innovation rate sampling of pulse streams with application to ultrasound imaging signals comprised of a stream of short pulses appear in many applications including bioimaging and radar the recent finite rate of innovation framework has paved the way to low rate sampling of such pulses by noticing that only a small number of parameters per unit time are needed to fully describe these signals unfortunately for high rates of innovation existing sampling schemes are numerically unstable in this paper we propose a general sampling approach which leads to stable recovery even in the presence of many pulses we begin by deriving a condition on the sampling kernel which allows perfect reconstruction of periodic streams from the minimal number of samples we then design a compactly supported class of filters satisfying this condition the periodic solution is extended to finite and infinite streams and is shown to be numerically stable even for a large number of pulses high noise robustness is also demonstrated when the delays are sufficiently separated finally we process ultrasound imaging data using our techniques and show that substantial rate reduction with respect to traditional ultrasound sampling schemes can be achieved

identification of sparse linear operators we consider the problem of identifying a linear deterministic operator from its response to a given probing signal for a large class of linear operators we show that stable identifiability is possible if the total support area of the operator s spreading function satisfies 1 2 this result holds for an arbitrary possibly fragmented support region of the spreading function does not impose limitations on the total extent of the support region and most importantly does not require the support region to be known prior to identification furthermore we prove that stable identifiability of almost all operators is possible if 1 this result is surprising as it says that there is no penalty for not knowing the support region of the spreading function prior to identification algorithms that provably recover all operators with 1 2 and almost all operators with 1 are presented

algebraic multigrid amg experiences and comparisons algebraic multigrid methods are designed for the solution of sparse linear systems of equations using multigrid principles in contrast to standard multigrid methods amg does not take advantage of the origin of a particular system of equations at hand nor does it exploit any underlying geometrical situation fully automatically and based solely on algebraic information contained in the given matrix amg constructs a sequence of grids and corresponding operators a special amg algorithm will be presented for a wide range of problems including certain problems which do not have a continuous background this algorithm yields an iterative method which exhibits a convergence behavior typical for multigrid methods

adaptivity in space and time for reaction diffusion systems in electrocardiology the paper introduces and studies numerical methods that are fully adaptive in both three dimensional 3d space and time to challenging multiscale cardiac reaction diffusion models in these methods temporal adaptivity comes via stepsize control in function space oriented linearly implicit time integration while spatial adaptivity is realized within multilevel finite element methods controlled by a posteriori local error estimators in contrast to other recent adaptivity approaches to cardiac modeling that discretize first in space and then in time so called method of lines our method discretizes first in time and then in space so called rothe method an approach that has already proven to be highly efficient in a number of challenging multiscale problems in science and technology kardos code library with this method the evolution of a complete heartbeat from the excitation to the recovery phase is simulated both in the frame of the anisotropic monodomain models and in the more realistic anisotropic bidomain models coupled with either a variant of the simple fitzhugh nagumo model or the more complex phase i luo rudy ionic model the numerical results exhibit a rather satisfactory performance of our adaptive method for complex cardiac reaction diffusion models on 3d domains up to moderate sizes in particular the method accurately resolves the evolution of the intra and extracellular potentials gating variables and ion concentrations during the excitation plateau and recovery phases

time delay estimation from low rate samples a union of subspaces approach time delay estimation arises in many applications in which a multipath medium has to be identified from pulses transmitted through the channel previous methods for time delay recovery either operate on the analog received signal or require sampling at the nyquist rate of the transmitted pulse in this paper we develop a unified approach to time delay estimation from low rate samples this problem can be formulated in the broader context of sampling over an infinite union of subspaces although sampling over unions of subspaces has been receiving growing interest previous results either focus on unions of finite dimensional subspaces or finite unions the framework we develop here leads to perfect recovery of the multipath delays from samples of the channel output at the lowest possible rate even in the presence of overlapping transmitted pulses and allows for a variety of different sampling methods the sampling rate depends only on the number of multipath components and the transmission rate but not on the bandwidth of the probing signal this result can be viewed as a sampling theorem over an infinite union of infinite dimensional subspaces by properly manipulating the low rate samples we show that the time delays can be recovered using the well known esprit algorithm combining results from sampling theory with those obtained in the context of direction of arrival estimation we develop sufficient conditions on the transmitted pulse and the sampling functions in order to ensure perfect recovery of the channel parameters at the minimal possible rate

compact schemes for anisotropic reaction diffusion equations with adaptive time step many problems in biology and engineering are governed by anisotropic reaction diffusion equations with a very rapidly varying reaction term these characteristics of the system imply the use of very fine meshes and small time steps in order to accurately capture the propagating wave avoiding the appearance of spurious oscillations in the wave front this work develops a fourth order compact scheme for anisotropic reaction diffusion equations with stiff reactive terms as mentioned the scheme accounts for the anisotropy of the media and incorporates an adaptive time step for handling the stiff reactive term the high order scheme allows working with coarser meshes without compromising numerical accuracy rendering a more efficient numerical algorithm by reducing the total computation time and memory requirements the order of convergence of the method has been demonstrated on an analytical solution with neumann boundary conditions the scheme has also been implemented for the solution of anisotropic electrophysiology problems anisotropic square samples of normal and ischemic cardiac tissue have been simulated by means of the monodomain model with the reactive term defined by luo rudy ii dynamics the simulations proved the effectiveness of the method in handling anisotropic heterogeneous non linear reaction diffusion problems bidimensional tests also indicate that the fourth order scheme requires meshes about 45 coarser than the standard second order method in order to achieve the same accuracy of the results copyright 2009 john wiley sons ltd

identification of parametric underspread linear systems and super resolution radar identification of time varying linear systems which introduce both time shifts delays and frequency shifts doppler shifts is a central task in many engineering applications this paper studies the problem of identification of underspread linear systems ulss whose responses lie within a unit area region in the delay doppler space by probing them with a known input signal it is shown that sufficiently underspread parametric linear systems described by a finite set of delays and doppler shifts are identifiable from a single observation as long as the time bandwidth product of the input signal is proportional to the square of the total number of delay doppler pairs in the system in addition an algorithm is developed that enables identification of parametric ulss from an input train of pulses in polynomial time by exploiting recent results on sub nyquist sampling for time delay estimation and classical results on recovery of frequencies from a sum of complex exponentials finally application of these results to super resolution target detection using radar is discussed specifically it is shown that the proposed procedure allows to distinguish between multiple targets with very close proximity in the delay doppler space resulting in a resolution that substantially exceeds that of standard matched filtering based techniques without introducing leakage effects inherent in recently proposed compressed sensing based radar methods

spectral theory for linearizations of dynamical systems the description of the asymptotic growth rates for the solution trajectories of a system of differential equations is a central problem in the qualitative theory of dynamical systems these growth rates are described by the eigenvalues of the jacobian matrix for trajectories near a critical point by the floquet exponents for trajectories close to a periodic orbit and generally by the lyapunov characteristic exponents in a neighborhood of an arbitrary trajectory in the examples the asymptotic behavior of the solution trajectories is given by the association of a spectrum and this is a recurrent theme in the method of linearization sacker and sell io recently introduced a spectral theory for differential systems of quite general type the linear skew product flows see section 3 their work is in the spirit of the classical linearization theory and is closely tied to the idea of the lyapunov exponents we prefer to use the word spectrum literally identifying it with the spectrum of an operator on a banach space in general we consider a one parameter group 4 of homeomorphisms of a compact metric space m which for example may be the solution flow of a differential equation on a smooth manifold second we assume that a continuous vector bundle e m is defined i e a locally trivial continuous map n e m with linear fibers which are isomorphic to either real n dimensional space r or complex n dimensional space c in case is the flow of a vector field x on m this vector bundle could be the tangent bundle tm finally we assume that there is group of homeomorphisms on e with n 4 and such that i e t e c is a linear isomorphism for each x e m for instance in case 4 is smooth cp could be the tangent flow on tm given by the derivative of 4 using the compactness of m we define a finsler on e which continuously assigns a norm denoted i to each fiber e with this assignment the space r e of continuous sections of e functions q m e such that 7 o q x x forms a banach space with norm iiqll supxem u x 155 1 i the topology of t e is

boosting the accuracy of finite difference schemes via optimal time step selection and non iterative defect correction abstract in this article we present a unified analysis of the simple technique for boosting the order of accuracy of finite difference schemes for time dependent partial differential equations pdes by optimally selecting the time step used to advance the numerical solution and adding defect correction terms in a non iterative manner the power of the technique which is applicable to time dependent semilinear scalar pdes where the leading order spatial derivative has a constant coefficient is its ability to increase the accuracy of formally low order finite difference schemes without major modification to the basic numerical algorithm through straightforward numerical analysis arguments we explain the origin of the boost in accuracy and estimate the computational cost of the resulting numerical method we demonstrate the utility of optimal time step ots selection combined with non iterative defect correction nidc on several different types of finite difference schemes for a wide array of classical linear and semilinear pdes in one and more space dimensions on both regular and irregular domains

subspace based estimation of time delays and doppler shifts this paper considers the problem of estimating the time delays and doppler shifts of a known waveform received via several distinct paths by an array of antennas the general maximum likelihood estimator is presented and is shown to require a 2d dimensional nonlinear minimization where d is the number of received signal reflections two alternative solutions based on signal and noise subspace fitting are proposed requiring only a d dimensional minimization in particular we show how to decouple the required search into a two step procedure where the delays are estimated and the dopplers solved for explicitly initial conditions for the time delay search can be obtained by applying generalizations of the music and esprit algorithms which are also outlined in the paper simulation examples are included to illustrate the algorithms performance relative to the cramer rao bound

performance bounds and design criteria for estimating finite rate of innovation signals in this paper we consider the problem of estimating finite rate of innovation fri signals from noisy measurements and specifically analyze the interaction between fri techniques and the underlying sampling methods we first obtain a fundamental limit on the estimation accuracy attainable regardless of the sampling method next we provide a bound on the performance achievable using any specific sampling approach essential differences between the noisy and noise free cases arise from this analysis in particular we identify settings in which noise free recovery techniques deteriorate substantially under slight noise levels thus quantifying the numerical instability inherent in such methods this instability which is only present in some families of fri signals is shown to be related to a specific type of structure which can be characterized by viewing the signal model as a union of subspaces finally we develop a methodology for choosing the optimal sampling kernels for linear reconstruction based on a generalization of the karhunen loeve transform the results are illustrated for several types of time delay estimation problems

fri sampling on structured nonuniform grids application to super resolved optical imaging the annihilating filter forms the core of many state of the art techniques for finite rate of innovation fri signal reconstruction and spectral estimation in its standard manifestation the annihilating filter is setup in the discrete time domain with uniform sampling and is used to estimate the parameters of a signal that is expressible as a sum of exponentials in this paper we consider annihilation in the continuous domain using translation operators codenamed opera and show that fri signals can be reconstructed using measurements taken on a structured nonuniform sampling sns grid the parameters are estimable uniquely under certain conditions the transition from uniform sampling to sns comes with specific advantages the measurements could be made in bursts from high signal to noise ratio snr regions which directly enhances parameter estimation accuracy the sampling interval could be made larger than in the uniform case which leads to a reduction in the mean squared error mse and closely spaced exponential parameters could be resolved better we present theoretical developments related to the sns grid and demonstrate accurate parameter estimation monte carlo performance analysis in the presence of noise shows that the opera with the sns approach yields a significant decrease in mse up to 10 to 20 db for certain snr ranges compared with the uniform sampling scenario considering an application to frequency domain optical coherence tomography we show that the proposed methodology results in higher resolution and higher reconstruction accuracy than the standard techniques

sampling moments and reconstructing signals of finite rate of innovation shannon meets strang fix consider the problem of sampling signals which are not bandlimited but still have a finite number of degrees of freedom per unit of time such as for example nonuniform splines or piecewise polynomials and call the number of degrees of freedom per unit of time the rate of innovation classical sampling theory does not enable a perfect reconstruction of such signals since they are not bandlimited recently it was shown that by using an adequate sampling kernel and a sampling rate greater or equal to the rate of innovation it is possible to reconstruct such signals uniquely these sampling schemes however use kernels with infinite support and this leads to complex and potentially unstable reconstruction algorithms in this paper we show that many signals with a finite rate of innovation can be sampled and perfectly reconstructed using physically realizable kernels of compact support and a local reconstruction algorithm the class of kernels that we can use is very rich and includes functions satisfying strang fix conditions exponential splines and functions with rational fourier transform this last class of kernels is quite general and includes for instance any linear electric circuit we thus show with an example how to estimate a signal of finite rate of innovation at the output of an rc circuit the case of noisy measurements is also analyzed and we present a novel algorithm that reduces the effect of noise by oversampling

esprit estimation of signal parameters via rotational invariance techniques an approach to the general problem of signal parameter estimation is described the algorithm differs from its predecessor in that a total least squares rather than a standard least squares criterion is used although discussed in the context of direction of arrival estimation esprit can be applied to a wide variety of problems including accurate detection and estimation of sinusoids in noise it exploits an underlying rotational invariance among signal subspaces induced by an array of sensors with a translational invariance structure the technique when applicable manifests significant performance and computational advantages over previous algorithms such as mem capon s mlm and music

operator splitting and adaptive mesh refinement for the luo rudy i model we apply second order operator splitting to the luo rudy i model for electrical wave propagation in the heart the purpose of the operator splitting is to separate the nonlinear but local reaction computations from the linear but globally coupled diffusion computations this approach allows us to use local nonlinear iterations for the stiff nonlinear reactions and to solve global linear systems for the implicit treatment of diffusion for computational efficiency we use dynamically adaptive mesh refinement amr involving hierarchies of unions of grid patches on distinct levels of refinement the linear system for the discretization of the diffusion on the composite amr grid is formulated via standard conforming finite elements on unions grid patches within a level of refinement and aligned mortar elements along interfaces between levels of refinement the linear systems are solved iteratively by preconditioned conjugate gradients our preconditioner uses multiplicative domain decomposition between levels of refinement the smoother involves algebraic additive domain decomposition between patches within a level of refinement and gauss seidel iteration within grid patches numerical results are presented in 1d and 2d including spiral waves

esprit a subspace rotation approach to estimation of parameters of cisoids in noise the application of a subspace invariance approach esprit to the estimation of parameters frequencies and powers of cisoids in noise is described esprit exploits an underlying rotational invariance of signal subspaces spanned by two temporally displaced data sets the new approach has several advantages including improved resolution over pisarenko s technique for harmonic retrieval

fourth order compact schemes with adaptive time step for monodomain reaction diffusion equations multigrid applied to fourth order compact schemes for monodomain reaction diffusion equations in two dimensions has been developed the scheme accounts for the anisotropy of the medium allows for any cellular activation model to be used and incorporates an adaptive time step algorithm numerical simulations show up to a 40 reduction in computational time for complex cellular models as compared to second order schemes for the same solution error these results point to high order schemes as valid alternatives for the efficient solution of the cardiac electrophysiology problem when complex cellular activation models are used

sampling and reconstruction of operators we study the recovery of operators with a bandlimited kohn nirenberg symbol from the action of such operators on a weighted impulse train a procedure we refer to as sampling of operators kailath bello and later kozek and the authors have shown that the sampling of operators is possible if the symbol of the operator is bandlimited to a set with area less than one in this paper we develop the theory of the sampling of operators in analogy with the classical theory of sampling of bandlimited functions we define the notions of sampling set and sampling rate for operators and give necessary and sufficient conditions on the sampling rate that depend on the size and geometry of the bandlimiting set we develop explicit reconstruction formulas for operator sampling that generalize reconstruction formulas for bandlimited functions we give necessary and sufficient conditions on the bandlimiting set under which sampling of operators is possible by their action on a given periodically weighted delta train we show that under mild geometric conditions on the bandlimiting set classes of operators are bandlimited to an unknown set of area less than one half permit sampling and reconstruction we generalize two results of the heckel and bolcskei concerning sampling of operators with area not greater than one half and less than one respectively by finding a larger class of operators to which they apply operators with bandlimited symbols have been used to model doubly dispersive communication channels with slowly time varying impulse response the results in this paper are rooted in work by bello and kailath in the 1960s

sampling signals with finite rate of innovation the authors consider classes of signals that have a finite number of degrees of freedom per unit of time and call this number the rate of innovation examples of signals with a finite rate of innovation include streams of diracs e g the poisson process nonuniform splines and piecewise polynomials even though these signals are not bandlimited we show that they can be sampled uniformly at or above the rate of innovation using an appropriate kernel and then be perfectly reconstructed thus we prove sampling theorems for classes of signals and kernels that generalize the classic bandlimited and sinc kernel case in particular we show how to sample and reconstruct periodic and finite length streams of diracs nonuniform splines and piecewise polynomials using sinc and gaussian kernels for infinite length signals with finite local rate of innovation we show local sampling and reconstruction based on spline kernels the key in all constructions is to identify the innovative part of a signal e g time instants and weights of diracs using an annihilating or locator filter a device well known in spectral analysis and error correction coding this leads to standard computational procedures for solving the sampling problem which we show through experimental results applications of these new sampling results can be found in signal processing communications systems and biological systems

paley wiener characterization of kernels for finite rate of innovation sampling exact reconstruction of finite rate of innovation signals can be achieved by employing customized sampling kernels that satisfy certain frequency domain properties we impose compact support in time as an additional constraint considering frequency domain reconstruction we derive conditions for admissible sampling kernels and corresponding sampling rates our constructive kernel design methodology is based on the paley wiener theorem for compactly supported functions the new kernels satisfy generalized strang fix conditions and have specific polynomial modulated exponential reproducing properties unlike exponential splines which have a support that is directly proportional to the number of exponentials they can generate the proposed kernels have a support that is independent of that number to analyze noise robustness we consider a special member of the class that has a sum of modulated splines sms form in the time domain and optimize its parameters to minimize the noise variance the sum of sincs sos kernel reported in the literature is an instance of this construction in noise robustness analysis sms kernels show improvement in mean squared error mse compared with the state of the art alternatives in continuous time noise the improvement in mse is about 2 db for low signal to noise ratio snr and 7 db for high snr in the case of discrete white gaussian noise the mse is lower by as much as 25 db by using a higher order sms kernel compared with the sos kernel for snrs in the range of 10 15 db

measurement of time variant linear channels the goal of channel measurement or operator identification is to obtain complete knowledge of a channel operator by observing the image of a finite number of input signals in this paper it is shown that the spreading support of the operator that is the support of the symplectic fourier transform of the kohn nirenberg symbol of the operator has area less than one then the operator is identifiable if the spreading support is larger than one then the operator is not identifiable the shape of the support region is essentially arbitrary thereby proving a conjecture of bello the input signal considered is a weighted delta train where the weights are the window function of a finite gabor system whose elements satisfy a certain robust completeness property

an efficient parametric technique for doppler delay estimation receiving an output signal that is a weighted sum of doppler and delay shifted versions of an input signal arises in various engineering systems including radar sonar and communications it is often desired to estimate the doppler and delay parameters from noisy measurements of the output signal an estimation algorithm is developed herein for the doppler frequencies delays and amplitudes for a specific choice of the input the algorithm is based on a computationally efficient search free frequency estimation technique for the sum of complex exponentials asymptotic performance bounds are developed for the estimated parameters simulation results show that the variance of the estimation error is close to the bound above a threshold signal to noise ratio indicating that the estimation algorithm is statistically efficient

matrix pencil method for estimating parameters of exponentially damped undamped sinusoids in noise a study of a matrix pencil method for estimating frequencies and damping factors of exponentially damped and or undamped sinusoids in noise is presented comparison of this method to a polynomial method svd prony method shows that the matrix pencil method and the polynomial method are two special cases of a matrix prediction approach and that the pencil method is more efficient in computation and less restrictive about signal probes it is found through perturbation analysis and simulation that for signals with unknown damping factors the pencil method is less sensitive to noise than the polynomial method an expression of the cramer rao bound for the exponential signals is presented

sub nyquist sampling of short pulses we develop sub nyquist sampling systems for analog signals comprised of several possibly overlapping finite duration pulses with unknown shapes and time positions efficient sampling schemes when either the pulse shape or the locations of the pulses are known have been previously developed to the best of our knowledge stable and low rate sampling strategies for continuous signals that are superpositions of unknown pulses without knowledge of the pulse locations have not been derived the goal in this paper is to fill this gap we propose a multichannel scheme based on gabor frames that exploits the sparsity of signals in time and enables sampling multipulse signals at sub nyquist rates moreover if the signal is additionally essentially multiband then the sampling scheme can be adapted to lower the sampling rate without knowing the band locations we show that with proper preprocessing the necessary gabor coefficients can be recovered from the samples using standard methods of compressed sensing in addition we provide error estimates on the reconstruction and analyze the proposed architecture in the presence of noise

operator algebras and the fredholm spectrum of advective equations of linear hydrodynamics abstract in this paper we give a complete description of the fredholm spectrum of the linearized 3d euler equation in terms of the dynamical spectrum of the cocycle governing the evolution of shortwave perturbations the argument is based on the pseudo differential representation of the euler group and an application of an abstract isomorphism theorem for operator c algebras we further show that for all but countably many times the fredholm spectrum of the group is rotationally invariant and thus may consist of one or two concentric annuli the results are obtained for a variety of other equations of ideal hydrodynamics