structural stability of solutions to the riemann problem for a scalar conservation law the aim of this paper is to study the structural stability of solutions to the riemann problem for a scalar conservation law with a linear flux function involving discontinuous coefficients it is proved that the riemann solution is possibly instable when one of the riemann initial data is at the vacuum furthermore we point out that the riemann solution is also possibly instable even when the riemann initial data stay far away from vacuum in order to deal with it we perturb the riemann initial data by taking three piecewise constant states and then the global structures and large time asymptotic behaviors of the solutions are obtained constructively it is also proved that the riemann solutions are unstable in some certain situations under the local small perturbations of the riemann initial data by letting the perturbed parameter e tend to zero in addition the interaction of the delta standing wave and the contact vacuum state is considered which appear in the riemann solutions

a numerical scheme for singular shock solutions and a study of its consistence in the sense of distributions in this paper we present a numerical scheme for the approximation of singular shocks as long as some properties such as boundedness of the velocity are verified when the space step h tends to 0 we prove that the scheme provides approximate solutions that tend to satisfy the equations more precisely when the approximate solutions are plugged into the equations the result tends to 0 when h 0 in the familiar definition of weak solutions with the requirement of a smooth test function these properties can be fully proved for general classes of systems extending the korchinski system that do not have distribution solutions in the usual sense in other cases such as the keyfitz kranzer singular shocks these properties have been checked up to very small values of h these results explain numerical observations on very different systems

generalized sampling a variational approach i theory we consider the problem of reconstructing a multidimensional vector function f sub in spl ropf sup m spl rarr spl ropf sup n from a finite set of linear measures these can be irregularly sampled responses of several linear filters traditional approaches reconstruct in an a priori given space e g the space of bandlimited functions instead we have chosen to specify a reconstruction that is optimal in the sense of a quadratic plausibility criterion j first we present the solution of the generalized interpolation problem later we also consider the approximation problem and we show that both lead to the same class of solutions imposing generally desirable properties on the reconstruction largely limits the choice of the criterion j linearity leads to a quadratic criterion based on bilinear forms specifically we show that the requirements of translation rotation and scale invariance restrict the form of the criterion to essentially a one parameter family we show that the solution can be obtained as a linear combination of generating functions we provide analytical techniques to find these functions and the solution itself

global structure of riemann solutions to a system of two dimensional hyperbolic conservation laws abstract the riemann problem for a two dimensional nonstrictly hyperbolic system of conservation laws is considered without the restriction that each jump of the initial data projects one planar elementary wave ten topologically distinct solutions are obtained by applying the method of generalized characteristic analysis some of these solutions involve the nonclassical waves i e the delta shock wave and the delta contact discontinuity for which we explicitly give the expressions of their strengths locations and propagation speeds moreover we demonstrate that the nature of our solutions is identical with that of solutions to the corresponding one dimensional cauchy problem which provides a verification that our construction produces the correct unique global solutions

a fast adaptive parameter estimation for total variation image restoration estimation of the regularization parameter which strikes a balance between the data fidelity and regularity is essential for successfully solving ill posed image restoration problems based on the classical total variation tv model and prevalent alternating direction method of multipliers we hammer out a fast algorithm being able to simultaneously estimate the regularization parameter and restore the degraded image by applying variable splitting technique to both the regularization term and data fidelity term we overcome the nondifferentiability of tv and achieve a closed form to update the regularization parameter in each iteration the solution is guaranteed to satisfy morozov s discrepancy principle furthermore we present a convergence proof for the proposed algorithm on the premise of a variable regularization parameter experimental results demonstrate that the proposed algorithm is superior in speed and competitive in accuracy compared with several state of the art methods besides the proposed method can be smoothly extended to the multichannel image restoration

image restoration via the adaptive tvp regularization abstract to keep structures in the restoration problem is very important via coupling the local information of the image with the proposed model in this paper we propose a local self adaptive l p regularization model for p 0 2 based on the total variation scheme where the choice of p depends on the local structures described by the eigenvalues of the structure tensor since the proposed model as the classic l p problem unifies two classes of optimization problems such as the nonconvex and nonsmooth problem when p 0 1 and the convex and smooth problem when p 1 2 it is generally challenging to find a ready algorithmic framework to solve it here we propose a new and robust numerical method via coupling with the half quadratic scheme and the alternating direction method of multipliers admm the convergence of the proposed algorithm is established and the numerical experiments illustrate the possible advantages of the proposed model and numerical methods over some existing variational based models and methods

generalized higher degree total variation hdtv regularization we introduce a family of novel image regularization penalties called generalized higher degree total variation hdtv these penalties further extend our previously introduced hdtv penalties which generalize the popular total variation tv penalty to incorporate higher degree image derivatives we show that many of the proposed second degree extensions of tv are special cases or are closely approximated by a generalized hdtv penalty additionally we propose a novel fast alternating minimization algorithm for solving image recovery problems with hdtv and generalized hdtv regularization the new algorithm enjoys a tenfold speed up compared with the iteratively reweighted majorize minimize algorithm proposed in a previous paper numerical experiments on 3d magnetic resonance images and 3d microscopy images show that hdtv and generalized hdtv improve the image quality significantly compared with tv

the perturbed riemann problem and delta contact discontinuity in chromatography equations abstract in this paper we discuss the perturbed riemann problem for the nonlinear chromatography equations the solutions to the problem are constructed with the method of the splitting delta function in solutions we find another kind of delta contact discontinuity on which both state variables simultaneously contain the dirac delta functions it is quite different from the previous ones on which only one state variable contains the dirac delta function this phenomenon is also predicted theoretically by mazzotti 2009 and mazzotti et al 2010 moreover by studying the limits of the solutions as the perturbed parameter e 0 it can be found that the riemann solutions are stable for such perturbations with the initial data

low dose cbct reconstruction using hessian schatten penalties cone beam computed tomography cbct has been widely used in radiation therapy for accurate patient setup and treatment target localization it is important to obtain high quality reconstruction images the total variation tv penalty has shown the state of the art performance in suppressing noise and preserving edges for statistical iterative image reconstruction but it sometimes leads to the so called staircase effect in this paper we proposed to use a new family of penalties the hessian schatten hs penalties for the cbct reconstruction consisting of the second order derivatives the hs penalties are able to reflect the smooth intensity transitions of the underlying image without introducing the staircase effect we discussed and compared the behaviors of several convex hs penalties with orders 1 2 and infty for cbct reconstruction we used the majorization minimization approach with a primal dual formulation for the corresponding optimization problem experiments on two digital phantoms and two physical phantoms demonstrated the proposed penalty family s outstanding performance over tv in suppressing the staircase effect and the hs penalty with order 1 had the best performance among the hs penalties tested

interactions of delta shock waves for the transport equations with split delta functions this paper is concerned with the interactions of shock waves and the vacuum states between the two contact discontinuities for the transport equations the solutions are obtained constructively when the initial data are three piecewise constant states the global structure and large time asymptotic behaviors of the solutions are analyzed case by case moreover it can be found that the riemann solutions are stable for such small perturbations with initial data by studying the limits of the solutions when the perturbed parameter e tends to zero

non selfsimilar solutions for a hyperbolic system of conservation laws in two space dimensions the non selfsimilar riemann problem for a two dimensional nonstrictly hyperbolic system of conservation laws is considered where the initial data are two constant states separated by a smooth curve without dimension reduction or coordinate transformation the two dimensional global solutions are constructed for six cases according to the normal velocities on both sides of the initial discontinuity moreover the interactions of non selfsimilar elementary waves are discussed by respectively taking the initial discontinuity as a parabola and a circle which enable us to provide the global structures of the entropy solutions explicitly c 2012 elsevier inc all rights reserved

noise removal using fourth order partial differential equation with applications to medical magnetic resonance images in space and time we introduce a new method for image smoothing based on a fourth order pde model the method is tested on a broad range of real medical magnetic resonance images both in space and time as well as on nonmedical synthesized test images our algorithm demonstrates good noise suppression without destruction of important anatomical or functional detail even at poor signal to noise ratio we have also compared our method with related pde models

wavelet theory demystified we revisit wavelet theory starting from the representation of a scaling function as the convolution of a b spline the regular part of it and a distribution the irregular or residual part this formulation leads to some new insights on wavelets and makes it possible to rederive the main results of the classical theory including some new extensions for fractional orders n a self contained accessible fashion in particular we prove that the b spline component is entirely responsible for five key wavelet properties order of approximation reproduction of polynomials vanishing moments multiscale differentiation property and smoothness regularity of the basis functions we also investigate the interaction of wavelets with differential operators giving explicit time domain formulas for the fractional derivatives of the basis functions this allows us to specify a corresponding dual wavelet basis and helps us understand why the wavelet transform provides a stable characterization of the derivatives of a signal additional results include a new peeling theory of smoothness leading to the extended notion of wavelet differentiability in the l sub p sense and a sharper theorem stating that smoothness implies order

fourth order partial differential equations for noise removal a class of fourth order partial differential equations pdes are proposed to optimize the trade off between noise removal and edge preservation the time evolution of these pdes seeks to minimize a cost functional which is an increasing function of the absolute value of the laplacian of the image intensity function since the laplacian of an image at a pixel is zero if the image is planar in its neighborhood these pdes attempt to remove noise and preserve edges by approximating an observed image with a piecewise planar image piecewise planar images look more natural than step images which anisotropic diffusion second order pdes uses to approximate an observed image so the proposed pdes are able to avoid the blocky effects widely seen in images processed by anisotropic diffusion while achieving the degree of noise removal and edge preservation comparable to anisotropic diffusion although both approaches seem to be comparable in removing speckles in the observed images speckles are more visible in images processed by the proposed pdes because piecewise planar images are less likely to mask speckles than step images and anisotropic diffusion tends to generate multiple false edges speckles can be easily removed by simple algorithms such as the one presented in this paper

higher degree total variation hdtv regularization for image recovery we introduce novel image regularization penalties to overcome the practical problems associated with the classical total variation tv scheme motivated by novel reinterpretations of the classical tv regularizer we derive two families of functionals involving higher degree partial image derivatives we term these families as isotropic and anisotropic higher degree tv hdtv penalties respectively the isotropic penalty is the mixed norm of the directional image derivatives while the anisotropic penalty is the separable norm of directional derivatives these functionals inherit the desirable properties of standard tv schemes such as invariance to rotations and translations preservation of discontinuities and convexity the use of mixed norms in isotropic penalties encourages the joint sparsity of the directional derivatives at each pixel thus encouraging isotropic smoothing in contrast the fully separable norm in the anisotropic penalty ensures the preservation of discontinuities while continuing to smooth along the line like features this scheme thus enhances the linenlike image characteristics analogous to standard tv we also introduce efficient majorize minimize algorithms to solve the resulting optimization problems the numerical comparison of the proposed scheme with classical tv penalty current second degree methods and wavelet algorithms clearly demonstrate the performance improvement specifically the proposed algorithms minimize the staircase and ringing artifacts that are common with tv and wavelet schemes while better preserving the singularities we also observe that anisotropic hdtv penalty provides consistently improved reconstructions compared with the isotropic hdtv penalty

new developments of delta shock waves and its applications in systems of conservation laws a theory of delta shock waves with dirac delta functions developing in both state variables for a class of nonstrictly hyperbolic systems of conservation laws is established in this paper we solve constructively the riemann problems for the system under consideration in solutions we find another kind of delta shock waves on which both state variables simultaneously contain the dirac delta functions it is quite different from the previous ones on which only one state variable contains the dirac delta function we strictly prove it satisfies the system in the sense of distributions the generalized rankine hugoniot relation and entropy condition are proposed to solve the delta shock waves furthermore we show all of the existence and stability of solutions including the delta shock waves to reasonable viscous perturbations the generalized rankine hugoniot relation is also confirmed in particular our theory on the delta shock waves possesses the generality and practicability which can be conveniently and successfully applied to those systems investigated by korchinski 1977 tan zhang and zheng 1991 ercole 2000 cheng and yang 2011 etc and we also give a simplified approach to solve a 2 d riemann problem for the system studied by tan and zhang 1990 for the case 2j 2j and obtain the explicit formulae of the delta shock waves finally the numerical simulations completely coinciding with theoretical analysis are presented

shadow waves entropies and interactions for delta and singular shocks this paper deals with two key problems for delta or singular shock solutions to systems of conservation laws that of their entropy admissibility conditions which is connected to the notorious uniqueness problem and that of their interaction we choose to represent systems of conservation laws by nets of functions which are piecewise constant or constant with respect only to the space variable here called shadow waves all the calculations can then be done on each element of such nets using only the usual rankine hugoniot conditions a 3 3 pressureless gas dynamics model is the main example in the paper

the pseudodifferential operator square root of the klein gordon equation a nonlocal square root of the klein gordon equation is proposed this nonlocal equation is a special relativistic equation for a scalar field of first order in the time derivative its space derivative part is described by a pseudodifferential operator the usual quantum mechanical formalism can be set up the nonrelativistic limit and the classical limit in the form of plane wave solutions and the ehrenfest theorem are correctly included the nonlocality of the wave equation does not disturb the light cone structure and the relativity principle of special relativity is fulfilled uniqueness and existence of solutions of the cauchy problem for this equation can be proved the second quantized version of this theory turns out to be macrocausal

hessian based norm regularization for image restoration with biomedical applications we present nonquadratic hessian based regularization methods that can be effectively used for image restoration problems in a variational framework motivated by the great success of the total variation tv functional we extend it to also include second order differential operators specifically we derive second order regularizers that involve matrix norms of the hessian operator the definition of these functionals is based on an alternative interpretation of tv that relies on mixed norms of directional derivatives we show that the resulting regularizers retain some of the most favorable properties of tv i e convexity homogeneity rotation and translation invariance while dealing effectively with the staircase effect we further develop an efficient minimization scheme for the corresponding objective functions the proposed algorithm is of the iteratively reweighted least square type and results from a majorization minimization approach it relies on a problem specific preconditioned conjugate gradient method which makes the overall minimization scheme very attractive since it can be applied effectively to large images in a reasonable computational time we validate the overall proposed regularization framework through deblurring experiments under additive gaussian noise on standard and biomedical images

spaces of weighted measures for conservation laws with singular shock solutions abstract we study a model system of two strictly hyperbolic conservation laws which is genuinely nonlinear but for which the riemann problem has no global solution singular solutions are defined by means of a generalized rankine hugoniot relation and an overcompressive condition on the discontinuity we show that approximate solutions which can be constructed by several standard methods converge in a weighted measure space and that the error in the approximation converges to zero viscous approximations satisfy approximate entropy inequalities which imply the overcompressive condition

splines in higher order tv regularization splines play an important role as solutions of various interpolation and approximation problems that minimize special functionals in some smoothness spaces in this paper we show in a strictly discrete setting that splines of degree m 1 solve also a minimization problem with quadratic data term and m th order total variation tv regularization term in contrast to problems with quadratic regularization terms involving m th order derivatives the spline knots are not known in advance but depend on the input data and the regularization parameter more precisely the spline knots are determined by the contact points of the m th discrete antiderivative of the solution with the tube of width 2 around the m th discrete antiderivative of the input data we point out that the dual formulation of our minimization problem can be considered as support vector regression problem in the discrete counterpart of the sobolev space w 2 0 m from this point of view the solution of our minimization problem has a sparse representation in terms of discrete fundamental splines

interactions of delta shock waves in a strictly hyperbolic system of conservation laws we study interaction of generalized solutions to a strictly hyperbolic system of two conservation laws from magnetohydrodynamics the riemann problem of this system is known to be solvable by a combination of shock waves contact discontinuities rarefaction waves and delta shock waves we derive how the solutions continue beyond points of interaction of these types of solutions we introduce a type of generalized solution a delta contact discontinuity a complete description of all global solutions with initial data given by two jump points and three constant states is obtained

reasons for a physical field to obey linear partial differential equations fields showing a deterministic evolution which is linear and local can be described by a linear system of partial differential equations

on the dimensionality of space time within constructive axiomatics of space time geometry based on elements of quantum mechanics it can be shown that by means of the study of the dynamical behavior of general matter fields in a geometry free way one can give reasons for the dimension of space time to be four

delta shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws abstract for simple models of hyperbolic systems of conservation laws we study a new type of nonlinear hyperbolic wave a delta shock wave which is a dirac delta function supported on a shock we prove that delta shock waves are w limits in l1 of solutions to some reasonable viscous perturbations as the viscosity vanishes further we solve the multiplication problem of a delta function with a discontinuous function to show that delta shock waves satisfy the equations in the sense of distributions under suitable generalized rankine hugoniot and entropy conditions we establish the existence and uniqueness of solutions involving delta shock waves for the riemann problems the existence of solutions to the cauchy problem is also investigated

stability of the riemann solutions for a nonstrictly hyperbolic system of conservation laws abstract we prove that the riemann solutions are stable for a nonstrictly hyperbolic system of conservation laws under local small perturbations of the riemann initial data the proof is based on the detailed analysis of the interactions of delta shock waves with shock waves and rarefaction waves during the interaction process of the delta shock wave with the rarefaction wave a new kind of nonclassical wave namely a delta contact discontinuity is discovered here which is a dirac delta function supported on a contact discontinuity and has already appeared in the interaction process for the magnetohydrodynamics equations m nedeljkov and m oberguggenberger interactions of delta shock waves in a strictly hyperbolic system of conservation laws j math anal appl 344 2008 1143 1157 moreover the global structures and large time asymptotic behaviors of the solutions are constructed and analyzed case by case

on the propagation of polarization sets for systems of real principal type we define and study the polarization sets for vector valued distributions systems of real principal type are defined and we prove that for such systems of pseudodifferential operators we have a unique propagation of the polarization sets for the solutions

high order total variation based image restoration the total variation tv denoising method is a pde based technique that preserves edges well but has the sometimes undesirable staircase effect namely the transformation of smooth regions ramps into piecewise constant regions stairs in this paper we present an improved model constructed by adding a nonlinear fourth order diffusive term to the euler lagrange equations of the variational tv model our technique substantially reduces the staircase effect while preserving sharp jump discontinuities edges we show numerical evidence of the power of resolution of this novel model with respect to the tv model in some 1d and 2d numerical examples

hessian schatten norm regularization for linear inverse problems we introduce a novel family of invariant convex and non quadratic functionals that we employ to derive regularized solutions of ill posed linear inverse imaging problems the proposed regularizers involve the schatten norms of the hessian matrix which are computed at every pixel of the image they can be viewed as second order extensions of the popular total variation tv semi norm since they satisfy the same invariance properties meanwhile by taking advantage of second order derivatives they avoid the staircase effect a common artifact of tv based reconstructions and perform well for a wide range of applications to solve the corresponding optimization problems we propose an algorithm that is based on a primal dual formulation a fundamental ingredient of this algorithm is the projection of matrices onto schatten norm balls of arbitrary radius this operation is performed efficiently based on a direct link we provide between vector projections onto norm balls and matrix projections onto schatten norm balls finally we demonstrate the effectiveness of the proposed methods through experimental results on several inverse imaging problems with real and simulated data

is there a stable hydrogen atom in higher dimensions the schrodinger equation in higher dimensions is considered it consists of the kinetic energy part given by the corresponding laplace operator and a term describing the interaction with the electrostatic field of a point charge from rutherford type scattering experiments one can conclude that the potential of a point charge is 1 r irrespective of the dimension of the space where the experiment is carried through also the structure of the kinetic energy is unchanged in higher dimensions so that one is lead to the result that there exist stable atoms in higher spatial dimensions d 4 the solutions and energy eigenvalues to this schrodinger equation in higher dimensions are presented as a consequence the dimensionality of space can be read off from the spectral scheme of atoms the three dimensionality of space is a consequence of the existence of the lyman series another consequence is that the maxwell equations in higher dimensions must be modified in order to have the 1 r potential as solution for