the largest singular value of e sup x a sub 0 e sup x is convex on convex sets of commuting matrices a short and direct proof of the convexity property is given it is shown that the theorem applies to any convex commuting set of matrices in r sup nxn where a sub 0 in r sup nxn is fixed it is also shown that the result does not hold if x is permitted to be a general square matrix a counterexample is supplied for noncommuting matrices

on the estimation of transfer functions this paper treats the close conceptual relationships between basic approaches to the estimation of transfer functions of linear systems the classical methods of frequency and spectral analysis are shown to be related to the well known time domain methods of prediction error type via a common empirical transfer function estimate asymptotic properties of the estimates obtained by the respective methods are also described and discussed an important feature that is displayed by this treatment is a frequency domain weighting function that determines the distribution of bias in case the true system cannot be exactly described within the chosen model set the choice of this weighting function is made in terms of noise models for time domain methods the noise model thus has a dual character from the system approximation point of view

fractal first order partial differential equations the present paper is concerned with semi linear partial differential equations involving a particular pseudo differential operator it investigates both fractal conservation laws and non local hamilton jacobi equations the idea is to combine an integral representation of the operator and duhamel s formula to prove on the one hand the key a priori estimates for the scalar conservation law and the hamilton jacobi equation and on the other hand the smoothing effect of the operator as far as hamilton jacobi equations are concerned a non local vanishing viscosity method is used to construct a viscosity solution when existence of regular solutions fails and a rate of convergence is provided turning to conservation laws global in time existence and uniqueness are established we also show that our formula allows us to obtain entropy inequalities for the non local conservation law and thus to prove the convergence of the solution as the non local term vanishes toward the entropy solution of the pure conservation law

paper unprejudiced optimal open loop input design for identification of transfer functions the problem to estimate transfer functions of linear systems is considered the quality of the resulting estimate depends among other things on the input used during the identification experiment we measure the quality using a quadratic norm in the frequency domain the problem to determine optimal inputs i e inputs that minimize the chosen norm subject to constrained input variance has long been studied we point out that such procedures may involve a prejudice that the system is to be found in a certain model set that may have some surprising effects we discuss how such a prejudice can be reduced by allowing the possibility that the true system cannot be exactly described in the chosen model set we also calculate explicit expressions for the resulting unprejudiced optimal inputs these expressions relate the signal to noise ratio as a function of frequency to the chosen weighting function in the quadratic norm we also point out the role of the employed noise model for the design

a numerical method for fractal conservation laws we consider a fractal scalar conservation law that is to say a conservation law modified by a fractional power of the laplace operator and we propose a numerical method to approximate its solutions we make a theoretical study of the method proving in the case of an initial data belonging to l bv that the approximate solutions converge in l weak and in l p strong for p and we give numerical results showing the efficiency of the scheme and illustrating qualitative properties of the solution to the fractal conservation law

wreath product of matrices abstract we introduce a new matrix product that we call the wreath product of matrices the name is inspired by the analogous product for graphs and the following important correspondence is proven the wreath product of the adjacency matrices of two graphs provides the adjacency matrix of the wreath product of the graphs this correspondence is exploited in order to study the spectral properties of the famous lamplighter random walk the spectrum is explicitly determined for the walk or switch model on a complete graph of any size with two lamp colors the investigation of the spectrum of the matrix wreath product is actually developed for the more general case where the second factor is a circulant matrix finally an application to the study of the uniqueness of the solution of generalized sylvester matrix equations is treated

numerical approximation of entropy solutions for hyperbolic integro differential equations scalar hyperbolic integro differential equations arise as models for e g radiating or self gravitating fluid flow we present finite volume schemes on unstructured grids applied to the cauchy problem for such equations for a rather general class of integral operators we show convergence of the approximate solutions to a possibly discontinuous entropy solution of the problem for a specific model problem in radiative hydrodynamics we introduce a convergent fully discrete finite volume scheme under the assumption of sufficiently fast spatial decay of the entropy solution we can even establish the convergence rate h1 4 ln h where h denotes the grid parameter the convergence proofs rely on appropriate variants of the classical kruzhkov method for local balance laws together with a truncation technique to cope with the nonlocal character of the integral operator

fractional burgers equation with nonlinear non locality we consider the viscous burgers equation with a fractional nonlinear term as a model involving non local nonlinearities in conservation laws which surprisingly has an analytical solution obtained by a fractional extension of the hopfcole transformation we use this model and its inviscid limit to develop stable spectral and discontinuous galerkin spectral element methods by employing the concept of spectral vanishing viscosity svv for the global spectral method svv is very effective and the computational cost is o n2 which is essentially the same as for the standard burgers equation we also develop a local discontinuous galerkin ldg spectral element method to improve the accuracy around discontinuities and we again stabilize the ldg method with the svv operator finally we solve numerically the inviscid fractional burgers equation both with the spectral and the spectral element ldg methods we study systematically the stability and convergence of both methods and determine the effectiveness of each method for different parameters

numerical methods for the fractional laplacian a finite difference quadrature approach the fractional laplacian delta alpha 2 is a nonlocal operator which depends on the parameter alpha and recovers the usual laplacian as alpha to 2 a numerical method for the fractional laplacian is proposed based on the singular integral representation for the operator the method combines finite differences with numerical quadrature to obtain a discrete convolution operator with positive weights the accuracy of the method is shown to be o h 3 alpha convergence of the method is proven the treatment of far field boundary conditions using an asymptotic approximation to the integral is used to obtain an accurate method numerical experiments on known exact solutions validate the predicted convergence rates computational examples include exponentially and algebraically decaying solutions with varying regularity the generalization to nonlinear equations involving the operator is discussed the obstacle problem for the fractional laplacian is computed

black box identification of multivariable transfer functions asymptotic properties and optimal input design abstract the problem of estimating the transfer function of a multivariable linear stochastic system is considered the transfer function is parametrized as a black box and no order is chosen a priori this means that the model order may increase to infinity as the number of observed data tends to infinity the asymptotic covariance of the transfer function estimate is calculated and is found to be independent of the noise model when the model order increases to infinity the derived expressions are also used to determine optimal unprejudiced input signals for the identification experiment

wreath product of a complete graph with a cyclic graph topological indices and spectrum abstract in this manuscript we continue the investigations related to the wreath product of graphs by considering the compound graph of a clique with a circuit this product shows nice combinatorial and algebraic properties which permit with reasonable effort to compute some topological indices and the adjacency spectrum

paper on the estimation of transfer functions this paper treats the close conceptual relationships between basic approaches to the estimation of transfer functions of linear systems the classical methods of frequency and spectral analysis are shown to be related to the well known time domain methods of prediction error type via a common empirical transfer function estimate asymptotic properties of the estimates obtained by the respective methods are also described and discussed an important feature that is displayed by this treatment is a frequency domain weighting function that determines the distribution of bias in case the true system cannot be exactly described within the chosen model set the choice of this weighting function is made in terms of noise models for time domain methods the noise model thus has a dual character from the system approximation point of view

the rate of convergence of spectral viscosity methods for periodic scalar conservation laws a rate of convergence is proven for spectral viscosity methods for periodic scalar conservation laws this rate is obtained by showing the discretization error to be small enough that the difference between the solutions of the spectral viscosity method and the ordinary viscosity method tends to zero in l 1 as the number of discrete modes tends to infinity and the viscosity simultaneously tends to zero at the appropriate rate the method is also used to obtain an l infty bound for the spectral viscosity approximations to the elasticity equations convergence although without a rate then follows from compensated compactness theory

numerical study of interacting particles approximation for integro differential equations the paper develops a numerical method based on the interacting particles approximation propagation of chaos for the solution of a large class of evolution problems involving the fractional laplacian operator and a non local quadratic type non linearity coupled stochastic differential equations driven by levy symmetric a stable processes are integrated numerically using euler s method and the solutions of the governing equations are obtained from their statistics the method is tested on several one and two dimensional examples and established analytical properties of the solutions are verified for the numerical approximates when they are available for initial conditions that are either integrable or monotone bounded functions it is shown that these methods represent viable tools for constructing the solution to the cauchy problem

on identification of stable systems and optimal approximation abstract approximative modelling of stable continuous time possibly infinite dimensional systems is studied based on an optimal approximation approach both approximation of analytical system representations system approximation as well as approximation of input output data based system estimates system identification are considered while special emphasis is given to approximative modelling in the h and hankel norms the l1 and l2 norm cases are also discussed the model sets considered here are finite dimensional systems and time shifted systems simple delay systems the theory of approximation numbers is shown to provide a convenient tool to study problems of identification of stable continuous time systems in a deterministic framework with close connections to complexity considerations laguerre fourier series methods and hankel operator techniques can be utilized to develop fully practical identification methods for continuous time possibly infinite dimensional systems

on numerical methods and error estimates for degenerate fractional convection diffusion equations first we introduce and analyze a convergent numerical method for a large class of nonlinear nonlocal possibly degenerate convection diffusion equations secondly we develop a new kuznetsov type theory and obtain general and possibly optimal error estimates for our numerical methods even when the principal derivatives have any fractional order between 1 and 2 the class of equations we consider includes equations with nonlinear and possibly degenerate fractional or general levy diffusion special cases are conservation laws fractional conservation laws certain fractional porous medium equations and new strongly degenerate equations

robust parametric transfer function estimation using complex logarithmic frequency response data the statistical properties of a logarithmic least squares frequency domain estimator are analyzed in an error in variables framework it is shown that contrary to most frequency domain estimators the logarithmic least squares estimator remains practically consistent when the variances of the frequency domain data are not a priori known if available the variance at each spectral line of the logarithmic frequency response data can be used to weight the logarithmic least squares estimator resulting in parameter estimates with a smaller variability an estimate of these variances can be derived from the residual errors at the excited spectral lines besides its robustness to lack of prior noise information it is demonstrated that the logarithmic estimator behaves remarkably well in the presence of outliers

mean square error of the empirical transfer function estimator for stochastic input signals the accuracy of the empirical transfer function estimator etfe with stochastic input has been described with a deterministic and a stochastic approach the results for bias and variance are different but the mean square error is infinite in both approaches in the deterministic approach where the input is considered as a single realization of a stochastic process a number of frequencies has no finite etfe estimate the average number of problematic frequencies will be constant if the number of observations increases

transfer function identification with filtering techniques this paper addresses the problem of identifying the transfer function of a plant with the aid of filtering techniques conditions for obtaining fair estimates of a transfer function are derived it is shown that to be able to reduce the effect of noise and other stochastic effects highly tuned filters are required it is also found that proper excitation signals are necessary for effective implementation of the proposed scheme some methods for designing the required highly tuned filters are proposed the relationship between the proposed scheme and existing methods is discussed it is shown that for some specific cases the proposed scheme is in effect a computationally efficient implementation of existing methods however for some other cases the proposed scheme gives more accurate results the theoretical predictions are substantiated with computer simulations

on the estimation of transfer functions regularizations and gaussian processes revisited intrigued by some recent results on impulse response estimation by kernel and nonparametric techniques we revisit the old problem of transfer function estimation from input output measurements we formulate a classical regularization approach focused on finite impulse response fir models and find that regularization is necessary to cope with the high variance problem this basic regularized least squares approach is then a focal point for interpreting other techniques like bayesian inference and gaussian process regression the main issue is how to determine a suitable regularization matrix bayesian prior or kernel several regularization matrices are provided and numerically evaluated on a data bank of test systems and data sets our findings based on the data bank are as follows the classical regularization approach with carefully chosen regularization matrices shows slightly better accuracy and clearly better robustness in estimating the impulse response than the standard approach the prediction error method maximum likelihood pem ml approach if the goal is to estimate a model of given order as well as possible a low order model is often better estimated by the pem ml approach and a higher order model is often better estimated by model reduction on a high order regularized fir model estimated with careful regularization moreover an optimal regularization matrix that minimizes the mean square error matrix is derived and studied the importance of this result lies in that it gives the theoretical upper bound on the accuracy that can be achieved for this classical regularization approach

robust identification of continuous systems with dead time from step responses in this paper a simple yet robust method is proposed for identification of linear continuous time delay processes from step responses new linear regression equations are derived from the solution and its various order integrals of the process differential equation the regression parameters are then estimated without iterations and explicit relationship between the regression parameters and those in the process are given due to use of the process output integrals in the regression equations the resulting parameter estimation is very robust in the face of large measurement noise in the output the proposed method is detailed for a second order plus dead time model with one zero which can approximate most practical industrial processes covering monotonic or oscillatory dynamics of minimum phase or non minimum phase processes such a model can be obtained without any iteration the effectiveness of the identification method has been demonstrated through simulation and real time implementation

dislocation dynamics short time existence and uniqueness of the solution we study a mathematical model describing dislocation dynamics in crystals we consider a single dislocation line moving in its slip plane the normal velocity is given by the peach koehler force created by the dislocation line itself the mathematical model is an eikonal equation with a velocity which is a non local quantity depending on the whole shape of the dislocation line we study the special case where the dislocation line is assumed to be a graph or a closed loop in the framework of discontinuous viscosity solutions for hamilton jacobi equations we prove the existence and uniqueness of a solution for small time we also give physical explanations and a formal derivation of the mathematical model finally we present numerical results based on a level sets formulation of the problem these results illustrate in particular the fact that there is no general inclusion principle for this model

finite element method for a space fractional anti diffusive equation the numerical solution of a nonlinear and space fractional anti diffusive equation used to model dune morphodynamics is considered spatial discretization is effected using a finite element method whereas the crank nicolson scheme is used for temporal discretization the fully discrete scheme is analyzed to determine stability condition and also to obtain error estimates for the approximate solution numerical examples are presented to illustrate convergence results

a frequency domain estimator for use in adaptive control systems abstract this paper presents a frequency domain estimator that can identify both a parametrized nominal model of a plant as well as a frequency domain bounding function on the modeling error associated with this nominal model this estimator which we call a robust estimator can be used in conjunction with a robust control law redesign algorithm to form a robust adaptive controller

fractal burgers equations abstract the paper studies local and global in time solutions to a class of multidimensional generalized burgers type equations with a fractional power of the laplacian in the principal part and with general algebraic nonlinearity such equations naturally appear in continuum mechanics our results include existence uniqueness regularity and asymptotic behavior of solutions to the cauchy problem as well as a construction of self similar solutions the role of critical exponents is also explained

transfer function and transient estimation by gaussian process regression in the frequency domain inspired by the recent promising developments of bayesian learning techniques in the context of system identification this paper proposes a transfer function estimator based on gaussian process regression contrary to existing kernel based impulse response estimators a frequency domain approach is adopted this leads to a formulation and implementation which is seamlessly valid for both continuous and discrete time systems and which conveniently enables the selection of the frequency band of interest a pragmatic approach is proposed in an output error framework from sampled input and output data the transient is dealt with by estimating it simultaneously with the transfer function modelling the transfer function and the transient as gaussian processes allows for the incorporation of relevant prior knowledge on the system in the form of suitably designed kernels the ss stable spline and dc diagonal correlated kernels from the literature are translated to the frequency domain and are proven to impose the stability of the estimated transfer function specifically the estimates are shown to be stable rational functions in the frequency variable the hyperparameters of the kernel are tuned via marginal likelihood maximisation the comparison between the proposed method and three existing methods from the literature the regularised finite impulse response rfir estimator the local polynomial method lpm and the local rational method for frequency response function estimation is illustrated on simulations on multiple case studies

convergence analysis of a mixed finite volume scheme for an elliptic parabolic system modeling miscible fluid flows in porous media we study a finite volume discretization of a strongly coupled elliptic parabolic pde system describing miscible displacement in a porous medium we discretize each equation by a finite volume scheme which allows a wide variety of unstructured grids in any space dimension and gives strong enough convergence for handling the nonlinear coupling of the equations we prove the convergence of the scheme as the time and space steps go to 0 finally we provide numerical results to demonstrate the efficiency of the proposed numerical scheme

optimal control with state space constraint i optimal control of piecewise deterministic processes with state space constraint is studied under appropriate assumptions it is shown that the optimal value function is the only viscosity solution on the open domain which is also a supersolution on the closed domain finally the uniform continuity of the value function is obtained under a condition on the deterministic drift

spectral viscosity approximations to multidimensional scalar conservation laws the authors study the spectral viscosity sv method in the context of multidimensional scalar conservation laws with periodic boundary conditions they show that the spectral viscosity which is sufficiently small to retain the formal spectral accuracy of the underlying fourier approximation is large enough to enforce the correct amount of entropy dissipation which is otherwise missing in the standard fourier method moreover they prove that because of the presence of the spectral viscosity the truncation error in this case becomes spectrally small independent of whether the underlying solution is smooth or not consequently the sv approximation remains uniformly bounded and converges to a measure valued solution satisfying the entropy condition that is the unique entropy solution they also show that the sv solution has a bounded total variation provided that the total variation of the initial data is bounded thus confirming its strong convergence to the entropy solution they obtain an l sup 1 convergence rate of the usual optimal order one half 22 refs

total variation and error estimates for spectral viscosity approximations we study the behavior of spectral viscosity approximations to nonlinear scalar conservation laws we show how the spectral viscosity method compromises between the total variation bounded viscosity approximations which are restricted to first order accuracy and the spectrally accurate yet unstable fourier method in particular we prove that the spectral viscosity method is l sup 1 stable and hence total variation bounded moreover the spectral viscosity solutions are shown to be lip sup stable in agreement with oleinik s e entropy condition this essentially nonoscillatory behavior of the spectral viscosity method implies convergence to the exact entropy solution and we provide convergence rate estimates of both global and local types 16 refs

design variables for bias distribution in transfer function estimation estimation of transfer functions of linear systems is one of the most common system identification problems several different design variables chosen by the user for the identification procedure affect the properties of the resulting estimate in this paper it is investigated how the choices of prefilters noise models sampling interval and prediction horizon i e the use of k step ahead prediction methods influence the estimate an important aspect is thai the true system is not assumed to be exactly represented within the chosen model set the estimate will thus be biased it is shown how the distribution of bias in the frequency domain is governed by a weighting function which emphasizes different frequency bands the weighting function in turn is a result of the previously listed design variables it is shown e g thai the common least squares method has a tendency to emphasize high frequencies and that this can be counteracted by prefiltering it is also shown that asymptotically it is only the prediction horizon itself and not how it is split up into sampling interval times number of predicted sampling instants that affects this weighting function

robust parameter adjustment with nonparametric weighted ball in h sup infinity uncertainty using a parameter adjustment law robust monotonic parameter error reduction is proved for a linear parameter estimation problem in which the signal is corrupted by the presence of nonparametric dynamical uncertainty the nonparameterized dynamics are characterized by a known frequency domain magnitude bound a nonconservative bound on the finite time energy of the nonparametric dynamics output is constructed in real time when the error single energy is small enough to be due to nonparametric dynamics alone the parameter adjustment is shut off to avoid misadjustment the approach is an extension of existing adaptive control error dead zone ideas to the case of frequency domain bounded uncertainty

entropy solutions for nonlinear degenerate problems we consider a class of elliptic hyperbolic degenerate equations g u delta b u divg phi u f with dirichlet homogeneous boundary conditions and a class of elliptic parabolic hyperbolic degenerate equations g u t delta b u divg phi u f with homogeneous dirichlet conditions and initial conditions existence of entropy solutions for both problems is proved for nondecreasing continuous functions g and b vanishing at zero and for a continuous vectorial function satisfying rather general conditions comparison and uniqueness of entropy solutions are proved for g and b continuous and nondecreasing and for continuous

analysis of the spectral vanishing viscosity method for periodic conservation laws the convergence of the spectral vanishing method for both the spectral and pseudospectral discretizations of the inviscid burgers equation is analyzed it is proved that this kind of vanishing viscosity is responsible for spectral decay of those fourier coefficients located toward the end of the computed spectrum consequently the discretization error is shown to be spectrally small independently of whether or not the underlying solution is smooth this in turn implies that the numerical solution remains uniformly bounded and convergence follows by compensated compactness arguments

asymptotic variance expressions for identified black box transfer function models identification of black box transfer function models is considered it is assumed that the transfer function models possess a certain shift property which is satisfied for example by all polynomial type models expressions for the variances of the transfer function estimates are derived that are asymptotic both in the number of observed data and in the model orders the result is that the joint covariance matrix of the transfer functions from input to output and from driving white noise source to the additive output disturbance respectively is proportional to the inverse of the joint spectrum matrix for the input and driving noise multiplied by the spectrum of the additive output noise the factor of proportionality is the ratio of model order to number of data this result is independent of the particular model structure used the result is applied to evaluate the performance degradation due to variance for a number of typical model uses some consequences for input design are also drawn

robust approximate modelling of stable linear systems robust approximation and worst case approximate modelling of stable shift invariant systems from corrupted transfer function estimates are studied in the h sense connections between the problem formulations of the present work and certain problems of worst case system identification notably the helmicki jacobson nett problem formulation for identification in h are established issues of model set selection are addressed using the n width concept a concrete result establishes a priori knowledge for which a certain rational model set is optimal in then width sense a notion of robust convergence is defined so that any untuned approximation method satisfying it has a generic well posedness property for systems in the disk algebra the existence of robustly convergent approximation methods based on any complete model set in the disk algebra is shown in a constructive way a framework is given in which approximate models can be obtained as stable perturbations of the true system these can be combined

efficient numerical schemes for fractional water wave models in this paper efficient numerical schemes are proposed for solving the fractional water wave models that describe the propagation of surface water wave by using the weighted and shifted grunwald letnikov wsgl formula to approximate the nonlocal fractional operators we design a series of second order accurate difference schemes for the considered models the existence stability and convergence of numerical solutions of the proposed numerical schemes are established rigorously the analysis shows that the proposed numerical schemes are unconditionally stable with second order accuracy for both temporal and spatial discretizations several numerical results are provided to verify the efficiency and accuracy of our theoretical analysis

convergence of spectral methods for nonlinear conservation laws we discuss the convergence of fourier methods for scalar nonlinear conservation laws that exhibit spontaneous shock discontinuities numerical tests indicate that the convergence may and in fact in some cases we prove it must fail with or without post processing of the numerical solution instead we introduce here a new kind of spectrally accurate vanishing viscosity to augment the fourier approximation of such nonlinear conservation laws using compensated compactness arguments augmented by the assumption of l infty stability we show for the inviscid burgers model equation that this spectral viscosity method prevents oscillations and convergence to the unique entropy solution follows

asymptotic properties of black box identification of transfer functions the problem of estimating the transfer function of a linear stochastic system is considered the transfer function is parametrized as a black box and no given order is chosen a priori this means that the model orders may increase to infinity when the number of observed data tends to infinity the consistency and convergence properties of the resulting transfer function estimates are investigated asymptotic expressions for the variances and distributions of these estimates are also derived for the case that the model orders increase it is shown that the variance of the transfer function estimate at a certain frequency is asymptotically given by the noise to signal ratio at that frequency mulliplied by the model order to number of data points ratio