an appropriate subdifferential for quasiconvex functions in this paper we introduce and study a subdifferential that is related to the quasi convex functions much as the fenchel moreau subdifferential is related to the convex ones it is defined for any lower semicontinuous function through an appropriate combination of an abstract subdifferential and the normal cone to sublevel sets we show that this quasiconvex subdifferential is always a cyclically quasimonotone operator that coincides with the fenchel moreau subdifferential whenever the function is convex and that under mild assumptions the density of its domain in the domain of the function is equivalent to the quasiconvexity of the function we also show that the quasiconvex subdifferential of a lower semicontinuous function contains the derivatives of its differentiable quasiaffine supports as a consequence it contains the subdifferential introduced by martinez legaz and sach in a recent paper j convex anal 6 1999 pp 1 12 several other properties and calculus rules are also established

on the convergence of the two dimensional second grade fluid model to the navier stokes equation we consider the equations governing the motion of incompressible second grade fluids in a bounded two dimensional domain with navier slip boundary conditions we first prove that the corresponding solutions are uniformly bounded with respect to the normal stress modulus in the l h1 and the l2 h2 time space norms next we study their asymptotic behavior when tends to zero prove that they converge to regular solutions of the navier stokes equations and give the rate of convergence in terms of

from second grade fluids to the navier stokes equations abstract we consider the limit 0 for a second grade fluid on a bounded domain with dirichlet boundary conditions we show convergence towards a solution of the navier stokes equations under two different types of hypothesis on the initial velocity u 0 if the product u 0 l 2 u 0 h 1 is sufficiently small we prove global in time convergence if there is no smallness assumption we obtain local in time convergence up to the time c u 0 h 1 4

a finite element approximation for the steady solution of a second grade fluid model abstract the aim of this work is to present a finite element method for the approximation of the steady solution of an incompressible second grade fluid model in two dimensions the equations for second grade fluids form a system of nonlinear partial differential equations of mixed elliptic hyperbolic type in the steady state using a fixed point argument associated with the decomposition of the system into a transport equation and a stokes system existence and uniqueness of the approximate solution are proved and error estimates are obtained this technique allows the construction of a decoupled fixed point algorithm converging to the discrete solution of the original problem

smoothness of bounded solutions of nonlinear evolution equations abstract it is shown that in many cases globally defined bounded solutions of evolution equations are as smooth in time as the corresponding operator even if a general solution of the initial value problem is much less smooth i e initial values for bounded solutions are selected in such a way that optimal smoothness is attained in particular solutions which bifurcate from certain steady states such as periodic orbits almost periodic orbits and also homo and heteroclinic orbits have this property as examples a neutral functional differential equation a slightly damped non linear wave equation and a heat equation are considered in the latter case the space variable is included into the discussion of smoothness finally generalized hopf bifurcation in infinite dimensions is considered here smoothness of the bifurcation function is discussed and known results on the order of a focus are generalized

compressive sensing via nonlocal low rank regularization sparsity has been widely exploited for exact reconstruction of a signal from a small number of random measurements recent advances have suggested that structured or group sparsity often leads to more powerful signal reconstruction techniques in various compressed sensing cs studies in this paper we propose a nonlocal low rank regularization nlr approach toward exploiting structured sparsity and explore its application into cs of both photographic and mri images we also propose the use of a nonconvex log det x as a smooth surrogate function for the rank instead of the convex nuclear norm and justify the benefit of such a strategy using extensive experiments to further improve the computational efficiency of the proposed algorithm we have developed a fast implementation using the alternative direction multiplier method technique experimental results have shown that the proposed nlr cs algorithm can significantly outperform existing state of the art cs techniques for image recovery

random attractors for the stochastic benjamin bona mahony equation on unbounded domains abstract we prove the existence of a compact random attractor for the stochastic benjamin bona mahony equation defined on an unbounded domain this random attractor is invariant and attracts every pulled back tempered random set under the forward flow the asymptotic compactness of the random dynamical system is established by a tail estimates method which shows that the solutions are uniformly asymptotically small when space and time variables approach infinity

the global attractor of g navier stokes equations with linear dampness on r2 in this paper the long time behaviors of g navier stokes equations with linear dampness on r 2 were investigated by using the energy equation method the existence of the global attractor for the equations was proved without the restriction of the forcing term belonging to some weighted sobolev space moreover the estimation of the hausdorff and fractal dimensions of such attractors were also obtained

d pullback attractor for a non autonomous wave equation with additive noise on unbounded domains this paper is devoted to investigating the asymptotic behavior of solutions of the wave equation with additive noise as well as a non autonomous term defined on unbounded domains firstly by uniform estimates a closed d pullback absorbing set of dynamical system f produced by the mentioned wave equation is obtained secondly the asymptotical compactness of f is established by the splitting technique finally the existence of ad pullback attractor for corresponding dynamical system f in h 1 r 3 xl 2 r 3 is proved where n 3 particularly we can show that the d pullback attractor is also periodic when the forcing term is periodic

bilinear factor matrix norm minimization for robust pca algorithms and applications the heavy tailed distributions of corrupted outliers and singular values of all channels in low level vision have proven effective priors for many applications such as background modeling photometric stereo and image alignment and they can be well modeled by a hyper laplacian however the use of such distributions generally leads to challenging non convex non smooth and non lipschitz problems and makes existing algorithms very slow for large scale applications together with the analytic solutions to ell p norm minimization with two specific values of p i e p 1 2 and p 2 3 we propose two novel bilinear factor matrix norm minimization models for robust principal component analysis we first define the double nuclear norm and frobenius nuclear hybrid norm penalties and then prove that they are in essence the schatten 1 2 and 2 3 quasi norms respectively which lead to much more tractable and scalable lipschitz optimization problems our experimental analysis shows that both our methods yield more accurate solutions than original schatten quasi norm minimization even when the number of observations is very limited finally we apply our penalties to various low level vision problems e g text removal moving object detection image alignment and inpainting and show that our methods usually outperform the state of the art methods

pullback attractors for the non autonomous fitzhugh nagumo system on unbounded domains abstract the existence of a pullback attractor is established for the singularly perturbed fitzhugh nagumo system defined on the entire space r n when external terms are unbounded in a phase space the pullback asymptotic compactness of the system is proved by using uniform a priori estimates for far field values of solutions although the limiting system has no global attractor we show that the pullback attractors for the perturbed system with bounded external terms are uniformly bounded and hence do not blow up as a small parameter approaches zero

on the backward behavior of some dissipative evolution equations abstract we prove that every solution of a kdv burgers sivashinsky type equation blows up in the energy space backward in time provided the solution does not belong to the global attractor this is a phenomenon contrast to the backward behavior of the periodic 2d navier stokes equations studied by constantin et al 1997 but analogous to the backward behavior of the kuramoto sivashinsky equation discovered by kukavica and malcok 2005 also we study the backward behavior of solutions to the damped driven nonlinear schrodinger equation the complex ginzburg landau equation and the hyperviscous navier stokes equations in addition we provide some physical interpretation of various backward behaviors of several perturbations of the kdv equation by studying explicit cnoidal wave solutions furthermore we discuss the connection between the backward behavior and the energy spectra of the solutions the study of backward behavior of dissipative evolution equations is motivated by the investigation of the bardos tartar conjecture on the navier stokes equations stated in bardos and tartar 1973

on the hyperbolicity of maxwell s equations with a local constitutive law maxwell s equations are considered in metric free form with a local but otherwise arbitrary constitutive law after splitting maxwell s equations into evolution equations and constraints we derive the characteristic equation and we discuss its properties in detail we present several results that are relevant for the question of whether the evolution equations are hyperbolic strongly hyperbolic or symmetric hyperbolic in particular we give a convenient characterization of all constitutive laws for which the evolution equations are symmetric hyperbolic the latter property is sufficient but not necessary for well posedness of the initial value problem by way of example we illustrate our results with the constitutive laws of biisotropic media and of born infeld theory

global attractor for weakly damped nonlinear schrodinger equations in l2 r abstract we prove that the weakly damped nonlinear schrodinger flow in l 2 r provides a dynamical system which possesses a global attractor the proof relies on the continuity of the schrodinger flow for the weak topology in l 2 r

beamforming via nonconvex linear regression impulsive processes frequently occur in many fields such as radar sonar communications audio and speech processing and biomedical engineering in this paper we propose a nonconvex linear regression nlr based minimum dispersion beamforming technique for impulsive signals to achieve significant performance improvement over the conventional minimum variance beamformer the proposed beamformer minimizes the ell p norm of the output with p subject to a linear distortionless response constraint resulting in a difficult nonconvex and nonsmooth optimization problem the constrained optimization problem is first reduced to a multivariate linear regression via constraint elimination as a major contribution of this paper a coordinate descent algorithm cda is devised for solving the resultant nlr problem of ell p minimization with p at a computational complexity of cal o mn 2 where m is the number of sensors and n is the sample size at each inner iteration of the cda an efficient algorithm is designed to find the global minimum of each subproblem of univariate linear regression the convergence of the cda is analyzed the nlr beamformer with a single constraint is further generalized to the case of multiple linear constraints which is robust against model mismatch simulation results demonstrate the superior performance of nonconvex optimization based beamformer

efficient tensor completion for color image and video recovery low rank tensor train this paper proposes a novel approach to tensor completion which recovers missing entries of data represented by tensors the approach is based on the tensor train tt rank which is able to capture hidden information from tensors thanks to its definition from a well balanced matricization scheme accordingly new optimization formulations for tensor completion are proposed as well as two new algorithms for their solution the first one called simple low rank tensor completion via tt silrtc tt is intimately related to minimizing a nuclear norm based on tt rank the second one is from a multilinear matrix factorization model to approximate the tt rank of a tensor and is called tensor completion by parallel matrix factorization via tt tmac tt a tensor augmentation scheme of transforming a low order tensor to higher orders is also proposed to enhance the effectiveness of silrtc tt and tmac tt simulation results for color image and video recovery show the clear advantage of our method over all other methods

decomposable medium conditions in four dimensional representation the well known te tm decomposition of time harmonic electromagnetic fields in uniaxial anisotropic media is generalized in terms of four dimensional differential form formalism by requiring that the field two form satisfies an orthogonality condition with respect to two given bivectors conditions for the electromagnetic medium in which such a decomposition is possible are derived and found to define three subclasses of media it is shown that the previously known classes of generalized q media and generalized p media are particular cases of the proposed decomposable media dcm associated to a quadratic equation for the medium dyadic as a novel solution another class of special decomposable media sdcm is defined by a linear dyadic equation the paper further discusses the properties of medium dyadics and plane wave propagation in all the identified cases of dcm and sdcm

asymptotic smoothing effect for a weakly damped nonlinear schrodinger equation in t2 abstract we prove that the global attractor for a weakly damped two dimensional nonlinear schrodinger equation in the usual energy space is in fact included and compact in a more regular energy space the method relies on a suitable approximation when time goes to infinity of the high frequency modes of the solutions

existence of global attractors for the benjamin bona mahony equation in unbounded domains in this paper we investigate the asymptotic behavior of the benjamin bona mahony equation in unbounded domains we prove the existence of a global attractor when the equation is defined in a three dimensional channel the asymptotic compactness of the solution operator is obtained by the uniform estimates on the tails of solutions

long time behavior of solution for coupled ginzburg landau equations describing bose einstein condensates and nonlinear optical waveguides and cavities abstract in this paper the cauchy problem of coupled ginzburg landau gl equations describing bose einstein condensates and nonlinear optical waveguides and cavities is considered the long time behavior of solution is investigated using a series of sharp a priori estimates in phase space e 1 and e 2 respectively the global strong attractor in e 2 is proved by energy equation method and its bound of dimension is shown

the uniform attractor for the 2d non autonomous navier stokes flow in some unbounded domain abstract by using the energy equation method the existence of the uniform attractor for the 2d non autonomous navier stokes equations in some unbounded domains with usual settings is obtained without the restriction of the forcing term belonging to some weighted sobolev space and for the quasi periodic forcing term the estimation of the hausdorff dimension of such attractors is also obtained

regularity of the global attractor and finite dimensional behavior for the second grade fluid equations abstract this paper is devoted to the large time behavior and especially to the regularity of the global attractor of the second grade fluid equations in the two dimensional torus we first recall that for any size of the material coefficient 0 these equations are globally well posed and admit a compact global attractor a in h 3 t 2 2 we prove that for any 0 there exists 0 such that a belongs to h 3 t 2 2 if the forcing term is in h 1 t 2 2 we also show that this attractor is contained in any sobolev space h 3 m t 2 2 provided that is small enough and the forcing term is regular enough these arguments lead also to a new proof of the existence of the compact global attractor a furthermore we prove that on a the second grade fluid system can be reduced to a finite dimensional system of ordinary differential equations with an infinite delay moreover the existence of a finite number of determining modes for the equations of the second grade fluid is established

global attractors for the klein gordon schrodinger equation in unbounded domains abstract in this paper we study the long time behavior of solutions for the klein gordon schrodinger equation in the whole space r n with n 3 we first prove the continuity of the solutions on initial data and then establish the asymptotic compactness of solutions finally we show the existence of the global attractor for this model in the space hk r n hk r n for each integer k 1

determining the structure of the jordan normal form of a matrix by symbolic computation abstract an algorithm for determining the jordan normal form of a matrix is presented the algorithm implements the geometric approach given by gantmacher using this algorithm structures of the jordan blocks of the matrix can be completely determined by means of symbolic computation the approximation of eigenvalues of the matrix has no influence on this structure

heteroclinic dynamics in the nonlocal parametrically driven nonlinear schrodinger equation faraday waves are described under appropriate conditions by a damped nonlocal parametrically driven nonlinear schrodinger equation as the strength of the applied forcing increases this equation undergoes a sequence of transitions to chaotic dynamics the origin of these transitions is explained using a careful study of a two mode galerkin truncation and linked to the presence of heteroclinic connections between the trivial state and spatially periodic standing waves these connections are associated with cascades of gluing and symmetry switching bifurcations such bifurcations are located in the partial differential equations as well

maximum likelihood array processing in non gaussian noise with gaussian mixtures many approaches have been studied for the array processing problem when the additive noise is modeled with a gaussian distribution but these schemes typically perform poorly when the noise is non gaussian and or impulsive this paper is concerned with maximum likelihood array processing in non gaussian noise we present the cramer rao bound on the variance of angle of arrival estimates for arbitrary additive independent identically distributed iid symmetric non gaussian noise then we focus on non gaussian noise modeling with a finite gaussian mixture distribution which is capable of representing a broad class of non gaussian distributions that include heavy tailed impulsive cases arising in wireless communications and other applications based on the gaussian mixture model we develop an expectation maximization em algorithm for estimating the source locations the signal waveforms and the noise distribution parameters the important problems of detecting the number of sources and obtaining initial parameter estimates for the iterative em algorithm are discussed in detail the initialization procedure by itself is an effective algorithm for array processing in impulsive noise novel features of the em algorithm and the associated maximum likelihood formulation include a nonlinear beamformer that separates multiple source signals in non gaussian noise and a robust covariance matrix estimate that suppresses impulsive noise while also performing a model based interpolation to restore the low rank signal subspace the em approach yields improvement over initial robust estimates and is valid for a wide snr range the results are also robust to pdf model mismatch and work well with infinite variance cases such as the symmetric stable distributions simulations confirm the optimality of the em estimation procedure in a variety of cases including a multiuser communications scenario we also compare with existing array processing algorithms for non gaussian noise

on perturbations of delay differential equations with periodic orbits abstract equations of retarded type and simple neutral type equations are considered the study concerns both autonomous and non autonomous perturbations of an autonomous equation which possesses a non trivial periodic orbit the main tool is a local coordinate system around the periodic orbit which is obtained from the phase space decomposition via floquet multipliers under the assumption that the perturbation function is lipschitz the existence of an integral manifold with periodic structure for the system in the new coordinates is shown this implies that under autonomous perturbations periodic orbits are continued furthermore we give a description of the flow on the center manifold of the periodic orbit

fast and accurate matrix completion via truncated nuclear norm regularization recovering a large matrix from a small subset of its entries is a challenging problem arising in many real applications such as image inpainting and recommender systems many existing approaches formulate this problem as a general low rank matrix approximation problem since the rank operator is nonconvex and discontinuous most of the recent theoretical studies use the nuclear norm as a convex relaxation one major limitation of the existing approaches based on nuclear norm minimization is that all the singular values are simultaneously minimized and thus the rank may not be well approximated in practice in this paper we propose to achieve a better approximation to the rank of matrix by truncated nuclear norm which is given by the nuclear norm subtracted by the sum of the largest few singular values in addition we develop a novel matrix completion algorithm by minimizing the truncated nuclear norm we further develop three efficient iterative procedures tnnr admm tnnr apgl and tnnr admmap to solve the optimization problem tnnr admm utilizes the alternating direction method of multipliers admm while tnnr agpl applies the accelerated proximal gradient line search method apgl for the final optimization for tnnr admmap we make use of an adaptive penalty according to a novel update rule for admm to achieve a faster convergence rate our empirical study shows encouraging results of the proposed algorithms in comparison to the state of the art matrix completion algorithms on both synthetic and real visual datasets

finite dimensional global attractor for a dissipative anisotropic fourth order schrodinger equation abstract we study the long time behavior of the solutions to a nonlinear damped anisotropic fourth order schrodinger type equation in r 2 we prove that this behavior is described by the existence of regular finite dimensional global attractor in the energy space

global existence and compact attractors for the discrete nonlinear schrodinger equation abstract we study the asymptotic behavior of solutions of discrete nonlinear schrodinger type dnls equations for a conservative system we consider the global in time solvability and the question of existence of standing wave solutions similarities and differences with the continuous counterpart nls partial differential equation are pointed out for a dissipative system we prove existence of a global attractor and its stability under finite dimensional approximations similar questions are treated in a weighted phase space finally we propose possible extensions for various types of dnls equations

normal hyperbolicity of center manifolds and saint venant s principle the concept of normal hyperbolicity of center manifolds is generalized to infinite dimensional differential equations in particular to elliptic problems in cylindrical domains it is shown that all solutions u staying close to the center manifold for t l l satisfy an estimate of the form left u t tilde u t right leqslant ce alpha l t where c and are independent of l and is a solution on the center manifold these results are applied to saint venant s principle for the static deformation of nonlinearly elastic prismatic bodies the use of the center manifold permits the effective treatment of the general case of non zero resultant forces and moments acting on each cross section

weighted nuclear norm minimization and its applications to low level vision as a convex relaxation of the rank minimization model the nuclear norm minimization nnm problem has been attracting significant research interest in recent years the standard nnm regularizes each singular value equally composing an easily calculated convex norm however this restricts its capability and flexibility in dealing with many practical problems where the singular values have clear physical meanings and should be treated differently in this paper we study the weighted nuclear norm minimization wnnm problem which adaptively assigns weights on different singular values as the key step of solving general wnnm models the theoretical properties of the weighted nuclear norm proximal wnnp operator are investigated albeit nonconvex we prove that wnnp is equivalent to a standard quadratic programming problem with linear constrains which facilitates solving the original problem with off the shelf convex optimization solvers in particular when the weights are sorted in a non descending order its optimal solution can be easily obtained in closed form with wnnp the solving strategies for multiple extensions of wnnm including robust pca and matrix completion can be readily constructed under the alternating direction method of multipliers paradigm furthermore inspired by the reweighted sparse coding scheme we present an automatic weight setting method which greatly facilitates the practical implementation of wnnm the proposed wnnm methods achieve state of the art performance in typical low level vision tasks including image denoising background subtraction and image inpainting

a note on the strong convergence towards attractors of damped forced kdv equations abstract this note is a continuation of a previous article which appeared in this journal 1988 74 369 390 in the previous work we showed that a weak dissipation is suffiient to impose a finite dimensional behavior for korteweg de vries equations with space periodic boundary conditions more precisely it was shown that all the trajectories are captured by a finite dimensional attractor however we were only able to show that the convergence of the trajectories holds good in the sense of weak topology the goal of this note is to show that in fact this convergence holds true in the sense of strong topology

random attractor for stochastic reaction diffusion equation with multiplicative noise on unbounded domains abstract in this paper we study the asymptotic dynamics for stochastic reaction diffusion equation with multiplicative noise defined on unbounded domains we investigate the existence of a random attractor for the random dynamical system associated with the equation the asymptotic compactness of the random dynamical system is established by using uniform a priori estimates for far field values of solutions and a cut off technique

asymptotic behavior of stochastic wave equations with critical exponents on mathbb r 3 the existence of a random attractor in h 1 3 x l 2 3 is proved for the damped semilinear stochastic wave equation defined on the entire space 3 the nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent the uniform pullback estimates on the tails of solutions for large space variables are established the pullback asymptotic compactness of the random dynamical system is proved by using these tail estimates and the energy equation method

low dimensional chaos in a driven damped nonlinear schro dinger equation abstract we study bifurcations of attractors in a driven damped nonlinear schrodinger equation which models nonlinear responses of a plasma driven by a rf field and a sequence of period doubling bifurcations is shown to lead to a chaotic attractor in a low dimensional subspace spanned by a soliton and long wavelength radiation

asymptotic behavior of a class of non autonomous degenerate parabolic equations abstract we investigate the asymptotic behavior of solutions of a class of non autonomous degenerate parabolic equations on r n with unbounded external forcing terms the existence of a pullback global attractor is proved in l 2 r n l p r n w 1 r r n for any p r 2

random attractors for stochastic reaction diffusion equations on unbounded domains the existence of a pullback attractor is established for a stochastic reaction diffusion equation on all n dimensional space the nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears as spatially distributed temporal white noise the reaction diffusion equation is recast as a random dynamical system and asymptotic compactness for this is demonstrated by using uniform a priori estimates for far field values of solutions

asymptotic smoothing and the global attractor of a weakly damped kdv equation on the real line abstract the existence of the global attractor of a weakly damped forced korteweg de vries equation in the phase space l 2 r is proved an optimal asymptotic smoothing effect of the equation is also shown namely that for forces in l 2 r the global attractor in the phase space l 2 r is actually a compact set in h 3 r the energy equation method is used in conjunction with a suitable splitting of the solutions the dispersive regularization properties of the equation in the context of bourgain spaces are extensively exploited

characterisation and representation of non dissipative electromagnetic medium with two lorentz null cones we study maxwell s equations on a 4 manifold n with a medium that is non dissipative and has a linear and pointwise response in this setting the medium can be represented by a suitable 22 tensor on the 4 manifold n moreover in each cotangent space on n the medium defines a fresnel surface essentially the fresnel surface is a tensorial analogue of the dispersion equation that describes the response of the medium for signals in the geometric optics limit for example in an isotropic medium the fresnel surface is at each point a lorentz null cone in a recent paper lindell favaro and bergamin introduced a condition that constrains the polarisation for plane waves in this paper we show under suitable assumptions that a slight strengthening of this condition gives a complete pointwise characterisation of all medium tensors for which the fresnel surface is the union of two distinct lorentz null cones this is for example the behaviour in uniaxial media such as calcite moreover using the repres

a determining form for the damped driven nonlinear schrodinger equation fourier modes case in this paper we show that the global attractor of the 1d damped driven nonlinear schrodinger equation nls is embedded in the long time dynamics of a determining form the determining form is an ordinary differential equation in a space of trajectories x cb1 r pmh2 where pm is the l2 projector onto the span of the first m fourier modes there is a one to one identification with the trajectories in the global attractor of the nls and the steady states of the determining form we also give an improved estimate for the number of the determining modes

an energy equation for the weakly damped driven nonlinear schro dinger equations and its application to their attractors abstract in this note we present an energy equation concerning the h 2 norm of a forced one dimensional nonlinear schrodinger equation as an application we are able to corroborate recent results on attractors of weakly damped forced nonlinear schrodinger equation by proving that the attractor is an attractor in the strong topology of h 2 and its hausdorff and fractal dimensions are finite in h 2 the energy equation for the h 1 norm can be derived in the same fashion and it implies corresponding results for the attractor in h 1

asymptotic smoothing effect of solutions to weakly dissipative klein gordon schrodinger equations abstract in this paper the authors consider the cauchy problem of weakly dissipative klein gordon schrodinger equations through yukawa coupling in r 3 making use of a strichartz type inequality and a suitable decomposition of the solution semigroup they prove the asymptotic smoothing effect of the solutions

computationally efficient truncated nuclear norm minimization for high dynamic range imaging matrix completion is a rank minimization problem to recover a low rank data matrix from a small subset of its entries since the matrix rank is nonconvex and discrete many existing approaches approximate the matrix rank as the nuclear norm however the truncated nuclear norm is known to be a better approximation to the matrix rank than the nuclear norm exploiting a priori target rank information about the problem in rank minimization in this paper we propose a computationally efficient truncated nuclear norm minimization algorithm for matrix completion which we call tnnm alm we reformulate the original optimization problem by introducing slack variables and considering noise in the observation the central contribution of this paper is to solve it efficiently via the augmented lagrange multiplier alm method where the optimization variables are updated by closed form solutions we apply the proposed tnnm alm algorithm to ghost free high dynamic range imaging by exploiting the low rank structure of irradiance maps from low dynamic range images experimental results on both synthetic and real visual data show that the proposed algorithm achieves significantly lower reconstruction errors and superior robustness against noise than the conventional approaches while providing substantial improvement in speed thereby applicable to a wide range of imaging applications

grade two fluids on non smooth domain driven by multiplicative noise existence uniqueness and regularity the author presented a lecture on the subject of this paper at the workshop on numerics for stochastic partial differential equations and their applications held at the johann radon institute for computational and applied mathematics ricam linz austria during the period 04 16 december 2016

tensor train decomposition a simple nonrecursive form of the tensor decomposition in d dimensions is presented it does not inherently suffer from the curse of dimensionality it has asymptotically the same number of parameters as the canonical decomposition but it is stable and its computation is based on low rank approximation of auxiliary unfolding matrices the new form gives a clear and convenient way to implement all basic operations efficiently a fast rounding procedure is presented as well as basic linear algebra operations examples showing the benefits of the decomposition are given and the efficiency is demonstrated by the computation of the smallest eigenvalue of a 19 dimensional operator

convergence of the 2d euler α to euler equations in the dirichlet case indifference to boundary layers abstract in this article we consider the euler system as a regularization of the incompressible euler equations in a smooth two dimensional bounded domain for the limiting euler system we consider the usual non penetration boundary condition while for the euler regularization we use velocity vanishing at the boundary we also assume that the initial velocities for the euler system approximate in a suitable sense as the regularization parameter 0 the initial velocity for the limiting euler system for small values of this situation leads to a boundary layer which is the main concern of this work our main result is that under appropriate regularity assumptions and despite the presence of this boundary layer the solutions of the euler system converge as 0 to the corresponding solution of the euler equations in l 2 in space uniformly in time we also present an example involving parallel flows in order to illustrate the indifference to the boundary layer of the 0 limit which underlies our work

random attractors for the stochastic fitzhugh nagumo system on unbounded domains abstract the existence of a random attractor for the stochastic fitzhugh nagumo system defined on an unbounded domain is established the pullback asymptotic compactness of the stochastic system is proved by uniform estimates on solutions for large space and time variables these estimates are obtained by a cut off technique

pullback asymptotic behavior of solutions for a non autonomous non newtonian fluid on two dimensional unbounded domains this paper studies the pullback asymptotic behavior of solutions for a non autonomous non newtonian fluid on r l l for some l 0 we first use the technique of truncation functions together with the decomposition of spatial domain to prove the existence of a pullback attractor in then we discuss the upper semicontinuity of the pullback attractors when the spatial domains vary from m to where m m 1 is an expanding sequence of simply connected bounded and smooth subdomains of such that m 1 m let a a r and am am r be the pullback attractors corresponding to and m respectively we establish for each r that for any neighborhood o a of a am enters o a if m is large enough

existence and uniqueness of the flow of second grade fluids with slip boundary conditions this paper treats the solvability of the equations of motion for an incompressible fluid of grade two subject to nonlinear partial slip boundary conditions in a bounded simply connected domain the existence of a unique classical solution global in time is proved under suitable regularity and growth restrictions on the initial data the slip law and the body and surface forces the method is based on a fixed point formulation of the problem

the global attractor for the 2d navier stokes flow on some unbounded domains we extend previous results on the existence of the global attractor for the 2d navier stokes equations on some unbounded domains in the sense that the forcing term need not lie in any weighted space nor is the boundary of the domain required to be smooth the existence of the global attractor is obtained on arbitrary open sets such that the poincare inequality holds and for forces in the natural dual space v the proof is based on the energy equation and the concept of asymptotic compactness an estimate of the hausdorff and fractal dimensions of the attractor is also given

long time behavior for the weakly damped driven long wave short wave resonance equations abstract in this paper we consider the cauchy problem of the long wave short wave resonance equations by making use of a strichartz type inequality for the solutions decomposing suitably the solution semigroup into a decay parts and a more regular parts and ruling out the vanishing and dichotomy of the solutions we prove the existence of the global attractor and the asymptotic smoothing effect of the solutions

estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems we show that the long time behavior of the projection of the exact solutions to the navier stokes equations and other dissipative evolution equations on the finite dimensional space of interpolant polynomials determines the long time behavior of the solution itself provided that the spatial mesh is fine enough we also provide an explicit estimate on the size of the mesh moreover we show that if the evolution equation has an inertial manifold then the dynamics of the evolution equation is equivalent to the dynamics of the projection of the solutions on the finite dimensional space spanned by the approximating polynomials our results suggest that certain numerical schemes may capture the essential dynamics of the underlying evolution equation

nonconvex sparse logistic regression with weakly convex regularization in this paper we propose to fit a sparse logistic regression model by a weakly convex regularized nonconvex optimization problem the idea is based on the finding that a weakly convex function as an approximation of the l 0 pseudo norm is able to better induce sparsity than the commonly used l 1 norm for a class of weakly convex sparsity inducing functions we prove the nonconvexity of the corresponding problem and study its local optimality conditions and the choice of the regularization parameter despite the nonconvexity a method based on proximal gradient descent is used to solve the general weakly convex sparse logistic regression and its convergence behavior is studied theoretically then the general framework is applied to a specific weakly convex function and a local optimality condition and a bound on the logistic loss at a local optimum are provided the solution method is instantiated in this case as an iterative firm shrinkage algorithm and a nesterov acceleration is used with a convergence guarantee its effectiveness is demonstrated in numerical experiments by both randomly generated and real datasets

robust low rank tensor recovery via nonconvex singular value minimization tensor robust principal component analysis via tensor nuclear norm tnn minimization has been recently proposed to recover the low rank tensor corrupted with sparse noise outliers tnn is demonstrated to be a convex surrogate of rank however it tends to over penalize large singular values and thus usually results in biased solutions to handle this issue we propose a new definition of tensor logarithmic norm tln as the nonconvex surrogate of rank which can decrease the penalization on larger singular values and increase that on smaller ones simultaneously to preserve the low rank structure of a tensor then the strategy of tensor factorization is combined into the minimization of tln to improve computational performance to handle impulsive scenarios we propose a nonconvex ell p ball projection scheme with 0 instead of the conventional convex scheme with p 1 which enhances the robustness against outliers by incorporating the tln minimization and the ell p ball projection we finally propose two low rank recovery algorithms whose resulting optimization problems are efficiently solved by the alternating direction method of multipliers admm with convergence guarantees the proposed algorithms are applied to the synthetic data recovery and image and video restorations in real world experimental results demonstrate the superior performance of the proposed methods over several state of the art algorithms in terms of tensor recovery accuracy and computational efficiency