slow fast n dimensional piecewise linear differential systems abstract in this article we analyze n dimensional slow fast systems in a piecewise linear framework in particular we prove a fenichel s like theorem where we give an explicit expression for the invariant slow manifold that leads to the proof of the existence and location of maximal canards orbits we show that these orbits perturb from singular orbits through contact points of order greater than or equal to two between the reduced flow and the fold manifold in the particular case n 3 we show that the unique contact point is a visible two fold singularity

extrapolation algorithms for discrete signals with application in spectral estimation in this paper we present some new algorithms useful for extrapolation and spectral estimation of band limited sequences in one and two dimensions first we show that many of the existing extrapolation algorithms for noiseless observations are unified under the criterion of minimum norm least squares mnls extrapolation for example the iterative algorithms proposed in 2 and 8 10 are shown to be special cases of a one step gradient algorithm which has linear convergence convergence and other numerical properties are improved by going to a conjugate gradient algorithm for noisy observations these algorithms could be extended by considering a mean square extrapolation criterion which gives rise to a mean square extrapolation filter and also to a recursive extrapolation filter examples and application of these methods are given extension of these algorithms is made for problems where the signal is known to be periodic a new set of functions called the periodic discrete prolate spheroidal sequences p dpss analogous to dpss 21 22 are introduced and their properties are studied finally several of these algorithms are generalized to two dimensions and the relevant equations are given

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

spectral extrapolation of spatially bounded images mri application a spectral extrapolation algorithm for spatially bounded images is presented an image is said to be spatially bounded when it is confined to a closed region and is surrounded by a background of zeros with prior knowledge of the spatial domain zeros the extrapolation algorithm extends the image s spectrum beyond a known interval of low frequency components the result which is referred to as the finite support solution has space variant resolution features near the edge of the support region are better resolved than those in the center the resolution of the finite support solution is discussed as a function of the number of known spatial zeros and known spectral components a regularized version of the finite support solution is included for handling the case where the known spectral components are noisy for both the noiseless and noisy cases the resolution of the finite support solution is measured in terms of its impulse response characteristics and compared to the resolution of the zerofilled and nyquist solutions the finite support solution is superior to the zerofilled solution for both the noisy and noiseless data cases when compared to the nyquist solution the finite support solution may be preferred in the noisy data case examples using medical image data are provided

min max linear estimation of band limited sequences in the presence of random noise corresp a problem in band limited signal estimation is presented the signal is assumed to be a band limited sequence subject to an energy constraint the observed signal components are assumed to be corrupted by the components of a discrete time stochastic noise process with zero mean and known antocorrelations the error metric is the energy in a fixed linear combination of the error components linear estimators are considered and an explicit expression for the optimal in a min max sense estimation scheme is derived an interesting feature of the optimal estimation scheme derived in this correspondence is that it is identical to one based on stochastic models for both the signal and the noise

the eigenvalues of matrices that occur in certain interpolation problems the eigenvalues of the matrices that occur in certain finite dimensional interpolation problems are directly related to their well posedness and strongly depend on the distribution of the interpolation knots that is on the sampling set we study this dependency as a function of the sampling set itself and give accurate bounds for the eigenvalues of the interpolation matrices the bounds can be evaluated in as few as four arithmetic operations and therefore they greatly simplify the assessment of sampling sets regarding numerical stability the accuracy and usefulness of the bounds are illustrated with examples

on the optimality of the gridding reconstruction algorithm gridding reconstruction is a method to reconstruct data onto a cartesian grid from a set of nonuniformly sampled measurements this method is appreciated for being robust and computationally fast however it lacks solid analysis and design tools to quantify or minimize the reconstruction error least squares reconstruction lsr on the other hand is another method which is optimal in the sense that it minimizes the reconstruction error this method is computationally intensive and in many cases sensitive to measurement noise hence it is rarely used in practice despite their seemingly different approaches the gridding and lsr methods are shown to be closely related the similarity between these two methods is accentuated when they are properly expressed in a common matrix form it is shown that the gridding algorithm can be considered an approximation to the least squares method the optimal gridding parameters are defined as the ones which yield the minimum approximation error these parameters are calculated by minimizing the norm of an approximation error matrix this problem is studied and solved in the general form of approximation using linearly structured matrices this method not only supports more general forms of the gridding algorithm it can also be used to accelerate the reconstruction techniques from incomplete data the application of this method to a case of two dimensional 2 d spiral magnetic resonance imaging shows a reduction of more than 4 db in the average reconstruction error

extrapolation and spectral estimation with iterative weighted norm modification an algorithm is developed to define from the data samples themselves a frequency weighted norm to use in minimum weighted norm extrapolation the iterative procedure developed consists of using a periodogram spectrum estimate obtained from some samples of the signal estimate extrapolation found at one iteration to define the weight that is used to estimate at the next iteration this algorithm usually converges in less than 10 iterations to an extrapolation which is characterized as a nonparametric frequency stationary extension of the data the frequency resolution and extrapolation length are controlled by the length of a time domain window used to obtain smooth spectral estimates between iterations examples are provided to illustrate the use of the algorithm for interpolation extrapolation the examples give comparable results to nonadaptive extrapolation methods without the need for a priori knowledge for a certain spectral estimation example the algorithm provides comparable resolution to the parametric methods with more accurate values of the relative strengths of the narrowband components

accelerating dynamic spiral mri by algebraic reconstruction from undersampled k hbox t space the temporal resolution of dynamic magnetic resonance imaging mri can be increased by sampling a fraction of k space in an interleaved fashion which introduces spatial and temporal aliasing we describe algebraically and graphically the aliasing process caused by dynamic undersampled spiral imaging within 3d xy f space the fourier transform of kxkyt space and formulate the unaliasing problem as a set of independent linear inversions since each linear system is numerically underdetermined the use of prior knowledge in the form of bounded support regions is proposed to overcome the excessive memory requirements for handling large matrices a fast implementation of the conjugate gradient cg method is used numerical simulation and in vivo experiments using spiral twofold undersampling demonstrate reduced motion artifacts and the improved depiction of fine cardiac structures the achieved reduction of motion artifacts and motion blur is comparable to simple filtering which is computationally more efficient while the proposed algebraic framework offers greater flexibility to incorporate additional algebraic acceleration techniques and to handle arbitrary sampling schemes

a strategy to identify exciting forces acting on structures the strategy for identifying the external forces acting on structures sketched by turco comput method appl mech eng 1998 165 291 306 is extended to the dynamic case starting from the measurement of the stress or strain history in a suitable number of points of the structure the exciting forces are reconstructed this inverse problem is tackled in the frequency domain by using the standard tools of computational mechanics along with a classical approach to treat ill conditioned problems some numerical results relating to pin jointed trusses are discussed in order to validate the proposed strategy copyright 2005 john wiley sons ltd

optimal estimation for band limited signals including time domain considerations the problem of estimation for band limited signals is investigated the desired estimate is based on a finite number of time samples taken from a measurement interval of length t m with no additional information the estimate must be based on these samples alone when the signal is band limited with normalized bandwidth this provides additional information which is used to improve the estimate by extrapolating the signal an extrapolation based on the measured time samples and the known bandwidth is concentrated in time about the measurement interval the length of this effective extrapolation interval is t l approx t m 2 beta several band limited estimation methods of this type have been described a new estimation method for band limited signals is described which is also based on a time concentrated extrapolation of the measurements this new method improves estimates by specifically incorporating the interval t l into the problem formulation and thereby improving the extrapolation on this relevant interval estimation error bounds and examples are presented in comparing these two types of band limited estimation procedures estimation with imperfect data is considered sensitivity of the band limited methods to errors in the data is discussed estimation with smoothing is utilized in reducing the effects of this noise

phase portraits of piecewise linear continuous differential systems with two zones separated by a straight line abstract this paper provides the classification of the phase portraits in the poincare disc of all piecewise linear continuous differential systems with two zones separated by a straight line having a unique finite singular point which is a node or a focus the sufficient and necessary conditions for existence and uniqueness of limit cycles are also given

a linear time varying system framework for noniterative discrete time band limited signal extrapolation a technique for the analysis and design of noniterative algorithms for discrete time band limited signal extrapolation is described the approach involves modeling the extrapolation process as a linear time varying ltv system or filter together with a previously developed fourier theory for ltv systems this model provides a frequency domain transfer function representation for the extrapolation system this representation serves as a powerful tool for characterizing and comparing the reconstruction properties of several well known least squares optimal algorithms for band limited extrapolation moreover the frequency domain setting provides a conceptually attractive means for understanding the process of extrapolation itself additionally a least squares approximation methodology for designing ltv filters for band limited extrapolation is developed the design technique is shown to unify a broad class of algorithms for extrapolating discrete time data and further to provide a means for designing new and improved extrapolation algorithms

mixed mode oscillations in a piecewise linear system with multiple time scale coupling abstract in this work we analyze a four dimensional slow fast piecewise linear system with three time scales presenting mixed mode oscillations the system possesses an attractive limit cycle along which oscillations of three different amplitudes and frequencies can appear namely small oscillations pulses medium amplitude and one surge largest amplitude in addition to proving the existence and attractiveness of the limit cycle we focus our attention on the canard phenomena underlying the changes in the number of small oscillations and pulses we analyze locally the existence of secondary canards leading to the addition or subtraction of one small oscillation and describe how this change is globally compensated for or not with the addition or subtraction of one pulse

hermitian plane wavelet finite element method wave propagation and load identification abstract the two dimensional hermitian interpolation wavelet is constructed by using the tensor product of the modified hermitian wavelets expanded at each coordinate then the two dimensional hermitian interpolation wavelet is substituted into finite element formulations to address the wave propagation and load identification problems hermitian wavelet finite element can be used to describe the wave propagation and to reveal the rule of the wave propagation in plane the wave propagation response is used to solve the load identification inverse problem results show that the identified load value is similar to the applied load when the location of the response node is close to the applied load position the proposed method can accurately identify the location waveform and amplitude of the applied load

canards folded nodes and mixed mode oscillations in piecewise linear slow fast systems canard induced phenomena have been extensively studied in the last three decades from both the mathematical and the application viewpoints canards in slow fast systems with at least two slow variables especially near folded node singularities give an essential generating mechanism for mixed mode oscillations mmos in the framework of smooth multiple timescale systems there is a wealth of literature on such slow fast dynamical systems and many models displaying canard induced mmos particularly in neuroscience in parallel since the late 1990s several papers have shown that the canard phenomenon can be faithfully reproduced with piecewise linear pwl systems in two dimensions although very few results are available in the three dimensional case the present paper aims to bridge this gap by analyzing canonical pwl systems that display folded singularities primary and secondary canards with a similar control of the maximal winding number as in the smooth case we also show that the singular phas

system parameter identification from projection of inverse analysis abstract the output of a system due to a change of its parameters is often approximated with the sensitivity matrix from the first order taylor series the system output can be measured in practice but the perturbation in the system parameters is usually not available inverse sensitivity analysis can be adopted to estimate the unknown system parameter perturbation from the difference between the observation output data and corresponding analytical output data calculated from the original system model the inverse sensitivity analysis is re visited in this paper with improvements based on the principal component analysis on the analytical data calculated from the known system model the identification equation is projected into a subspace of principal components of the system output and the sensitivity of the inverse analysis is improved with an iterative model updating procedure the proposed method is numerical validated with a planar truss structure and dynamic experiments with a seven storey planar steel frame results show that it is robust to measurement noise and the location and extent of stiffness perturbation can be identified with better accuracy compared with the conventional response sensitivity based method

analysis of discrete ill posed problems by means of the l curve when discrete ill posed problems are analyzed and solved by various numerical regularization techniques a very convenient way to display information about the regularized solution is to plot the norm or seminorm of the solution versus the norm of the residual vector in particular the graph associated with tikhonov regularization plays a central role the main purpose of this paper is to advocate the use of this graph in the numerical treatment of discrete ill posed problems the graph is characterized quantitatively and several important relations between regularized solutions and the graph are derived it is also demonstrated that several methods for choosing the regularization parameter are related to locating a characteristic l shaped corner of the graph

an explicit method with improved stability property a new explicit method is presented this method has an enhanced stability property when compared with the previously published explicit method j eng mech 2007 133 7 748 760 which is unconditionally stable for linear elastic and any instantaneous stiffness softening systems based on linearized analysis whereas it is only conditionally stable for any instantaneous stiffness hardening system in contrast to the previously published explicit method the most important improvement of the new explicit method is that it can have unconditional stability for general instantaneous stiffness hardening systems in addition to linear elastic and instantaneous stiffness softening systems since these stability results are obtained from a linearized stability analysis they are applicable to the non linear systems that satisfied the simplifications used for the analysis this explicit method is also shown to be second order accurate it is computationally efficient in the solution of a general structural dynamic problem where the total response is dominated by low frequency modes when compared with explicit methods such as the newmark explicit method and the family of explicit methods developed by tamma et al comput methods appl mech eng 2003 192 257 290 and implicit methods such as the constant average acceleration method some numerical examples are used to confirm the enhanced stability property and the efficiency in computing copyright 2008 john wiley sons ltd

a state space force identification method based on markov parameters precise computation and regularization technique a time domain force identification approach for linear system is proposed this approach can found a highly precise force identification model within the scope of general computer precision while it does not cost much computing time although the force identification model is accurate the force identification process like other inverse methods is still ill posed due to the inversion process and the white noise in measured structural responses the singular value decomposition is used to reveal the intrinsically matter of the ill posedness of force identification problem and a regularization technique is utilized to deal with this issue finally the proposed method with the aid of regularization technique is successfully applied to identify the input forces in two numerical simulations

extension of finite support extrapolation using the generalized series model for mr spectroscopic imaging in magnetic resonance mr imaging limited data sampling in spl kappa space leads to the well known fourier truncation artifact which includes ringing and blurring this problem is particularly severe for mr spectroscopic imaging where only 16 24 points are typically acquired along each spatial dimension several methods have been proposed to overcome this problem by incorporating prior information in the image reconstruction these include the generalized series gs model and the finite support extrapolation method this paper shows the connection between finite support extrapolation and the gs model in particular finite support extrapolation is a limiting case of the gs model when the only available prior information is the support region the support region refers to those image portions with nonzero intensities and it can be estimated in practice as the nonbackground region of an image by itself the support region constitutes a rather weak constraint that may not lead to considerable resolution gain this situation can be improved by using additional prior information which can be incorporated systematically with the gs model examples of such additional prior information include intensity estimates of anatomical structures inside the support region

interpolation and extrapolation of an ideal band limited random process this paper considers the problem of interpolation and extrapolation of a band limited random process from a finite set of n equally spaced samples using a linear mean square estimation approach various theoretical properties of the solutions are discussed such as the effect of sample spacing number of samples and bandwidth computer solutions are presented that are useful for both illustrating the theoretical behavior and providing quantitative measures of the interpolation and extrapolation error it is demonstrated that nyquist sampling at twice the highest frequency is not an appropriate criterion for finite n undersampling can give satisfactory performance for a band pass process while oversampling can be required for a low pass process these conclusions are similar to known results for sampling over infinite intervals

low rank modeling of local k space neighborhoods loraks for constrained mri recent theoretical results on low rank matrix reconstruction have inspired significant interest in low rank modeling of mri images existing approaches have focused on higher dimensional scenarios with data available from multiple channels timepoints or image contrasts the present work demonstrates that single channel single contrast single timepoint k space data can also be mapped to low rank matrices when the image has limited spatial support or slowly varying phase based on this we develop a novel and flexible framework for constrained image reconstruction that uses low rank matrix modeling of local k space neighborhoods loraks a new regularization penalty and corresponding algorithm for promoting low rank are also introduced the potential of loraks is demonstrated with simulated and experimental data for a range of denoising and sparse sampling applications loraks is also compared against state of the art methods like homodyne reconstruction l1 norm minimization and total variation minimization and is demonstrated to have distinct features and advantages in addition while calibration based support and phase constraints are commonly used in existing methods the loraks framework enables calibrationless use of these constraints

interpolation and the discrete papoulis gerchberg algorithm analyze the performance of an iterative algorithm similar to the discrete papoulis gerchberg algorithm and which can be used to recover missing samples in finite length records of band limited data no assumptions are made regarding the distribution of the missing samples in contrast with the often studied extrapolation problem in which the known samples are grouped together indeed it is possible to regard the observed signal as a sampled version of the original one and to interpret the reconstruction result studied as a sampling result the authors show that the iterative algorithm converges if the density of the sampling set exceeds a certain minimum value which naturally increases with the bandwidth of the data they give upper and lower bounds for the error as a function of the number of iterations together with the signals for which the bounds are attained also they analyze the effect of a relaxation constant present in the algorithm on the spectral radius of the iteration matrix from this analysis they infer the optimum value of the relaxation constant they also point out among all sampling sets with the same density those for which the convergence rate of the recovery algorithm is maximum or minimum for low pass signals it turns out that the best convergence rates result when the distances among the missing samples are a multiple of a certain integer the worst convergence rates generally occur when the missing samples are contiguous

explicit form of an implicit method for inverse force identification abstract most existing time domain force identification methods are based on the state space method which has the disadvantage of large discretization error with a low sampling frequency or a long sampling duration this work transforms the conventional implicit newmark algorithm into an explicit form for the solution of the ax b equations which is an equivalent transformation and it exhibits the same characteristics of newmark algorithm as an implicit method the accuracy of this method for the dynamic force identification is illustrated with two shear frame buildings and a planar truss structure numerical results are compared with reference analytical responses the new method gives more accurate identified force time histories compared with those from the conventional state space method for multiple sinusoidal and white noise excitations with 10 measurement noise in the responses

mixed mode oscillations with multiple time scales mixed mode oscillations mmos are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes mmos have been observed and studied for over thirty years in chemical physical and biological systems few attempts have been made thus far to classify different patterns of mmos in contrast to the classification of the related phenomena of bursting oscillations this paper gives a survey of different types of mmos concentrating its analysis on mmos whose small amplitude oscillations are produced by a local multiple time scale mechanism recent work gives substantially improved insight into the mathematical properties of these mechanisms in this survey we unify diverse observations about mmos and establish a systematic framework for studying their properties numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper

bandlimited extrapolation using time bandwidth dimension the problem of extrapolating discrete index bandlimited signals from a finite number of samples is addressed in this paper the algorithm presented in this paper exploits the fact that the set of bandlimited signals that are also essentially time limited is approximated well by a low dimensional linear subspace this fact which is well known for one dimensional 1 d signals with contiguous passbands and time concentration intervals is established for a more general class of multidimensional m d signals with discontiguous passbands and discontiguous time concentration regions a criterion is presented for determining the dimension of the approximating subspace and the minimax optimal subspace itself based on knowledge of the passband time concentration regions energy concentration factor and bounds on the tolerable extrapolation error the extrapolation is constrained to lie in this subspace and parameters characterizing the extrapolation are obtained from the data by solving a linear system of equations for certain sampling patterns the system is ill conditioned and a second rank reduction is needed to reduce the deleterious effects of observation noise and modeling error a novel criterion for rank selection based on known bounds on noise power and modeling error is presented the effectiveness of the new algorithm and the rank selection criterion are demonstrated by means of computer simulations

reconstruction from partial information with applications to tomography the problem of reconstruction from projections in hilbert space is treated an axiomatic basis is considered which leads to techniques which provide improved reconstructions by incorporating prior knowledge to tailor the hilbert space to the problem at hand when applied to reconstruction of the fourier transform of a function sampled at finitely many discrete points the procedures lead to previously derived optimal estimation techniques when applied to x ray tomography the procedures lead to new reconstruction techniques which are shown to include as a special case the minimum energy reconstruction of logan and shepp duke math j 42 1975 pp 645 659

moving force identification optimal state estimation approach existing techniques to identify moving forces exhibit the common weakness of having large fluctuations in the identified results a new method of moving force identification is developed in this paper making use of the dynamic programming technique to overcome this weakness the forces in the state space formulation of the dynamic system are identified in the time domain using a recursive formula based on several distributed sensor measurements and responses of the structure are reconstructed using the identified forces for comparison like all inverse problems the computation is ill conditioned however the dynamic programming technique inherently provides bounds to the ill conditioned forces and results from the simulation studies and laboratory work show great improvements over existing methods in the accuracy of identification

dynamic regularization of input estimation problems by explicit block inversion input estimation problems in structural dynamics often lead to solving damped least squares problems explicit block inversion algorithms are developed to invert the associated upper block triangular toeplitz matrix moreover the derived inversion algorithms enable the introduction of a new type of sequential regularization technique for optimal regularization parameters it is demonstrated by a numerical example that the solution using the proposed regularization technique has an upper bound equal to zeroth order tikhonov regularization