three transmit antenna space time codes based on su 3 fully diverse constellations i e a set of unitary matrices whose pairwise differences are nonsingular are useful in multiantenna communications especially in multiantenna differential modulation since they have good pairwise error properties recently group theoretic ideas especially fixed point free fpf groups have been used to design fully diverse constellations of unitary matrices here we give systematic design methods of space time codes which are appropriate for three transmit antenna differential modulation the structures of the codes are motivated by the special unitary lie group su 3 one of the codes which is called the ab code has a fast maximum likelihood ml decoding algorithm using complex sphere decoding diversity products of the codes can be easily calculated and simulated performance shows that they are better than group based codes especially at high rates and as good as the elaborately designed nongroup code

a space time petrov galerkin certified reduced basis method application to the boussinesq equations we present a space time certified reduced basis method for long time integration of parametrized parabolic equations with quadratic nonlinearity which admit an affine decomposition in parameter but with no restriction on coercivity of the linearized operator we first consider a finite element discretization based on discontinuous galerkin time integration and introduce associated petrov galerkin space time trial and test space norms that yield optimal and asymptotically mesh independent stability constants we then employ an hp petrov galerkin or minimum residual space time reduced basis approximation we provide the brezzi rappaz raviart a posteriori error bounds which admit efficient offline online computational procedures for the three key ingredients the dual norm of the residual an inf sup lower bound and the sobolev embedding constant the latter are based respectively on a more round off resistant residual norm evaluation procedure a variant of the successive constraint method and

nd convolutional codes i performance analysis a noncoherent coded system which incorporates convolutional codes in conjunction of multiple symbol noncoherent detection is presented in this two part paper where part i focuses on the performance analysis of the system and part ii deals with the structural properties of the underlying convolutional codes these convolutional codes are referred to as nd convolutional codes it is shown that nd convolutional codes provide a general framework for various noncoherent coding systems including differential systems two models of the carrier phase are examined and the relationships between them is established for the first one the carrier phase remains constant for l channels signals whereas for the second one it unvaries throughout the transmission period the regular structure of nd codes facilitates the evaluation of a simple upper bound on the pairwise and bit error probabilities as well as a simple expression for the generalized cutoff rate the exponential rate of the error probability which is the single parameter governing the error performance at large signal to noise ratios is identified via large deviations techniques this parameter leads to the interesting conclusion that increasing l does not necessarily monotonically improve the error performance of the noncoherent system the same conclusion is reached by examining upper bounds and computer simulation results of several interesting examples these examples also reveal that optimal codes for coherent detection are not necessarily optimal for noncoherent detection and a search for good codes some of which are tabulated in part ii of the paper is required

divergence free wavelet bases on the hypercube free slip boundary conditions and applications for solving the instationary stokes equations we construct wavelet riesz bases for the usual sobolev spaces of divergence free functions on that have vanishing normals at the boundary we give a simultaneous space time variational formulation of the instationary stokes equations that defines a boundedly invertible mapping between a bochner space and the dual of another bochner space by equipping these bochner spaces by tensor products of temporal and divergence free spatial wavelets the stokes problem is rewritten as an equivalent well posed bi infinite matrix vector equation this equation can be solved with an adaptive wavelet method in linear complexity with best possible rate that under some mild besov smoothness conditions is nearly independent of the space dimension for proving one of the intermediate results we construct an eigenfunction basis of the stationary stokes operator

optimal space time adaptive wavelet methods for degenerate parabolic pdes we analyze parabolic pdes with certain type of weakly singular or degenerate time dependent coefficients and prove existence and uniqueness of weak solutions in an appropriate sense a localization of the pdes to a bounded spatial domain is justified for the numerical solution a space time wavelet discretization is employed an optimality result for the iterative solution of the arising systems can be obtained finally applications to fractional brownian motion models in option pricing are presented

simultaneous space time adaptive wavelet solution of nonlinear parabolic differential equations dynamically adaptive numerical methods have been developed to efficiently solve differential equations whose solutions are intermittent in both space and time these methods combine an adjustable time step with a spatial grid that adapts to spatial intermittency at a fixed time the same time step is used for all spatial locations and all scales this approach clearly does not fully exploit space time intermittency we propose an adaptive wavelet collocation method for solving highly intermittent problems e g turbulence on a simultaneous space time computational domain which naturally adapts both the space and time resolution to match the solution besides generating a near optimal grid for the full space time solution this approach also allows the global time integration error to be controlled the efficiency and accuracy of the method is demonstrated by applying it to several highly intermittent 1d t dimensional and 2d t dimensional test problems in particular we found that the space time method uses roughly 18 times fewer space time grid points and is roughly 4 times faster than a dynamically adaptive explicit time marching method while achieving similar global accuracy

adaptive solution of operator equations using wavelet frames in adaptive wavelet methods ii beyond the elliptic case of cohen dahmen and devore found comput math 2 2002 pp 203 245 an adaptive method has been developed for solving general operator equations using a riesz basis of wavelet type for the energy space the operator equation is transformed into an equivalent matrix vector system this system is solved iteratively where the application of the infinite stiffness matrix is replaced by an adaptive approximation assuming that the stiffness matrix is sufficiently compressible i e that it can be sufficiently well approximated by sparse matrices it was proved that the adaptive method has optimal computational complexity in the sense that it converges with the same rate as the best n term approximation for the solution assuming that the latter would be explicitly available the condition concerning compressibility requires that dependent on their order the wavelets have sufficiently many vanishing moments and that they be sufficiently smooth however except on tensor product domains wavelets that satisfy this smoothness requirement are not easy to construct in this paper we write the domain or manifold on which the operator equation is posed as an overlapping union of subdomains each of them being the image under a smooth parametrization of the hypercube by lifting wavelets on the hypercube to the subdomains we obtain a em frame for the energy space with this frame the operator equation is transformed into a matrix vector system after which this system is solved iteratively by an adaptive method similar to the one from the work of cohen dahmen and devore with this approach frame elements that have sufficiently many vanishing moments and are sufficiently smooth something needed for the compressibility are easily constructed by handling additional difficulties due to the fact that a frame gives rise to an underdetermined matrix vector system we prove that this adaptive method has optimal computational complexity

a posteriori error control and adaptivity for crank nicolson finite element approximations for the linear schrodinger equation we derive optimal order a posteriori error estimates for fully discrete approximations of linear schrodinger type equations in the l infty l 2 l l 2 norm for the discretization in time we use the crank nicolson method while for the space discretization we use finite element spaces that are allowed to change in time the derivation of the estimators is based on a novel elliptic reconstruction that leads to estimates which reflect the physical properties of schrodinger equations the final estimates are obtained using energy techniques and residual type estimators various numerical experiments for the one dimensional linear schrodinger equation in the semiclassical regime verify and complement our theoretical results the numerical implementations are performed with both uniform partitions and adaptivity in time and space for adaptivity we further develop and analyze an existing time space adaptive algorithm to the cases of schrodinger equations the adaptive algorithm reduces the computational cost substantially and provides efficient error control for the solution and the observables of the problem especially for small values of the planck constant

an optimal adaptive wavelet method without coarsening of the iterands in this paper an adaptive wavelet method for solving linear operator equations is constructed that is a modification of the method from math comp 70 2001 pp 27 75 by cohen dahmen and devore in the sense that there is no recurrent coarsening of the iterands despite this it will be shown that the method has optimal computational complexity numerical results for a simple model problem indicate that the new method is more efficient than an existing alternative adaptive wavelet method

a fast sparse grid based space time boundary element method for the nonstationary heat equation this article presents a fast sparse grid based space time boundary element method for the solution of the nonstationary heat equation we make an indirect ansatz based on the thermal single layer potential which yields a first kind integral equation this integral equation is discretized by galerkin s method with respect to the sparse tensor product of the spatial and temporal ansatz spaces by employing the mathcal h matrix and toeplitz structure of the resulting discretized operators we arrive at an algorithm which computes the approximate solution in a complexity that essentially corresponds to that of the spatial discretization nevertheless the convergence rate is nearly the same as in case of a traditional discretization in full tensor product spaces

computation of differential operators in wavelet coordinates in found comput math 2 2002 pp 203 245 cohen dahmen and devore proposed an adaptive wavelet algorithm for solving general operator equations assuming that the operator defines a boundedly invertible mapping between a hilbert space and its dual and that a riesz basis of wavelet type for this hilbert space is available the operator equation is transformed into an equivalent well posed infinite matrix vector system this system is solved by an iterative method where each application of the infinite stiffness matrix is replaced by an adaptive approximation it was shown that if the errors of the best linear combinations from the wavelet basis with n terms are o n s for some s 0 which is determined by the besov regularity of the solution and the order of the wavelet basis then approximations yielded by the adaptive method with n terms also have errors of o n s moreover their computation takes only 0 n operations provided s s assuming that each entry of the stiffness matrix is exactly available at unit cost generally these entries have to be approximated using numerical quadrature in this paper restricting ourselves to differential operators we develop a numerical integration scheme that computes these entries giving an additional error that is consistent with the approximation error whereas in each column the average computational cost per entry is 0 1 as a consequence we can conclude that the adaptive wavelet algorithm has optimal computational complexity

trace orthonormal full diversity cyclotomic space time codes in this paper we consider the design of full diversity space time codes for a coherent multiple input multiple output mimo communication system starting from both the information theoretic and detection error viewpoints we first establish that a desirable property for general linear dispersion ld codes is to have an interunitary as well as an intraunitary structure a structure we call trace orthonormality by imposing the trace orthonormal structure on an ld code and applying cyclotomic number theory we establish for an arbitrary number of transmitter and receiver antennas a systematic and simple method to jointly design a unitary cyclotomic matrix the diophantine number and the corresponding constellation for an ld code as a result this enables us to construct full diversity rectangular cyclotomic ld codes with any symbol transmission rate less than or equal to the number of transmitter antennas in addition for the case when the number of transmitter antennas is greater than the number of receiver antennas by taking advantage of the delay we also arrive at the design of a special trace orthonormal full diversity cyclotomic space time block code which for the number of transmitter antenna being equal to 2m can be proved to minimize the worst case pairwise error probability of a maximum likelihood ml detector for a q ary quadrature amplitude modulation qam signal constellation and therefore achieves optimal coding gain computer simulations show that these codes have bit error performance advantages over currently available codes

differential space time modulation space time coding and modulation exploit the presence of multiple transmit antennas to improve the performance on multipath radio channels thus far most work on space time coding has assumed that perfect channel estimates are available at the receiver in certain situations however it may be difficult or costly to estimate the channel accurately in which case it is natural to consider the design of modulation techniques that do not require channel estimates at the transmitter or receiver we propose a general approach to differential modulation for multiple transmit antennas based on group codes this approach ran be applied to any number of transmit and receive antennas and any signal constellation we also derive low complexity differential receivers error bounds and modulator design criteria which we use to construct optimal differential modulation schemes for two transmit antennas these schemes can be demodulated with or without channel estimates this permits the receiver to exploit channel estimates when they are available the performance degrades by approximately 3 db when estimates are not available

multilevel sparse grids collocation for linear partial differential equations with tensor product smooth basis functions abstract radial basis functions have become a popular tool for approximation and solution of partial differential equations pdes the recently proposed multilevel sparse interpolation with kernels musik algorithm proposed in georgoulis et al 2013 shows good convergence in this paper we use a sparse kernel basis for the solution of pdes by collocation we will use the form of approximation proposed and developed by kansa 1986 we will give numerical examples using a tensor product basis with the multiquadric mq and gaussian basis functions this paper is novel in that we consider space time pdes in four dimensions using an easy to implement algorithm with smooth approximations the accuracy observed numerically is as good with respect to the number of data points used as other methods in the literature see langer 2016 and wang et al 2016

wavelets on manifolds an optimized construction a key ingredient of the construction of biorthogonal wavelet bases for sobolev spaces on manifolds which is based on topological isomorphisms is the hestenes extension operator here we firstly investigate whether this particular extension operator can be replaced by another extension operator our main theoretical result states that an important class of extension operators based on interpolating boundary values cannot be used in the construction setting required by dahmen and schneider in the second part of this paper we investigate and optimize the hestenes extension operator the results of the optimization process allow us to implement the construction of biorthogonal wavelets from dahmen and schneider as an example we illustrate a wavelet basis on the 2 sphere

an adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems an efficient and reliable a posteriori error estimate is derived for linear parabolic equations which does not depend on any regularity assumption on the underlying elliptic operator an adaptive algorithm with variable time step sizes and space meshes is proposed and studied which at each time step delays the mesh coarsening until the final iteration of the adaptive procedure allowing only mesh and time step size refinements before it is proved that at each time step the adaptive algorithm is able to reduce the error indicators and thus the error below any given tolerance within a finite number of iteration steps the key ingredient in the analysis is a new coarsening strategy numerical results are presented to show the competitive behavior of the proposed adaptive algorithm

space time adaptive wavelet methods for parabolic evolution problems with respect to space time tensor product wavelet bases parabolic initial boundary value problems are equivalently formulated as bi infinite matrix problems adaptive wavelet methods are shown to yield sequences of approximate solutions which converge at the optimal rate in case the spatial domain is of product type the use of spatial tensor product wavelet bases is proved to overcome the so called curse of dimensionality i e the reduction of the convergence rate with increasing spatial dimension

grassmannian packings from operator reed muller codes this paper introduces multidimensional generalizations of binary reed muller codes where the codewords are projection operators and the corresponding subspaces are widely separated with respect to the chordal distance on grassmannian space parameters of these grassmannian packings are derived and a low complexity decoding algorithm is developed by modifying standard decoding algorithms for binary reed muller codes the subspaces are associated with projection operators determined by pauli matrices appearing in the theory of quantum error correction and this connection with quantum stabilizer codes may be of independent interest the grassmannian packings constructed here find application in noncoherent wireless communication with multiple antennas where separation with respect to the chordal distance on grassmannian space guarantees closeness to the channel capacity it is shown that the capacity of the noncoherent multiple input multiple output mimo channel at both low and moderate signal to noise ratio snr under the constraint that only isotropically distributed unitary matrices are used for information transmission is closely approximated by these packings

unitary space time modulation for multiple antenna communications in rayleigh flat fading motivated by information theoretic considerations we propose a signaling scheme unitary space time modulation for multiple antenna communication links this modulation is ideally suited for rayleigh fast fading environments since it does not require the receiver to know or learn the propagation coefficients unitary space time modulation uses constellations of t spl times m space time signals spl phi sub i l 1 l where t represents the coherence interval during which the fading is approximately constant and m t is the number of transmitter antennas the columns of each spl phi sub i are orthonormal when the receiver does not know the propagation coefficients which between pairs of transmitter and receiver antennas are modeled as statistically independent this modulation performs very well either when the signal to noise ratio snr is high or when t spl gt m we design some multiple antenna signal constellations and simulate their effectiveness as measured by bit error probability with maximum likelihood decoding we demonstrate that two antennas have a 6 db diversity gain over one antenna at 15 db snr

space time isogeometric analysis of parabolic evolution problems abstract we present and analyze a new stable space time isogeometric analysis iga method for the numerical solution of parabolic evolution equations in fixed and moving spatial computational domains the discrete bilinear form is elliptic on the iga space with respect to a discrete energy norm this property together with a corresponding boundedness property consistency and approximation results for the iga spaces yields an a priori discretization error estimate with respect to the discrete norm the theoretical results are confirmed by several numerical experiments with low and high order iga spaces

a novel signaling scheme for blind unique identification of alamouti space time block coded channel in this paper we present a simple signaling scheme to blindly and uniquely identify the alamouti space time block coded channel first under a noise free environment and then under a complex gaussian noise environment in which pth and qth order statistics p and q are coprime of the received signals are available in both cases closed form solutions to determine the channel coefficients are obtained by exploiting specific properties of the alamouti space time block code stbc and the linear diophantine equation theory under gaussian noise when the length of the received data is finite we propose to use the semidefinite relaxation sdr algorithm to approximate maximum likelihood ml detection so that the joint estimation of the channel and symbols can be efficiently implemented simulation results show that while other existing blind methods fail our signaling scheme works well

an improved error bound for reduced basis approximation of linear parabolic problems we consider a space time variational formulation for linear para bolic partial dierential equations we introduce an associated petrov galerkin truth nite element discretization with favorable discrete inf sup constant the inverse of which enters into error estimates is unity for the heat equa tion decreases only linearly in time for non coercive but asymptotically stable convection operators the latter in turn permits eective long time a posteriori error bounds for reduced basis approximations in sharp contrast to classical pessimistic exponentially growing energy estimates the paper contains a full analysis and various extensions for the formulation introduced briey in 13 as well as numerical results for a model reaction convection diusion equation

an optimal adaptive wavelet method for nonsymmetric and indefinite elliptic problems in this paper we modify the adaptive wavelet algorithm from gantumur et al an optimal adaptive wavelet method without coarsening of the iterands technical report 1325 department of mathematics utrecht university march 2005 math comp to appear so that it applies directly i e without forming the normal equation not only to self adjoint elliptic operators but also to operators of the form l a b where a is self adjoint elliptic and b is compact assuming that the resulting operator equation is well posed we show that the algorithm has optimal computational complexity

fast evaluation of nonlinear functionals of tensor product wavelet expansions for a nonlinear functional f and a function u from the span of a set of tensor product interpolets it is shown how to compute the interpolant of f u from the span of this set of tensor product interpolets in linear complexity assuming that the index set has a certain multiple tree structure applications are found in the field of adaptive tensor product solution methods for semilinear operator equations by collocation methods or after transformations between the interpolet and bi orthogonal wavelet bases by galerkin methods

adaptive finite element methods for parabolic problems i a linear model problem this paper is the first part in a series of papers on adaptive finite element methods for parabolic problems in this paper an adaptive algorithm is presented and analyzed for choosing the space a

adaptive wavelet algorithms for elliptic pde s on product domains with standard isotropic approximation by piecewise polynomials of fixed order in a domain d subset of r d the convergence rate in terms of the number n of degrees of freedom is inversely proportional to the space dimension d this so called curse of dimensionality can be circumvented by applying sparse tensor product approximation when certain high order mixed derivatives of the approximated function happen to be bounded in l 2 it was shown by nitsche 2006 that this regularity constraint can be dramatically reduced by considering best n term approximation from tensor product wavelet bases when the function is the solution of some well posed operator equation dimension independent approximation rates can be practically realized in linear complexity by adaptive wavelet algorithms assuming that the infinite stiffness matrix of the operator with respect to such a basis is highly compressible applying piecewise smooth wavelets we verify this compressibility for general non separable elliptic pdes in tensor domains applications of the general theory developed include adaptive galerkin discretizations of multiple scale homogenization problems and of anisotropic equations which are robust i e independent of the scale parameters resp of the size of the anisotropy

efficient algorithms for robust feature matching abstract one of the basic building blocks in any point based registration scheme involves matching feature points that are extracted from a sensed image to their counterparts in a reference image this leads to the fundamental problem of point matching given two sets of points find the affine transformation that transforms one point set so that its distance from the other point set is minimized because of measurement errors and the presence of outlying data points it is important that the distance measure between the two point sets be robust to these effects we measure distances using the partial hausdorff distance point matching can be a computationally intensive task and a number of theoretical and applied approaches have been proposed for solving this problem in this paper we present two algorithmic approaches to the point matching problem in an attempt to reduce its computational complexity while still providing a guarantee of the quality of the final match our first method is an approximation algorithm which is loosely based on a branch and bound approach due to huttenlocher and rucklidge technical report 1321 dept of computer science cornell university ithaca 1992 proc ieee conf on computer vision and pattern recognition new york 1993 pp 705 706 we show that by varying the approximation error bounds it is possible to achieve a tradeoff between the quality of the match and the running time of the algorithm our second method involves a monte carlo method for accelerating the search process used in the first algorithm this algorithm operates within the framework of a branch and bound procedure but employs point to point alignments to accelerate the search we show that this combination retains many of the strengths of branch and bound search but provides significantly faster search times by exploiting alignments with high probability this method succeeds in finding an approximately optimal match we demonstrate the algorithms performances on both synthetically generated data points and actual satellite images

optimal signaling schemes and capacities of non coherent correlated miso channels under per antenna power constraints this paper investigates the optimal signaling schemes and capacities of non coherent correlated multiple input single output miso channels in fast rayleigh fading we consider both channels under per antenna power constraints as well as channels under joint per antenna and sum power constraints for per antenna power constraint channels we first establish the convex and compact properties of the feasible sets and demonstrate the existence of optimal input distribution and the uniqueness of optimal effective magnitude input distribution by exploiting the solutions of a quadratic optimization problem we show that the kuhn tucker condition on the optimal inputs can be simplified to a single dimension as a result we can apply the identity theorem to show the discrete and finite nature of the optimal effective magnitude distribution with a mass point located at the origin by using this distribution we then construct a finite and discrete optimal input vector distribution the use of this input allows us to determine the capacity gain of miso over siso via the phase solutions of a constrained quadratic optimization problem on a sphere which can be obtained using a proposed penalized optimization algorithm we also extend the results to miso channels subject to the joint per antenna and sum power constraints under this consideration it is shown that not all per antenna constraints are active while the finiteness and discreteness of the optimal effective magnitude and the optimal input vector distributions still hold the optimal phases and the optimal power allocation among the transmit antennas need to be determined simultaneously via a quadratic optimization problem under inequality constraints these solutions can finally be used to obtain the miso capacity gain

wavelets with patchwise cancellation properties we construct wavelets on general n dimensional domains or manifolds via a domain decomposition technique resulting in so called composite wavelets with this construction wavelets with supports that extend to more than one patch are only continuous over the patch interfaces normally this limited smoothness restricts the possibility for matrix compression and with that the application of these wavelets in adaptive methods for solving operator equations by modifying the scaling functions on the interval and with that on the n cube that serves as parameter domain we obtain composite wavelets that have patchwise cancellation properties of any required order meaning that the restriction of any wavelet to each patch is again a wavelet this is also true when the wavelets are required to satisfy zeroth order homogeneous dirichlet boundary conditions on part of the boundary as a result compression estimates now depend only on the patchwise smoothness of the wavelets that one may choose also taking stability into account our composite wavelets have all the properties for the application to the adaptive solution of well posed operator equations of orders 2t for t 3 2

multiple antenna signaling over fading channels with estimated channel state information performance analysis multiple antenna concepts for wireless communication systems promise high spectral efficiency and low error rates by proper exploitation of the randomness in multipath propagation in this paper we investigate the impact of channel uncertainty caused by channel estimation errors on the error rate performance we consider a training based multiple antenna system that reserves a portion of time to sound the channel training symbols are used to estimate the channel by means of an arbitrary linear filter at the receiver no channel state information csi is assumed at the transmitter we present a new framework to analyze training based multiple antenna systems by introducing an equivalent system model that specifies the channel by the estimated and hence known channel coefficients and an uncorrelated data dependent multiplicative noise we derive the maximum likelihood ml detector and highlight its behavior in the limiting cases of perfect csi and no csi and its relation to several mismatched detectors we deduce new exact expressions and chernoff bounds of the pairwise error probability pep used to assess word error and bit error rate bounds for ml and mismatched detection finally we review the code design guidelines in terms of the deleterious effect of channel uncertainty for coherent and noncoherent signaling schemes and present numerical results

isotropic fading vector broadcast channels the scalar upper bound and loss in degrees of freedom we propose a scalar upper bound on the capacity region of the isotropic fading vector broadcast channel in terms of the capacity region of a scalar fading broadcast channel the scalar upper bound is applicable to the broad class of isotropic fading broadcast channels regardless of the distribution of the users channel magnitudes the distribution of the additive noise experienced by each user or the amount of channel knowledge available at the receiver using this upper bound we prove the optimality of the alamouti scheme in a broadcast setting extend the recent results on the capacity of nondegraded fading scalar broadcast channels to nondegraded fading vector broadcast channels and determine the capacity region of a fading vector gaussian broadcast channel with channel magnitude feedback we also provide an example of a rayleigh fading broadcast channel with no channel state information available to the receiver csir where the bound on the capacity region obtained by a naive application of the scalar upper bound is provably loose because it fails to account for the additional loss in degrees of freedom due to lack of channel knowledge at the receiver a tighter upper bound is obtained by separately accounting for the loss in degrees of freedom due to lack of csir before applying the scalar upper bound

solving chemical master equations by adaptive wavelet compression solving chemical master equations numerically on a large state space is known to be a difficult problem because the huge number of unknowns is far beyond the capacity of traditional methods we present an adaptive method which compresses the problem very efficiently by representing the solution in a sparse wavelet basis that is updated in each step the step size is chosen adaptively according to estimates of the temporal and spatial approximation errors numerical examples demonstrate the reliability of the error estimation and show that the method can solve large problems with bimodal solution profiles

adaptive finite elements for a linear parabolic problem a posteriori error estimates for the heat equation in two space dimensions are presented a classical discretization is used euler backward in time and continuous piecewise linear triangular finite elements in space the error is bounded above and below by an explicit error estimator based on the residual numerical results are presented for uniform triangulations and constant time steps the quality of our error estimator is discussed an adaptive algorithm is then proposed successive delaunay triangulations are generated so that the estimated relative error is close to a preset tolerance again numerical results demonstrate the efficiency of our approach c 1998 elsevier science s a all rights reserved

a space time variational approach to hydrodynamic stability theory we present a hydrodynamic stability theory for incompressible viscous fluid flows based on a space time variational formulation and associated generalized singular value decomposition of the linearized navier stokes equations we first introduce a linear framework applicable to a wide variety of stationary or time dependent base flows we consider arbitrary disturbances in both the initial condition and the dynamics measured in a data space time norm the theory provides a rigorous sharp realizable and eciently computed bound for the velocity perturbation measured in a solution spacetime norm we next present a generalization of the linear framework in which the disturbances and perturbation are now measured in respective selected space time seminorms the semi norm theory permits rigorous and sharp quantification of for example the growth of initial disturbances or functional outputs we then develop a brezzirappaz raviart nonlinear theory which provides for disturbances which satisfy a certain rather stringent amplitude condition rigorous finite amplitude bounds for the velocity and output perturbations finally we demonstrate the application of our linear and nonlinear hydrodynamic stability theory to unsteady moderate reynolds number flow in an eddy promoter channel

on noncoercive variational inequalities we consider variational inequalities with different trial and test spaces and a possibly noncoercive bilinear form well posedness is shown under general conditions that are e g valid for the space time variational formulation of parabolic variational inequalities moreover we prove an estimate for the error of a petrov galerkin approximation in terms of the residual for parabolic variational inequalities the arising estimate is independent of the final time

an adaptive multiresolution scheme with local time stepping for evolutionary pdes we present a fully adaptive numerical scheme for evolutionary pdes in cartesian geometry based on a second order finite volume discretization a multiresolution strategy allows local grid refinement while controlling the approximation error in space for time discretization we use an explicit runge kutta scheme of second order with a scale dependent time step on the finest scale the size of the time step is imposed by the stability condition of the explicit scheme on larger scales the time step can be increased without violating the stability requirement of the explicit scheme the implementation uses a dynamic tree data structure numerical validations for test problems in one space dimension demonstrate the efficiency and accuracy of the local time stepping scheme with respect to both multiresolution scheme with global time stepping and finite volume scheme on a regular grid fully adaptive three dimensional computations for reaction diffusion equations illustrate the additional speed up of the local time stepping for a thermo diffusive flame instability

adaptive wavelet schemes for parabolic problems sparse matrices and numerical results a simultaneous space time variational formulation of a parabolic evolution problem is solved with an adaptive wavelet method this method is shown to converge with the best possible rate in linear complexity thanks to the use of tensor product bases there is no penalty in complexity due to the additional time dimension special wavelets are designed such that the bi infinite system matrix is sparse this sparsity largely simplifies the implementation and improves the quantitative properties of the adaptive wavelet method numerical results for an ode and the heat equation are presented

generalized quadratic receivers for unitary space time modulation over rayleigh fading channels we propose the generalized quadratic receivers gqrs for unitary space time modulation over flat rayleigh fading channels the gqrs realize the performance improvement potential known to be approximately 2 4 db between the quadratic receiver qr and the coherent receiver cr by performing channel estimation without the help of additional training signals that consume additional bandwidth they are designed for various unitary space time constellations ustc in which signal matrices may or may not contain explicit inherent training blocks and may be orthogonal or nonorthogonal to one another as the channel memory span exploited for channel estimation increases the error probability of the gqrs reduces from that of the qr to that of the cr the gqrs work well for both slow and fast fading channels and the performance improvement increases as the channel fade rate decreases for a class of ustc with the orthogonal design structure the gqr is simplified to a form whose complexity can be less than the complexity of the qr or even that of the simplified form of the qr

adaptive finite element methods for parabolic problems ii optimal error estimates in l l 2 and l l optimal error estimates are derived for a complete discretization of linear parabolic problems using space time finite elements the discretization is done first in time using the discontinuous galerkin method and then in space using the standard galerkin method the underlying partitions in time and space need not be quasi uniform and the partition in space may be changed from time step to time step the error bounds show in particular that the error may be controlled globally in time on a given tolerance level by controlling the discretization error on each individual time step on the same given level i e without error accumulation effects the derivation of the estimates is based on the orthogonality of the galerkin procedure and the use of strong stability estimates the particular and precise form of these error estimates makes it possible to design efficient adaptive methods with reliable automatic error control for parabolic problems in the norms under consideration

leveraging coherent space time codes for noncoherent communication via training training codes are introduced for the multiple antenna noncoherent multiple block rayleigh fading channel in which the fading coefficients which are constant over a fixed number of dimensions coherence interval for each block and then change independently to a new realization are known neither at the transmitter nor at the receiver each codeword of a training code consists of a part known to the receiver used to form a minimum mean squared error mmse estimate of the channel and a part that contains codeword s of a space time block or trellis code designed for the coherent channel in which the receiver has perfect knowledge of the channel the channel estimate is used as if it were error free for decoding the information bearing part of the training codeword training codes are hence easily designed to have high rate and low decoding complexity by choosing the underlying coherent code to have high rate and to be efficiently decodable conditions for which the estimator detector e d receiver is equivalent to the optimal noncoherent receiver are established a key performance analysis result of this paper is that the training codes when decoded with the e d receiver achieve a diversity order of the error probability that is equal to the diversity order of the underlying coherent code in some cases the performance of training codes can be measured relative to coherent reception via training efficiency which is then optimized over the energy allocation between the training and data phases in the limit of increasing block lengths training codes always achieve the performance of coherent reception the examples of training codes provided in this work have polynomial complexity in rate but an error rate comparable to the best performing unitary designs available even though the latter require exponential decoding complexity

first order k th moment finite element analysis of nonlinear operator equations with stochastic data we develop and analyze a class of efficient galerkin approximation methods for uncertainty quantification of nonlinear operator equations the algorithms are based on sparse galerkin discretizations of tensorized linearizations at nominal parameters specifically we consider abstract nonlinear parametric operator equations j alpha u 0 for random input alpha omega with almost sure realizations in a neighborhood of a nominal input parameter alpha 0 under some structural assumptions on the parameter dependence we prove existence and uniqueness of a random solution u omega s alpha omega we derive a multilinear tensorized operator equation for the deterministic computation of k th order statistical moments of the random solution s fluctuations u omega s alpha 0 we introduce and analyse sparse tensor galerkin discretization schemes for the efficient deterministic computation of the k th statistical moment equation we prove a shift theorem for the k point correlation equation in anisotropic smoothness scales and deduce that sparse tensor galerkin discretizations of this equation converge in accuracy vs complexity which equals up to logarithmic terms that of the galerkin discretization of a single instance of the mean field problem we illustrate the abstract theory for nonstationary diffusion problems in random domains

optimality of adaptive galerkin methods for random parabolic partial differential equations galerkin discretizations of a class of parametric and random parabolic partial differential equations pdes are considered the parabolic pdes are assumed to depend on a vector y y 1 y 2 of possibly countably many parameters y j which are assumed to take values in 1 1 well posedness of weak formulations of these parametric equations in suitable bochner spaces is established adaptive galerkin discretizations of the equation based on a tensor product of a generalized polynomial chaos in the parameter domain 1 1 n and of suitable wavelet bases in the time interval i 0 t and the spatial domain d r d are proposed and their optimality is established

space time frequency shift keying frequency shift keying fsk is a popular modulation scheme in power limited communication links this paper introduces space time fsk st fsk which does not require any channel state information at the transmitter and the receiver as in conventional noncoherent fsk st fsk can be viewed as a special unitary st modulation design however st fsk has a number of advantages over existing unitary st modulation designs st fsk is easier to design and enjoys lower decoding complexity furthermore st fsk guarantees full diversity finally st fsk can be adopted in the digital as well as in the analog domain and merges very naturally with frequency hopping multiple access as expected all these advantages come at the cost of a decrease in spectral efficiency

numerical approximations to nonlinear conservation laws with locally varying time and space grids an explicit time differencing technique is introduced to approximate nonlinear conservation laws this differencing technique links together an arbitrary number of space regimes containing fine and coarse time increments previous stability requirements i e the cfl condition placed a global bound on the size of the time increments for scalar monotone approximations in one space dimension using this variable step time differencing convergence to the correct physical solution is proven given only a local cfl condition

a posteriori error analysis for a class of integral equations and variational inequalities we consider elliptic and parabolic variational equations and inequalities governed by integro differential operators of order 2s in 0 2 our main motivation is the pricing of european or american options under levy processes in particular pure jump processes or jump diffusion processes with tempered stable processes the problem is discretized using piecewise linear finite elements in space and the implicit euler method in time we construct a residual type a posteriori error estimator which gives a computable upper bound for the actual error in h s norm the estimator is localized in the sense that the residuals are restricted to the discrete non contact region numerical experiments illustrate the accuracy of the space and time estimators and show that they can be used to measure local errors and drive adaptive algorithms

adaptive petrov galerkin methods for first order transport equations we propose stable variational formulations for certain linear unsymmetric operators with first order transport equations in bounded domains serving as the primary focus of this paper the central objective is to develop for such classes adaptive solution concepts with provable error reduction to adaptively resolve anisotropic solution features such as propagating singularities the presently proposed variational formulations allow in particular the employment of trial spaces spanned by directional representation systems since such systems typically given as frames are known to be stable only in l 2 special emphasis is placed on l 2 stable formulations the proposed stability concept is based on perturbations of certain ideal test spaces in petrov galerkin formulations see also l f demkowicz and j gopalakrishnan comput methods appl mech engrg 199 2010 pp 1558 1572 l demkowicz and j gopalakrishnan numer methods partial differential equations 27 2011 pp 70 105 j

wavelet in time multigrid in space preconditioning of parabolic evolution equations two space time variational formulations of linear parabolic evolution equations are considered one is symmetric and elliptic on the trial space while the other is not in each case a space time petrov galerkin discretization using suitable tensor product trial and test functions leads to a large linear system of equations the well posedness of this system with respect to parabolic norms induces a canonical preconditioner for the algebraic equations that arise after a choice of basis for the iterative resolution of this algebraic system with parallelization in the temporal direction we propose a sparse algebraic wavelet in time transformation on possibly nonuniform temporal meshes this transformation approximately block diagonalizes the preconditioner and the individual spatial blocks can then be inverted for instance by standard spatial multigrid methods in parallel the performance of the preconditioner is documented in a series of numerical experiments

covariance structure of parabolic stochastic partial differential equations with multiplicative levy noise the characterization of the covariance function of the solution process to a stochastic partial differential equation is considered in the parabolic case with multiplicative levy noise of affine type for the second moment of the mild solution a well posed deterministic space time variational problem posed on projective and injective tensor product spaces is derived which subsequently leads to a deterministic equation for the covariance function

constellation design for trellis coded unitary space time modulation systems we consider the problem of creating signal constellations for trellis coded unitary space time communication links where neither the transmitter nor the receiver knows the fading gains of the channel our study includes constellation design techniques for trellis coded schemes with and without parallel paths which allows us to find a tradeoff between low complexity and high performance we present a new formulation of the constellation design problem for trellis coded unitary space time modulation tcustm schemes the two key differences in our approach against those of other authors are that we not only combine the constellation design and mapping by set partitioning into one step but we also use directly the chernoff bound of the pairwise error probability as a design metric by novelly employing a theorem for the clarke subdifferential of the sum of the k largest singular values of the unitary matrix we also present a numerical optimization procedure for finding signal constellations resulting in high performance communications systems to demonstrate the advantages of our new design method we report the best constellations found for tcustm systems simulation results show that these constellations achieve a 1 db coding gain at a bit error rate of 10 4 against usually used constellations

on the construction of sparse tensor product spaces let 1 rn1 and 2 rn2 be two given domains and consider on each domain a multiscale sequence of ansatz spaces of polynomial exactness r1 and r2 respectively in this paper we study the optimal construction of sparse tensor products made from these spaces in particular we derive the resulting cost complexities to approximate functions with anisotropic and isotropic smoothness on the tensor product domain 1 2 numerical results validate our theoretical findings

application of discrete wavelet transform to the solution of boundary value problems for quasi linear parabolic equations the wavelet method for solving the linear and quasi linear parabolic equations under initial and boundary conditions is set out by applying regular multi resolution analysis and received formula for differentiating wavelet decompositions of functions of many variables the problem is reduced to a finite set of linear and accordingly nonlinear algebraic equations for the wavelet coefficients of the problem solution the general scheme for finite dimensional approximation in the regularization method is combined with the discrepancy principle for quasi linear parabolic equations the convergence rate of an approximate weak solution to a classical one is estimated the proposed method is used for constructing stable approximate wavelet decompositions of weak solutions to boundary value problems for the unsteady porous medium flow equation with discontinuous coefficients and inexact data

adaptive wavelet methods for elliptic partial differential equations with random operators we apply adaptive wavelet methods to boundary value problems with random coefficients discretized by wavelets in the spatial domain and tensorized polynomials in the parameter domain greedy algorithms control the approximate application of the fully discretized random operator and the construction of sparse approximations to this operator we suggest a power iteration for estimating errors induced by sparse approximations of linear operators

sparse grid reduced basis bayesian inversion abstract we analyze reduced basis rb for short short acceleration of recently proposed sparse bayesian inversion algorithms for partial differential equations with uncertain distributed parameter for observation data subject to additive gaussian observation noise specifically bayesian inversion of affine parametric linear operator families on possibly high dimensional parameter spaces we consider high fidelity petrov galerkin pg discretizations of these countably parametric operator families we allow general families of inf sup stable pg finite element methods covering most conforming primal and mixed finite element discretizations of standard problems in mechanics rb acceleration of the high dimensional parametric forward response maps which need to be numerically solved numerous times in bayesian inversion is proposed and convergence rate bounds for the error in the bayesian estimate incurred by the use of rb are derived as consequence of recent theoretical results on dimension independent sparsity of parametric responses and preservation of sparsity for holomorphic parametric problems we establish new convergence rates of greedy rb compressions for both the parametric forward maps as well as for the countably parametric posterior densities which arise in bayesian inversion we show that the convergence rates for the rb compressions of the parametric forward maps as well as of the countably parametric sparse bayesian posterior densities are free from the curse of dimensionality and depend only on the sparsity of the uncertain input data in particular we establish the quadratic convergence of the rb compression for the posterior densities with respect to that for the parametric forward maps numerical experiments for linear elliptic affine parametric model problems in two space dimensions with hundreds of parameters are reported which confirm that the proposed adaptive sparse grid reduced basis algorithms indeed exploit sparsity of both the parametric forward maps as well as the bayesian posterior density

communication on the grassmann manifold a geometric approach to the noncoherent multiple antenna channel we study the capacity of multiple antenna fading channels we focus on the scenario where the fading coefficients vary quickly thus an accurate estimation of the coefficients is generally not available to either the transmitter or the receiver we use a noncoherent block fading model proposed by marzetta and hochwald see ibid vol 45 p 139 57 1999 the model does not assume any channel side information at the receiver or at the transmitter but assumes that the coefficients remain constant for a coherence interval of length t symbol periods we compute the asymptotic capacity of this channel at high signal to noise ratio snr in terms of the coherence time t the number of transmit antennas m and the number of receive antennas n while the capacity gain of the coherent multiple antenna channel is min m n bits per second per hertz for every 3 db increase in snr the corresponding gain for the noncoherent channel turns out to be m 1 m t bits per second per hertz where m min m n t 2 the capacity expression has a geometric interpretation as sphere packing in the grassmann manifold

adaptive wavelet methods for elliptic operator equations convergence rates this paper is concerned with the construction and analysis of wavelet based adaptive algorithms for the numerical solution of elliptic equations these algorithms approximate the solution u of the equation by a linear combination of n wavelets therefore a benchmark for their performance is provided by the rate of best approximation to u by an arbitrary linear combination of n wavelets so called n term approximation which would be obtained by keeping the n largest wavelet coefficients of the real solution which of course is unknown the main result of the paper is the construction of an adaptive scheme which produces an approximation to u with error o n s in the energy norm whenever such a rate is possible by n term approximation the range of s 0 for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator moreover it is shown that the number of arithmetic operations needed to compute the approximate solution stays proportional to n the adaptive algorithm applies to a wide class of elliptic problems and wavelet bases the analysis in this paper puts forward new techniques for treating elliptic problems as well as the linear systems of equations that arise from the wavelet discretization