discontinuous galerkin finite element methods for time dependent hamilton jacobi bellman equations with cordes coefficients we propose and analyse a fully discrete discontinuous galerkin time stepping method for parabolic hamilton jacobi bellman equations with cordes coefficients the method is consistent and unconditionally stable on rather general unstructured meshes and time partitions error bounds for both rough and regular solutions in terms of temporal regularity show that the method is arbitrarily high order with optimal convergence rates with respect to the mesh size time interval length and temporal polynomial degree and possibly suboptimal by an order and a half in the spatial polynomial degree numerical experiments on problems with strongly anisotropic diffusion coefficients and early time singularities demonstrate the accuracy and computational efficiency of the method with exponential convergence rates achieved under combined hp and tau q q refinement

convergent difference schemes for degenerate elliptic and parabolic equations hamilton jacobi equations and free boundary problems convergent numerical schemes for degenerate elliptic partial differential equations are constructed and implemented simple conditions are identified which ensure that nonlinear finite difference schemes are monotone and nonexpansive in the maximum norm explicit schemes endowed with an explicit cfl condition are built for time dependent equations and are used to solve stationary equations iteratively explicit and implicit formulations of monotonicity for first and second order equations are unified bounds on orders of accuracy are established an example of a scheme which is stable but nonmonotone and nonconvergent is presented schemes for hamilton jacobi equations obstacle problems one phase free boundary problems and stochastic games are built and computational results are presented

construction of robust frames in erasure recovery abstract erasures occur often in signal transmission in 10 the authors introduced robust frames with which one can recover erased frame coefficients even if the locations are unknown in this paper we give some explicit construction of robust frames

burst erasures and the mean square error for cyclic parseval frames this paper investigates the performance of frames for the linear redundant encoding of vectors when consecutive frame coefficients are lost due to the occurrence of random burst errors we assume that the distribution of bursts is invariant under cyclic shifts and that the burst length statistics are known in analogy with rate distortion theory we wish to find frames of a given size which minimize the mean square reconstruction error for the encoding of vectors in a complex finite dimensional hilbert space we obtain an upper bound for the mean square reconstruction error for a given parseval frame and in the case of cyclic parseval frames we find a family of frames which minimizes this upper bound under certain conditions these minimizers are identical to complex bose chaudhuri hocquenghem codes discussed in the literature the accuracy of our upper bounds for the mean square error is substantiated by complementary lower bounds all estimates are based on convexity arguments and a discrete rearrangement inequality

energy concentration in band limited extrapolation the authors present a novel bandlimited extrapolation method that incorporates qualitative prior knowledge of the distribution of signal energy in both the time domain spatial domain and the frequency domain the extrapolation is easily computed by simple noniterative algorithm the prior information is incorporated by means of a weighted energy measure and the resulting solution is derived by straightforward hilbert space optimization the algorithm accommodates complex valued sequences nonuniform sampling and arbitrary spectral support and is applicable in one or several dimensions many existing extrapolation algorithms are noted as special cases examples of signal extrapolation and spectrum estimation using the new algorithm are given in one and two dimensions algorithm modification for noisy data is investigated and error bounds are presented

c 1 comonotone hermite interpolation via parametric cubics abstract we present a local scheme for c 1 hermite interpolation the interpolant constructed is a parametric piecewise cubic and it is comonotone with the data the approximation order is also investigated under appropriate assumptions the method is fourth order accurate

linear dispersion codes for mimo systems based on frame theory multiple input multiple output mimo wireless communication systems provide high capacity due to the plurality of modes available in the channel existing signaling techniques for mimo systems have focused primarily on multiplexing for high data rate or diversity for high link reliability in this paper we present a new linear dispersion code design for mimo rayleigh fading channels the proposed design bridges the gap between multiplexing and diversity and yields codes that typically perform well both in terms of ergodic capacity as well as error probability this is important because as we show designs performing well from an ergodic capacity point of view do not necessarily perform well from an error probability point of view various techniques are presented for finding codes with good error probability performance monte carlo simulations illustrate performance of some example code designs in terms of ergodic capacity codeword error probability and bit error probability

duality based a posteriori error estimates for some approximation schemes for optimal investment problems abstract we consider a markov chain approximation scheme for utility maximization problems in continuous time which uses in turn a piecewise constant policy approximation euler maruyama time stepping and a gaus hermite approximation of the gausian increments the error estimates previously derived in picarelli and reisinger 2018 are asymmetric between lower and upper bounds due to the control approximation and improve on known results in the literature in the lower case only in the present paper we use duality results to obtain a posteriori upper error bounds which are empirically of the same order as the lower bounds the theoretical results are confirmed by our numerical tests

joint source channel coding using real bch codes for robust image transmission in this paper a new still image coding scheme is presented in contrast with standard tandem coding schemes where the redundancy is introduced after source coding it is introduced before source coding using real bch codes a joint channel model is first presented the model corresponds to a memoryless mixture of gaussian and bernoulli gaussian noise it may represent the source coder the channel coder the physical channel and their corresponding decoder decoding algorithms are derived from this channel model and compared to a state of art real bch decoding scheme a further comparison with two reference tandem coding schemes and the proposed joint coding scheme for the robust transmission of still images has been presented when the tandem scheme is not accurately tuned the joint coding scheme outperforms the tandem scheme in all situations compared to a tandem scheme well tuned for a given channel situation the joint coding scheme shows an increased robustness as the channel conditions worsen the soft performance degradation observed when the channel worsens gives an additional advantage to the joint source channel coding scheme for fading channels since a reconstruction with moderate quality may be still possible even if the channel is in a deep fade

comparison of the asymptotic stability for multirate rosenbrock methods systems of ordinary differential equations containing different time scales can be treated by multirate extensions of singlerate integration methods since the stability properties of a singlerate method are usually not carried over to the corresponding multirate method it is necessary to investigate the asymptotic stability of multirate methods we apply different linearly implicit rosenbrock methods in a recursive multirate procedure for the interpolation continuous extension hermite interpolation and a special monotone hermite interpolation are considered we give stability regions and illustrate their importance for a two component test problem

an optimal basis of band limited functions for signal analysis and design this paper studies signal concentration in the time and frequency domains using the general constrained variational method of franks the minimum k th k 0 2 4 moment time duration measure for band limited signals is formulated a complete orthonormal set of band limited functions in l2 w w with the minimum fourth moment time duration measure is obtained numerical investigations of our optimal 4th moment functions show 1 less energy concentration in the main lobe than the prolate spheroidal wave functions pswf 2 higher energy concentration in the main lobe than gabor s function and 3 a larger main lobe than both pswf and gabor applications for our basis functions include 1 radar systems and high resolution communication systems and 2 representation and approximation to any band limited signal in a given time interval

frames graphs and erasures two uniform frames and their use for the coding of vectors are the main subject of this paper these frames are known to be optimal for handling up to two erasures in the sense that they minimize the largest possible error when up to two frame coefficients are set to zero here we consider various numerical measures for the reconstruction error associated with a frame when an arbitrary number of the frame coefficients of a vector are lost we derive general error bounds for two uniform frames when more than two erasures occur and apply these to concrete examples we show that among the 227 known equivalence classes of two uniform 36 15 frames arising from hadamard matrices there are 5 that give smallest error bounds for up to 8 erasures

numerical approximation for a superreplication problem under gamma constraints we study a superreplication problem of european options with gamma constraints in mathematical finance the initially unbounded control problem is set back to a problem involving a viscosity pde solution with a set of bounded controls then a numerical approach is introduced unconditionally stable with respect to the mesh steps a generalized finite difference scheme is used since basic finite differences cannot work in our case numerical tests illustrate the validity of our approach

error bounds for monotone approximation schemes for parabolic hamilton jacobi bellman equations we obtain nonsymmetric upper and lower bounds on the rate of convergence of general monotone approximation numerical schemes for parabolic hamilton jacobi bellman equations by introducing a new notion of consistency our results are robust and general they improve and extend earlier results by krylov barles and jakobsen we apply our general results to various schemes including crank nicholson type finite difference schemes splitting methods and the classical approximation by piecewise constant controls in the first two cases our results are new and in the last two cases the results are obtained by a new method which we develop here

two approximations of solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games the associated initial value problems almost never have global time classical solutions and one must deal with suitable generalized solutions the correct class of generalized solutions has only recently been established by the authors this article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates analogous results are proved by similar means for the method of vanishing viscosity author

optimum waveform signal sets with amplitude and energy constraints it is desirable to choose the waveforms making up a signaling alphabet so that they are maximally separated one from another this problem is considered in the space of square integrable functions for signals which have finite duration and are constrained in the ranges of their values as well as in energy corresponding to each of the following cases we establish sharp bounds for the minimum distance and for the average distance between elements of a fixed size signal set and construct sets of signals that attain both bounds simultaneously begin list item em case a energy constraint only the average energy of the waveforms in the signal set is at most sigma where 0 leq sigma item em case b energy and peak amplitude constraints the average energy of the waveforms in the signal set is leq sigma 0 leq sigma and the absolute value of each waveform is at most 1 item em case c energy and value constraints the average energy of the waveforms in the signal set is at most b 2 sigma a 2 1 sigma and each waveform takes values in the set a b where 0 leq a and 0 leq sigma leq 1 end list cases a and b are applicable to signal design for communication in channels with additive noise say gaussian and case c is applicable to signal design for optical channels where the signal represents the intensity of a photon stream the general character of the results is that the minimum distance behaves like gamma sigma in cases a and b and like gamma sigma 1 sigma in case c with gamma a suitable constant

optimal frames for erasures we study frames from the viewpoint of coding theory we introduce a numerical measure of how well a frame reconstructs vectors when some of the frame coefficients of a vector are lost and then attempt to find and classify the frames that are optimal in this setting

on quadratic approximations for hamilton jacobi bellman equations we study simple quadratic approximations for general hamilton jacobi bellman equations the theoretical error bounds are shown to be composed of the time discretization errors and quadratic approximation ones at each time step several numerical experiments show that the quadratic approximations work for the equations with the boundary and hamiltonian well approximated by the quadratic functions

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

efficient reconstruction from frame based multiple descriptions redundant bases or frames are a promising technique for multiple description signal coding frames can provide sufficient redundancy for recovery in case of coefficient losses due for instance to packet transmission errors in this paper we consider the problem of dual frame computation in the context of finite impulse response fir analysis and synthesis filterbanks in particular we propose an algorithm for dual frame computation which recovers the missing frame coefficients followed by synthesis with the original filterbank it is shown that the algorithm operates locally and allows for reduced complexity and low delay reconstruction of the original signal the case of excessive losses and incomplete bases is also considered experiments are provided to evaluate the algorithm performance and test its applicability in practical contexts

linearly connected sequences and spectrally optimal dual frames for erasures abstract in the case that a frame is prescribed for applications and erasures occur in the process of data transmissions we examine optimal dual frames for the recovery from single erasures in contrast to earlier papers we consider the spectral radius of the error operator instead of its operator norm as a measure of optimality this notion of optimality is natural when the neumann series is used to recover the original data in an iterative manner we obtain a complete characterization of spectrally one erasure optimal dual frames in terms of the redundancy distribution of the prescribed frame our characterization relies on the connection between erasure optimal frames and the linear connectivity property of the frame we prove that the linear connectivity property is equivalent to the intersection dependent property and is also closely related to the well known concept of a k independent set additionally we also establish several necessary and sufficient conditions for the existence of an alternate dual frame to make the iterative reconstruction work

convergence of meshfree collocation methods for fully nonlinear parabolic equations we prove the convergence of meshfree collocation methods for the terminal value problems of fully nonlinear parabolic partial differential equations in the framework of viscosity solutions provided that the basis function approximations of the terminal condition and the nonlinearities are successful at each time step a numerical experiment with a radial basis function demonstrates the convergence property

generalized and extended subspace algorithms for error correction with quantized dft codes discrete fourier transform dft codes have been used to provide robustness against errors and erasures in various applications this paper focuses on improving error localization of the bose chaudhuri hocquenghem bch dft codes first we analyze how the subspace based error localization outperforms the coding theoretic one then we propose an extension of the subspace based error localization based on additional syndrome that improves the existing one and is naturally suitable for rate adaptive distributed source coding dsc further we propose a new generic subspace based algorithm to decode bch dft codes the proposed approach generalizes the encoding and decoding of this important class of dft codes it introduces many different decoding matrices for a dft code this diversity is then used to diminish the effect of the quantization noise and thus to improve the decoding finally the extended and generalized approaches are combined to maximize the decoding gain simulation results demonstrate the capability of the proposed algorithms to perform significantly better than the existing subspace based error localization in the presence of quantization noise

hypercontractivity for the heat semigroup for ultraspherical polynomials and on the n sphere a logarithmic sobolev inequality with respect to the probability measure a 1 x2 12 dx on 1 1 is proved for all 12 from this inequality sharp hypercontractive estimates are derived for the heat semigroups for ultraspherical polynomials and on the n sphere

piecewise cubic curve fitting algorithm we give a piecewise cubic curve fitting algorithm which preserves monotonicity of the data the algorithm has a higher order of convergence than the fritsch carlson algorithm and is simpler than the eisenstat jackson lewis algorithm

the order of monotone piecewise cubic interpolation fritsch and carlson siam j numer anal 17 1980 pp 238 246 developed an algorithm which produces a monotone c 1 piecewise cubic interpolant to a monotone function we show that their algorithm yields a third order approximation while a modification is fourth order accurate

consistency of generalized finite difference schemes for the stochastic hjb equation we analyze a class of numerical schemes for solving the hjb equation for stochastic control problems which enters the framework of markov chain approximations and generalizes the usual finite difference method the latter is known to be monotonic and hence valid only if the scaled covariance matrix is dominant diagonal we generalize this result by given the set of neighboring points allowed to enter the scheme showing how to compute effectively the class of covariance matrices that is consistent with this set of points we perform this computation for several cases in dimensions 2 3 and 4

an optimum complete orthonormal basis for signal analysis and design derives a new set of basis functions which have high energy concentration in both time and frequency the approach is based on transforming a signal energy maximization problem into an equivalent eigenvalue problem for a certain integral operator and then showing that the eigenvectors of this integral operator form a complete orthonormal basis this basis set is in some sense optimal since each basis function has the highest energy concentration within a certain subspace of spl lscr sup 2 furthermore this basis can be employed to analyze and design communication signals for which the energy concentration within the essential duration and essential bandwidth is important and some other optimum characteristics would also be required

on the rate of convergence of finite difference approximations for bellmans equations with variable coefficients the estimates presented here for parabolic bellman s equations with variable coefficients extend the ones earlier obtained for constant coefficients

semi lagrangian schemes for linear and fully non linear diffusion equations for linear and fully non linear diffusion equations of bellmanisaacs type we introduce a class of monotone approximation schemes relying on monotone interpolation as opposed to classical numerical methods these schemes converge for degenerate diffusion equations having general nondiagonal dominant coefficient matrices such schemes have to have a wide stencil in general besides providing a unifying framework for several known first order accurate schemes our class of schemes also includes more efficient versions and a new second order scheme that converges only for essentially monotone solutions the methods are easy to implement and analyze and they are more efficient than some other known schemes we prove stability and convergence of the schemes in the general case and provide error estimates in the convex case which are robust in the sense that they apply to degenerate equations and non smooth solutions the methods are extensively tested

reconstruction of signals from frame coefficients with erasures at unknown locations we propose new approaches to the problems of recovering signals from the rearranged frame coefficients or frame coefficients with erasures at either known or unknown locations these problems naturally arise from applications where the encoded information needs to be transmitted for example in signal image processing information and coding theory and communications we show that with the appropriate choices of the frames that are used for encoding the signal with erasures occurring at known locations can be easily recovered without inverting the sub frame operators each time our new easy to implement and cost efficient algorithm provides perfect reconstruction of the original signal to address the problem of recovering erased coefficients from unknown locations we propose to use a class of frames that are almost robust with respect to m erasures we prove that every frame with uniform excess can be rescaled to an almost robust frame and the locations of erased data can be perfectly recovered for almost all the signals similar results are obtained for recovering the original order of a disordered rearranged set of frame coefficients numerical examples are presented to test the main results whenever the received data are noise free we can recover the original signal exactly from frame coefficients with erasures at unknown locations or from a disordered set of frame coefficients

frame theoretic analysis of dft codes with erasures discrete fourier transform dft codes have been used as joint source channel codes in order to provide robustness against channel errors and erasures this paper presents a frame theoretic analysis of lowpass dft codes with erasures first it is shown that such codes make a special class of frames then the message reconstructions by the frame theoretic signal space projection and coding theoretic syndrome decoding of erasures are analyzed and their equivalence is proved the error performance of lowpass dft codes is analyzed using the frame theory the analysis helps in classifying the subframes of a dft frame into several classes this classification leads to the development of packetization criteria in order to guarantee minimum mean square reconstruction error and thus to minimize the effect of the quantization error simulation results substantiate the presented theory and the proposed packetization schemes

filter bank frame expansions with erasures we study frames for robust transmission over the internet in our previous work we used quantized finite dimensional frames to achieve resilience to packet losses here we allow the input to be a sequence in l sub 2 z and focus on a filter bank implementation of the system we present results in parallel r sup n or c sup n versus l sub 2 z and show that uniform tight frames as well as newly introduced strongly uniform tight frames provide the best performance

some regularity and convergence results for parabolic hamilton jacobi bellman equations in bounded domains abstract we study parabolic hamilton jacobi bellman hjb equations in bounded domains with strong dirichlet boundary conditions we work under the assumption of the existence of a sufficiently regular barrier function for the problem to obtain well posedness and regularity of a related switching system and the convergence of its components to the hjb equation in particular we show existence of a viscosity solution to the switching system by a novel construction of sub and supersolutions and application of perron s method error bounds for monotone schemes for the hjb equation are then derived from estimates near the boundary where the standard regularisation procedure for viscosity solutions is not applicable and are found to be of the same order as known results for the whole space we deduce error bounds for some common finite difference and truncated semi lagrangian schemes

convergence of the embedded mean variance optimal points with discrete sampling a numerical technique based on the embedding technique proposed in math finan 10 387 406 2000 appl math optim 42 19 33 2000 for dynamic mean variance mv optimization problems may yield spurious points i e points which are not on the efficient frontier in siam j control optim 52 1527 1546 2014 it is shown that spurious points can be eliminated by examining the left upper convex hull of the solution of the embedded problem however any numerical algorithm will generate only a discrete sampling of the solution set of the embedded problem in this paper we formally establish that under mild assumptions every limit point of a suitably defined sequence of upper convex hulls of the sampled solution of the embedded problem is on the original mv efficient frontier for illustration we discuss an mv asset liability problem under jump diffusions which is solved using a numerical hamilton jacobi bellman partial differential equation approach

convergent difference schemes for nonlinear parabolic equations and mean curvature motion explicit finite difference schemes are given for a collection of parabolic equations which may have all of the following complex features degeneracy quasilinearity full nonlinearity and singularities in particular the equation of motion by mean curvature is included the schemes are monotone and consistent so that convergence is guaranteed by the general theory of approximation of viscosity solutions of fully nonlinear problems in addition an intriguing new type of nonlocal problem is analyzed which is related to the schemes and another very different sort of approximation is presented as well

complex equiangular cyclic frames and erasures we derive various interesting properties of complex equiangular cyclic frames for many pairs n k using gauss sums and number theory we further use these results to study the random and burst errors of some special classes of complex equiangular cyclic n k frames

a finite element like scheme for integro partial differential hamilton jacobi bellman equations we construct a finite element like scheme for fully nonlinear integro partial differential equations arising in optimal control of jump processes special cases of these equations include optimal portfolio and option pricing equations in finance the schemes are monotone and robust we prove that they converge in very general situations including degenerate equations multiple dimensions relatively low regularity of the data and for most if not all types of jump models used in finance in all cases we provide probably optimal error bounds these bounds apply when grids are unstructured and integral terms are very singular two features that are new or highly unusual in this setting

some estimates for finite difference approximations some estimates for the approximation of optimal stochastic control problems by discrete time problems are obtained in particular an estimate for the solutions of the continuous time versus the discrete time hamilton jacobi bellman equations is given the technique used is more analytic than probabilistic

nonnegativity monotonicity or convexity preserving cubic and quintic hermite interpolation the hermite polynomials are simple effective interpolants of discrete data these interpolants can preserve local positivity monotonicity and convexity of the data if we restrict their derivatives to satisfy constraints at the data points this paper de scribes the conditions that must be satisfied for cubic and quintic hermite interpolants to preserve these properties when they exist in the discrete data we construct algorithms to ensure that these constraints are satisfied and give numerical examples to illustrate the effectiveness of the algorithms on locally smooth and rough data 1 introduction piecewise polynomial interpolants especially those based on hermite polynomials polynomials determined by their values and values of one or more derivatives at both ends of an interval have a number of desirable properties they are easy to compute once the derivative values are chosen if the derivative values are chosen locally e g by finite difference methods then the interpolant at a given point will depend only on the given data at nearby mesh points if the derivatives are computed by spline methods then the interpolant will have an extra degree of continuity at the mesh points in either case the interpolant is linear in the given function values and has excellent convergence properties as the mesh spacing decreases these methods however do not necessarily preserve the shape of the given data when the data arise from a physical experiment it may be vital that the interpolant preserve nonnegativity f x 0 nonpositivity f x 0 or f x 0 or concavity f x 0 in this and other cases geometric considerations such as preventing spurious behavior near rapid changes in the data may be more important than the asymptotic accuracy of the interpolation method one can construct a shape preserving interpolant by constraining the derivatives for the hermite polynomials to meet conditions which imply the desired properties 4 5 8 11 15 20 by adding new mesh points

probabilistic error analysis for some approximation schemes to optimal control problems abstract we introduce a class of numerical schemes for optimal stochastic control problems based on a novel markov chain approximation which uses in turn a piecewise constant policy approximation euler maruyama time stepping and a gaus hermite approximation of the gausian increments we provide lower error bounds of order arbitrarily close to 1 2 in time and 1 3 in space for lipschitz viscosity solutions coupling probabilistic arguments with regularization techniques as introduced by krylov the corresponding order of the upper bounds is 1 4 in time and 1 5 in space for sufficiently regular solutions the order is 1 in both time and space for both bounds finally we propose techniques for further improving the accuracy of the individual components of the approximation

frames of uniform subframe bounds with applications to erasures abstract we introduce a class of frames called usb frames frames of uniform subframe bounds which mean that any subsequence of the frame is a frame for the subspace it spans and the frame bounds depend only on the cardinality of the subsequence we give several necessary and sufficient conditions for a frame to be a usb frame as an application we show that a tight usb frame is optimal for erasures in the sense that the reconstruction error is independent of the position of erasures we give several explicit construction of usb frames