equipped with an increasing filtration mn n 1 of von neumann subalgebras we also obtain the corresponding results for the real method of interpolation we discuss the appropriate operator space matrix norms and show that these interpolation results hold in the category of operator spaces

linear bounds for calderon zygmund operators with even kernel on umd spaces it is well known that several classical results about calderon zygmund singular integral operators can be extended to x valued functions if and only if the banach space x has the umd property the dependence of the norm of an x valued calderon zygmund operator on the umd constant of the space x is conjectured to be linear we prove that this is indeed the case for sufficiently smooth calderon zygmund operators with cancellation associated to an even kernel our method uses the bellman function technique to obtain the right estimates for the norm of dyadic haar shift operators we then apply the representation theorem of t hytonen to extend the result to general calderon zygmund operators

gundy s decomposition for non commutative martingales and applications we provide an analogue of gundy s decomposition for l 1 bounded non commutative martingales an important difference from the classical case is that for any l 1 bounded non commutative martingale the decomposition consists of four martingales this is strongly related with the row column nature of non commutative hardy spaces of martingales as applications we obtain simpler proofs of the weak type 1 1 boundedness for non commutative martingale transforms and the non commutative analogue of burkholder s weak type inequality for square functions a sequence x n n ge 1 in a normed space mathrm x is called 2 co lacunary if there exists a bounded linear map from the closed linear span of x n n ge 1 to l 2 taking each x n to the n th vector basis of l 2 we prove using our decomposition that any relatively weakly compact martingale difference sequence in l 1 mathcal m tau whose sequence of norms is bounded away from zero is 2 co lacunary generalizing a result of aldous and fremlin to non commutative l 1 spaces

a geometrical characterization of banach spaces in which martingale difference sequences are unconditional we study banach space valued martingale transforms and in particular characterize those banach spaces for which the classical theorems of the real valued case carry over for example if b is a banach space and 1 0 and zeta x y leq x y text if x leq 1 leq y

algebraic multigrid for stabilized finite element discretizations of the navier stokes equations abstract an algebraic multigrid method for the solution of stabilized finite element discretizations of the euler and navier stokes equations on generalized unstructured grids is described the method is based on an elemental agglomeration multigrid strategy employing a semi coarsening scheme designed to reduce grid anisotropy the viscous terms are discretized in a consistent manner on coarse grids using an algebraic galerkin coarse grid approximation in which higher order grid transfer operators are constructed from the underlying triangulation however the combination of higher order transfer operators and galerkin rediscretization diminishes the stability of stabilized inviscid operators on coarse grids and a modification is proposed to alleviate this problem a generalized line implicit relaxation scheme is also described where the lines are constructed to follow the direction of strongest coupling applications are demonstrated for convection diffusion euler and laminar navier stokes the results show that the convergence rate is largely unaffected by mesh size over a wide range of reynolds peclet numbers

adaptive discontinuous evolution galerkin method for dry atmospheric flow we present a new adaptive genuinely multidimensional method within the framework of the discontinuous galerkin method the discontinuous evolution galerkin deg method couples a discontinuous galerkin formulation with approximate evolution operators the latter are constructed using the bicharacteristics of multidimensional hyperbolic systems such that all of the infinitely many directions of wave propagation are considered explicitly in order to take into account multiscale phenomena that typically appear in atmospheric flows nonlinear fluxes are split into a linear part governing the acoustic and gravitational waves and a nonlinear part that models advection time integration is realized by the imex type approximation using the semi implicit second order backward differentiation formula bdf2 moreover in order to approximate efficiently small scale phenomena adaptive mesh refinement using the space filling curves via the amatos function library is employed four standard meteorological test cases are used to validate the new discontinuous evolution galerkin method for dry atmospheric convection comparisons with the rusanov flux a standard one dimensional approximate riemann solver used for the flux integration demonstrate better stability and accuracy as well as the reliability of the new multidimensional deg method

weak type commutator and lipschitz estimates resolution of the nazarov peller conjecture let m be a semi finite von neumann algebra and let f r c be a lipschitz function if a b m are self adjoint operators such that a b l1 m then

multilevel preconditioning of two dimensional elliptic problems discretized by a class of discontinuous galerkin methods we present optimal order preconditioners for certain discontinuous galerkin dg finite element discretizations of elliptic boundary value problems we consider two variants of hierarchical splittings where the underlying finite element meshes are geometrically nested a specific assembling process is proposed which facilitates the local analysis of the angle between the related subspaces by applying the corresponding two level basis transformation recursively a sequence of algebraic problems is generated that can be associated with a hierarchy of coarse versions of dg approximations of the original problem new bounds for the constant gamma in the strengthened cauchy bunyakowski schwarz inequality are derived the presented numerical results support the theoretical analysis and demonstrate the potential of this approach

a second order class d audio amplifier class d audio amplifiers are particularly efficient and this efficiency has led to their ubiquity in a wide range of modern electronic appliances their output takes the form of a high frequency square wave whose duty cycle ratio of on time to off time is modulated at low frequency according to the audio signal a mathematical model is developed here for a second order class d amplifier design i e containing one second order integrator with negative feedback we derive exact expressions for the dominant distortion terms corresponding to a general audio input signal and confirm these predictions with simulations we also show how the observed phenomenon of pulse skipping arises from an instability of the analytical solution upon which the distortion calculations are based and we provide predictions of the circumstances under which pulse skipping will take place based on a stability analysis these predictions are confirmed by simulations

smoothed aggregation multigrid for the discontinuous galerkin method the aim of this paper is to investigate theoretically as well as experimentally an algebraic multilevel algorithm for the solution of the linear systems that arise from the discontinuous galerkin method the smoothed aggregation multigrid introduced by vanek for the conforming finite element method is applied to low order discretizations of convection diffusion equations for the elliptic model problem the algorithm is shown to be quasi optimal adjustments for the case of nonvanishing advection such as directionally implicit smoothing and a suitable splitting of the operator are discussed several numerical experiments are presented for two dimensional problems including a newton type linearization of the compressible navier stokes equations

weak boundedness of calderon zygmund operators on noncommutative l 1 spaces abstract in 2008 j parcet showed the 1 1 weak boundedness of calderon zygmund operators acting on functions taking values in a von neumann algebra we propose a simplified version of his proof using the same tools cuculescu s projections and a pseudo localisation theorem this will unable us to recover the l p boundedness of calderon zygmund operators with hilbert valued kernels acting on operator valued functions for 1 p and an l p pseudo localisation result of p hytonen

scalable two level preconditioning and deflation based on a piecewise constant subspace for sip dg systems for diffusion problems this paper is focused on the preconditioned conjugate gradient cg method for linear systems resulting from symmetric interior penalty discontinuous galerkin sipg discretizations for stationary diffusion problems in particular it concerns two level preconditioning strategies where the coarse space is based on piecewise constant dg basis functions in this paper we show that both the two level preconditioner and the corresponding bnn or adef2 deflation variant yield scalable convergence of the cg method independent of the mesh element diameter these theoretical results are illustrated by numerical experiments we consider a solver for the discontinuous galerkin method we use the symmetric interior penalty variant the deflated solver appears to be scalable our solver is independent of the jump in the coefficients

a multilevel discontinuous galerkin method a variable v cycle preconditioner for an interior penalty finite element discretization for elliptic problems is presented an analysis under a mild regularity assumption shows that the preconditioner is uniform the interior penalty method is then combined with a discontinuous galerkin scheme to arrive at a discretization scheme for an advection diffusion problem for which an error estimate is proved a multigrid algorithm for this method is presented and numerical experiments indicating its robustness with respect to diffusion coefficient are reported

two level additive schwarz methods for a discontinuous galerkin approximation of second order elliptic problems we present some two level nonoverlapping and overlapping additive schwarz methods for a discontinuous galerkin method for solving second order elliptic problems it is shown that the condition numbers of the preconditioned systems are of the order o frac h h for the nonoverlapping schwarz methods and of the order o frac h delta for the overlapping schwarz methods where h and h stand for the fine mesh size and the coarse mesh size respectively and delta denotes the size of the overlaps between subdomains numerical experiments are provided to gauge the efficiency of the methods and to validate the theory

rosenthal type inequalities for free chaos let a denote the reduced amalgamated free product of a family a 1 a 2 an of von neumann algebras over a von neumann subalgebra b with respect to normal faithful conditional expectations e k a k b we investigate the norm in lp a of homogeneous polynomials of a given degree d we first generalize voiculescu s inequality to arbitrary degree d 1 and indices 1 p this can be regarded as a free analogue of the classical rosenthal inequality our second result is a length reduction formula from which we generalize recent results of pisier ricard and the authors all constants in our estimates are independent of n so that we may consider infinitely many free factors as applications we study square functions of free martingales more precisely we show that in contrast with the khintchine and rosenthal inequalities the free analogue of the burkholder gundy inequalities does not hold in l a at the end of the paper we also consider khintchine type inequalities for shlyakhtenko s generalized circular systems

noncommutative khintchine inequalities in interpolation spaces of lp spaces abstract we prove noncommutative khintchine inequalities for all interpolation spaces between l p and l 2 with p 2 in particular it follows that khintchine inequalities hold in l 1 using a similar method we find a new deterministic equivalent for the rc norm in all interpolation spaces between l p spaces which unifies the cases p 2 and p 2 it produces a new proof of khintchine inequalities for p 1 for free variables to complete the picture we exhibit counter examples which show that neither of the usual closed formulas for khintchine inequalities can work in l 2 we also give an application to martingale inequalities

noncommutative maximal ergodic theorems this paper is devoted to the study of various maximal ergodic theorems in noncommutative spaces in particular we prove the noncommutative analogue of the classical dunford schwartz maximal ergodic inequality for positive contractions on and the analogue of stein s maximal inequality for symmetric positive contractions we also obtain the corresponding individual ergodic theorems we apply these results to a family of natural examples which frequently appear in von neumann algebra theory and in quantum probability

amg e based on element agglomeration this paper contains the main ideas for an amge algebraic multigrid for finite elements method based on element agglomeration in the method coarse grid elements are formed by agglomerating fine grid elements compatible interpolation operators are constructed which yield coarse grid basis functions with a minimal energy property heuristics based on interpolation quality measures are used to guide the agglomeration procedure the performance of the resulting method is demonstrated in two level numerical experiments

the analysis of smoothers for multigrid algorithms the purpose of this paper is to provide a general technique for defining and analyzing smoothing operators for use in multigrid algorithms the smoothing operators considered are based on subspace decomposition and include point line and block versions of jacobi and gauss seidel iteration as well as generalizations we shall show that these smoothers will be effective in multigrid algorithms provided that the subspace decomposition satisfies two simple conditions in many applications these conditions are trivial to verify

multilevel preconditioners for discontinuous galerkin approximations of elliptic problems with jump coefficients in this article we develop and analyze two level and multi level methods for the family of interior penalty ip discontinuous galerkin dg discretizations of second order elliptic problems with rough coecients exhibiting large jumps across interfaces in the domain these methods are based on a decomposition of the dg nite element space that inherently hinges on the diusion coecient of the problem our analysis of the proposed preconditioners is presented for both symmetric and non symmetric ip schemes and we establish both robustness with respect to the jump in the coecient and near optimality with respect to the mesh size following the analysis we present a sequence of detailed numerical results which verify the theory and illustrate the performance of the methods

harmonic analysis on quantum tori this paper is devoted to the study of harmonic analysis on quantum tori we consider several summation methods on these tori including the square fej r means square and circular poisson means and bochner riesz means we first establish the maximal inequalities for these means then obtain the corresponding pointwise convergence theorems in particular we prove the noncommutative analogue of the classical stein theorem on bochner riesz means the second part of the paper deals with fourier multipliers on quantum tori we prove that the completely bounded l p fourier multipliers on a quantum torus are exactly those on the classical torus of the same dimension finally we present the littlewood paley theory associated with the circular poisson semigroup on quantum tori we show that the hardy spaces in this setting possess the usual properties of hardy spaces as one can expect these include the quantum torus analogue of fefferman s h 1 bmo duality theorem and interpolation theorems our analysis is based on the recent developments of noncommutative martingale ergodic inequalities and littlewood paley stein theory

twisted hilbert transforms vs kakeya sets of directions given a discrete group g and an orthogonal action g o n we study lp convergence of fourier integrals which are frequency supported on the semidirect product ro g given a unit u rn and 1 p 6 2 our main result shows that the twisted directional hilbert transform hu o idg is lp bounded iff the orbit o u is finite this is in sharp contrast with twisted riesz transforms ru o idg which are always bounded our result characterizes fourier summability in lp for this class of groups we also extend de leeuw s compactification theorem to this setting and obtain stronger estimates for functions with lacunary frequency support

singular convolution integrals with operator valued kernel we study operators f mapsto kf of the form kf t int bf r n k t s f s rm d s where f is a vector valued function and k an operator valued singular kernel sufficient conditions for boundedness on lp spaces of umd valued functions are given in terms of a hormander type condition involving r boundedness the results cover large classes of kernels and also provide new proofs of recent operator valued fourier multiplier theorems moreover they give new information about families of singular integral operators

short note an explicit expression for the penalty parameter of the interior penalty method we derive an explicit expression for the penalty parameter of the interior penalty method for elliptic problems the expression yields a coercive bilinear form and is valid for general meshes comprising of geometrically nonconforming simplical elements

noncommutative burkholder rosenthal inequalities we investigate martingale inequalities in noncommutative lp spaces associated with a von neumann algebra equipped with a faithful normal state we prove the noncommutative analogue of the classical burkholder inequality on the conditioned or little square function and extend the noncommutative burkholder gundy inequalities from comm math phys 189 1997 667 698 to this nontracial setting we include several related results

on the order of prolongations and restrictions in multigrid procedures abstract it is well known in the world of multigrid that the order of the prolongation and the order of the restriction in a multigrid method should satisfy certain conditions a rule of thumb is that the sum of the orders of the prolongation and of the restriction should at least be equal to the order of the differential equation solved in this note we show the correctness of this rule we notice that we have to distinguish between low frequency and high frequency orders for the transfer operators for the restriction the low frequency order is related with its accuracy whereas for the interpolation operator both orders are related with the accuracy of the result of the interpolation if an interpolation rule leaves all polynomials of degree k 1 invariant then both the low and the high frequency order are equal to k it is the high frequency order that plays a role in the above mentioned rule of thumb

convergence of algebraic multigrid based on smoothed aggregation we prove aconvergence estimate for the algebraic multigrid method with prolongations defined by aggregation using zero energy modes followed by a smoothing the method input is the problem matrix and a matrix of the zero energy modes the estimate depends only polylogarithmically on the mesh size and requires only a weak approximation property for the aggregates which can be a priori verified computationally construction of the prolongator in the case of a general second order system is described and the assumptions of the theorem are verified for a scalar problem discretized by linear conforming finite elements

conditioned square functions for noncommutative martingales we prove a weak type 1 1 inequality involving conditioned versions of square functions for martingales in noncommutative l p spaces associated with finite von neumann algebras as application we determine the optimal orders for the best constants in the noncommutative burkholder rosenthal inequalities from ann probab 31 2003 948 995 we also discuss bmo norms of sums of noncommuting order independent operators

a noncommutative version of the john nirenberg theorem we prove a noncommutative version of the john nirenberg theorem for nontracial filtrations of von neumann algebras as an application we obtain an analogue of the classical large deviation inequality for elements of the associated bmo space

a t1 theorem for integral transformations with operator valued kernel we consider generalized calderon zygmund operators whose kernel k x y takes values in l x continuous linear operators on the banach space x and satis es variants of the classical standard estimates involving rboundedness which has recently become a crucial notion in connection with operator valued singular integrals boundedness criteria in the spirit of the t1 theorem of david and journe are proved for such operators on the bochner spaces lp rn x where 1 p and x is a umd space for some results x is also required to have pisier s property in the special case t1 t 1 0 we obtain an essentially complete analogue of the scalar valued theorem we also provide su cient conditions for the general case but they are stronger in general than the necessary t1 t 1 bmo type conditions although they reduce to them in the classical situation a counterexample is given to show that the natural necessary conditions are not su cient in general

a note on dyadic coverings and nondoubling calderon zygmund theory abstract we construct a family of n 1 dyadic filtrations in r n so that every euclidean ball b is contained in some cube q of our family satisfying diam q c n diam b for some dimensional constant c n our dyadic covering is optimal on the number of filtrations and improves previous results of christ and garnett jones by extending a construction of mei for the n torus based on this covering and motivated by applications to matrix valued functions we provide a dyadic nondoubling calderon zygmund decomposition which avoids besicovitch type coverings in tolsa s decomposition we also use a recent result of hytonen and kairema to extend our dyadic nondoubling decomposition to the more general setting of upper doubling metric spaces

class d audio amplifiers with negative feedback there are many different designs for audio amplifiers class d or switching amplifiers generate their output signal in the form of a high frequency square wave of variable duty cycle ratio of on time to off time the square wave nature of the output allows a particularly efficient output stage with minimal losses the output is ultimately filtered to remove components of the spectrum above the audio range mathematical models are derived here for a variety of related class d amplifier designs that use negative feedback these models use an asymptotic expansion in powers of a small parameter related to the ratio of typical audio frequencies to the switching frequency to develop a power series for the output component in the audio spectrum these models confirm that there is a form of distortion intrinsic to such amplifier designs the models also explain why two approaches used commercially succeed in largely eliminating this distortion a new means of overcoming the intrinsic distortion is revealed by the analysis copyright 2006 society for industrial and applied mathematics

robust multigrid for high order discontinuous galerkin methods we present a polynomial multigrid method for nodal interior penalty and local discontinuous galerkin formulations of the poisson equation on cartesian grids for smoothing we propose two classes of overlapping schwarz methods the first class comprises element centered and the second face centered methods within both classes we identify methods that achieve superior convergence rates prove robust with respect to the mesh spacing and the polynomial order at least up to p 32 consequent structure exploitation yields a computational complexity of o pn where n is the number of unknowns further we demonstrate the suitability of the face centered method for element aspect ratios up to 32 p multigrid for discontinuous formulations of poisson problems on cartesian grids robust with respect to the mesh size and polynomial order at least up to p 32 convergence rates between 0 02 and 0 003 improving with increasing p computational complexity of pn for n unknowns linear complexity in runtime excellent performance up to aspect ratios of 1 16

non commutative martingale inequalities we prove the analogue of the classical burkholder gundy inequalites for non commutative martingales as applications we give a characterization for an ito clifford integral to be an lp martingale via its integrand and then extend the ito clifford integral theory in l2 developed by barnett streater and wilde to lp for all 1 p we include an appendix on the non commutative analogue of the classical fefferman duality between h1 and bmo

h multigrid agglomeration based solution strategies for discontinuous galerkin discretizations of incompressible flow problems abstract in this work we exploit agglomeration based h multigrid preconditioners to speed up the iterative solution of discontinuous galerkin discretizations of the stokes and navier stokes equations as a distinctive feature h coarsened mesh sequences are generated by recursive agglomeration of a fine grid admitting arbitrarily unstructured grids of complex domains and agglomeration based discontinuous galerkin discretizations are employed to deal with agglomerated elements of coarse levels both the expense of building coarse grid operators and the performance of the resulting multigrid iteration are investigated for the sake of efficiency coarse grid operators are inherited through element by element l 2 projections avoiding the cost of numerical integration over agglomerated elements specific care is devoted to the projection of viscous terms discretized by means of the br2 dg method we demonstrate that enforcing the correct amount of stabilization on coarse grids levels is mandatory for achieving uniform convergence with respect to the number of levels the numerical solution of steady and unsteady linear and non linear problems is considered tackling challenging 2d test cases and 3d real life computations on parallel architectures significant execution time gains are documented

an overlapping domain decomposition preconditioner for a class of discontinuous galerkin approximations of advection diffusion problems we consider a scalar advection diffusion problem and a recently proposed discontinuous galerkin approximation which employs discontinuous finite element spaces and suitable bilinear forms containing interface terms that ensure consistency for the corresponding sparse non symmetric linear system we propose and study an additive two level overlapping schwarz preconditioner consisting of a coarse problem on a coarse triangulation and local solvers associated to suitable problems defined on a family of subdomains this is a generalization of the corresponding overlapping method for approximations on continuous finite element spaces related to the lack of continuity of our approximation spaces some interesting new features arise in our generalization which have no analog in the conforming case we prove an upper bound for the number of iterations obtained by using this preconditioner with gmres which is independent of the number of degrees of freedom of the original problem and the number of subdomains the performance of the method is illustrated by several numerical experiments for different test problems using linear finite elements in two dimensions

a weak type inequality for non commutative martingales and applications we prove a weak type 1 1 inequality for square functions of non commutative martingales that are simultaneously bounded in l 2 and l 1 more precisely the following non commutative analogue of a classical result of burkholder holds there exists an absolute constant k 0 such that if m is a semi finite von neumann algebra and mn 1 1 is an increasing filtration of von neumann subalgebras of m then for any given martingale x xn 1 1 that is bounded in l 2 m l 1 m adapted to mn 1 1 there exist two martingale difference sequences a an 1 1 and b bn 1 1 with dxn an bn for every n 1 1 x n 1 aan 1 2

algebraic multilevel preconditioning methods ii a recursive way of constructing preconditioning matrices for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems is proposed it is based on a sequence of nested finite element spaces with the usual nodal basis functions using a nodeordering corresponding to the nested meshes the finite element stiffness matrix is recursively split up into two level block structures and is factored approximately in such a way that any successive schur complement is replaced approximated by a matrix defined recursively and thereform only implicitely given to solve a system with this matrix we need to perform a fixed number v of iterations on the preceding level using as an iteration matrix the preconditioning matrix already defined on that level it is shown that by a proper choice of iteration parameters it suffices to use v left 1 gamma 2 right tfrac 1 2 iterations for the so constructedv foldv cycle wherev 2 corresponds to aw cycle preconditioning matrices to be spectrally equivalent to the stiffness matrix the conditions involve only the constant in the strengthened c b s inequality for the corresponding two level hierarchical basis function spaces and are therefore independent of the regularity of the solution for instance if we use successive uniform refinements of the meshes the method is of optimal order of computational complexity if gamma 2 tfrac 8 9 under reasonable assumptions of the finite element mesh the condition numbers turn out to be so small that there are in practice few reasons to use an accelerated iterative method like the conjugate gradient method for instance