the stability for a one dimensional wave equation with nonlinear uncertainty on the boundary abstract in this work we are concerned with the boundary stabilization of a one dimensional wave equation subject to boundary nonlinear uncertainty the nonlinear uncertainty is first estimated in terms of the output and then canceled by its estimates we show that this strategy works well when the derivative of the uncertainty is bounded

adaptive wavelet schemes for nonlinear variational problems we develop and analyze wavelet based adaptive schemes for nonlinear variational problems we derive estimates for convergence rates and corresponding work counts that turn out to be asymptotically optimal our approach is based on a new paradigm that has been put forward recently for a class of linear problems the original problem is transformed first into an equivalent one which is well posed in the euclidean metric ell 2 then conceptually one seeks iteration schemes for the infinite dimensional problem that exhibits at least a fixed error reduction per step this iteration is then realized approximately through an adaptive application of the involved operators with suitable dynamically updated accuracy tolerances the main conceptual ingredients center around nonlinear tree approximation and the sparse evaluation of nonlinear mappings of wavelet expansions we prove asymptotically optimal complexity for adaptive realizations of first order iterations and of newton s method

multivariate approximation theory and applications this introductory paper describes the main topics of this special issue dedicated to leonardo traversoni known at the international level as the promoter of the conference series multivariate approximation theory and applications to celebrate his 60th birthday

the efficient implementation of a finite element multi resolution viscosity method for hyperbolic conservation laws it is well known that the classic galerkin finite element method is unstable when applied to hyperbolic conservation laws such as the euler equations for compressible flows adding a diffusion term to the equations stabilizes the method but sacrifices too much accuracy to be of any practical use an elegant solution developed in the context of spectral methods by eitan tadmor and coworkers is to add diffusion only to the high frequency modes of the solution and can lead to stabilization without sacrificing accuracy we incorporate this idea into the finite element framework by using hierarchical functions as a multi frequency basis the result is a new finite element method for solving hyperbolic conservation laws for this method convergence for a one dimensional scalar conservation law has previously been proved here the method is described in detail several issues connected with its efficient implementation are considered and numerical results for several examples involving one and two dimensional hyperbolic conservation laws are provided several advantageous features of the method are discussed including the ease for which discontinuities can be detected and artificial diffusion can be applied anisotropically and locally in physical as well as frequency space

detecting and handling unreliable points for camera parameter estimation the popularly used dlt method sometimes fails to give reliable camera parameter estimation it is therefore important to detect the unreliability and provide the corresponding solutions based on a complete framework of invariance for six points we construct two evaluation functions to detect the unreliability the two evaluation functions do not involve any computations for the camera projective matrix or optical center and thus are efficient to perform the detection then the guidelines corresponding to the different detection results are presented in particular a filtering ransac method to remove the detected unreliable points is provided the filtering ransac proves to be successful in removing the unreliable points even if these points are of a large proportion

stability analysis of the anti stable heat equation with uncertain disturbance on the boundary abstract in this paper we study the stability of the anti stable 1 d heat equation with uncertain disturbance on the boundary we first estimate the disturbance in terms of the output then we design some control strategy to stabilize the anti stable system

invariants of six points and projective reconstruction from three uncalibrated images there are three projective invariants of a set of six points in general position in space it is well known that these invariants cannot be recovered from one image however an invariant relationship does exist between space invariants and image invariants this invariant relationship is first derived for a single image then this invariant relationship is used to derive the space invariants when multiple images are available this paper establishes that the minimum number of images for computing these invariants is three and the computation of invariants of six points from three images can have as many as three solutions algorithms are presented for computing these invariants in closed form the accuracy and stability with respect to image noise selection of the triplets of images and distance between viewing positions are studied both through real and simulated images applications of these invariants are also presented both the results of faugeras 1992 and hartley et al 1992 for projective reconstruction and sturm s method 1869 for epipolar geometry determination from two uncalibrated images with at least seven points are extended to the case of three uncalibrated images with only six points

the invariant representations of a quadric cone and a twisted cubic up to now the shortest invariant representation of a quadric has 138 summands and there has been no invariant representation of a twisted cubic in 3d projective space which limit to some extent the applications of invariants in 3d space we give a very short invariant representation of a quadric cone a special quadric which has only two summands similar to the invariant representation of a planar conic and give a short invariant representation of a twisted cubic then a completely linear algorithm for generating the parametric equations of a twisted cubic is provided also finally we exemplify some applications of our proposed invariant representations in the fields of computer vision and automated geometric theorem proving

active disturbance rejection control based on weighed moving average state observer abstract in this work we are concerned with the boundary stabilization of a one dimensional anti stable wave equation corrupted by a boundary disturbance we firstly propose by weighed moving average technique a state observer to make an estimation of the disturbance secondly we design a control by the active disturbance rejection strategy to stabilize the system

on existence and asymptotic stability of solutions of the degenerate wave equation with nonlinear boundary conditions abstract we study the global existence of solutions of the nonlinear degenerate wave equation 0 x y y 0 in 0 y 0 on 1 0 y y f y g y 0 on 0 0 y x 0 y 0 y x 0 y 1 x in where y denotes the derivative of y with respect to parameter t f s c0 s s and g is a nondecreasing c1 function such that k1 s 2 g s s k2 s 2 for some k1 k2 0 with 0 0 if n 1 2 the existence of solutions is proved by means of the faedo galerkin method furthermore when 0 the uniform decay is obtained by making use of the perturbed energy method

uniform stabilization of the wave equation by nonlinear boundary feedback the question of uniformly stabilizing the solution of the wave equation y delta y 0 in omega times 0 infty omega is a bounded domain of bf r n by means of a nonlinear feedback law of the following form is studied partial y partial v k x g y on gamma 0 times 0 infty y 0 on gamma 1 times 0 infty gamma 0 gamma 1 being a suitable partition of the boundary of omega and g a continuous nondecreasing function such that g 0 0 we choose k x in l infty k x geqq 0 such that k x vanishes linearly at the interface points x in bar gamma 0 cap bar gamma 1 then if g s behaves like s p 1 s as s to 0 with p 1 and linearly as s to infty it is proved that the energy of every solution decays like t 2 p 1 as t to infty in the case where p 1 the exponential decay rate is proved

existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term the wave equation with a source term is considered utt u u uin 0 we prove the existence and uniform decay rates of the energy by assuming a nonlinear feedback ut acting on the boundary provided that has necessarily not a polynomial growth near the origin to obtain the existence of global solutions we make use of the potential well method combined with the faedo galerkin procedure and constructing a special basis furthermore we prove that the energy of the system decays uniformly to zero and we obtain an explicit decay rate estimate adapting the ideas of lasiecka and tataru differential integral equations 6 3 1993 507 and patrick martinez esain control optimisation calc var 4 1999 419 the resulting problem generalizes martinez results and complements the works of lasiecka and tataru 1993 and vitillaro glasgow math j 44 2002 375

globally optimal estimates for geometric reconstruction problems we introduce a framework for computing statistically optimal estimates of geometric reconstruction problems while traditional algorithms often suffer from either local minima or non optimality or a combination of both we pursue the goal of achieving global solutions of the statistically optimal cost function our approach is based on a hierarchy of convex relaxations to solve non convex optimization problems with polynomials these convex relaxations generate a monotone sequence of lower bounds and we show how one can detect whether the global optimum is attained at a given relaxation the technique is applied to a number of classical vision problems triangulation camera pose homography estimation and last but not least epipolar geometry estimation experimental validation on both synthetic and real data is provided in practice only a few relaxations are needed for attaining the global optimum

a note on the boundary stabilization of the wave equation we study the energy decay rates of the wave equation in a domain where boundary damping is present we generalize the geometrical conditions obtained earlier in j math purer appl 58 1979 pp 249 273 by using some more general multipliers of strauss comm pure appl math 28 1975 pp 265 278 the interaction between distributed damping and boundary damping is discussed a regulator problem is also formally discussed by the synthesis method

a finite element multiresolution viscosity method for hyperbolic conservation laws it is well known that the classic galerkin finite element method is unstable when applied to hyperbolic conservation laws such as the euler equations for compressible flow it is also well known that naively adding artificial diffusion to the equations stabilizes the method but sacrifices too much accuracy to be of any practical use an elegant approach referred to as spectral viscosity methods has been developed for spectral methods in which one adds diffusion only to the high frequency modes of the solution the result being that stabilization is effected without sacrificing accuracy we extend this idea into the finite element framework by using hierarchical finite element functions as a multifrequency basis the result is a new finite element method for solving hyperbolic conservation laws in which artificial diffusion can be applied selectively only to the high frequency modes of the approximation as for spectral viscosity methods this results in stability without compromising accuracy in the context of a one dimensional scalar hyperbolic conservation law we prove the convergence of approximate solutions obtained using the new method to the entropy solution of the conservation law to illustrate the method the results of a computational experiment for a one dimensional hyperbolic conservation law are provided

well posedness and optimal decay rates for the wave equation with nonlinear boundary damping source interaction we establish subject to some natural additional assumptions imposed on the relation between the source and the damping both well posedness and effective optimal decay rates for the solutions of a semilinear model of the wave equation the theory presented allows to consider both superlinear and sublinear behaviours of the dissipation in the presence of unstructured sources

entropy viscosity method for nonlinear conservation laws a new class of high order numerical methods for approximating nonlinear conservation laws is described entropy viscosity method the novelty is that a nonlinear viscosity based on the local size of an entropy production is added to the numerical discretization at hand this new approach does not use any flux or slope limiters applies to equations or systems supplemented with one or more entropy inequalities and does not depend on the mesh type and polynomial approximation various benchmark problems are solved with finite elements spectral elements and fourier series to illustrate the capability of the proposed method

equidistant fish eye calibration and rectification by vanishing point extraction in this paper we describe a method to photogrammetrically estimate the intrinsic and extrinsic parameters of fish eye cameras using the properties of equidistance perspective particularly vanishing point estimation with the aim of providing a rectified image for scene viewing applications the estimated intrinsic parameters are the optical center and the fish eye lensing parameter and the extrinsic parameters are the rotations about the world axes relative to the checkerboard calibration diagram

on the convergence of an extended state observer for nonlinear systems with uncertainty abstract the extended state observer first proposed by jingqing han in j q han a class of extended state observers for uncertain systems control decis 10 1 1995 85 88 in chinese is the key link toward the active disturbance rejection control that is taking off as a technology after numerous successful applications in engineering unfortunately there is no rigorous proof of convergence to date in this paper we attempt to tackle this long unsolved extraordinary problem the main idea is to transform the error equation of objective system with its extended state observer into a asymptotical stable system with a small disturbance for which the effect of total disturbance error is eliminated by the high gain

an adaptive wavelet viscosity method for systems of hyperbolic conservation laws for the burgers equation as an example of a hyperbolic conservation law we have considered in our previous paper castano diez et al 2008 8 a weak formulation with a stabilization for handling discontinuities commonly called a viscosity approach numerically this was realized by locally introducing degrees of freedom around the discontinuities by means of an adaptive wavelet method in an a posteriori fashion in the present paper we apply this method to systems of conservation laws specifically euler s equations for gas dynamics moreover as the viscosity stabilization produces some gibbs phenomena we discuss different post processing techniques known from data and image processing together with a number of numerical comparisons

uniform decay rate estimates for schrodinger and plate equations with nonlinear locally distributed damping on a compact n dimensional riemannian manifold m g we establish uniform decay rate estimates for the linear schrodinger and plate equations subject to an internal nonlinear damping locally distributed on the manifold our approach can be also employed for other equations provided that inverse inequality for the linear model occurs in the particular case of the wave equation where the well known geometric control condition gcc is equivalent to the observability inequality our method generalizes the results due to cavalcanti et al 2010 2009 9 10 regarding the optimal choice of dissipative regions

the application of cubic trigonometric b spline to the numerical solution of the hyperbolic problems in this paper a collocation finite difference scheme based on new cubic trigonometric b spline is developed and analyzed for the numerical solution of a one dimensional hyperbolic equation wave equation with non local conservation condition the usual finite difference scheme is used to discretize the time derivative while a cubic trigonometric b spline is utilized as an interpolation function in the space dimension the scheme is shown to be unconditionally stable using the von neumann fourier method the accuracy of the proposed scheme is tested by using it for several test problems the numerical results are found to be in good agreement with known exact solutions and with existing schemes in literature

reconstruction of wavelet coefficients using total variation minimization we propose a model to reconstruct wavelet coefficients using a total variation minimization algorithm the approach is motivated by wavelet signal denoising methods where thresholding small wavelet coefficients leads to pseudo gibbs artifacts by replacing these thresholded coefficients by values minimizing the total variation our method performs a nearly artifact free signal denoising in this paper we detail the algorithm based on a subgradient descent combining a projection on a linear space the convergence of the algorithm is established and numerical experiments are reported

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

output feedback stabilization of an unstable wave equation with general corrupted boundary observation we consider boundary output feedback stabilization for an unstable wave equation with boundary observation subject to a general disturbance we adopt for the first time the active disturbance rejection control approach to stabilization for a system described by the partial differential equation with corrupted output feedback by the approach the disturbance is first estimated by a relatively independent estimator it is then canceled in the feedback loop as a result the control law can be designed almost as that for the system without disturbance we show that with a time varying gain properly designed the observer driven by the disturbance estimator is convergent and that all subsystems in the closed loop are asymptotically stable in the energy state space we also provide numerical simulations which demonstrate the convergence results and underline the effect of the time varying gain estimator on peaking value reduction

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